Thảo luận:Định lý Hurewicz

Bài đã hơn 1 tháng, bỏ phần chưa dịch vào đây. Đây thuộc về chuyên môn tôpô trong toán học, ai có chuyên môn xin mời dịch tiếp. conbo 00:42, ngày 22 tháng 8 năm 2007 (UTC)Trả lời

For any n-connected CW-complex or Kan complex X and integer k ≥ 1 such that n ≥ 0, there exists a homomorphism

${\displaystyle h_{*}:\pi _{k}(X)\rightarrow {\tilde {H}}_{k}(X)}$

called the Hurewicz homomorphism from homotopy to reduced homology (with integer coefficients), which turns out to be isomorphic to the canonical abelianization map

${\displaystyle \pi _{1}(X)\rightarrow \pi _{1}(X)/[\pi _{1}(X),\pi _{1}(X)]\,}$

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:

${\displaystyle \pi _{1}(X)/[\pi _{1}(X),\pi _{1}(X)]\cong {\tilde {H}}_{1}(X).}$

The first reduced homology group therefore vanishes if π₁ is perfect and X is connected.