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 Hurewicz homomorphism is an isomorphism.

• For $n\geq 2$, if X is $(n-1)$-connected (that is: $\pi _{i}(X)=0$ for all $i<n$), then ${\tilde {H_{i}}}(X)=0$ 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 $h_{*}\colon \pi _{n+1}(X)\to H_{n+1}(X)$ is an epimorphism in this case.
• For $n=1$, the Hurewicz homomorphism induces an isomorphism ${\tilde {h}}_{*}\colon \pi _{1}(X)/[\pi _{1}(X),\pi _{1}(X)]\to H_{1}(X)$, 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 $k<n$ 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, Whitehead (1978) by induction, proving in turn the absolute version and the Homotopy Addition Lemma.

This relative Hurewicz theorem is reformulated by Brown & Higgins (1981) 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 $k<p+q-2$ and $H_{p+q-1}(X;A)$ is obtained from $\pi _{p+q-1}(X;A,B)$ 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 $\operatorname {cat} ^{n}$-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 $\pi _{i}(X)\otimes \mathbb {Q} =0$ for $i\leq r$. Then the Hurewicz map

$h\otimes \mathbb {Q} \colon \pi _{i}(X)\otimes \mathbb {Q} \longrightarrow H_{i}(X;\mathbb {Q} )$

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

References

• Brown, Ronald (1989), "Triadic Van Kampen theorems and Hurewicz theorems", Algebraic topology (Evanston, IL, 1988), Contemporary Mathematics, vol. 96, Providence, RI: American Mathematical Society, pp. 39–57, doi:10.1090/conm/096/1022673, ISBN 9780821851029, MR 1022673
• Brown, Ronald; Higgins, P. J. (1981), "Colimit theorems for relative homotopy groups", Journal of Pure and Applied Algebra, 22: 11–41, doi:10.1016/0022-4049(81)90080-3, ISSN 0022-4049
• Brown, R.; Loday, J.-L. (1987), "Homotopical excision, and Hurewicz theorems, for n-cubes of spaces", Proceedings of the London Mathematical Society, Third Series, 54: 176–192, CiteSeerX 10.1.1.168.1325, doi:10.1112/plms/s3-54.1.176, ISSN 0024-6115
• Brown, R.; Loday, J.-L. (1987), "Van Kampen theorems for diagrams of spaces", Topology, 26 (3): 311–334, doi:10.1016/0040-9383(87)90004-8, ISSN 0040-9383

• Rotman, Joseph J. (1988), An Introduction to Algebraic Topology, Graduate Texts in Mathematics, vol. 119, Springer-Verlag (published 1998-07-22), ISBN 978-0-387-96678-6
• Whitehead, George W. (1978), Elements of Homotopy Theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, ISBN 978-0-387-90336-1