Suspension pushes up connectivity by one: Difference between revisions

Latest revision as of 03:18, 25 December 2010

Define, for a topological space $X$ , the connectivity of $X$ as follows:

If $X$ is not path-connected, it is $- 1$ .
If $X$ is path-connected but not simply connected (i.e., the fundamental group is nontrivial), it is $0$ .
Otherwise, it is the largest $n$ such that the homotopy group $π_{k} (X)$ is a trivial group for $1 \leq k \leq n$ . If no such largest $n$ exists, set it as $+ \infty$ (when this occurs, we say that $X$ is a weakly contractible space).

The connectivity of the Suspension (?) $S X$ is exactly one more than the connectivity of $X$ .

In particular, $X$ is a weakly contractible space if and only if $S X$ is.

The proof essentially follows from facts (1)-(4).

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 * If <math>X</math> is not [[path-connected space|path-connected]], it is <math>-1</math>.
 * If <math>X</math> is path-connected but not [[simply connected space|simply connected]] (i.e., the [[fundamental group]] is nontrivial), it is <math>0</math>.
-* Otherwise, it is the largest <math>n</math> such that the [[homotopy group]] <math>\pi_k(X)</math> is a trivial group for <matH>1 \le k \le n</math>. If no such largest <math>n</math> exists, set it as <math>+\infty</math>.
+* Otherwise, it is the largest <math>n</math> such that the [[homotopy group]] <math>\pi_k(X)</math> is a trivial group for <matH>1 \le k \le n</math>. If no such largest <math>n</math> exists, set it as <math>+\infty</math> (when this occurs, we say that <math>X</math> is a [[weakly contractible space]]).
 The connectivity of the [[fact about::suspension]] <math>SX</math> is exactly one more than the connectivity of <math>X</math>.
-In particular, <math>X</math> is a [[weakly contractible space]] if and only if <math>SX</math> is.
+In particular, <math>X</math> is a [[uses property satisfaction of::weakly contractible space]][[proves property satisfaction of::weakly contractible space| ]][[fact about::weakly contractible space| ]] if and only if <math>SX</math> is.
 ==Facts used==