Định lý Hurewicz
Bách khoa toàn thư mở Wikipedia
Trong toán học, định lý Hurewicz là một kết quả cơ bản của tôpô đại số, liên hệ lý thuyết đồng luân với lý thuyết đồng điều qua đồng cấu Hurewicz. Định lý này do Witold Hurewicz đưa ra.
[sửa] Phát biểu định lý
For any n-connected CW-complex or Kan complex X and integer k ≥ 1 such that n ≥ 0, there exists a homomorphism
called the Hurewicz homomorphism from homotopy to reduced homology (with integer coefficients), which turns out to be isomorphic to the canonical abelianization map
![\pi_1(X) \rightarrow \pi_1(X)/[\pi_1(X), \pi_1(X)]\,](../../../math/9/6/4/964b212804ab8047359ebab2fab8e43b.png)
if k = 1. The Hurewicz theorem states that under the above conditions, the Hurewicz map is an isomorphism if k ≤ n and an epimorphism if k = n + 1.
In particular, if the first homotopy group (the fundamental group) is nonabelian, this theorem says that its abelianization is isomorphic to the first reduced homology group:
![\pi_1(X)/[\pi_1(X), \pi_1(X)] \cong \tilde{H}_1(X).](../../../math/5/d/f/5dffcbd3578fd35c9648ed41c2e59fe9.png)
The first reduced homology group therefore vanishes if π1 is perfect and X is connected.
