Hurewicz theorem

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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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