title: The Horror Show at Extreme Rules

text: In the main event, Universal Champion Braun Strowman faced Bray Wyatt in a non-title Wyatt Swamp Fight. At the swamp, Wyatt sat in his old rocking chair waiting for Strowman to arrive. After arriving in a vehicle, Strowman walked towards Wyatt, however, it suddenly went dark and Wyatt vanished, after which, an enraged Strowman obliterated the rocking chair. In search of Wyatt, Strowman fought off a couple of masked men, however, another attacked Strowman with a spade, who was revealed as Strowman's own doppelganger of his past self from The Wyatt Family. The doppelganger attacked Strowman again with the spade, knocking him unconscious. After regaining consciousness, Strowman found himself chained to a chair in a house adorned with Wyatt's eerie dolls. Wyatt then approached Strowman to his old theme music, carrying his lantern, and welcomed Strowman home. Wyatt spoke about how he created Strowman and that he would destroy him to make him anew. He also stated that his old self had been trapped inside his mind thanks to his alter ego "The Fiend". A female figure with a black veil then appeared with a snake that bit Strowman, rendering him unconscious.


title: The Horror Show at Extreme Rules

text: Sometime later, Strowman awoke in a field where he was attacked by another man whom Strowman set ablaze. The female figure appeared again and unveiled herself as an illusion of Alexa Bliss, Strowman's old tag team partner from Season 1 of "Mixed Match Challenge". The illusion implied Strowman's love interest for her and implored him to come home. Wyatt appeared once again and the two fought onto a dock until Strowman performed a chokeslam on Wyatt into a boat that sailed off into the darkness. Thinking that the fight was over, Strowman began to depart only for the boat to return. Strowman went to investigate, but found it empty. Wyatt then suddenly emerged from the swampy water and attacked Strowman with an oar before attempting to drown him. As Wyatt again disappeared, Strowman emerged from the water, gasping for air, and made his way back onto the dock. Wyatt appeared once again and attacked Strowman further. After attacking Strowman with a kendo stick, Wyatt spoke reluctantly about what he had just done to Strowman, who rose to his feet and kicked Wyatt through a barricade into the swampy water. Strowman surveyed the water and after seeing no signs of Wyatt, he declared it was over. However, Wyatt suddenly emerged from the water and used the Mandible Claw to pull Strowman into the water, not to be seen again. Wyatt attempted to come out of the water but was pulled back in. The water then turned red and began to bubble. Wyatt's alter ego "The Fiend" then peered over the dock, laughing manically before proclaiming "Let Me In" as the screen went to black to end the show.


title: The Horror Show at Extreme Rules

text: On the following "Raw", Seth Rollins prided himself in defeating Rey Mysterio and blamed the WWE Universe for his horrendous actions before Aleister Black interrupted. Black condemned Rollins for the events of the past few weeks, leading to a match between the two which Rollins won, capitalizing on Black's injured arm. After the match, Rollins and his disciple Murphy continued their assault on Black, continuing to target his injured arm. For what Rollins did to his father, Mysterio's son Dominik challenged Rollins to a match at SummerSlam, which Rollins accepted after he and Murphy were attacked by Dominik with a kendo stick on the August 3 episode. Their match was made a Street Fight.


title: The Horror Show at Extreme Rules

text: Following his unrecognized forfeit win over Apollo Crews, The Hurt Business (MVP, Bobby Lashley, and new faction member Shelton Benjamin, who had just won the 24/7 Championship from R-Truth), challenged Ricochet and Cedric Alexander to a 3-on-2 handicap match due to them not having a third partner, who would have been Crews. Alexander and Ricochet revealed that they did have a partner, the returning Mustafa Ali, who was traded from SmackDown. The newly-formed trio then defeated MVP, Lashley, and Benjamin in the ensuing six-man tag team match. Crews, sporting the older United States Championship belt, returned on the August 3 episode and he and MVP formally had their title match where Crews retained. Crews later said he would give the old belt to his kids while he kept the new belt MVP introduced. MVP then demanded a rematch for SummerSlam and Crews accepted.


title: The Horror Show at Extreme Rules

text: WWE Champion Drew McIntyre cut a promo on his win over Dolph Ziggler, who interrupted and demanded another title opportunity to which McIntyre initially denied. Ziggler tried to cheap shot McIntyre only for McIntyre to perform a Glasgow Kiss on Ziggler. A weakened Ziggler continued his plea, allowing McIntyre to choose the time, place, as well as the stipulation for the title match, which intrigued McIntyre enough to accept the challenge. McIntyre then said that he needed time to decide on a stipulation and that he would reveal the stipulation just prior to the match, just as Ziggler had done at Extreme Rules. The rematch occurred the following week, but for unexplained reasons as a non-title match. Backstage, Ziggler cut a promo stating that if he defeated McIntyre, he would face him for the title at SummerSlam, however, McIntyre, who chose an Extreme Rules match as the stipulation, defeated Ziggler once again. Randy Orton, who earlier that night issued a challenge for the title at SummerSlam that McIntyre accepted, performed an "RKO" from behind and stood over McIntyre while hoisting the title.


title: The Horror Show at Extreme Rules

text: While Sasha Banks and Bayley were celebrating Banks' unofficial title win over Asuka for the Raw Women's Championship, they were interrupted by Asuka, who claimed that Banks did not win. WWE Chief Brand Officer Stephanie McMahon then appeared on the TitanTron and stated that Banks nor Asuka won at Extreme Rules and then announced a rematch between the two for the Raw Women's Championship for the following week in a match where the title could be won by pinfall, submission, disqualification, or count out. In the ensuing match, Bayley attacked Asuka's tag team partner, Kairi Sane, backstage. This distracted Asuka, who went to check on Sane and was counted out, thus Banks officially won the Raw Women's Championship for a record fifth time. Asuka then defeated Bayley to earn a rematch against Banks at SummerSlam, while Asuka also won a triple brand battle royal, featuring wrestlers from Raw, SmackDown, and NXT on the August 14 episode of "SmackDown" to also face Bayley for her SmackDown Women's Championship at the event.


title: The Horror Show at Extreme Rules

text: On the following "SmackDown", Kofi Kingston was evaluated backstage by medical personnel and informed Big E that he was injured and would be out for at least 6–8 weeks. He then encouraged Big E to pursue a solo run, which he said himself and other injured teammate Xavier Woods fully supported. The two embraced afterwards.


title: The Horror Show at Extreme Rules

text: Also on the following "SmackDown" and despite what Stephanie McMahon had stated on the previous episode of "Raw", Bayley and Sasha Banks continued celebrating as if they held all the women's championships for Raw and SmackDown, and labeled themselves as the definition of "greatness" (this occurring before Banks officially won the Raw Women's Championship on the next "Raw"). An irate Nikki Cross, accompanied by her tag team partner Alexa Bliss, demanded a rematch for the SmackDown Women's Championship. Bayley agreed to the rematch, however, she stated that Cross must defeat a worthy adversary in Bliss, setting up a match between Bliss and Cross where the winner would face Bayley for the title on the following week's "SmackDown". Cross defeated Bliss to earn a rematch, but was again unsuccessful in winning the title from Bayley. Following the match, Bliss tried to console Cross, who shoved her away.


title: The Horror Show at Extreme Rules

text: Bray Wyatt appeared in the Firefly Fun House and stated that his old Eater of Worlds self was done for now and assured that Braun Strowman had come back home, thus "reuniting" with his family. He then declared that The Fiend had been unleashed. The following week, Wyatt, who stated that Strowman was still lost somewhere at the swamp, issued a warning to Strowman, stating that The Fiend wanted the Universal Championship and that no one was safe until he got it. The Fiend then made his first TV appearance in nearly four months and taunted Strowman by attacking his old "Mixed Match Challenge" tag team partner Alexa Bliss with the Mandible Claw. The Fiend attempted to attack Bliss again the following week, however, Bliss showed affection to The Fiend, who backed off. Strowman then appeared on the TitanTron, stated he did not care for Bliss, and that the swamp fight had made him into a monster. Strowman then accepted The Fiend's challenge for the Universal Championship at SummerSlam. The following week, Strowman, now sporting a bald look, attacked Bliss.


title: The Horror Show at Extreme Rules

text: On August 17, it was announced that all future episodes of "Raw" and "SmackDown" and all pay-per-views would be held at Orlando's Amway Center until further notice, beginning with the August 21 episode of "SmackDown". The move also brought a new virtual fan viewing experience in a bio-secure bubble called the WWE ThunderDome. This effectively made The Horror Show at Extreme Rules the final WWE pay-per-view for Raw and SmackDown to be held at the Performance Center during the COVID-19 pandemic; NXT's events would later be moved to the Performance Center in October, with NXT TakeOver events becoming available on pay-per-view starting with NXT TakeOver 31.


title: 2022 Toronto International Film Festival

text: The 2022 festival was staged primarily in-person; a small selection of films were offered on the Digital TIFF Bell Lightbox platform, but this represented a much smaller proportion of the overall lineup than in 2020 and 2021. The festival also saw the return of its networking and gala events, which were also suspended or held virtually in 2020 and 2021 due to the COVID-19 pandemic in Toronto.


text: As of mid-July, the festival had diverged from its usual practice of releasing a list of ten or twenty films to start its programming announcements, and instead issued a number of single-film announcements, before finally releasing its first extended list of the full gala and special presentations programs on July 28. Additional films were announced throughout August, until the full schedule was released on August 23. Festival CEO Cameron Bailey indicated that the full lineup would comprise around 200 feature films and about 40 short films, the largest lineup the festival has booked since the pre-pandemic 2019 edition.


text: Sally El Hosaini's "The Swimmers" was the opening film, and the closing film was Mary Harron's "Dalíland". "The Fabelmans", which screened as a special presentation and won the People's Choice Award, marked the first time in history that a Steven Spielberg film premiered at the festival. Jordan Peele's "Nope", although already released in theaters prior to the festival, was also screened in IMAX at the Cinesphere as a special presentation in the main festival slate.


text: The festival also indicated that in light of the 2022 Russian invasion of Ukraine, state-backed Russian films and organizations would be banned from the festival, although participation by independent Russian filmmakers would still be permitted.


text: The TIFF Tribute Awards, the festival's program of honouring film personnel for their overall achievements in cinema, were presented early during the festival run.


