Hurewicz theorem

Hurewicz theorem In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré.

Contents 1 Statement of the theorems 1.1 Absolute version 1.2 Relative version 1.3 Triadic version 1.4 Simplicial set version 1.5 Rational Hurewicz theorem 2 Notes 3 References Statement of the theorems The Hurewicz theorems are a key link between homotopy groups and homology groups.

Absolute version For any path-connected space X and positive integer n there exists a group homomorphism {displaystyle h_{*}colon pi _{n}(X)to H_{n}(X),} called the Hurewicz homomorphism, from the n-th homotopy group to the n-th homology group (with integer coefficients). It is given in the following way: choose a canonical generator {displaystyle u_{n}in H_{n}(S^{n})} , then a homotopy class of maps {displaystyle fin pi _{n}(X)} is taken to {displaystyle f_{*}(u_{n})in H_{n}(X)} .

The Hurewicz theorem states cases in which the Hurewitz homomorphism is an isomorphism.

For {displaystyle ngeq 2} , if X is {displaystyle (n-1)} -connected (that is: {displaystyle pi _{i}(X)=0} for all i1} there exists a homomorphism {displaystyle h_{*}colon pi _{k}(X,A)to H_{k}(X,A)} from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if both {displaystyle X} and {displaystyle A} are connected and the pair is {displaystyle (n-1)} -connected then {displaystyle H_{k}(X,A)=0} for {displaystyle k2} (crossed modules if {displaystyle n=2} ), which itself is deduced from a higher homotopy van Kampen theorem for relative homotopy groups, whose proof requires development of techniques of a cubical higher homotopy groupoid of a filtered space.

Triadic version For any triad of spaces {displaystyle (X;A,B)} (i.e., a space X and subspaces A, B) and integer {displaystyle k>2} there exists a homomorphism {displaystyle h_{*}colon pi _{k}(X;A,B)to H_{k}(X;A,B)} from triad homotopy groups to triad homology groups. Note that {displaystyle H_{k}(X;A,B)cong H_{k}(Xcup (C(Acup B))).} The Triadic Hurewicz Theorem states that if X, A, B, and {displaystyle C=Acap B} are connected, the pairs {displaystyle (A,C)} and {displaystyle (B,C)} are {displaystyle (p-1)} -connected and {displaystyle (q-1)} -connected, respectively, and the triad {displaystyle (X;A,B)} is {displaystyle (p+q-2)} -connected, then {displaystyle H_{k}(X;A,B)=0} for {displaystyle k

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