# 胡列维茨定理

## 定理陈述

### 绝对版本

${\displaystyle h_{\ast }\colon \,\pi _{k}(X)\to H_{k}(X)\,\!}$

${\displaystyle h_{\ast }\colon \,\pi _{1}(X)\to \pi _{1}(X)/[\pi _{1}(X),\pi _{1}(X)]\to H_{1}(X).\,\!}$

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

### 有理胡列维茨定理

${\displaystyle X}$ 为单连通拓扑空间，并对于所有 ${\displaystyle i\leq r}$ 满足 ${\displaystyle \pi _{i}(X)\otimes \mathbb {Q} =0}$。那么胡列维茨映射

${\displaystyle h\otimes \mathbb {Q} :\pi _{i}(X)\otimes \mathbb {Q} \longrightarrow H_{i}(X;\mathbb {Q} )}$

## 参考资料

1. ^ Goerss, P. G.; Jardine, J. F., Simplicial Homotopy Theory, Progress in Mathematics 174, Basel, Boston, Berlin: Birkhäuser, 1999, ISBN 978-3-7643-6064-1, III.3.6, 3.7
2. ^ Klaus, S.; Kreck, M., 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, 2004, 136: 617–623, doi:10.1017/s0305004103007114
3. ^ Cartan, H.; Serre, J. P., Espaces fibres et groupes d'homotopie, II, Applications, C. R. Acad. Sci. Paris, 1952, 2 (34): 393–395