Hurewicz theorem

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 fact about::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 fact about::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.