Let $B \leftarrow A \to C$ be a span of spaces, and consider the homotopy pushout $B \cup\_A C$.

**Question:** When is $B \cup\_A C$ contractible?

This is a pretty open-ended question. I'm interested in necessary conditions or sufficient conditions or interesting examples or special cases, etc.

**Some additional conditions I'd be happy to take as blanket assumptions:**

1. I don't think too much is lost if we assume that $A,B,C$ are connected.
2. I'm happy to assume that $A \to B$ is 1-connected (conventions vary about what this means; I mean that the homotopy fiber of $A \to B$ is 1-connected, i.e. that $A \to B$ is $\pi\_1$-surjective).
3. For somewhat obscure reasons, I'm particularly interested in the case where $A \to C$ is the projection $A\_0 \times C \to C$ out of a binary product.
4. For similarly obscure reasons, I'm particularly interested in the case where $C$ is a loopspace, or even where $C = \Omega \Sigma C\_0$ is a free loopspace.

**Observation:**

* By pasting on a few more pushout squares, we obtain the identity $\Sigma(B \vee C) = \Sigma A$.

I'm not sure what to make of this, though.