Eilenberg–Zilber theorem

Eilenberg–Zilber theorem In mathematics, specifically in algebraic topology, the Eilenberg–Zilber theorem is an important result in establishing the link between the homology groups of a product space {Anzeigestil X mal Y} and those of the spaces {Anzeigestil X} und {Anzeigestil Y} . The theorem first appeared in a 1953 paper in the American Journal of Mathematics by Samuel Eilenberg and Joseph A. Zilber. One possible route to a proof is the acyclic model theorem.

Inhalt 1 Aussage des Theorems 2 Statement in terms of composite maps 3 The coproduct 4 Statement in cohomology 5 Verallgemeinerungen 6 Konsequenzen 7 Siehe auch 8 References Statement of the theorem The theorem can be formulated as follows. Vermuten {Anzeigestil X} und {Anzeigestil Y} are topological spaces, Then we have the three chain complexes {Anzeigestil C_{*}(X)} , {Anzeigestil C_{*}(Y)} , und {Anzeigestil C_{*}(X mal Y)} . (The argument applies equally to the simplicial or singular chain complexes.) We also have the tensor product complex {Anzeigestil C_{*}(X)otimes C_{*}(Y)} , whose differential is, per Definition, {Anzeigestil teilweise _{C_{*}(X)otimes C_{*}(Y)}(sigma otimes tau )=partial _{X}sigma otimes tau +(-1)^{p}sigma otimes partial _{Y}Jawohl } zum {displaystyle sigma in C_{p}(X)} und {Anzeigestil teilweise _{X}} , {Anzeigestil teilweise _{Y}} the differentials on {Anzeigestil C_{*}(X)} , {Anzeigestil C_{*}(Y)} .

Then the theorem says that we have chain maps {displaystyle Fcolon C_{*}(X mal Y)rightarrow C_{*}(X)otimes C_{*}(Y),quad Gcolon C_{*}(X)otimes C_{*}(Y)rightarrow C_{*}(X mal Y)} so dass {Anzeigestil FG} is the identity and {displaystyle GF} is chain-homotopic to the identity. Darüber hinaus, the maps are natural in {Anzeigestil X} und {Anzeigestil Y} . Consequently the two complexes must have the same homology: {Anzeigestil H_{*}(C_{*}(X mal Y))cong H_{*}(C_{*}(X)otimes C_{*}(Y)).} Statement in terms of composite maps The original theorem was proven in terms of acyclic models but more mileage was gotten in a phrasing by Eilenberg and Mac Lane using explicit maps. The standard map {Anzeigestil F} they produce is traditionally referred to as the Alexander–Whitney map and {Anzeigestil G} the Eilenberg–Zilber map. The maps are natural in both {Anzeigestil X} und {Anzeigestil Y} and inverse up to homotopy: hat man {displaystyle FG=mathrm {Ich würde} _{C_{*}(X)otimes C_{*}(Y)},qquad GF-mathrm {Ich würde} _{C_{*}(X mal Y)}=partial _{C_{*}(X)otimes C_{*}(Y)}H+Hpartial _{C_{*}(X)otimes C_{*}(Y)}} for a homotopy {Anzeigestil H} natural in both {Anzeigestil X} und {Anzeigestil Y} such that further, each of {displaystyle HH} , {displaystyle FH} , und {displaystyle HG} ist Null. This is what would come to be known as a contraction or a homotopy retract datum.

The coproduct The diagonal map {displaystyle Delta colon Xto Xtimes X} induces a map of cochain complexes {Anzeigestil C_{*}(X)to C_{*}(Xtimes X)} die, followed by the Alexander–Whitney {Anzeigestil F} yields a coproduct {Anzeigestil C_{*}(X)to C_{*}(X)otimes C_{*}(X)} inducing the standard coproduct on {Anzeigestil H_{*}(X)} . With respect to these coproducts on {Anzeigestil X} und {Anzeigestil Y} , the map {Anzeigestil H_{*}(X)otimes H_{*}(Y)to H_{*}{groß (}C_{*}(X)otimes C_{*}(Y){groß )} {overset {sim }{zu }} H_{*}(X mal Y)} , also called the Eilenberg–Zilber map, becomes a map of differential graded coalgebras. The composite {Anzeigestil C_{*}(X)to C_{*}(X)otimes C_{*}(X)} itself is not a map of coalgebras.

