# When is a homotopy pushout contractible?

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.

• You've probably already thought of this, but you can get some algebraic necessary conditions by using Mayer-Vietoris in your favorite (co)homology theory. – John Greenwood Jan 22 at 2:56
• @John I think this is covered already by Tim's observation, which is an almost unstable version of what MV gives you – Phil Tosteson Jan 22 at 4:07
• @Tim Knots are one rich source of examples. If $K \subset S^3$ is a knot there is a pushout square with $C = S^3 - K$, $B$ a tubular neighborhood of the knot in $\mathbb R^3$ (contractible), and $A$ is $B - K$ (equivalent to $S^1$). However, this doesn't produce many examples which satisfy $4$, because $\pi_1 C$ is rarely abelian. – Phil Tosteson Jan 22 at 4:23
• The paper Structure theorems for homotopy pushouts. I. Contractible pushouts. by John Klein (MR1490203) is all about contractible pushouts. – Jeff Strom Jan 22 at 17:08

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 $$i \geq 1$$.

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.