Suspension pushes up connectivity by one: Difference between revisions

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 $+ \infty$ .

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).

@@ Line 9: / Line 9: @@
 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==