“Hurewicz theorem”的意思、由来-开放百科全书

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

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 π₁(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 version

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, {{Harvtxt|Whitehead|1978}} by induction, proving in turn the absolute version and the Homotopy Addition Lemma.

This relative Hurewicz theorem is reformulated by {{Harvtxt|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 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

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

for

. Then the Hurewicz map

Notes

1. ^* {{citation |last= Hatcher |first= Allen |author-link= Allen Hatcher |title= Algebraic Topology |publisher= Cambridge University Press |year= 2001 |series= |volume= |isbn= 978-0-521-79160-1 |page=390 }}
2. ^{{Citation | last1=Goerss | first1=Paul G. | last2=Jardine | first2=John Frederick | authorlink2=Rick Jardine| title=Simplicial Homotopy Theory | publisher=Birkhäuser | location=Basel, Boston, Berlin | series=Progress in Mathematics | isbn=978-3-7643-6064-1 | year=1999 | volume=174}}, III.3.6, 3.7
3. ^{{Citation | last1=Klaus | first1=Stephan | last2=Kreck | first2=Matthias |authorlink2=Matthias Kreck | title=A quick proof of the rational Hurewicz theorem and a computation of the rational homotopy groups of spheres | journal= Mathematical Proceedings of the Cambridge Philosophical Society | year=2004 | volume=136 | issue=3 | pages=617–623 | doi=10.1017/s0305004103007114}}
4. ^{{Citation | last1=Cartan | first1=Henri |authorlink1=Henri Cartan| last2=Serre | first2=Jean-Pierre | authorlink2=Jean-Pierre Serre| title= Espaces fibrés et groupes d'homotopie, II, Applications | journal= C. R. Acad. Sci. Paris | year=1952 | volume=2 | number=34 |pages=393–395}}