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 path-connected space X and positive integer n there exists a group homomorphism

 

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  , then a homotopy class of maps   is taken to  .

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

  • For  , if X is  -connected (that is:   for all  ), then   for all  , and the Hurewicz map   is an isomorphism.[1]: 366, Thm.4.32  This implies, in particular, that the homological connectivity equals the homotopical connectivity when the latter is at least 1. In addition, the Hurewicz map   is an epimorphism in this case.[1]: 390, ? 
  • For  , the Hurewicz homomorphism induces an isomorphism  , between the abelianization of the first homotopy group (the fundamental group) and the first homology group.

Relative version

edit

For any pair of spaces   and integer   there exists a homomorphism

 

from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if both   and   are connected and the pair is  -connected then   for   and   is obtained from   by factoring out the action of  . 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

 

where   denotes the cone of  . This statement is a special case of a homotopical excision theorem, involving induced modules for   (crossed modules if  ), 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   (i.e., a space X and subspaces A, B) and integer   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   are connected, the pairs   and   are  -connected and  -connected, respectively, and the triad   is  -connected, then   for   and   is obtained from   by factoring out the action of   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  -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.[2]

Rational Hurewicz theorem

edit

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  .

Notes

edit
  1. ^ a b Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, ISBN 978-0-521-79160-1
  2. ^ Goerss, Paul G.; Jardine, John Frederick (1999), Simplicial Homotopy Theory, Progress in Mathematics, vol. 174, Basel, Boston, Berlin: Birkhäuser, ISBN 978-3-7643-6064-1, III.3.6, 3.7
  3. ^ 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, Bibcode:2004MPCPS.136..617K, doi:10.1017/s0305004103007114, S2CID 119824771
  4. ^ Cartan, Henri; Serre, Jean-Pierre (1952), "Espaces fibrés et groupes d'homotopie, II, Applications", Comptes rendus de l'Académie des Sciences, 2 (34): 393–395

References

edit