text: The cast of "My Policeman"—Harry Styles, Emma Corrin, Gina McKee, Linus Roache, David Dawson, and Rupert Everett—were named as the recipients of one of the two awards for actors, marking the first time the award has been presented to an ensemble cast rather than an individual, while Sam Mendes has been named as the recipient of the Director Award. Brendan Fraser received a Tribute Award for Performance in "The Whale".


text: Composer Hildur Guðnadóttir was named the recipient of the "Variety" Artisan Award, Sally El Hosaini was named the winner of the Emerging Talent Award, Buffy Sainte-Marie was the recipient of the Jeff Skoll Award in Impact Media, and Michelle Yeoh was named the winner of the inaugural Share Her Journey Groundbreaker Award.


text: The festival's main awards for films in the festival program, including the People's Choice Award, were presented on September 18.


text: Ulrich Seidl's "Sparta" was withdrawn by the festival after allegations of child exploitation on set surfaced.


text: Following its premiere on September 14, TIFF was forced to cancel followup screenings of Vera Drew's film "The People's Joker" following a rights conflict with Warner Bros. over the film's unauthorized use of The Joker.


text: A selection of films from the above programs that was available for home viewing through the Digital TIFF Bell Lightbox platform.


text: Films that were available for screening by film buyers and industry professionals, but not open to the general public.


text: Older movies that were shown at the OLG Cinema Park, an open-air cinema run by TIFF at David Pecaut Square. Many featured actors and/or directors involved with TIFF 2022 selections.


text: The festival's annual year-end Canada's Top Ten list, collecting the films named as the top Canadian films of the year by critics and film festival programmers from across Canada, was released on December 8, 2022.


title: Auditing (Scientology)

text: In Dianetics and Scientology, auditing is a process whereby the 'auditor' takes an individual through times in their current or past lives with the ostensible purpose of ridding the individual of negative influences from past events or behaviors. Auditing is meant to bring the individual to "Clear" status; thus, an individual being audited is known as a PC or "preclear".


title: Auditing (Scientology)

text: Auditing was invented by L. Ron Hubbard as an integral part of Dianetics, first introduced in 1950. In 1951, auditing also became a core practice of Scientology. The E-meter, a device to measure electrodermal activity, became an integral part of auditing in scientology. According to the Church of Scientology, "one formal definition of auditing is the action of asking a person a question (which he can understand and answer), getting an answer to that question and acknowledging him or her for that answer".


title: Auditing (Scientology)

text: Hubbard claimed auditing provided many benefits including unsupported medical and psychological health effects. Since 1971, Scientology now publishes disclaimers in its books and publications declaring that the E-Meter "by itself does nothing", and that it is used specifically for spiritual purposes, not for mental or physical health.


title: Auditing (Scientology)

text: In the context of Dianetics or Scientology auditing, Dianetics addresses the body, Scientology the spirit or being. Auditing is an activity where an auditor, trained in the task of communication, listens and gives auditing commands to a subject, who is referred to as a "preclear", or more often as a "PC". Auditing sessions are confidential.


title: Auditing (Scientology)

text: Auditing involves the use of "processes", which are sets of questions asked or directions given by an auditor. Based on a prior interview looking for "charged" subjects—"charge" being that which prevents the PC from thinking on a subject or getting rid of a subject or approaching a subject—on the e-meter, found by asking questions to the PC in regard to them and their fancied case. When the specific objective of any one process is achieved, the process is ended, and another can then be started. Through auditing, the subjects are said to free themselves from barriers that inhibit their natural abilities. Charged areas can be viewed as areas of misinformation or lies. Once uncovered, they dissipate as their truth becomes apparent and the charge is eliminated once viewed for what it really is, an untruth.


title: Auditing (Scientology)

text: The auditor is obliged to maintain a strict code of conduct, called the Auditor's Code. Auditing is said to be successful only when the auditor conducts himself in accordance with the Code. A violation of the Auditor's Code is considered a high crime in Scientology.


title: Auditing (Scientology)

text: The main intention of an auditing session is to remove "charged incidents" that have caused trauma, which are believed in Scientology to be stored in the 'reactive mind'. These incidents must then be eliminated for proper functioning.


title: Auditing (Scientology)

text: In 1952, auditing techniques "began to focus on the goal of exteriorizing the thetan" with the goal of providing complete spiritual awareness.


title: Auditing (Scientology)

text: Most auditing sessions employ a device called the Hubbard Electropsychometer or E-Meter. The E-Meter is not a custom electrodermal activity measurement device, instead it measures the resistance of the body (flesh, bones, liquids and all; skin resistance is only a small part of the total resistance being measured) from one hand through the breast to the other hand. It measures changes in the electrical resistance of the 'preclear' by passing a small electric current (typically in the range from 50 µA to 120 µA) through the preclear's body by means of a pair of tin-plated tubes originally much like empty soup cans, attached to the meter by wires and held by the preclear during auditing. These changes in electrical resistance are allegedly a reliable and precise indication of changes in the 'reactive mind' of the preclear.


title: Auditing (Scientology)

text: According to L. Ron Hubbard the development of the E-Meter enabled auditing techniques and made it more precise. Later, the E-Meter was used to identify which processes should (and could) be run and equally crucially, to determine when to stop running a particular action. As a repair tool, the E-Meter reacts to a list of possible difficulties and relevant phrases, called out by the auditor, helping to guide the auditor to the difficulty. Hubbard clarified how the E-Meter should be used in conjunction with auditing:


title: Auditing (Scientology)

text: Hubbard claimed that the device also has such sensitivity that it can measure whether or not fruits can experience pain, claiming in 1968 that tomatoes "scream when sliced".


title: Auditing (Scientology)

text: Scientology teaches that individuals are immortal souls or spirits (called "thetans" by Scientology) and are not limited to a single lifetime. Scientologists state that the E-Meter aids the auditor in locating subliminal memories ("engrams", "incidents", and "implants") of past events in a thetan's current life and in previous ones. In such Scientology publications as "Have You Lived Before This Life", Hubbard wrote about past life experiences dating back billions and even trillions of years.


title: Auditing (Scientology)

text: When various foundations of Dianetics were formed in the 1950s, auditing sessions were a hybrid of confession, counseling and psychotherapy. According to Passas and Castillo, the E-Meter was used to "disclose truth to the individual who is being processed and thus free him spiritually".


title: Auditing (Scientology)

text: Back in 1950, at the very end of his book, "Dianetics: the Modern Science of Mental Health", Hubbard talks about a bridge from one plateau of existence to another, higher plateau.


title: Auditing (Scientology)

text: Hubbard wanted to make the processes structured in such a way that one could take a new person and walk them through standardized steps, one after another, to cross this hypothesized "Bridge". This intent led to the development of the Standard Operating Procedure for Theta Clearing by 1952.


title: Auditing (Scientology)

text: In 1970, the Standard Operating Procedure was used to create the Classification and Gradation Chart. This chart, first published in 1965 and revised in 1966, 1968 and 1969, had the steps of the bridge plotted out from a beginner at the bottom to the highest states attainable at the top. The left-hand side of the chart contains auditor skill levels, while the right-hand side contains pre-clear grades and OT (Operating Thetan) levels.


title: Auditing (Scientology)

text: The 1970 version of the chart is entitled "Classification Gradation and Awareness Chart of Levels and Certificates". By 1974 the above title had slipped down a little to make way for "THE BRIDGE" as the top line, with "TO A NEW WORLD" underneath in a smaller font. A more recent () chart is entitled "THE BRIDGE TO TOTAL FREEDOM" and subtitled "SCIENTOLOGY CLASSIFICATION GRADATION AND AWARENESS CHART."


title: Auditing (Scientology)

text: Each Grade on the Bridge has a list of processes that auditors should run. Some auditing actions use commands, for example "Recall a time you knew you understood someone," and some auditing actions use questions such as, "What are you willing for me to talk to others about?" Below are sample commands from processes run in each Grade.


title: Auditing (Scientology)

text: Each Grade is targeted at a specific area of potential difficulty a person might have. The working hypothesis is that if the subject matter is not "charged"; in other words, if it is not causing any difficulty, then it will not read on the E-meter, and therefore will not be run. (Reference see above)


title: Auditing (Scientology)

text: The Scientology and Dianetics auditing process has raised concerns from a number of quarters, as auditing sessions are permanently recorded in the form of handwritten notes in preclear folders, which are supposed to be kept private. Judge Paul Breckenridge, in Church of Scientology of California vs. Gerald Armstrong, noted that Mary Sue Hubbard (the plaintiff in that case) "authored the infamous order 'GO 121669', which directed culling of supposedly confidential P.C. Preclear files/folders for the purposes of internal security". This directive was later canceled because it was not part of Scientology as written by L. Ron Hubbard. Bruce Hines has noted in an interview with Hoda Kotb that Scientology's collecting of personal and private information through auditing can possibly leave an adherent vulnerable to potential "blackmail" should they ever consider disaffecting from the cult. A number of sources have claimed that preclear folders have indeed been used for intimidation and harassment.


title: Auditing (Scientology)

text: In 1965 the Anderson Report, an official inquiry conducted for the state of Victoria, Australia, found that auditing involved a form of "authoritative" or "command" hypnosis, in which the hypnotist assumes "positive authoritative control" over the subject. "It is the firm conclusion of this Board that most scientology and dianetic techniques are those of authoritative hypnosis and as such are dangerous. ... the scientific evidence which the Board heard from several expert witnesses of the highest repute ... which was virtually unchallenged - leads to the inescapable conclusion that it is only in name that there is any difference between authoritative hypnosis and most of the techniques of scientology. "


title: Auditing (Scientology)

text: L. Ron Hubbard claimed benefits from auditing including improved IQ, improved ability to communicate, enhanced memory and alleviation of issues such as psychosis, dyslexia and attention deficit disorder. Some people have alleged that auditing amounts to medical treatment without a license, and in the 1950s, some auditors were arrested on the charge. The Church disputes that it is practicing medicine, and it has successfully established in United States courts of law that auditing addresses only spiritual relief. According to the Church, the psychotherapist treats mental health and the Church treats the spiritual being. Hubbard clarified the difference between the two:


title: Auditing (Scientology)

text: In 1971, a ruling of the United States District Court, District of Columbia (333 F. Supp. 357), specifically stated that the E-meter "has no proven usefulness in the diagnosis, treatment or prevention of any disease, nor is it medically or scientifically capable of improving any bodily function." As a result of this ruling, Scientology now publishes disclaimers in its books and publications declaring that the E-meter "by itself does nothing" and that it is used specifically for spiritual purposes.


