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 {stile di visualizzazione Xtimes Y} and those of the spaces {stile di visualizzazione X} e {stile di visualizzazione 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.

Contenuti 1 Enunciato del teorema 2 Statement in terms of composite maps 3 The coproduct 4 Statement in cohomology 5 generalizzazioni 6 Conseguenze 7 Guarda anche 8 References Statement of the theorem The theorem can be formulated as follows. Supponiamo {stile di visualizzazione X} e {stile di visualizzazione Y} are topological spaces, Then we have the three chain complexes {stile di visualizzazione C_{*}(X)} , {stile di visualizzazione C_{*}(Y)} , e {stile di visualizzazione C_{*}(X volte Y)} . (The argument applies equally to the simplicial or singular chain complexes.) We also have the tensor product complex {stile di visualizzazione C_{*}(X)otimes C_{*}(Y)} , whose differential is, per definizione, {displaystyle parziale _{C_{*}(X)otimes C_{*}(Y)}(sigma otimes tau )=partial _{X}sigma otimes tau +(-1)^{p}sigma otimes partial _{Y}sì } per {displaystyle sigma in C_{p}(X)} e {displaystyle parziale _{X}} , {displaystyle parziale _{Y}} the differentials on {stile di visualizzazione C_{*}(X)} , {stile di visualizzazione C_{*}(Y)} .

Then the theorem says that we have chain maps {displaystyle Fcolon C_{*}(X volte Y)rightarrow C_{*}(X)otimes C_{*}(Y),quad Gcolon C_{*}(X)otimes C_{*}(Y)rightarrow C_{*}(X volte Y)} tale che {stile di visualizzazione FG} is the identity and {displaystyle GF} is chain-homotopic to the identity. Inoltre, the maps are natural in {stile di visualizzazione X} e {stile di visualizzazione Y} . Consequently the two complexes must have the same homology: {stile di visualizzazione H_{*}(C_{*}(X volte 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 {stile di visualizzazione F} they produce is traditionally referred to as the Alexander–Whitney map and {stile di visualizzazione G} the Eilenberg–Zilber map. The maps are natural in both {stile di visualizzazione X} e {stile di visualizzazione Y} and inverse up to homotopy: uno ha {displaystyle FG=mathrm {id} _{C_{*}(X)otimes C_{*}(Y)},qquad GF-mathrm {id} _{C_{*}(X volte Y)}=partial _{C_{*}(X)otimes C_{*}(Y)}H+Hpartial _{C_{*}(X)otimes C_{*}(Y)}} for a homotopy {stile di visualizzazione H} natural in both {stile di visualizzazione X} e {stile di visualizzazione Y} such that further, each of {displaystyle HH} , {displaystyle FH} , e {displaystyle HG} è zero. 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 {stile di visualizzazione C_{*}(X)to C_{*}(Xtimes X)} quale, followed by the Alexander–Whitney {stile di visualizzazione F} yields a coproduct {stile di visualizzazione C_{*}(X)to C_{*}(X)otimes C_{*}(X)} inducing the standard coproduct on {stile di visualizzazione H_{*}(X)} . With respect to these coproducts on {stile di visualizzazione X} e {stile di visualizzazione Y} , the map {stile di visualizzazione H_{*}(X)otimes H_{*}(Y)ad H_{*}{grande (}C_{*}(X)otimes C_{*}(Y){grande )} {overset {sim }{a }} H_{*}(X volte Y)} , also called the Eilenberg–Zilber map, becomes a map of differential graded coalgebras. The composite {stile di visualizzazione 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 {stile di visualizzazione k} with unity) to a pair of maps {stile di visualizzazione G^{*}colon C^{*}(X volte Y)rightarrow {grande (}C_{*}(X)otimes C_{*}(Y){grande )}^{*},quad F^{*}colon {grande (}C_{*}(X)otimes C_{*}(Y){grande )}^{*}rightarrow C^{*}(X volte Y)} which are also homotopy equivalences, as witnessed by the duals of the preceding equations, using the dual homotopy {stile di visualizzazione 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)a volte C^{*}(Y)a {grande (}C_{*}(X)otimes C_{*}(Y){grande )}^{*}} dato da {displaystyle alpha otimes beta mapsto (sigma otimes tau mapsto alpha (sigma )beta (sì ))} , the product being taken in the coefficient ring {stile di visualizzazione k} . This {stile di visualizzazione i} induces an isomorphism in cohomology, so one does have the zig-zag of differential graded algebra maps {stile di visualizzazione C^{*}(X)a volte C^{*}(X) {overset {io}{a }} {grande (}C_{*}(X)otimes C_{*}(X){grande )}^{*} {overset {G^{*}}{leftarrow }} C^{*}(Xtimes X){overset {C^{*}(Delta )}{a }}C^{*}(X)} inducing a product {displaystyle smile colon H^{*}(X)otimes H^{*}(X)to H^{*}(X)} in cohomology, known as the cup product, perché {stile di visualizzazione H^{*}(io)} e {stile di visualizzazione H^{*}(G)} are isomorphisms. Sostituzione {stile di visualizzazione G^{*}} insieme a {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 {Grande (}sigma mapsto (alpha otimes beta )(F^{*}Delta ^{*}sigma )=somma _{p=0}^{dim sigma }alfa (sigma |_{Delta ^{[0,p]}})cdot beta (sigma |_{Delta ^{[p,dim sigma ]}}){Grande )}} , quale, since cochain evaluation {stile di visualizzazione C^{p}(X)otimes C_{q}(X)to k} vanishes unless {stile di visualizzazione p=q} , reduces to the more familiar expression.

Note that if this direct map {stile di visualizzazione C^{*}(X)a volte C^{*}(X)to C^{*}(X)} of cochain complexes were in fact a map of differential graded algebras, then the cup product would make {stile di visualizzazione 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 {stile di visualizzazione H_{*}(X volte Y)} in termini di {stile di visualizzazione H_{*}(X)} e {stile di visualizzazione 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", Giornale americano di matematica, vol. 75, No. 1, pp. 200–204, doi:10.2307/2372629, JSTOR 2372629, SIG 0052767. Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 978-0-521-79540-1. Tonks, Andrea (2003), "On the Eilenberg–Zilber theorem for crossed complexes", Diario di algebra pura e applicata, vol. 179, No. 1–2, pp. 199–230, doi:10.1016/S0022-4049(02)00160-3, SIG 1958384. Marrone, Ronald; Higgins, Filippo J. (1991), "The classifying space of a crossed complex", Atti matematici della Cambridge Philosophical Society, vol. 110, pp. 95–120, CiteSeerX 10.1.1.145.9813, doi:10.1017/S0305004100070158. Categorie: Homological algebraTheorems in algebraic topology