# 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.

## Contents

## Statement of the theoremsEdit

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

### Absolute versionEdit

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

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

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 *k* = 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 path-connected 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 versionEdit

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 versionEdit

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 *n*-cube of spaces.

### Simplicial set versionEdit

The Hurewicz theorem for topological spaces can also be stated for *n*-connected simplicial sets satisfying the Kan condition.^{[2]}

### Rational Hurewicz theoremEdit

**Rational Hurewicz theorem: ^{[3]}^{[4]}** Let

*X*be a simply connected topological space with for . Then the Hurewicz map

induces an isomorphism for and a surjection for .

## NotesEdit

**^*** Hatcher, Allen (2001),*Algebraic Topology*, Cambridge University Press, p. 390, ISBN 978-0-521-79160-1**^**Goerss, Paul G.; Jardine, John Frederick (1999),*Simplicial Homotopy Theory*, Progress in Mathematics,**174**, Basel, Boston, Berlin: Birkhäuser, ISBN 978-3-7643-6064-1, 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, Jean-Pierre (1952), "Espaces fibrés et groupes d'homotopie, II, Applications",*C. R. Acad. Sci. Paris*,**2**(34): 393–395

## ReferencesEdit

- 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:10.1016/0022-4049(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,**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