title: Auditing (Scientology)

text: Dutch investigative reporter Rinke Verkerk reported that she was given an auditing session by an 11-year-old in the Netherlands. This has been criticized by clinical psychologists and child psychologists, on the grounds that secondary stress can affect children more strongly than adults. The fact that the child was working full days for a whole weekend was also considered to be problematic.


title: Auditing (Scientology)

text: An auditor is only allowed to audit processes (on others) which are below his own case (level of auditing which he has undergone).


title: Erick Sermon

text: Erick Sermon (born November 25, 1968) is an American rapper, musician, and record producer. He is best known as one-third—alongside PMD & DJ Scratch—of 1980s/1990s hip hop group EPMD and for his production work.


title: Erick Sermon

text: Sermon started professionally in 1986 as a producer and artist of the hip hop group EPMD. He began recording solo albums for Def Jam in 1993; in 1997, he rejoined EPMD. The following year, Sermon, Murray and Redman recorded a cover version of "Rapper's Delight" by the Sugarhill Gang. EPMD disbanded a second time in 1999.


title: Erick Sermon

text: In 2000, Sermon moved over to J Records, and released the album "Music" the following year. The album's first single, "Music", featured guest vocals from Marvin Gaye, which Sermon reportedly culled from unreleased recordings found in a small record shop in London. "Music" went on to become Sermon's highest-charting song, peaking at number 22 on the "Billboard" Hot 100 and number 2 on the R&B chart. Sermon's second album on J Records, "React", was released in 2002. "React's" title track peaked at number 36 on the Billboard Hot 100, but the album sold poorly and Sermon was dropped from the label in 2003. In a June 30, 2004 interview with "HipHopDX.com", Sermon told music journalist Bayer Mack, "Things weren't right at J Records. Clive Davis and them don't believe in promotion. When Puffy had Bad Boy at Arista, it was him doing all the [promotional] work." He also stated Busta Rhymes and Wyclef Jean had similar issues with J Records.


title: Erick Sermon

text: Sermon went on to establish his Def Squad imprint with Universal Records and released his sixth solo album, "Chilltown, New York", in 2004. The album was powered by the single "Feel It" (which contained a sample of reggae/R&B singer Sean Paul), a song which became a success in the United States.


title: Erick Sermon

text: In an interview, he stated that he was going to step aside and try to get upcoming artists in the spotlight. However, Sermon has not stopped in the music industry, as he produced the song "Goldmine" on Busta Rhymes' album, "The Big Bang" in 2006. Soon after, Sermon has recorded "Don't Make No Sense" with Def Squad. He also collaborated with Redman and produced a few songs on the album "Red Gone Wild" while also making an appearance with Def Squad member Keith Murray.


title: Erick Sermon

text: In early 2008, Sermon and Smith started their own record label called EP Records, distributed by RBS/Universal Music Group. The seventh EPMD album, "We Mean Business", came out in December 2008.


title: Erick Sermon

text: Sermon was featured in the final episode of "Yo! MTV Raps" in a freestyle session featuring artists such as Rakim, KRS-One, Chubb Rock, MC Serch and Craig Mack.


title: Erick Sermon

text: In 2018, Sermon teamed up with Lost Boyz' own Mr Cheeks & broadcaster Ryan Verneuille to become the executive producer on their FM radio program, "The Ryan Show".


title: Erick Sermon

text: On September 25, 2001, Sermon was injured when he fell from the third floor of an apartment building. Police claimed Sermon attempted suicide, however he later denied and claimed it was accidental.


title: United States Army Futures Command

text: The United States Army Futures Command (AFC) is a United States Army command, designed as a public-private initiative, that runs modernization projects for the Army. It is headquartered in Austin, Texas, and was first commanded by General John Murray, formerly the Army's G-8; the second and current commander was formerly the Army's G-3/5/7.


title: United States Army Futures Command

text: As of 2018 Futures Command was focused on six priorities: 1) Long-range precision fires, 2) Next Generation Combat Vehicle, 3) Future Vertical Lift platforms, 4) a mobile & expeditionary Army network, 5) air and missile defense capabilities, and 6) soldier lethality.


title: United States Army Futures Command

text: AFC's cross-functional teams (CFTs) are Futures Command's vehicle for sustainable reform of the acquisition process for the future.


title: United States Army Futures Command

text: By October 2021, the 40th Chief of Staff of the Army was able to project that 24 of the top 35 priority programs for modernization would be fielded in fiscal year 2023 (FY2023).


title: United States Army Futures Command

text: Overmatch of the capability of a competitor or adversary is one of the goals of AFC. More specifically, the imposition of multiple simultaneous dilemmas upon a competitor or adversary is a goal of the US Army: to get into a position of relative advantage. By 2021, Army leadership recognized that new Army formations (the multi-domain operations task force —MDTF) had the ability to simultaneously compete with, and also threaten an adversary, with its new capability, across domains (space, cyber, disinformation) of the conflict continuum. By 2022 or 2023, a new concept for command and control (JADC2) will have been largely prototyped.


title: United States Army Futures Command

text: U.S. Army Futures Command was activated in the summer of 2018. The Decker-Wagner report on the 2010 Army Acquisition Review (Jan 2011) listed numerous changes to the acquisition process; the recommendation to disestablish RDECOM was not followed. Instead a unitary Futures Command, to unify development over the life cycle was moved forward by an Acting Secretary of the Army (Ryan McCarthy), and the 36th Vice Chief of Staff of the Army (James McConville), who established a task force for modernization in 2016-2017 using cross-functional teams of subject matter experts to drive initial actions."See § Need for modernization reform"


title: United States Army Futures Command

text: AFC declared its full operational capability in July 2019, after an initial one-year period. The FY2020 military budget allocated $30 billion for the top six modernization priorities over the next five years. The $30 billion came from $8 billion in cost avoidance and $22 billion in terminations. Over 30 projects are envisioned to become the materiel basis needed for overmatching any potential competitors in the 'continuum of conflict' over the next ten years, in multi-domain operations (MDO). By 2018 a fundamental strategy was formulated, involving simultaneous integrated operations across domains. This strategy involves pushing adversaries to standoff, by presenting them with multiple simultaneous dilemmas. By 2028, the ability to project rapid, responsive power across domains will have become apparent to potential adversaries. "See Power projection"


title: United States Army Futures Command

text: From an initial 12 people at its headquarters in 2018, AFC grew to 24,000 across 25 states and 15 countries in 2019. The apparent rapid expansion came by research facilities and personnel (including ARCIC and RDECOM) migrated from other commands and parts of the Army such as the United States Army Research Laboratory.


title: United States Army Futures Command

text: The AFC was created in 2018 as a peer of Forces Command (FORSCOM), Training and Doctrine Command (TRADOC), and Army Materiel Command (AMC), the Army commands that provide forces, training and doctrine, and materiel respectively. The other Army commands focus on their readiness to "fight tonight" when called upon. In contrast, AFC is focused on future readiness for competition with near-peers, who have updated their capabilities. The command is supported by United States Army Reserve Innovation Command (aka. 75th Innovation Command).


title: United States Army Futures Command

text: According to the 24th Secretary of the Army Ryan McCarthy, the three elements in Futures Command are to be:


title: United States Army Futures Command

text: 23rd Secretary of the Army, Mark Esper emphasized that the 2018 administrative infrastructure for the Futures and Concepts Center (formerly ARCIC) and United States Army Combat Capabilities Development Command (CCDC, now called DEVCOM, (formerly RDECOM)) remains in place at their existing locations. What has changed or will change is the layers of command (operational control, or OPCON) needed to make a decision. He said, "You've got to remain open to change, you've got to remain flexible, you've [got] to remain accessible. That is the purpose of this command." "See § Army of 2040"


title: United States Army Futures Command

text: When he was 33rd Under Secretary of the Army, Ryan McCarthy characterized a Cross-Functional Team (CFT) as a team of teams, led by a requirements leader, program manager, sustainer and tester. Each CFT must strike a balance for itself amid constraints: the realms of requirements, acquisition, science and technology, test, resourcing, costing, and sustainment. A balance is needed in order for a CFT in order to produce a realizable concept before a competitor achieves it. The Army Requirements Oversight Council (AROC) itself serves as a kind of CFT, operating at a higher level as response to Congressional oversight, budgeting, funding, policy, and authorization for action.


title: United States Army Futures Command

text: CFTs for materiel and capabilities were first structured in a task force, in order to de-layer the Army Commands. Each CFT addresses a capability gap, which the Army must now match for its future: there can be a Capability Development Integration Directorate (CDID), for each CFT. Initially, the CFTs were placed as needed; eventually they might each co-locate at a Center of Excellence (CoE) listed below. For example, the Aviation CoE at Fort Rucker, in coordination with the Aviation Program Executive Officer (PEO), also contains the Vertical Lift CFT and the Aviation CDID.


title: United States Army Futures Command

text: The CFTs will be involved in all three of AFC's elements: Futures and concepts, Combat development, and Combat systems. "We were never above probably a total of eight people" —BG Wally Rugen, Aviation CFT. Four of the eight CFT leads have now shifted from dual-hat jobs to full-time status. Each CFT lead is mentored by a 4-star general.


title: United States Army Futures Command

text: Although AFC and the CFTs are a top priority of the Department of the Army, as AFC and the CFTs are expected to unify control of the $30 billion-dollar modernization budget, "The new command will not tolerate a zero-defects mentality. 'But if you fail, we'd like you to fail early and fail cheap,' because progress and success often builds on failure." —Ryan McCarthy Holland notes that prototyping applies to the conceptual realm ('harvestable content') as much as prototyping applies to the hardware realm.


title: United States Army Futures Command

text: A 2019 Government Accountability Office (GAO) report cautions that lessons learned from the CFT pilot are yet to be applied; Holland notes that this organizational critique applies to prototyping hardware, a different realm than concept refinement ("scientific research is a fundamentally different activity than technology development"). Also in 2019 the GAO recommended that the government establish a process to ensure that CFTs implement their intended business reforms; however by 2021 the office of the Chief Management officer (CMO) had been disestablished.


