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 space X and positive integer k there exists a group homomorphism
called the Hurewicz homomorphism from the kth homotopy group to the kth homology group (with integer coefficients), which for k = 1 and X pathconnected is equivalent to the canonical abelianization map
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:
The first homology group therefore vanishes if X is pathconnected and π_{1}(X) is a perfect group.
In addition, the Hurewicz homomorphism is an epimorphism from whenever X is (n − 1)connected, for .[1]
The group homomorphism is given in the following way. Choose canonical generators . Then a homotopy class of maps is taken to .
Relative version
For any pair of spaces (X,A) and integer k > 1 there exists a homomorphism
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 H_{k}(X,A) = 0 for k < n and H_{n}(X,A) is obtained from π_{n}(X,A) by factoring out the action of π_{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
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. space X and subspaces A,B) and integer k > 2 there exists a homomorphism
from triad homotopy groups to triad homology groups. Note that
The Triadic Hurewicz Theorem states that if X, A, B, and C = A∩B 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 H_{k}(X;A,B) = 0 for k < p+q−2 and H_{p+q−1}(X;A) is obtained from π_{p+q−1}(X;A,B) by factoring out the action of π_{1}(A∩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 cat^{n}group of an ncube of spaces.
Simplicial set version
The Hurewicz theorem for topological spaces can also be stated for nconnected simplicial sets satisfying the Kan condition.[2]
Notes

 Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, p. 390, ISBN 9780521791601
 Goerss, Paul G.; Jardine, John Frederick (1999), Simplicial Homotopy Theory, Progress in Mathematics, 174, Basel, Boston, Berlin: Birkhäuser, ISBN 9783764360641, III.3.6, 3.7
 Klaus, Stephan; Kreck, Matthias (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 (3): 617–623, doi:10.1017/s0305004103007114
 Cartan, Henri; Serre, JeanPierre (1952), "Espaces fibrés et groupes d'homotopie, II, Applications", C. R. Acad. Sci. Paris, 2 (34): 393–395
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, MR 1022673
 Brown, Ronald; Higgins, P. J. (1981), "Colimit theorems for relative homotopy groups", Journal of Pure and Applied Algebra, 22: 11–41, doi:10.1016/00224049(81)900803, ISSN 00224049
 Brown, R.; Loday, J.L. (1987), "Homotopical excision, and Hurewicz theorems, for ncubes of spaces", Proceedings of the London Mathematical Society, Third Series, 54: 176–192, CiteSeerX 10.1.1.168.1325, doi:10.1112/plms/s354.1.176, ISSN 00246115
 Brown, R.; Loday, J.L. (1987), "Van Kampen theorems for diagrams of spaces", Topology, 26 (3): 311–334, doi:10.1016/00409383(87)900048, ISSN 00409383
 Rotman, Joseph J. (1988), An Introduction to Algebraic Topology, Graduate Texts in Mathematics, 119, SpringerVerlag (published 19980722), ISBN 9780387966786
 Whitehead, George W. (1978), Elements of Homotopy Theory, Graduate Texts in Mathematics, 61, SpringerVerlag, ISBN 9780387903361