When is a homotopy pushout contractible?

Question

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.

Answer 1

Assume all the spaces are connected, $\pi_1 A \to \pi_1 B$ is surjective, and $\pi_1 C$ is abelian.

The pushout, $P = C \sqcup_A B$, is contractible if and only if $\pi_1(P)$ and all the $H_i(P)$ are trivial. However, by Seifert-Van-Kampen, and the hypothesis on $\pi_1 A \to \pi_1 B$, we have that $\pi_1 C$ surjects onto $\pi_1 P$. Thus $\pi_1 P$ is abelian, so $P$ is contractible if and only $H_i(P) = 0$ for all $i$.

For this, as you and John observed, it is necessary and sufficent for $H_i(A) \to H_i(B) \oplus H_i(C)$ to be an isomorphism for all $i \geq 1$.

Answer 2

Let me add some additional remarks on the enumeration question:

How many such spaces $A$ are there sitting over $B\times C$ such that the homotopy pushout $$ B \leftarrow A \to C $$ is contractible?

1. If $B\cup_A C$ is contractible, then the sum of the maps $\Sigma A \to \Sigma B$ and $\Sigma A \to \Sigma C$ gives a homology isomorphism $$ \Sigma A \to \Sigma B \vee \Sigma C\, . $$ If $A$ is connected, then we conclude that the map is also a weak homotopy equivalence, so $\Sigma A$ is a wedge of $\Sigma B$ and $\Sigma C$ in this case. The converse to this statement is also true.

2. Hence, we are reduced to classifying those spaces $A$ whose suspension splits as a wedge of $\Sigma B$ and $\Sigma C$. Hence, this problem can be formulated as a kind desuspension problem.

3. There is a classifying space ${\cal D}(B,C)$ which is the realization of (the nerve of) a category whose objects are spaces $A$ with structure map $A \to B\times C$ such that $\text{hocolim}(B \leftarrow A \to C)$ is contractible. A morphism of this category is a weak homotopy equivalence of spaces over $B\times C$.

4. Classification Result: Assume $A,B,C$ are $1$-connected. There is a function $$ \pi_0({\cal D}(B,C)) \to \{B\vee C, B\wedge C\} $$ (the target is the abelian group of homotopy classes of stable maps) which is a bijection in a metastable range $$ \max(b,c) \le 3\min(r,s)-1\, , $$ where $b$ is the homotopy dimension of $B$ (the CW complex of minimal dimension having the homotopy type of $B$), $c$ is the homotopy dimension of $C$, $r$ is the connectivity of $B$ and $s$ is the connectivity of $C$. Here I have given $B$ and $C$ basepoints.

5. The function in (4) is easily described as the stable homotopy class of the weak map $$ \Sigma B\vee \Sigma C \overset{\simeq}\leftarrow \Sigma A \to \Sigma (B \wedge C) $$ where the right arrow is given by suspending the composition $A \to B\times C \to B\wedge C$.

6. If we want to work beyond the metastable range, the homotopy type of ${\cal D}(B,C)$ can be determined "up to extensions" via the coefficients of the identity functor in homotopy functor calculus.

See the paper

Klein, John R.; Peter, John W. Fake wedges. Trans. Amer. Math. Soc. 366 (2014), no. 7, 3771–3786

which extends my earlier paper:

John R. Klein, Structure theorems for homotopy pushouts. I. Contractible pushouts, Math. Proc. Cambridge Philos. Soc. 123 (1998), no. 2, 301–324.