Statement in cohomology The Alexander–Whitney and Eilenberg–Zilber maps dualize (over any choice of commutative coefficient ring {Anzeigestil k} with unity) to a pair of maps {Anzeigestil G^{*}colon C^{*}(X mal Y)rightarrow {groß (}C_{*}(X)otimes C_{*}(Y){groß )}^{*},quad F^{*}colon {groß (}C_{*}(X)otimes C_{*}(Y){groß )}^{*}rightarrow C^{*}(X mal Y)} which are also homotopy equivalences, as witnessed by the duals of the preceding equations, using the dual homotopy {Anzeigestil H^{*}} . The coproduct does not dualize straightforwardly, because dualization does not distribute over tensor products of infinitely-generated modules, but there is a natural injection of differential graded algebras {displaystyle icolon C^{*}(X)otimes C^{*}(Y)zu {groß (}C_{*}(X)otimes C_{*}(Y){groß )}^{*}} gegeben von {displaystyle alpha otimes beta mapsto (sigma otimes tau mapsto alpha (Sigma )Beta (Jawohl ))} , the product being taken in the coefficient ring {Anzeigestil k} . This {Anzeigestil i} induces an isomorphism in cohomology, so one does have the zig-zag of differential graded algebra maps {Anzeigestil C^{*}(X)otimes C^{*}(X) {overset {ich}{zu }} {groß (}C_{*}(X)otimes C_{*}(X){groß )}^{*} {overset {G^{*}}{leftarrow }} C^{*}(Xtimes X){overset {C^{*}(Delta )}{zu }}C^{*}(X)} inducing a product {displaystyle smile colon H^{*}(X)otimes H^{*}(X)to H^{*}(X)} in cohomology, known as the cup product, Weil {Anzeigestil H^{*}(ich)} und {Anzeigestil H^{*}(G)} are isomorphisms. Ersetzen {Anzeigestil G^{*}} mit {displaystyle F^{*}} so the maps all go the same way, one gets the standard cup product on cochains, given explicitly by {displaystyle alpha otimes beta mapsto {Groß (}sigma mapsto (alpha otimes beta )(F^{*}Delta ^{*}Sigma )= Summe _{p=0}^{dim sigma }Alpha (Sigma |_{Delta ^{[0,p]}})cdot beta (Sigma |_{Delta ^{[p,dim sigma ]}}){Groß )}} , die, since cochain evaluation {Anzeigestil C^{p}(X)otimes C_{q}(X)to k} vanishes unless {Anzeigestil p=q} , reduces to the more familiar expression.

Note that if this direct map {Anzeigestil C^{*}(X)otimes C^{*}(X)to C^{*}(X)} of cochain complexes were in fact a map of differential graded algebras, then the cup product would make {Anzeigestil C^{*}(X)} a commutative graded algebra, which it is not. This failure of the Alexander–Whitney map to be a coalgebra map is an example the unavailability of commutative cochain-level models for cohomology over fields of nonzero characteristic, and thus is in a way responsible for much of the subtlety and complication in stable homotopy theory.

Generalizations An important generalisation to the non-abelian case using crossed complexes is given in the paper by Andrew Tonks below. This give full details of a result on the (simplicial) classifying space of a crossed complex stated but not proved in the paper by Ronald Brown and Philip J. Higgins on classifying spaces.

Consequences The Eilenberg–Zilber theorem is a key ingredient in establishing the Künneth theorem, which expresses the homology groups {Anzeigestil H_{*}(X mal Y)} bezüglich {Anzeigestil H_{*}(X)} und {Anzeigestil H_{*}(Y)} . In light of the Eilenberg–Zilber theorem, the content of the Künneth theorem consists in analysing how the homology of the tensor product complex relates to the homologies of the factors.

See also Acyclic model References Eilenberg, Samuel; Zilber, Joseph A. (1953), "On Products of Complexes", Amerikanisches Journal für Mathematik, vol. 75, nein. 1, pp. 200–204, doi:10.2307/2372629, JSTOR 2372629, HERR 0052767. Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 978-0-521-79540-1. Tonks, Andreas (2003), "On the Eilenberg–Zilber theorem for crossed complexes", Zeitschrift für reine und angewandte Algebra, vol. 179, nein. 1–2, pp. 199–230, doi:10.1016/S0022-4049(02)00160-3, HERR 1958384. Braun, Roland; Higgins, Philipp J. (1991), "The classifying space of a crossed complex", Mathematische Verfahren der Cambridge Philosophical Society, vol. 110, pp. 95–120, CiteSeerX 10.1.1.145.9813, doi:10.1017/S0305004100070158. Kategorien: Homological algebraTheorems in algebraic topology