groups spheres mathematical field groups spheres describe spheres dimensions wrap examples reflect terms structure spheres viewed spaces precise geometry unlike groups groups surprisingly complex difficult dimensional unit sphere called sphere denoted familiar circle ordinary sphere sphere defined set points space dimension located unit distance origin group different ways dimensional sphere mapped continuously dimensional sphere summary distinguish continuously deformed classes summarized addition operation defined classes makes set classes group problem determining falls regimes depending equal greater mapping continuously constant mapping mapping maps single point map degree measures times sphere wrapped degree identifies group addition example point circle mapped continuously point circle point moved circle second point cycle times second circle depending particular mapping interesting surprising results occur surprise discovery mapping called sphere usual sphere non trivial fashion equivalent point mapping question computing group positive turned central question contributed development fundamental techniques served focus research main discoveries groups independent called stable groups spheres values stable groups form ring extraordinary theory called stable theory unstable groups erratic modern use sequences technique applied groups spheres important patterns established remains unknown unexplained background study groups spheres builds great deal background material briefly reviewed provides larger context built abstract groups basic example sphere ordinary sphere dimensional space surface solid ball example sphere means geometry defines sphere shape alternatives implicit surface set points dimensional space found exactly unit away origin called sphere reasons given idea applies dimension equation produces sphere geometric object space example sphere circle disk collapsed rim written construction moves geometry pure disk region contained circle described inequality rim boundary circle described equality balloon spread flat produces disk construction repairs like pulling slash pronounced means space left disk join points right circle region dimensional calls resulting space sphere produces example line segment construction joins ends circle equivalent description boundary dimensional disk point producing complex suspension equator written construction simple great theoretical importance circle equator sweep point point producing northern hemisphere point producing southern hemisphere positive sphere equator suspension produces theory requires selecting fixed point sphere calling pair sphere point pointed sphere spaces choice matters sphere points equivalent choice matter convenience point equator spheres works geometric spheres collapsed rim disk obvious choice group distinguishing feature space continuity structure terms open sets neighborhoods continuous map function spaces preserves continuity continuous path continuous maps maps connected said idea common concepts variations affect outcomes interest important practical example complex analysis closed curves continuous maps circle complex plane closed curves produce integral result space consisting plane minus points group fundamental group path connected space begins continuous maps pointed circle pointed space maps pair map maps closed curves grouped classes based keeping base point fixed maps class point distinguished class distinguished maps curves constant map called classes abstract group introduction addition defined equator pinch pinch maps equator pointed sphere circle distinguished point producing spheres pointed spheres joined distinguished point maps added map upper lower spheres separately agreeing distinguished point composition pinch gives sum map generally group begins pointed sphere follows procedure class acts identity group addition equal positive groups spheres groups generated maps group consists element called trivial group continuous map spaces group associated groups particular map continuous spaces groups real plane exactly groups solitary point space dimension real plane point removed groups circle groups distinguish spaces loss discrimination power unfortunate certain easier low dimensional examples low dimensional examples groups spheres provide sense subject special cases ordinary dimensional space mathematical capture possible complexity maps spheres simplest case concerns ways circle sphere wrapped circle wrapping rubber band finger wrapped twice times wrapping directions opposite directions cancel group infinite group group addition class identified counting number times mapping class circle thought winding number loop origin plane identification group group written equality sphere sphere wrapping plastic bag ball sealing sealed bag equivalent sphere surface ball bag wrapped twisting wrapping ball requirement continuous map bag allowed pass twist directions opposite twists cancel total number twists cancellation called degree mapping case circle circle degree identifies group group results continuous mapping circle ordinary sphere continuously deformed point mapping class trivial way imagine rubber band wrapped ball band slid ball group trivial group element identity element identified consisting number zero group denoted result higher dimensions lower dimensional sphere sphere higher dimension similarly trivial interesting cases groups spheres involve higher dimensional sphere lower dimension unfortunately example easily interesting ordinary sphere circle real line universal cover type point addition simply connected lifting criterion map lifted map real line descends space example concerns sphere ordinary sphere discovered constructed map known map generates group history late 19th century introduced notion notion group language group theory connor rigorous approach adopted set papers analysis related concepts fundamental group introduced connor higher groups defined paper withdrawn advice grounds groups right fundamental group credited introduction groups paper groups important method calculating groups concept stable finds properties independent dimensions typically hold larger dimensions result suspension published stable flourished important results showed range groups spheres sequences groups finite exceptions worked area included stable groups known unknown larger stable groups general theory noted trivial group reason continuous mapping sphere sphere deformed consequently image contained point removed space mapping space deformed point mapping case noted easy consequence links groups groups generally easier particular shows simply connected space group group sphere immediately implies groups trivial came great surprise historically corresponding groups trivial general case real importance higher groups surprisingly complex difficult effort generated significant new mathematics table following table gives idea complexity higher groups spheres dimension table entries trivial group infinite group finite groups order written direct products groups written example extended tables groups spheres given end article rows table straightforward groups dimensional sphere trivial base point preserving map sphere sphere point mapping similarly groups sphere trivial universal covering space higher groups rows higher groups appear chaotic fact patterns obvious subtle groups black line constant indicated red green blue coloring groups finite unstable groups main diagonal immediately line highlighted yellow fourth rows table starting column induced