# Suspension pushes up connectivity by one

From Topospaces

## Statement

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

- If is not path-connected, it is .
- If is path-connected but not simply connected (i.e., the fundamental group is nontrivial), it is .
- Otherwise, it is the largest such that the homotopy group is a trivial group for . If no such largest exists, set it as (when this occurs, we say that is a weakly contractible space).

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

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

## Facts used

- Suspension of any space is path-connected
- Suspension of path-connected space is simply connected
- Homology for suspension
- Hurewicz theorem

## Proof

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