title: United States Army Futures Command

text: The Secretaries of the Army, Air Force, and Navy meet regularly to take advantage of overlap in their programs:


title: United States Army Futures Command

text: AFC is actively seeking partners outside the military, including research funding to over 300 colleges and universities, but with one-year program cycles. "We will come to you. You don't have to come to us. —General Mike Murray, 24 August 2018"


title: United States Army Futures Command

text: especially Capital Factory, with offices of Defense Innovation Unit (DIU) and AFWERX (USAF tech hub). Gen. Murray will stand up an Army Applications Lab there to accelerate acquisition and deployment of materiel to the soldiers, using artificial intelligence (AI)


title: United States Army Futures Command

text: For the US, and its allies and partners, the basis of global surveillance and target acquisition is the US National Defense Space Architecture (NDSA) as realized by layered constellations of Earth satellites and Earth stations (such as TITAN). It will take machine learning (ML) and AI to handle the scale of data flows needed for MDO.


title: United States Army Futures Command

text: Futures Command will stand up Army Software Factory in August 2021, to immerse Soldiers and Army civilians of all ranks in modern software development, in Austin. Similar in spirit to the Training with industry program, participants are expected to take these practices back with them, to influence other Army people in their future assignments, and to build up the Army's capability in software development. The training program lasts three years, and will produce skill sets for trainees as product managers, user experience and user interface designers, software engineers, or platform engineers. The Al Work Force Development program and this Software Factory will complement the Artificial Intelligence Task Force.


title: United States Army Futures Command

text: AFC is seeking to design signature systems in a relevant time frame according to priorities of the Chief of Staff of the Army (CSA). AFC will partner with other organizations such as Defense Innovation Unit Experimental (DIUx) as needed.


title: United States Army Futures Command

text: If a team from industry presents a viable program idea to a CFT, that CFT connects to the Army's requirements developers, Secretary Esper said, and the program prototype is then put on a fast track.


title: United States Army Futures Command

text: The Secretary of the Army has approved an Intellectual Property Management Policy, to protect both the Army and the entrepreneur or innovator.


title: United States Army Futures Command

text: For example, the Network CFT and the Program Executive Office Command, Control, Communications—Tactical (PEO C3T) hosted a forum on 1 August 2018 for vendors to learn what might function as a testable/deployable in the near future. A few of the hundreds of white papers from the vendors, adjudged to be 'very mature ideas', were passed to the Army's acquisition community, while many others were passed to United States Army Communications-Electronics Research, Development and Engineering Center (CERDEC) for continuation in the Army's effort to modernize the network for combat. Although some test requirements were inappropriately applied, the Command post computing environment (CPCE) has passed a hurdle.


title: United States Army Futures Command

text: While seeking information, the Army is especially interested in ideas that accelerate an acquisition program —in for example the Future Vertical Lift Requests for Information (RFIs): "provide a detailed description of tailored, alternative or innovative approaches that streamlines the acquisition process to accelerate the program as much as possible" (4 April 2019); in January 2020 the current Optionally manned fighting vehicle (OMFV) solicitation was cancelled when the OMFV's requirements added up to an unobtainable project; in February 2020 Futures command was soliciting the industry for do-able ideas for an OMFV, whereas in August 2022 Army Network modernization was the target for being speeded-up.


title: United States Army Futures Command

text: In the Army's search for capabilities, 6th ASA(ALT) Bruce Jette initiated xTechsearch to reward private innovators. The program is ongoing.


title: United States Army Futures Command

text: The COVID-19 pandemic triggered the Army to run an Ventilator Challenge; entrants can submit their ideas online for immediate consideration and a possible cash prize to encourage participation for a $100,000 prize and possible Army contract. In 1964 Henrik H. Straub of Harry Diamond Labs, a predecessor to CCDC Army Research Laboratory, invented the Army Emergency Respirator (now termed a 'ventilator' in current terminology). This ventilator is one application of the fluidic amplifier (a 1957 Harry Diamond Labs invention), which allows the labored breathing of the patient to control the flow from an externally purified air stream, to augment the air flow into a patient's lungs.


title: United States Army Futures Command

text: TRX Systems won an award for technology which allows navigation in a GPS-denied environment, an A-PNT priority. The award was delayed by the COVID-19 pandemic, which allowed the company more time for business development.


title: United States Army Futures Command

text: Air-launched effects (ALEs) are drones which are launched from the Army's helicopters. In swarms, ALEs promise to multiply the combat effectiveness of the Army's helicopters; single ground-launched drones have already been used in combat during the 2022 Russian invasion of Ukraine, for intelligence, surveillance, and reconnaissance. ALEs have been used as munitions, as in the Nagorno-Karabakh conflicts.


title: United States Army Futures Command

text: In 2021 candidate Robotic combat vehicles (RCVs), both medium and light RCVs, along with surrogate heavy RCVs (modified M-113s) and proxy manned control vehicles (MET-Ds) were to marshal at Camp Grayling MI to test a company-sized tele-operated / unmanned formation. The light RCVs had their autonomous driving software installed in November and December 2020. The robotic vehicle formation begins a shakeout in April 2021. The RCVs (and the software, which is common to all 18 vehicles) enters ATEC (Army Test and Evaluation Command) safety testing through May 2022. Live-fire drills are scheduled to conclude in August 2022. By October 2021 experiments with RCVs, in concert with drones for intelligence, surveillance, and reconnaissance (ISR), were underway.


title: United States Army Futures Command

text: By June 2022 Army RCVs had demonstrated some of their disruptive capabilities, in preparation for Project Convergence 2022. Autonomous capabilities, for example in resupply by unmanned helicopters, by the US, Australia, and UK will be demonstrated at PC22.


title: United States Army Futures Command

text: By 13 October 2021 the 40th Chief of Staff of the Army could announce that the majority of the Army's Futures Command's 31 signature systems, and the four rapid capability projects of the § Rapid Capabilities and Critical Technologies Office would be fielded by fiscal year 2023 (FY2023).


title: United States Army Futures Command

text: By 2022, Futures Command was conducting the third annual iteration of Project Convergence (PC22), experiments and joint tests of 300 technologies by the Department of Defense and its allies and partners.


title: United States Army Futures Command

text: Futures Command partners with the ASA(ALT), who, in the role of the Army Acquisition Executive (AAE), has milestone decision authority (MDA) at multiple points in a Materiel development decision (MDD). Thus, from the perspective of AFC, which seeks to modernize, they consolidate the relevant expertise into the relevant CFT. The CFT balances the constraints needed to realize a prototype, beginning with realizable requirements, science and technology, test, etc. before entering the acquisition process (typically the Army prototypes on its own and, as of 2019, initiates acquisition at Milestone B in order to have the Acquisition Executive, with the concurrence of the Army Chief of Staff, decide on production as a Program of Record at Milestone C). Next, refine the prototype to address the factors needed to pass the Milestone decisions A, B, and C which require Milestone decision authority (MDA) in an acquisition process. This consolidation of expertise thus reduces the risks in a Materiel development decision (MDD), for the Army to admit a prototype into a program of record.) The existing processes (as of April 2018) for a Materiel development decision (MDD) have been updated to clarify their place in the Life Cycle of a program of record: over 1200 programs/projects were reviewed; by October 2019, over 600 programs of record have been moved from the acquisition (development for modernization) phase to the sustainment phase (for mature projects, to continue their manufacture and fielding to the brigades). An additional life cycle management action is underway, to re-examine which of these projects/programs should be divested. (Surplus materiel might well go to the Security Assistance Command, perhaps to Foreign Military Sales.)


title: United States Army Futures Command

text: The emphasis remains with Futures Command, which selects programs to develop. In order to achieve its mission of achieving overmatch, each Futures Command CFT partners with the acquisition community. This community (the Army acquisition workforce (AAW)) includes an entire Army branch (the Acquisition Corps), U.S. Army Acquisition Support Center (USAASC), Army Contracting Command, . The Principal Military Deputy to the ASA(ALT) is also deputy commanding general for Combat Systems, Army Futures Command, and leads the Program Executive Officers (PEO); he has directed each PEO who does not have a CFT to coordinate with, to immediately form one, at least informally.


title: United States Army Futures Command

text: Operationally, the CFTs offer "de-layering" (fewer degrees of separation between the echelons of the Army—Rugen estimates two degrees of separation), and provide a point of contact (POC) for Army reformers interested in adding value in the midst of constraints to be balanced while modernizing. "... and if we're really good, we'll continue to adapt. Year over year over year." —Secretary Esper ("See Value stream".)


title: United States Army Futures Command

text: "Our new approach is really to prototype as much as we can to help us identify requirements, so our reach doesn't exceed our grasp. ... A good example is Future Vertical Lift: The prototyping has been exceptional." —Secretary of the Army Mark Esper. The development process will be cyclic, consisting of prototype, demonstration/testing, and evaluation, in an iterative process designed to unearth unrealistic requirements early, before prematurely including that requirement in a program of record.


title: United States Army Futures Command

text: AFC activities include at least one Cross-functional team, its Capability development integration directorate (CDID), and the associated Battle Lab, for each (Army Center of Excellence (CoE)) respectively. Each CDID and associated Battle Lab work with their CFT to develop operational experiments and prototypes to test.


title: United States Army Futures Command

text: ASA(ALT), in coordination with AFC, has dotted-line relationships between its PEOs and the CFTs. In particular, the Rapid Capabilities and Critical Technologies Office of ASA(ALT) has a PEO who is charged with developing experimental prototype 'units of action' for rapid fielding to the Soldiers. The prototypes are currently for Long range hypersonic weapons, High energy laser defense, and Space, as of June 2019, Speed and range are the Army capabilities which are being augmented, with spending on these capabilities tripling between 2017 and 2019.


title: United States Army Futures Command

text: Tests are run by JMC and White Sands Missile Range, which hosts ATEC. As United States Army Test and Evaluation Command (ATEC) reports directly to the Army Chief of Staff, the test support level from ATEC is to be specified by the CFT, or PEO. Fort Bliss and WSMR together cover 3.06 million acres, large enough to test every non-nuclear weapon system in the Army inventory.