patterns follow different theoretical results stable unstable groups fact groups line table constant explained suspension implies suspension groups called stable groups spheres denoted finite groups numerous cases general pattern elusive stable groups groups called unstable groups spheres classical fiber bundle formula general theory fiber shows long exact sequence groups formula specific bundle group induced inclusion maps zero lower dimensional sphere deformed point inside higher dimensional corresponds long exact sequence breaks short exact sequences formula suspension sequences split suspension giving formula row shows observed constructed follows pairs complex numbers form sphere cover complex plane plus infinity sphere map sends pair ratio similarly formula formula constructed pairs instead complex numbers zero long exact sequences break families split short exact sequences implying families relations formula formula base space exist relations true fail example formula formula non trivial case problem imply failed relation true framed groups spheres closely related classes established group group classes framed normal bundle map map formula framed dimensional example group framed dimensional sum points corresponding degree maps formula projection formula represents generator corresponds framed dimensional defined standard formula normal plane bundle advent sophisticated methods early 1950s main tool computing groups spheres expressing groups classes groups spaces recent work argument usually reversed groups terms groups showed groups spheres finite form positive group product infinite group finite group particular groups determined components components hardest ways behave differently components odd paper found place occurs groups dimensional spheres showing unique order case dimensional spheres slightly different occurs case odd precise results case big difference odd dimensional spheres odd prime elements component order moore sense best possible result groups known elements order values furthermore stable range extended case odd double suspension components equality holds intermediate group strictly larger results odd hold odd dimensional spheres dimensional spheres james gives odd terms odd dimensional spheres formula means component exact sequence similar ones coming difference works dimensional spheres albeit expense ignoring combining results odd dimensional spheres shows odd unstable groups determined odd stable groups stable groups precise results example prime primary component stable group case order important image denotes special group adams stable range groups depend period pattern known reflected stable groups spheres image group order trivial group order equal number case accounts elements unusually large finite order values example stable groups order stable groups spheres direct sum image adams groups roughly speaking image understood easy elements stable groups understood elements account elements stable groups spheres small dimensions image considered hard stable groups spheres adams adams introduced certain order elements considered understood tables groups spheres easy save space ring structure direct sum formula stable groups spheres graded ring given composition representing maps element non zero degree complex implies example generator order generates times generator zero group trivial elements elements element stable group defined addition products certain elements composition product label elements groups higher elements defined suitable lower parallels theory products element stable groups spheres expressed composition products higher terms certain known elements called elements methods finite complex finite fundamental group particular sphere dimension groups generated groups groups components prime calculating groups separately groups spheres hoc variations ideas point methods computing groups spheres based sequences usually constructing suitable taking associated long exact sequences groups sequences systematic way organizing complicated information process generates method killing groups involves repeatedly non trivial group killing eliminating involving space principle gives effective computing groups finite simply connected complex practice use computing groups complex complicated time kills group sequence prove results mentioned previously fact taking loop space behaved space shifts groups group space group repeated loop space equal group loop space reduces calculation groups calculation groups repeated loop spaces sequence relates space loop space loop spaces sequence tends non zero hard control appear higher groups consequently superseded powerful sequences fewer non zero information sequence groups spheres based calculations groups classical adams sequence term given groups closely related component stable groups initial terms adams sequence hard auxiliary sequence called sequence adams sequence powerful version adams sequence replacing ordinary theory complex usually piece called brown initial term hard use sequence chapter variation approach uses backwards version adams sequence brown limit known initial terms involve unknown stable groups spheres trying find approach components stable groups unfortunately mistake calculations 54th stem corrected groups reduced group theory question identify groups certain groups correspondence element represented disk example map corresponds rings applications winding number corresponding prove fundamental states non constant complex zero fact implies fixed point continuous map dimensional ball fixed point stable groups spheres important theory studies structure singular points smooth maps varieties arise critical points smooth maps geometry near critical point map described element considering way small sphere critical point maps sphere critical value fact stable group spheres order proved implies signature compact smooth spin stable groups spheres describe group classes oriented spheres group smooth structures spheres orientation preserving non trivial elements group represented exotic spheres precisely map formula represented spheres bound stable group spheres image form case image index groups stable groups spheres classification possible smooth structures linear problem existence dimensions reduced question stable groups spheres example knowledge stable groups degree settle problem dimension smallest value question open time says stable groups spheres expressed terms plus construction applied classifying space group leading identification theory field element stable groups table groups tables groups spheres organized showing following table shows groups tables based table groups spheres stable groups highlighted blue unstable ones red group product groups orders given table following conventions entry denotes trivial group entry group group order generally written entry group infinite group entry product group product direct sum groups orders powers indicate repeated products note common factor example denoted table table stable groups stable groups product groups infinite prime power orders shown table largely historical reasons stable groups usually given products groups prime power order tables unstable groups products smallest number groups main complexity components components range table accounted order divides components found errors corrected components behavior table comes image general references allen isbn course chicago lectures mathematics revised isbn historical papers die der die berlin stable history science isbn
Homotopy groups of spheres