# Hurewicz theorem related to Galois group (or Tannakian categories)?

Is there a proof of the Hurewicz theorem $\pi_1(X)^{ab} = H_1(X, \mathbf Z)$ ($X$ a connected topological space) expressing $\pi_1(X)$ as the "Galois" group of $X$, i.e., group of deck transformations of the universal cover?

(as opposed to a proof using the construction of $\pi_1$ as the group of paths up to homotopy).

Thank you.

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I don't understand your question. It is certainly true that $\pi_1(X)$ is the group of deck-transformations, but what does this have to do with Hurewicz theorem? – J.C. Ottem Feb 23 '11 at 19:17
How do you want to define $H_1(X)$? If you want to use singular homology then it seems unlikely that you can avoid some kind of paths. You could describe $H^1(X)$ as $[X,S^1]$ or using Cech cochains and then consider pairings $\pi_1(X)^{ab}\otimes H^1(X)\to\mathbb{Z}$, but you would lose information about torsion in $H_1(X)$ that way. – Neil Strickland Feb 23 '11 at 19:18

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