title: United States Army Futures Command

text: JMC runs live developmental experiments to test and assess MDO concepts or capabilities that support the Army's six modernization priorities which are then analyzed by The Research and Analysis Center, denoted TRAC based out of Fort Leavenworth, or AMSAA, denoted the Data Analysis Center at APG. CCDC, now called DEVCOM (formerly RDECOM, at APG) includes the several Army research laboratory locations (ARLs), as well as research, development and engineering centers (RDECs) listed:


title: United States Army Futures Command

text: In internal partnerships, CCDC, now called DEVCOM (formerly RDECOM) has taken Long range precision fires (LRPF) as its focus in aligning its organizations (the six research, development and engineering centers (RDECs), and the Army Research Laboratory (ARL)); as of September 2018, RDECOM's 'concept of operation' is first to support the LRPF CFT, with ARDEC. AMRDEC is looking to improve the energetics and efficiency of projectiles. TARDEC Ground Vehicle Center is working on high-voltage components for Extended range cannon artillery (ERCA) that save on size and weight. Two dedicated RDECOM people support the LRPF CFT, with reachback support from two dozen more at RDECOM. In January 2019 RDECOM was reflagged as CCDC; General Mike Murray noted that CCDC will have to support more Soldier feedback, and that prototyping and testing will have to begin before a project ever becomes a program of record.


title: United States Army Futures Command

text: Although the Army Research Laboratory has not changed its name, Secretary Esper notes that the CCDC "objectives" supersede the "activities" of the Laboratory; the Laboratory remains in its support role for the top-six priorities for modernizing combat capabilities.


title: United States Army Futures Command

text: Acquisition specialists are being encouraged to accept lateral transfers to the several research, development and engineering centers (RDECs), where their skills are needed: Ground vehicle systems center (formerly TARDEC, at Detroit Arsenal. Michigan), Aviation and missile center (formerly AMRDEC, at Redstone Arsenal), C5ISR center (formerly CERDEC, at Aberdeen Proving Ground), Soldier center (formerly NSRDEC, Natick, MA), and Armaments center (formerly ARDEC, at Picatinny Arsenal) listed below.


title: United States Army Futures Command

text: The following activities for Futures Command are at 23 locations. (A US Army center of excellence (CoE), or TRADOC Center of Excellence, can be co-located near a CFT, along with the associated Capability Development Integration Directorate (CDID) and "Battle Lab") The interrelation between AFC and TRADOC can be seen by the role of a TRADOC Capability manager, who is responsible for DOTMLPF, and reports to the TRADOC commander.


title: United States Army Futures Command

text: Between 1995 and 2009, $32 billion was expended on programs such as the Future Combat System (2003-2009), with no harvestable content by the time of its cancellation. As of 2021, the Army had not fielded a new combat system in decades.


title: United States Army Futures Command

text: 23rd Secretary of the Army Mark Esper has remarked that AFC will provide the unity of command and purpose needed to reduce the requirements definition phase from 60 months to 12 months.


title: United States Army Futures Command

text: A simple statement of a problem (rather than a full-blown requirements definition) that the Army is trying to address may suffice for a surprising, usable solution. —General Mike Murray, paraphrasing Trae Stephens (One task will be to quantify the lead time for identifying a requirement; the next task would then be to learn how to reduce that lead time.—Gap analysis ) Process changes are expected. The development process will be cyclic, consisting of prototype, demonstration/testing, and evaluation, in an iterative process designed to unearth unrealistic requirements early, before prematurely including that requirement in a program of record. The 6th ASA(ALT) Bruce Jette has cautioned the acquisition community to 'call-out' unrealistic processes which commit a program to a drawn-out failure, rather than failing early, and seeking another solution.


title: United States Army Futures Command

text: Secretary Esper scrubbed through 800 modernization programs to reprioritize funding for the top six modernization priorities, which will consume 80% of the modernization funding, of eighteen systems. IVAS was slowed during the 2022 Russian invasion of Ukraine in March 2022. Secretary McCarthy had cautioned that a stopgap 2019 Continuing resolution (CR) would halt development of some of the critical modernization projects. Realistically, budget considerations will restrict the fielding of new materiel to one Armor BCT per year; at that rate, updates would take decades. The Budget Control Act (BCA) expires in 2022. The "night court" budget review process realigned $2.4 billion for modernization away from programs which were not tied to modernization or to the 2018 National Defense Strategy. The total FY2021 budget request of $178 billion is $2 billion less than the enacted FY2020 budget of $180 billion.


title: United States Army Futures Command

text: The CIO/G6 has targeted Futures Command (Austin) in 2019 as the first pilot for "enterprise IT-as-a-service"-style service contracts; General Murray now (July 2019) has a sensitive compartmented information facility in his headquarters, as a result of this pilot. Two other locations are to be announced for 2019. Six to eight other pilots are envisioned for 2020. However, 288 other enterprise network locations remain to be migrated away from the previous "big bang" migration concept from several years ago, as they are vulnerable to near-peer cyber threats. The CIO/G6 emphasizes that this enterprise migration is not the tactical network espoused in the top six priorities (a 'mobile & expeditionary Army network').


title: United States Army Futures Command

text: By February 2020 the 37th Vice Chief of Staff could assess that Army modernization was perceptibly speeding up.


title: United States Army Futures Command

text: Chief Milley noted that AFC would actively reach out into the community in order to learn, and that Senator John McCain's frank criticism of the acquisition process was instrumental for modernization reform at Futures command. In fact, AFC soldiers would blend into Austin by not wearing their uniforms [to work side by side with civilians in the tech hubs], Milley noted on 24 August 2018 press conference. Secretary Esper said he expected failures during the process of learning how to reform the acquisition and modernization process; the Network CFT and PEO have detected a process failure in the DOT&E requirements process: some test requirements were inappropriately applied.


title: United States Army Futures Command

text: In the Department of Defense, the materiel supply process was underwritten by the acquisition, logistics, and technology directorate of the Office of the Secretary of Defense (OSD), with a deputy secretary of defense (DSD) to oversee five areas, one of them being acquisition, logistics, and technology (ALT). ALT is overseen by an under secretary of defense (USD). (Each of the echelons at the level of DSD and USD serve at the pleasure of the president, as does the secretary of defense (SECDEF).) The Defense Acquisition University (DAU) trains acquisition professionals for the Army as well.


title: United States Army Futures Command

text: In 2016 when RDECOM reported to AMC (instead of to AFC, as it does as of 2018), AMC instituted Life cycle management command (LCMC) of three of RDECOM's centers for aviation and missiles, electronics, and tanks: AMRDEC, CERDEC, and TARDEC respectively, as well as the three contracting functions for the three centers.


title: United States Army Futures Command

text: This Life Cycle Management (formulated in 2004) was intended to exert the kind of operational control (OPCON) needed just for the sustainment function (AMC's need for Readiness today), rather than for its relevance to modernization for the future, which is the focus of AFC. AFC now serves as the deciding authority when moving a project in its Life Cycle, out of the Acquisition phase and into the Sustainment phase.


title: United States Army Futures Command

text: Due to the COVID-19 pandemic, the Acquisition Executive, and the AFC commander created a COVID-19 task force to try to project supplier problems 30, 60, and 90 days out; they are respectively tracking 800 programs, and 35 priorities on a daily basis.


title: United States Army Futures Command

text: The CFTs, as prioritized 1 through 6 by the Chief of Staff of the Army (CSA), each have to consider constraints: a balance of requirements, acquisition, science and technology, test, resourcing, costing, and sustainment.


title: United States Army Futures Command

text: The Doctrine, Organization, Training, Materiel, Leadership and education, Personnel and Facilities (DOTMLPF) method of mission planning was instituted to quantify tradeoffs in joint planning. TRADOC's Mission Command CoE uses DOTMLPF.


title: United States Army Futures Command

text: DOTMLPF will be used for modernization of the Army beyond materiel alone, which (as of 2019) is the current focus of the CFTs.


title: United States Army Futures Command

text: The updated modernization strategy, to move from concept to doctrine as well, will be unveiled by summer 2019.


title: United States Army Futures Command

text: DOTMLPF (doctrine, organization, training, materiel, leadership and education, personnel, and facilities) itself is planned as a driver for modernization. The plan is to have an MDO-capable Army by 2028, and an MDO-ready Army by 2035.


title: United States Army Futures Command

text: TRADOC, ASA (ALT), and AFC are tied together in this process, according to 36th Vice Chief McConville. AFC will have to be "a little bit disruptive [but not upsetting to the existing order]" in order to institute reforms within budget in a timely way.


title: United States Army Futures Command

text: The Assistant Secretary of the Army for acquisition, logistics, and technology —ASA (ALT)— is the civilian executive overseeing both the acquisition and the sustainment processes of the Department of the Army (as of 2018: Dr. Bruce Jette was 6th ASA(ALT)). The ASA(ALT) will coordinate the acquisition portion of modernization reform with AFC.


title: United States Army Futures Command

text: Congress has given the Army Other Transaction Authority (OTA), which allows the PEOs to enter into Full Rate Production quicker by permitting the services to control their own programs of record, rather than DoD. This strips out one layer of bureaucracy as of 2018.


title: United States Army Futures Command

text: MTA (middle tier acquisition authority) is another tool available to Program Managers and Contracting Officers.


title: United States Army Futures Command

text: Besides the AFC CFTS, the Army Requirements Oversight Council (AROC) could also play a part in acquisition reform; as of September 2018 the Deputy Chief of Staff G-8 (DCS G-8), who leads AROC and JROC (Joint Requirements Oversight Council) has aligned with the priorities of AFC.


title: United States Army Futures Command

text: In addition, the Program Executive Officers (PEOs) of ASA (ALT) are to maintain a dotted-line relationship (i.e., coordination) with Futures Command.


title: United States Army Futures Command

text: There is now a PEO for Rapid Capabilities, to get rapid turnaround. The Rapid Capabilities Office (RCO)'s PEO gets two program managers, one for rapid prototyping, and one for rapid acquisition, of a capability. The Rapid capabilities office (RCO) does not develop its own requirements; rather, the RCO gets the requirements from the Cross-functional team (CFT). Rapid Capabilities (RCO) was headed by Tanya Skeen as PEO RCO but Skeen moved to DoD, in late 2018. In 2019 RCO became the Rapid Capabilities and Critical Technologies Office (RCCTO) Redstone Arsenal, headed by LTG L. Neil Thurgood, lately of ASA (ALT)'s Army Hypersonics office.


