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

$h_{{\ast }}\colon \,\pi _{k}(X)\to H_{k}(X)\,\!$

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 $n\geq 2$.^[1]

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

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 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 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

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 H_k(X;A,B) ≅ H_k(X∪(C(A∪B)). 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 π₁(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 catⁿ-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.^[2]

Rational Hurewicz theorem

Rational Hurewicz theorem:^[3][4] 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 $i=2r+1$.

Notes

1. • Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, p. 390, ISBN 0-521-79160-X
2. 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
3. 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, doi:10.1017/s0305004103007114
4. Cartan, H.; Serre, J. P. (1952), "Espaces fibres et groupes d'homotopie, II, Applications", C. R. Acad. Sci. Paris, 2 (34): 393–395

References

• Brown, R. (1989), "Triadic Van Kampen theorems and Hurewicz theorems", Contemporary Mathematics, 96: 39–57, doi:10.1090/conm/096/1022673, ISSN 0040-9383
• 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

• Rotman, Joseph J. (1988), An Introduction to Algebraic Topology, Graduate Texts in Mathematics, 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, 61, Springer-Verlag, ISBN 978-0-387-90336-1