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 $n \ge 1$ (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 $1 \le k \le n$.

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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