Hurewicz theorem

This fact is related to: homotopy groups

Statement

In terms of the Hurewicz homomorphism: absolute version

If $X$ is a $n$-connected space with $n\geq 1$ (viz its first $n$ homotopy groups vanish) then the Hurewicz map on the $(n+1)^{th}$ homotopy group is an isomorphism:

$\pi _{n+1}(X,x_{0})\to {\tilde {H}}_{n+1}(X)=H_{n+1}(X)$

and moreover, all the reduced homology groups up to $n$ are zero. In particular, ${\tilde {H}}_{0}(X)=0$ and $H_{k}(X)=0$ for $1\leq k\leq n$.

In the case $n=0$, so that $X$ is a path-connected space but nothing more is known, the Hurewicz homomorphism from the Fundamental group (?) to the first homology group:

$\pi _{1}(X,x_{0})\to H_{1}(X)$

is surjective and has kernel precisely the derived subgroup of $\pi _{1}(X,x_{0})$, so $H_{1}(X)$ is isomorphic to the abelianization of $\pi _{1}(X,x_{0})$.

In terms of first non-vanishing member: absolute version

Suppose $X$ is a Path-connected space (?) that is simply connected. In particular, $\pi _{0}(X)$ and $\pi _{1}(X)$ are both trivial (the one-point set and the trivial group respectively). Then:

1. The smallest $k$ for which $\pi _{k}(X)$ is nontrivial is the same as the smallest $k$ for which ${\tilde {H}}_{k}(X)$ is nontrivial.
2. Both of these groups are isomorphic, and the Hurewicz homomorphism gives an isomorphism.

In the case that we are only given that $X$ is a path-connected space, $H_{1}(X)\cong \pi _{1}(X)/[\pi _{1}(X),\pi _{1}(X)]$ and the Hurewicz homomorphism descends to this natural identification.

Relative version

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