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 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

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

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

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 for . Then the Hurewicz map

induces an isomorphism for and a surjection for .

Notes

  1. 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

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