# Suspension pushes up connectivity by one

## Statement

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 $\pi_k(X)$ is a trivial group for $1 \le k \le n$. 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 (?) $SX$ is exactly one more than the connectivity of $X$.

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

## Proof

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