title: United States Army Futures Command

text: Then-CG of Army Futures Command (AFC) Gen. Murray announced full operational capability (FOC) 31 July 2019. By 2021 the Army's 40th Chief of Staff could lay out Waypoint 2028, and Aimpoint 2035 for the Army.


title: United States Army Futures Command

text: The Army G8 is monitoring just how producible (Milestone C) the upcoming materiel will be; for the moment, the G8 is funding the materiel. Follow-up on Modernization reviews is forthcoming, on a regular basis, according to the G8.


title: United States Army Futures Command

text: The progress in the top six priorities (long-range precision fires, Next Generation Combat Vehicle, Future Vertical Lift platforms, a mobile & expeditionary Army network, air & missile defense capabilities, and soldier lethality) being:


title: United States Army Futures Command

text: According to AFC, the mission of the Long Range Precision Fires (LRPF) CFT is to "deliver cutting-edge surface-to-surface (SSM) fires systems that will significantly increase range and effects over currently fielded US and adversary systems."


title: United States Army Futures Command

text: Based on Futures Command's development between July 2018 and December 2020, by 2023 the earliest versions of these weapons will be fielded:


title: United States Army Futures Command

text: The kill chains will take less than 1 minute, from detection of the target, to execution of the fires command; these operations will have the capability to precisely strike "command centers, air defenses, missile batteries, and logistics centers" nearly simultaneously.


title: United States Army Futures Command

text: The current M109A6 "Paladin" howitzer range is doubled in the M109A7 variant. An operational test of components of the Long range cannon was scheduled for 2020. The LRC is complementary to Extended range cannon artillery (ERCA), the M1299 Extended Range Cannon Artillery howitzer. Baseline ERCA is to enter service in 2023. Investigations for ERCA in 2025: rocket-boosted artillery shells: Tests of the Multiple launch rocket system (MLRS) XM30 rocket shell have demonstrated a near-doubling of the range of the munition, using the Tail controlled guided multiple launch rocket system, or TC-G. The TRADOC capability manager (TCM) Field Artillery Brigade - DIVARTY has been named a command position.


title: United States Army Futures Command

text: The LRHW has been named 'Dark Eagle' The first LRHW battery will start to receive its first operational rounds in early FY2023; all eight rounds for this battery will have been delivered by FY2023. By then, the PEO Missiles and Space will have picked up the LRHW program, for batteries two and three in FY'25 and FY'27, respectively. Battery one will first train, and then participate in the LRHW flight test launches in FY'22 and FY'23.


title: United States Army Futures Command

text: Firestorm was made possible by a mesh network—improvising a medium earth orbit (MEO, at 1200 mile altitude), and then a geosynchronous earth orbit (GEO, at 22,000 mile altitude) satellite link between Joint Base Lewis-McChord to Yuma Proving Ground.


title: United States Army Futures Command

text: Armored Multi-Purpose Vehicle (AMPV): in Limited User Tests General purpose variant supports Blue force tracking


title: United States Army Futures Command

text: An Advanced Powertrain Demonstrator, compact enough for AMPVs, Bradleys, OMFVs, or RCVs, can generate 1,000 horsepower from diesel. Alternatively, the demonstrator can generate electrical power: 160 kiloWatts for SHORAD high-energy lasers, or for propulsion of a 50-ton vehicle in quiet mode, for brief periods.


title: United States Army Futures Command

text: The JLTV was approved for full rate production in June 2019. Joint Modernization Command (JMC) is supporting a TCM Stryker study on the optimum number of JLTVs for light infantry brigades.


title: United States Army Futures Command

text: AFC's Futures and concepts center is proposing a strategy to guide the electrification of the GCVs, using the JLTV as an example for a step-by-step pathway and transition plan for electrification. Loren Thompson cautions that electrification per se could harm further fielding due to scope creep in specifications for the JLTV. The Army has not requested a hybrid electric JLTV.


title: United States Army Futures Command

text: The Maneuver CDID (MCDID) is undertaking the requirements development for electrification of Tactical and Combat Vehicles in September 2020; General Wesley had previously announced a plan in April 2020 for the modernization of Tactical and Combat Vehicles using the JLTV electrification plan as a prototype template of the electrification process.


title: United States Army Futures Command

text: After prototype JLTV electrification, the Army is seeking ideas for an electrified Light Reconnaissance Vehicle (LRV) by 2025. The LRVs would complement the Infantry Squad Vehicles (ISVs), and electrified versions of Stryker Infantry Carrier Vehicle - Dragoon which are already fielded. GM Defense has since converted one of its bid vehicles for the ISV to an all-electric version.


title: United States Army Futures Command

text: Mobile Protected Firepower approved by joint requirements oversight council. Two vendors were selected to build competing prototype light tanks (MPF), with contract award in 2022. A unit of 82nd Airborne Division will begin assessment of prototype MPFs beginning in March 2020. General Dynamics Land Systems will build 42 MPFs, a battalion of light infantry tanks by FY2025.


title: United States Army Futures Command

text: Optionally Manned Fighting Vehicle (OMFV): soliciting input, in requirements definition stage; the 2018 requirement was that 2 OMFVs fit in a C-17. A request for proposal for a vehicle prototype was placed 29 March 2019. On 16 January 2020 the Optionally Manned Fighting Vehicle solicitation was cancelled, as a middle tier acquisition in its early stage; the requirements and schedule are being revisited. The FY2021 budget request has been adjusted accordingly.


title: United States Army Futures Command

text: NGCV optionally manned fighting vehicle: OMFV is getting some industry silhouettes which may be incorporated in digital designs for 2023, prototypes by 2025. A fifth OMFV bidder (a small business) is still a contender in the competition, includes large consortia. However, Mark Cancian points out that OMFV might not be suitable for a pivot to the Pacific theater.


title: United States Army Futures Command

text: Robotic Combat Vehicles (RCVs): General Murray envisions that by FY2023 critical decisions will be made on RCVs after years of experimentation. Russia's Uran-9 (Уран-9) is not a robotic tank; rather it is an unmanned radio-controlled drone tank. A Next Generation main battle tank remains a § Future concept.


title: United States Army Futures Command

text: Future Vertical Lift (FVL) is a plan for a family of military helicopters for the United States Armed Forces using common elements such as sensors avionics and engines. Five different sizes of aircraft are to be developed, to replace the wide range of rotorcraft in use. The project began in 2009. By 2014, the SB-1 Defiant and V-280 Valor had been chosen as demonstrators. On 5 December 2022 Bell's V-280 was chosen for the Future Long-Range Assault Aircraft; the first phase of the contract award will be for a virtual prototype.


title: United States Army Futures Command

text: Future Vertical Lift will use the DoD modular open systems approach (MOSA), an integrated business and technical strategy in FARA, and in FLRAA Both FLRAA and FARA are to enter service by Fiscal Year 2030. By abstracting its requirements, the Army was able to request prototypes which used new technologies.


title: United States Army Futures Command

text: Joint Multi-Role Technology Demonstrator (JMR-TD) prototypes are to be built by two teams to replace Sikorsky UH-60 Blackhawks with Future Long-Range Assault Aircraft (FLRAA). Zephyr would have broken the world record for continuous flight, had it lasted just a few more hours. In April 2021 a Vanilla UAS demonstrated 45 hour 23 minute unrefueled flight from Point Mugu to the Pacific Fleet’s (PACFLT) UxS IBP 2021 exercise.


title: United States Army Futures Command

text: From 2022 to 2028, the Army, in multi-domain operation, has designated the Corps to be the Unit of Employment; the Division echelon to be the Unit of Action; Brigades are Units for Close combat. Divisions are purpose-built designs:


title: United States Army Futures Command

text: By 2022 the 25th Secretary of the Army, Christine Wormuth was able to announce the top six areas for the Army of 2030: 1) improved Intelligence, surveillance, and reconnaissance (ISR); 2) "Coordination at greater speed"; 3) "Win the Fires fight"; 4) §concealment via improved mobility and reduced signature; 5) "talk often and quickly"; and 6) logistics.


title: United States Army Futures Command

text: By October 2022 the 25th Secretary of the Army, Christine Wormuth could charge the 2nd commander of Futures Command with proponency for the Army of 2040.


title: United States Army Futures Command

text: AFC's headquarters is based in Austin, Texas where it spreads across three locations totaling 75,000 ft; One location is a University of Texas System building at 210 W. Seventh St. in downtown Austin, on the 15th and 19th floors; the UT Regents were not going to charge rent to AFC until December 2019. The command began initial operations on 1 July 2018.


title: United States Army Futures Command

text: In a hearing before Congress' House Armed Services Committee, the AFC commander projected that materiel will result from the value stream below, within a two-year time frame, from concept to Soldier. The commanding general is assisted by three deputy commanders.


title: United States Army Futures Command

text: Army Chief of Staff Milley is looking for AFC to attain full operational capability (FOC) by August 2019. By 2022 the Army's unclassified Multi-Domain Operations concept (which had been initially formulated by Futures Command) was disclosed to the public.


title: United States Army Futures Command

text: On 16 July 2018, Lieutenant General John M. Murray was nominated for promotion and appointment as Army Futures Command's first commanding general. and his appointment was confirmed on 20 August 2018 and he assumed command during the official activation ceremony of AFC on 24 August 2018, in Austin, Texas.


title: United States Army Futures Command

text: Murray relinquished command of AFC on 3 December 2021. Lieutenant General James E. Rainey (later promoted to general on 7 October 2022) became AFC's second commanding general on 4 October 2022.


title: Crackles

text: Crackles are the clicking, rattling, or crackling noises that may be made by one or both lungs of a human with a respiratory disease during inhalation. They are usually heard only with a stethoscope ("on auscultation"). Pulmonary crackles are abnormal breath sounds that were formerly referred to as rales.


title: Crackles

text: Bilateral crackles refers to the presence of crackles in both lungs. Basal crackles are crackles apparently originating in or near the base of the lung. Bibasal crackles refer to crackles at the bases of both the left and right lungs. Bilateral basal crackles also refers to the presence of "basal" crackles in both lungs.


title: Crackles

text: Crackles are caused by the "popping open" of small airways and alveoli collapsed by fluid, exudate, or lack of aeration during expiration.


title: Crackles

text: Crackles can be heard in patients with pneumonia, atelectasis, pulmonary fibrosis, acute bronchitis, bronchiectasis, acute respiratory distress syndrome (ARDS), interstitial lung disease or post thoracotomy or metastasis ablation. Pulmonary edema secondary to left-sided congestive heart failure can also cause crackles.


