Hurewicz theorem

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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[edit]

The Hurewicz theorems are a key link between homotopy groups and homology groups.

Absolute version[edit]

For any space X and positive integer k there exists a group homomorphism

h_{\ast}\colon\, \pi_k(X) \to H_k(X) \,\!

called the Hurewicz homomorphism from the k-th homotopy group to the k-th homology group (with integer coefficients), which for k = 1 and X path-connected is equivalent to the canonical abelianization map

h_{\ast}\colon\, \pi_1(X) \to \pi_1(X)/[ \pi_1(X), \pi_1(X)] . \,\!

The Hurewicz theorem states that if X is (n − 1)-connected, the Hurewicz map is an isomorphism for all k ≤ n when n ≥ 2 and abelianization for n = 1. In particular, this theorem says that the abelianization of the first homotopy group (the fundamental group) is isomorphic to the first homology group:

 H_1(X) \cong  \pi_1(X)/[ \pi_1(X), \pi_1(X)] . \,\!

The first homology group therefore vanishes if X is path-connected and π1(X) is a perfect group.

In addition, the Hurewicz homomorphism is an epimorphism from \pi_{n+1}(X) \to H_{n+1}(X) whenever X is (n − 1)-connected, for n \ge 2.

The group homomorphism is given in the following way. Choose canonical generators 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).

Relative version[edit]

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

h_{\ast}\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 each of X, A are connected and the pair (X,A) is (n−1)-connected then Hk(X,A) = 0 for k < n and Hn(X,A) is obtained from πn(X,A) by factoring out the action of π1(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) \,\!.

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[edit]

For any triad of spaces (X;A,B) (i.e. space X and subspaces A,B) and integer k > 2 there exists a homomorphism

h_{\ast}\colon \pi_k(X;A,B) \to H_k(X;A,B) \,\!

from triad homotopy groups to triad homology groups. Note that Hk(X;A,B) ≅ Hk(X∪(C(AB)). The Triadic Hurewicz Theorem states that if X, A, B, and C = AB are connected, the pairs (A,C), (B,C) are respectively (p−1)-, (q−1)-connected, and the triad (X;A,B) is p+q−2 connected, then Hk(X;A,B) = 0 for k < p+q−2 and Hp+q−1(X;A) is obtained from πp+q−1(X;A,B) by factoring out the action of π1(AB) 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 catn-group of an n-cube of spaces.

Simplicial set version[edit]

The Hurewicz theorem for topological spaces can also be stated for n-connected simplicial sets satisfying the Kan condition.[1]

Rational Hurewicz theorem[edit]

Rational Hurewicz theorem:[2][3] 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} : \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[edit]

  1. ^ Goerss, P. G.; Jardine, J. F. (1999), Simplicial Homotopy Theory, Progress in Mathematics 174, Basel, Boston, Berlin: Birkhäuser, ISBN 978-3-7643-6064-1 , III.3.6, 3.7
  2. ^ Klaus, S.; Kreck, M. (2004), "A quick proof of the rational Hurewicz theorem and a computation of the rational homotopy groups of spheres", Mathematical Proceedings of the Cambridge Philosophical Society 136: 617–623 
  3. ^ Cartan, H.; Serre, J. P. (1952), "Espaces fibres et groupes d'homotopie, II, Applications", C. R. Acad. Sci. Paris 2 (34): 393–395