Hurewicz theorem - Alchetron, The Free Social Encyclopedia

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 ∗ : π k ( X ) → 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 ∗ : π 1 ( X ) → π 1 ( X ) / [ π 1 ( X ) , π 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 ) ≅ π 1 ( X ) / [ π 1 ( X ) , π 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 π n + 1 ( X ) → H n + 1 ( X ) whenever X is (n − 1)-connected, for n ≥ 2 .

The group homomorphism is given in the following way. Choose canonical generators u n ∈ H n ( S n ) . Then a homotopy class of maps f ∈ π n ( X ) is taken to f ∗ ( u n ) ∈ H n ( X ) .

Relative version

For any pair of spaces (X,A) and integer k > 1 there exists a homomorphism

h ∗ : π k ( X , A ) → 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

π n ( X , A ) → π n ( X ∪ C A ) .

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 ∗ : π k ( X ; A , B ) → 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.

Rational Hurewicz theorem

Rational Hurewicz theorem: Let X be a simply connected topological space with π i ( X ) ⊗ Q = 0 for i ≤ r . Then the Hurewicz map

h ⊗ Q : π i ( X ) ⊗ Q ⟶ H i ( X ; Q )

induces an isomorphism for 1 ≤ i ≤ 2 r and a surjection for i = 2 r + 1 .

References

Hurewicz theorem Wikipedia

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