title: Crackles

text: René Laennec adopted the existing word "râles" (which has been translated as "rattles", 'groans" and otherwise) to describe the added breath sounds that are now referred to as "crackles". He described them using unusual daily examples, such as "whistling of little birds", "crackling of salt on a heated dish", "cooing of the woodpidgeon", etc., but he soon realized that he was unable to use the term in front of his patients because it conjured the association of "le râle de la mort", which translates to " the death rattle", the noise that people who are about to die make when they can no longer clear secretions. Therefore, at the bedside, he used the Latin word "rhonchus", which originally meant a 'snore'. That was not clearly understood by his translator, John Forbes, and the terminology became very confusing after the publication in the 1830s of Forbes's English translation of Laennec's "De L'Auscultation Mediate". The difficulty of translating "râle" itself had been remarked upon in a British review of Laennec's work in 1820.


title: Crackles

text: The terminology of "rales" and "rhonchi" in English remained variable until 1977, when a standardization was established by the American Thoracic Society and American College of Chest Physicians. As a result, the term "râles" was abandoned, and "crackles" became its recommended substitute. The term "rales" is still common in English-language medical literature, but cognizance of the ATS/CHEST guidelines calls for "crackles".


title: Crackles

text: In 2016, the European Respiratory Society reported on a study of various physicians listening to audiovisual recordings of auscultation findings and interobserver variation was analyzed. The study found that broad descriptions agreed better than detailed descriptions.


title: Crackles

text: Crackles are caused by explosive opening of small airways and are discontinuous, nonmusical, and brief. Crackles are much more common during the inspiratory than the expiratory phase of breathing, but they may be heard during the expiratory phase. Crackles are often associated with inflammation or infection of the small bronchi, bronchioles, and alveoli. Crackles that do not clear after a cough may indicate pulmonary edema or fluid in the alveoli due to heart failure, pulmonary fibrosis, or acute respiratory distress syndrome. Crackles that partially clear or change after coughing may indicate bronchiectasis.


title: Homology (mathematics)

text: In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.


title: Homology (mathematics)

text: The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes. Homology was originally a rigorous mathematical method for defining and categorizing holes in a manifold. Loosely speaking, a "cycle" is a closed submanifold, a "boundary" is a cycle which is also the boundary of a submanifold, and a "homology class" (which represents a hole) is an equivalence class of cycles modulo boundaries. A homology class is thus represented by a cycle which is not the boundary of any submanifold: the cycle represents a hole, namely a hypothetical manifold whose boundary would be that cycle, but which is "not there".


title: Homology (mathematics)

text: There are many different homology theories. A particular type of mathematical object, such as a topological space or a group, may have one or more associated homology theories. When the underlying object has a geometric interpretation as topological spaces do, the "n"th homology group represents behavior in dimension "n". Most homology groups or modules may be formulated as derived functors on appropriate abelian categories, measuring the failure of a functor to be exact. From this abstract perspective, homology groups are determined by objects of a derived category.


title: Homology (mathematics)

text: Homology theory can be said to start with the Euler polyhedron formula, or Euler characteristic. This was followed by Riemann's definition of genus and "n"-fold connectedness numerical invariants in 1857 and Betti's proof in 1871 of the independence of "homology numbers" from the choice of basis.


title: Homology (mathematics)

text: Homology itself was developed as a way to analyse and classify manifolds according to their "cycles" – closed loops (or more generally submanifolds) that can be drawn on a given "n" dimensional manifold but not continuously deformed into each other. These cycles are also sometimes thought of as cuts which can be glued back together, or as zippers which can be fastened and unfastened. Cycles are classified by dimension. For example, a line drawn on a surface represents a 1-cycle, a closed loop or formula_1 (1-manifold), while a surface cut through a three-dimensional manifold is a 2-cycle.


title: Homology (mathematics)

text: On the ordinary sphere formula_2, the cycle "b" in the diagram can be shrunk to the pole, and even the equatorial great circle "a" can be shrunk in the same way. The Jordan curve theorem shows that any arbitrary cycle such as "c" can be similarly shrunk to a point. All cycles on the sphere can therefore be continuously transformed into each other and belong to the same homology class. They are said to be homologous to zero. Cutting a manifold along a cycle homologous to zero separates the manifold into two or more components. For example, cutting the sphere along "a" produces two hemispheres.


title: Homology (mathematics)

text: This is not generally true of cycles on other surfaces. The torus formula_3 has cycles which cannot be continuously deformed into each other, for example in the diagram none of the cycles "a", "b" or "c" can be deformed into one another. In particular, cycles "a" and "b" cannot be shrunk to a point whereas cycle "c" can, thus making it homologous to zero.


title: Homology (mathematics)

text: If the torus surface is cut along both "a" and "b", it can be opened out and flattened into a rectangle or, more conveniently, a square. One opposite pair of sides represents the cut along "a", and the other opposite pair represents the cut along "b".


title: Homology (mathematics)

text: The edges of the square may then be glued back together in different ways. The square can be twisted to allow edges to meet in the opposite direction, as shown by the arrows in the diagram. Up to symmetry, there are four distinct ways of gluing the sides, each creating a different surface:


title: Homology (mathematics)

text: formula_4 is the Klein bottle, which is a torus with a twist in it (The twist can be seen in the square diagram as the reversal of the bottom arrow). It is a theorem that the re-glued surface must self-intersect (when immersed in Euclidean 3-space). Like the torus, cycles "a" and "b" cannot be shrunk while "c" can be. But unlike the torus, following "b" forwards right round and back reverses left and right, because "b" happens to cross over the twist given to one join. If an equidistant cut on one side of "b" is made, it returns on the other side and goes round the surface a second time before returning to its starting point, cutting out a twisted Möbius strip. Because local left and right can be arbitrarily re-oriented in this way, the surface as a whole is said to be non-orientable.


title: Homology (mathematics)

text: The projective plane formula_5 has both joins twisted. The uncut form, generally represented as the Boy surface, is visually complex, so a hemispherical embedding is shown in the diagram, in which antipodal points around the rim such as "A" and "A′" are identified as the same point. Again, "a" and "b" are non-shrinkable while "c" is. But this time, both "a" and "b" reverse left and right.


title: Homology (mathematics)

text: Cycles can be joined or added together, as "a" and "b" on the torus were when it was cut open and flattened down. In the Klein bottle diagram, "a" goes round one way and −"a" goes round the opposite way. If "a" is thought of as a cut, then −"a" can be thought of as a gluing operation. Making a cut and then re-gluing it does not change the surface, so "a" + (−"a") = 0.


title: Homology (mathematics)

text: But now consider two "a"-cycles. Since the Klein bottle is nonorientable, you can transport one of them all the way round the bottle (along the "b"-cycle), and it will come back as −"a". This is because the Klein bottle is made from a cylinder, whose "a"-cycle ends are glued together with opposite orientations. Hence 2"a" = "a" + "a" = "a" + (−"a") = 0. This phenomenon is called torsion. Similarly, in the projective plane, following the unshrinkable cycle "b" round twice remarkably creates a trivial cycle which "can" be shrunk to a point; that is, "b" + "b" = 0. Because "b" must be followed around twice to achieve a zero cycle, the surface is said to have a torsion coefficient of 2. However, following a "b"-cycle around twice in the Klein bottle gives simply "b" + "b" = 2"b", since this cycle lives in a torsion-free homology class. This corresponds to the fact that in the fundamental polygon of the Klein bottle, only one pair of sides is glued with a twist, whereas in the projective plane both sides are twisted.


title: Homology (mathematics)

text: A square is a contractible topological space, which implies that it has trivial homology. Consequently, additional cuts disconnect it. The square is not the only shape in the plane that can be glued into a surface. Gluing opposite sides of an octagon, for example, produces a surface with two holes. In fact, all closed surfaces can be produced by gluing the sides of some polygon and all even-sided polygons (2"n"-gons) can be glued to make different manifolds. Conversely, a closed surface with "n" non-zero classes can be cut into a 2"n"-gon. Variations are also possible, for example a hexagon may also be glued to form a torus.


title: Homology (mathematics)

text: The first recognisable theory of homology was published by Henri Poincaré in his seminal paper "Analysis situs", "J. Ecole polytech." (2) 1. 1–121 (1895). The paper introduced homology classes and relations. The possible configurations of orientable cycles are classified by the Betti numbers of the manifold (Betti numbers are a refinement of the Euler characteristic). Classifying the non-orientable cycles requires additional information about torsion coefficients.


title: Homology (mathematics)

text: A manifold with boundary or open manifold is topologically distinct from a closed manifold and can be created by making a cut in any suitable closed manifold. For example the disk or 2-ball formula_7 is bounded by a circle formula_1. It may be created by cutting a trivial cycle in any 2-manifold and keeping the piece removed, by piercing the sphere and stretching the puncture wide, or by cutting the projective plane. It can also be seen as filling-in the circle in the plane.


title: Homology (mathematics)

text: When two cycles can be continuously deformed into each other, then cutting along one produces the same shape as cutting along the other, up to some bending and stretching. In this case the two cycles are said to be or to lie in the same . Additionally, if one cycle can be continuously deformed into a combination of other cycles, then cutting along the initial cycle is the same as cutting along the combination of other cycles. For example, cutting along a figure 8 is equivalent to cutting along its two lobes. In this case, the figure 8 is said to be homologous to the sum of its lobes.


title: Homology (mathematics)

text: Two open manifolds with similar boundaries (up to some bending and stretching) may be glued together to form a new manifold which is their connected sum.


title: Homology (mathematics)

text: This geometric analysis of manifolds is not rigorous. In a search for increased rigour, Poincaré went on to develop the simplicial homology of a triangulated manifold and to create what is now called a chain complex. These chain complexes (since greatly generalized) form the basis for most modern treatments of homology.


title: Homology (mathematics)

text: In such treatments a cycle need not be continuous: a 0-cycle is a set of points, and cutting along this cycle corresponds to puncturing the manifold. A 1-cycle corresponds to a set of closed loops (an image of the 1-manifold formula_1). On a surface, cutting along a 1-cycle yields either disconnected pieces or a simpler shape. A 2-cycle corresponds to a collection of embedded surfaces such as a sphere or a torus, and so on.


