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.

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

:$h_* \colon \pi_n(X) \to H_n(X),$

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 $u_n \in H_n(S^n)$, then a homotopy class of maps $f \in \pi_n(X)$ is taken to $f_*(u_n) \in H_n(X)$.

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

* For $n\ge 2$, if ''X'' is $(n-1)$-connected (that is: $\pi_i(X)= 0$ for all ''i''<''n''), then $\tilde(X)= 0$ for all ''i''<''n'', and the Hurewicz map $h_* \colon \pi_n(X) \to H_n(X)$ is an isomorphism. This implies, in particular, that the homological connectivity equals the homotopical connectivity when the latter is at least 1. In addition, the Hurewicz map $h_* \colon \pi_(X) \to H_(X)$ is an epimorphism in this case.
* For $n=1$, the Hurewicz homomorphism induces an isomorphism \tilde_* \colon \pi_1(X)/ \pi_1(X), \pi_1(X)\to H_1(X), between the abelianization of the first homotopy group (the fundamental group) and the first homology group.

Relative version

For any pair of spaces $(X,A)$ and integer $k>1$ there exists a homomorphism

:$h_* \colon \pi_k(X,A) \to H_k(X,A)$

from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if both $X$ and $A$ are connected and the pair is $(n-1)$-connected then $H_k(X,A)=0$ for k and $H_n(X,A)$ is obtained from $\pi_n(X,A)$ by factoring out the action of $\pi_1(A)$. This is proved in, for example, by induction, proving in turn the absolute version and the Homotopy Addition Lemma.

This relative Hurewicz theorem is reformulated by as a statement about the morphism

:$\pi_n(X,A) \to \pi_n(X \cup CA),$

where $CA$ denotes the cone of $A$. This statement is a special case of a homotopical excision theorem, involving induced modules for $n>2$ (crossed modules if $n=2$), 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 $(X;A,B)$ (i.e., a space ''X'' and subspaces ''A'', ''B'') and integer $k>2$ there exists a homomorphism

:$h_*\colon \pi_k(X;A,B) \to H_k(X;A,B)$

from triad homotopy groups to triad homology groups. Note that

:$H_k(X;A,B) \cong H_k(X\cup (C(A\cup B))).$

The Triadic Hurewicz Theorem states that if ''X'', ''A'', ''B'', and $C=A\cap B$ are connected, the pairs $(A,C)$ and $(B,C)$ are $(p-1)$-connected and $(q-1)$-connected, respectively, and the triad $(X;A,B)$ is $(p+q-2)$-connected, then $H_k(X;A,B)=0$ for k and $H_(X;A)$ is obtained from $\pi_(X;A,B)$ by factoring out the action of $\pi_1(A\cap B)$ and the generalised Whitehead products. The proof of this theorem uses a higher homotopy van Kampen type theorem for triadic homotopy groups, which requires a notion of the fundamental $\operatorname^n$-group of an ''n''-cube of spaces.

Simplicial set version

The Hurewicz theorem for topological spaces can also be stated for ''n''-connected simplicial sets satisfying the Kan condition.

Rational Hurewicz theorem

Rational Hurewicz theorem: Let ''X'' be a simply connected topological space with $\pi_i(X)\otimes \Q = 0$ for $i\leq r$. Then the Hurewicz map

:$h\otimes \Q \colon \pi_i(X)\otimes \Q \longrightarrow H_i(X;\Q )$

induces an isomorphism for $1\leq i \leq 2r$ and a surjection for $i = 2r+1$.

Notes

References

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* {{citation
, last= Whitehead
, first= George W.
, author-link= George W. Whitehead
, title= Elements of Homotopy Theory
, publisher= Springer-Verlag
, year= 1978
, series= Graduate Texts in Mathematics
, volume= 61
, isbn= 978-0-387-90336-1

Theorems in homotopy theory
Homology theory
Theorems in algebraic topology