mayer sequence mathematics particularly theory mayer sequence tool help spaces known groups result austrian method consists splitting space groups easier sequence relates groups space groups natural long exact sequence entries groups space direct sum groups groups intersection mayer sequence holds variety theories including singular singular general sequence holds theories satisfying variations reduced relative spaces directly definitions uses tools mayer sequence hope obtaining partial information spaces encountered constructed simple patches carefully choosing covering intersection simpler space allow complete space respect mayer sequence analogous van fundamental group precise relation exists dimension background motivation history like fundamental group higher groups space groups important theories tools linear important theories especially singular directly definition spaces singular singular chains cycles groups big handle directly subtle indirect approaches necessary mayer sequence approach giving partial information groups space relating groups intersection natural convenient way express relation involves concept exact sequences sequences objects case groups case group image equals general allow groups space completely important spaces encountered complexes complexes constructed simple patches mayer potentially broad deep mayer introduced colleague attending lectures local university vienna told result way solution solved question numbers applied results considered union cylinders later proved result groups express exact sequence concept exact sequence appeared print book foundations results mayer expressed modern form basic versions singular let space interiors cover interiors need mayer sequence singular long exact sequence relating singular groups group spaces intersection reduced version version mayer sequence states following sequence exact formula maps inclusion maps formula denotes direct sum groups boundary map boundary maps lowering dimension explicit follows element class cycle subdivision example written sum chains images lie wholly respectively implies images boundary contained intersection class choosing affect implies lie class choosing different representative notice maps mayer sequence depend choosing order particular boundary map changes sign swapped reduced version reduced mayer sequence assumption non intersection sequence identical positive dimensions ends formula analogy van analogy mayer sequence especially groups dimension van path connected reduced mayer sequence yields formula formula precisely statement van compare fact fundamental group path connected basic applications sphere completely sphere let intersection equivalent sphere dimensional discs groups trivial mayer sequence reduced groups yields formula immediately implies map reduced sphere points base case follows formula delta complete understanding groups spheres stark contrast current knowledge groups spheres especially case little known bottle slightly difficult application mayer sequence calculation groups bottle uses union strips boundary circle illustration right intersection equivalent circles sequence yields formula trivial implies dimensions greater central map sends boundary circle band twice core circle particular dimension finally choosing basis follows formula wedge sums let wedge sum spaces suppose furthermore identified open neighborhoods letting follows construction reduced version sequence yields formula dimensions illustration right shows sum spheres specific case result spheres formula suspension space let double cone respectively union intersection equivalent mayer sequence yields formula illustration right shows sphere suspension sphere general sphere suspension easy derive groups sphere induction discussion relative form relative form mayer sequence exists union exact sequence formula groups natural sense continuous map canonical map groups composition composition formula mayer sequence natural sense mapping connecting mayer sequence following horizontal maps usual ones formula versions mayer long exact sequence singular groups group dual version following formula dimension preserving maps restriction maps induced boundary maps defined similar fashion version relative important special case group real numbers underlying space additional structure smooth mayer sequence formula open cover denotes restriction map difference map defined similarly map briefly described follows class represented closed form express difference forms partition unity subordinate open cover example exterior derivative agree define form consider long exact sequence associated short exact sequences chain groups constituent groups chain complexes formula chain group consisting sums chains chains fact singular images contained generate group words gives mayer sequence singular applied short exact sequences spaces differential forms formula yields mayer sequence formal point view mayer sequence derived theories long exact sequence theories mayer sequence require dimension addition existing ordinary theories holds extraordinary theories theory point view mayer sequence related specifically arises sequence relates called mayer sequence case open cover consists open sets sequence exists arbitrary
Mayer–Vietoris sequence