title: Homology (mathematics)

text: Emmy Noether and, independently, Leopold Vietoris and Walther Mayer further developed the theory of algebraic homology groups in the period 1925–28. The new combinatorial topology formally treated topological classes as abelian groups. Homology groups are finitely generated abelian groups, and homology classes are elements of these groups. The Betti numbers of the manifold are the rank of the free part of the homology group, and the non-orientable cycles are described by the torsion part.


title: Homology (mathematics)

text: The subsequent spread of homology groups brought a change of terminology and viewpoint from "combinatorial topology" to "algebraic topology". Algebraic homology remains the primary method of classifying manifolds.


title: Homology (mathematics)

text: The homology of a topological space "X" is a set of topological invariants of "X" represented by its "homology groups"


title: Homology (mathematics)

text: where the formula_11 homology group formula_12 describes, informally, the number of holes in "X" with a "k"-dimensional boundary. A 0-dimensional-boundary hole is simply a gap between two components. Consequently, formula_13 describes the path-connected components of "X".


title: Homology (mathematics)

text: A one-dimensional sphere formula_1 is a circle. It has a single connected component and a one-dimensional-boundary hole, but no higher-dimensional holes. The corresponding homology groups are given as


title: Homology (mathematics)

text: where formula_16 is the group of integers and formula_17 is the trivial group. The group formula_18 represents a finitely-generated abelian group, with a single generator representing the one-dimensional hole contained in a circle.


title: Homology (mathematics)

text: A two-dimensional sphere formula_2 has a single connected component, no one-dimensional-boundary holes, a two-dimensional-boundary hole, and no higher-dimensional holes. The corresponding homology groups are


title: Homology (mathematics)

text: A two-dimensional ball formula_7 is a solid disc. It has a single path-connected component, but in contrast to the circle, has no higher-dimensional holes. The corresponding homology groups are all trivial except for formula_24. In general, for an "n"-dimensional ball formula_25


title: Homology (mathematics)

text: The torus is defined as a product of two circles formula_27. The torus has a single path-connected component, two independent one-dimensional holes (indicated by circles in red and blue) and one two-dimensional hole as the interior of the torus. The corresponding homology groups are


title: Homology (mathematics)

text: The two independent 1-dimensional holes form independent generators in a finitely-generated abelian group, expressed as the product group formula_29


title: Homology (mathematics)

text: For the projective plane "P", a simple computation shows (where formula_30 is the cyclic group of order 2):


title: Homology (mathematics)

text: formula_32 corresponds, as in the previous examples, to the fact that there is a single connected component. formula_33 is a new phenomenon: intuitively, it corresponds to the fact that there is a single non-contractible "loop", but if we do the loop twice, it becomes contractible to zero. This phenomenon is called torsion.


title: Homology (mathematics)

text: The following text describes a general algorithm for constructing the homology groups. It may be easier for the reader to look at some simple examples first: graph homology and simplicial homology.


title: Homology (mathematics)

text: The general construction begins with an object such as a topological space "X", on which one first defines a "C"("X") encoding information about "X". A chain complex is a sequence of abelian groups or modules formula_34. connected by homomorphisms formula_35 which are called boundary operators. That is,


title: Homology (mathematics)

text: where 0 denotes the trivial group and formula_37 for "i" < 0. It is also required that the composition of any two consecutive boundary operators be trivial. That is, for all "n",


title: Homology (mathematics)

text: The statement that the boundary of a boundary is trivial is equivalent to the statement that formula_41, where formula_42 denotes the image of the boundary operator and formula_43 its kernel. Elements of formula_44 are called boundaries and elements of formula_45 are called cycles.


title: Homology (mathematics)

text: Since each chain group "C" is abelian all its subgroups are normal. Then because formula_43 is a subgroup of "C", formula_43 is abelian, and since formula_48 therefore formula_42 is a normal subgroup of formula_43. Then one can create the quotient group


title: Homology (mathematics)

text: called the n"th homology group of "X. The elements of "H"("X") are called homology classes. Each homology class is an equivalence class over cycles and two cycles in the same homology class are said to be homologous.


title: Homology (mathematics)

text: A chain complex is said to be exact if the image of the ("n"+1)th map is always equal to the kernel of the "n"th map. The homology groups of "X" therefore measure "how far" the chain complex associated to "X" is from being exact.


title: Homology (mathematics)

text: The reduced homology groups of a chain complex "C"("X") are defined as homologies of the augmented chain complex


title: Homology (mathematics)

text: for a combination formula_55 of points formula_56 which are the fixed generators of "C". The reduced homology groups formula_57 coincide with formula_58 for formula_59 The extra formula_16 in the chain complex represents the unique map formula_61 from the empty simplex to "X".


title: Homology (mathematics)

text: Computing the cycle formula_62 and boundary formula_63 groups is usually rather difficult since they have a very large number of generators. On the other hand, there are tools which make the task easier.


title: Homology (mathematics)

text: The "simplicial homology" groups "H"("X") of a "simplicial complex" "X" are defined using the simplicial chain complex "C"("X"), with "C"("X") the free abelian group generated by the "n"-simplices of "X". See simplicial homology for details.


title: Homology (mathematics)

text: The "singular homology" groups "H"("X") are defined for any topological space "X", and agree with the simplicial homology groups for a simplicial complex.


title: Homology (mathematics)

text: Cohomology groups are formally similar to homology groups: one starts with a cochain complex, which is the same as a chain complex but whose arrows, now denoted formula_64 point in the direction of increasing "n" rather than decreasing "n"; then the groups formula_65 of "cocycles" and formula_66 of follow from the same description. The "n"th cohomology group of "X" is then the quotient group


title: Homology (mathematics)

text: Homotopy groups are similar to homology groups in that they can represent "holes" in a topological space. There is a close connection between the first homotopy group formula_68 and the first homology group formula_69: the latter is the abelianization of the former. Hence, it is said that "homology is a commutative alternative to homotopy". The higher homotopy groups are abelian and are related to homology groups by the Hurewicz theorem, but can be vastly more complicated. For instance, the homotopy groups of spheres are poorly understood and are not known in general, in contrast to the straightforward description given above for the homology groups.


title: Homology (mathematics)

text: As an example, let "X" be the figure eight. Its first homotopy group formula_68 is the group of directed loops starting and ending at a predetermined point (e.g. its center). It is equivalent to the free group of rank 2, which is not commutative: looping around the leftmost cycle and then around the rightmost cycle is different than looping around the rightmost cycle and then looping around the leftmost cycle. In contrast, its first homology group formula_69 is the group of cuts made in a surface. This group is commutative, since (informally) cutting the leftmost cycle and then the rightmost cycle leads to the same result as cutting the rightmost cycle and then the leftmost cycle.


title: Homology (mathematics)

text: The different types of homology theory arise from functors mapping from various categories of mathematical objects to the category of chain complexes. In each case the composition of the functor from objects to chain complexes and the functor from chain complexes to homology groups defines the overall homology functor for the theory.


title: Homology (mathematics)

text: The motivating example comes from algebraic topology: the simplicial homology of a simplicial complex "X". Here the chain group "C" is the free abelian group or module whose generators are the "n"-dimensional oriented simplexes of "X". The orientation is captured by ordering the complex's vertices and expressing an oriented simplex formula_72 as an "n"-tuple formula_73 of its vertices listed in increasing order (i.e. formula_74 in the complex's vertex ordering, where formula_75 is the formula_76th vertex appearing in the tuple). The mapping formula_77 from "C" to "C" is called the and sends the simplex


title: Homology (mathematics)

text: which is considered 0 if formula_80 This behavior on the generators induces a homomorphism on all of "C" as follows. Given an element formula_81, write it as the sum of generators formula_82 where formula_83 is the set of "n"-simplexes in "X" and the "m" are coefficients from the ring "C" is defined over (usually integers, unless otherwise specified). Then define


title: Homology (mathematics)

text: The dimension of the "n"-th homology of "X" turns out to be the number of "holes" in "X" at dimension "n". It may be computed by putting matrix representations of these boundary mappings in Smith normal form.


title: Homology (mathematics)

text: Using simplicial homology example as a model, one can define a "singular homology" for any topological space "X". A chain complex for "X" is defined by taking "C" to be the free abelian group (or free module) whose generators are all continuous maps from "n"-dimensional simplices into "X". The homomorphisms ∂ arise from the boundary maps of simplexes.


title: Homology (mathematics)

text: In abstract algebra, one uses homology to define derived functors, for example the Tor functors. Here one starts with some covariant additive functor "F" and some module "X". The chain complex for "X" is defined as follows: first find a free module formula_85 and a surjective homomorphism formula_86 Then one finds a free module formula_87 and a surjective homomorphism formula_88 Continuing in this fashion, a sequence of free modules formula_89 and homomorphisms formula_90 can be defined. By applying the functor "F" to this sequence, one obtains a chain complex; the homology formula_91 of this complex depends only on "F" and "X" and is, by definition, the "n"-th derived functor of "F", applied to "X".


title: Homology (mathematics)

text: A common use of group (co)homology formula_92is to classify the possible extension groups "E" which contain a given "G"-module "M" as a normal subgroup and have a given quotient group "G", so that formula_93


title: Homology (mathematics)

text: Chain complexes form a category: A morphism from the chain complex (formula_94) to the chain complex (formula_95) is a sequence of homomorphisms formula_96 such that formula_97 for all "n". The "n"-th homology "H" can be viewed as a covariant functor from the category of chain complexes to the category of abelian groups (or modules).


title: Homology (mathematics)

text: If the chain complex depends on the object "X" in a covariant manner (meaning that any morphism formula_98 induces a morphism from the chain complex of "X" to the chain complex of "Y"), then the "H" are covariant functors from the category that "X" belongs to into the category of abelian groups (or modules).


title: Homology (mathematics)

text: The only difference between homology and cohomology is that in cohomology the chain complexes depend in a "contravariant" manner on "X", and that therefore the homology groups (which are called "cohomology groups" in this context and denoted by "H") form "contravariant" functors from the category that "X" belongs to into the category of abelian groups or modules.


title: Homology (mathematics)

text: If (formula_94) is a chain complex such that all but finitely many "A" are zero, and the others are finitely generated abelian groups (or finite-dimensional vector spaces), then we can define the "Euler characteristic"