Hurewicz theorem

Short description: Gives a homomorphism from homotopy groups to homology groups

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

[math]\displaystyle{ h_* \colon \pi_n(X) \to H_n(X), }[/math]

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 [math]\displaystyle{ u_n \in H_n(S^n) }[/math], then a homotopy class of maps [math]\displaystyle{ f \in \pi_n(X) }[/math] is taken to [math]\displaystyle{ f_*(u_n) \in H_n(X) }[/math].

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

• For [math]\displaystyle{ n\ge 2 }[/math], if X is [math]\displaystyle{ (n-1) }[/math]-connected (that is: [math]\displaystyle{ \pi_i(X)= 0 }[/math] for all [math]\displaystyle{ i \lt n }[/math]), then [math]\displaystyle{ \tilde{H_i}(X)= 0 }[/math] for all [math]\displaystyle{ i \lt n }[/math], and the Hurewicz map [math]\displaystyle{ h_* \colon \pi_n(X) \to H_n(X) }[/math] 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 [math]\displaystyle{ h_* \colon \pi_{n+1}(X) \to H_{n+1}(X) }[/math] is an epimorphism in this case.
• For [math]\displaystyle{ n=1 }[/math], the Hurewicz homomorphism induces an isomorphism [math]\displaystyle{ \tilde{h}_* \colon \pi_1(X)/[ \pi_1(X), \pi_1(X)] \to H_1(X) }[/math], between the abelianization of the first homotopy group (the fundamental group) and the first homology group.

Relative version

For any pair of spaces [math]\displaystyle{ (X,A) }[/math] and integer [math]\displaystyle{ k\gt 1 }[/math] there exists a homomorphism

[math]\displaystyle{ h_* \colon \pi_k(X,A) \to H_k(X,A) }[/math]

from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if both [math]\displaystyle{ X }[/math] and [math]\displaystyle{ A }[/math] are connected and the pair is [math]\displaystyle{ (n-1) }[/math]-connected then [math]\displaystyle{ H_k(X,A)=0 }[/math] for [math]\displaystyle{ k\lt n }[/math] and [math]\displaystyle{ H_n(X,A) }[/math] is obtained from [math]\displaystyle{ \pi_n(X,A) }[/math] by factoring out the action of [math]\displaystyle{ \pi_1(A) }[/math]. 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) as a statement about the morphism

[math]\displaystyle{ \pi_n(X,A) \to \pi_n(X \cup CA), }[/math]

where [math]\displaystyle{ CA }[/math] denotes the cone of [math]\displaystyle{ A }[/math]. This statement is a special case of a homotopical excision theorem, involving induced modules for [math]\displaystyle{ n\gt 2 }[/math] (crossed modules if [math]\displaystyle{ n=2 }[/math]), 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 [math]\displaystyle{ (X;A,B) }[/math] (i.e., a space X and subspaces A, B) and integer [math]\displaystyle{ k\gt 2 }[/math] there exists a homomorphism

[math]\displaystyle{ h_*\colon \pi_k(X;A,B) \to H_k(X;A,B) }[/math]

from triad homotopy groups to triad homology groups. Note that

[math]\displaystyle{ H_k(X;A,B) \cong H_k(X\cup (C(A\cup B))). }[/math]

The Triadic Hurewicz Theorem states that if X, A, B, and [math]\displaystyle{ C=A\cap B }[/math] are connected, the pairs [math]\displaystyle{ (A,C) }[/math] and [math]\displaystyle{ (B,C) }[/math] are [math]\displaystyle{ (p-1) }[/math]-connected and [math]\displaystyle{ (q-1) }[/math]-connected, respectively, and the triad [math]\displaystyle{ (X;A,B) }[/math] is [math]\displaystyle{ (p+q-2) }[/math]-connected, then [math]\displaystyle{ H_k(X;A,B)=0 }[/math] for [math]\displaystyle{ k\lt p+q-2 }[/math] and [math]\displaystyle{ H_{p+q-1}(X;A) }[/math] is obtained from [math]\displaystyle{ \pi_{p+q-1}(X;A,B) }[/math] by factoring out the action of [math]\displaystyle{ \pi_1(A\cap B) }[/math] 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 [math]\displaystyle{ \operatorname{cat}^n }[/math]-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 [math]\displaystyle{ \pi_i(X)\otimes \Q = 0 }[/math] for [math]\displaystyle{ i\leq r }[/math]. Then the Hurewicz map

[math]\displaystyle{ h\otimes \Q \colon \pi_i(X)\otimes \Q \longrightarrow H_i(X;\Q ) }[/math]

induces an isomorphism for [math]\displaystyle{ 1\leq i \leq 2r }[/math] and a surjection for [math]\displaystyle{ i = 2r+1 }[/math].

References

• Brown, Ronald (1989), "Triadic Van Kampen theorems and Hurewicz theorems", Algebraic topology (Evanston, IL, 1988), Contemporary Mathematics, 96, Providence, RI: American Mathematical Society, pp. 39–57, doi:10.1090/conm/096/1022673, ISBN 9780821851029 
• 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, 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 

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