Hurewicz theorem: Difference between revisions

Latest revision as of 01:32, 21 December 2010

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 $(n + 1)^{t h}$ homotopy group is an isomorphism:

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

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:

$π_{1} (X, x_{0}) \to H_{1} (X)$

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

In terms of first non-vanishing member: absolute version

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

The smallest $k$ for which $π_{k} (X)$ is nontrivial is the same as the smallest $k$ for which ${\tilde{H}}_{k} (X)$ is nontrivial.
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, $H_{1} (X) ≅ π_{1} (X) / [π_{1} (X), π_{1} (X)]$ and the Hurewicz homomorphism descends to this natural identification.

Relative version

Fill this in later

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+{{factrelatedtos|homotopy group}}
 ==Statement==
-If <math>X</math> is a <math>n</math>-[[multiply connected space|connected space]] with <math>n \ge 2</math> (viz its first <math>n</math> [[homotopy group]]s vanish) then the [[Hurewicz map]] on the <math>(n+1)^{th}</math> homotopy group is an isomorphism:
+===In terms of the Hurewicz homomorphism: absolute version===
+If <math>X</math> is a <math>n</math>-[[fact about::multiply connected space|connected space]] with <math>n \ge 1</math> (viz its first <math>n</math> [[homotopy group]]s vanish) then the [[Hurewicz map]] on the <math>(n+1)^{th}</math> homotopy group is an isomorphism:
+<math>\pi_{n+1}(X,x_0) \to \tilde{H}_{n+1}(X) = H_{n+1}(X)</math>
+and moreover, all the reduced homology groups up to <math>n</math> are zero. In particular, <math>\tilde{H}_0(X) = 0</math> and <math>H_k(X) = 0</math> for <math>1 \le k \le n</math>.
+In the case <math>n = 0</math>, so that <math>X</math> is a [[path-connected space]] but nothing more is known, the Hurewicz homomorphism from the [[fact about::fundamental group]] to the [[first homology group]]:
+<math>\pi_1(X,x_0) \to H_1(X)</math>
+is surjective and has kernel precisely the [[derived subgroup]] of <math>\pi_1(X,x_0)</math>, so <math>H_1(X)</math> is isomorphic to the abelianization of <math>\pi_1(X,x_0)</math>.
+===In terms of first non-vanishing member: absolute version===
+Suppose <math>X</math> is a [[fact about::path-connected space]] that is [[fact about::simply connected space|simply connected]]. In particular, <math>\pi_0(X)</math> and <math>\pi_1(X)</math> are both ''trivial'' (the one-point set and the trivial group respectively). Then:
+# The smallest <math>k</math> for which <math>\pi_k(X)</math> is nontrivial is the same as the smallest <math>k</math> for which <math>\tilde{H}_k(X)</math> is nontrivial.
+# Both of these groups are isomorphic, and the Hurewicz homomorphism gives an isomorphism.
+In the case that we are only given that <math>X</math> is a [[path-connected space]], <math>H_1(X) \cong \pi_1(X)/[\pi_1(X),\pi_1(X)]</math> and the Hurewicz homomorphism descends to this natural identification.
-<math>\pi_{n+1}(X,x_0) \to \tilde{H}_{n+1}(X)</math>
+===Relative version===
-and moreover, all the reduced homology groups upto <math>n</math> are zero.
+{{fillin}}