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2 weird error

I think that part of what you are looking for is from work of Eilenberg-Moore.

Suppose that all four spaces are simply-connected. On taking cohomology, you get a commutative diagram of graded-commutative rings $$\begin{array}{ccc} H^*(A)&\leftarrow &H^*(B)\\ \uparrow & & \uparrow\\ H^*(C)&\leftarrow& H^*(D). \end{array}$$ The simplest possible criterion would be that $H^*(A)$ is a pushout (of graded-commutative rings) in this diagram, or equivalently that the map $H^*(B) \otimes_{H^*(D)} H^*(C) \to H^*(A)$ is an isomorphism.

Unfortunately, the answer is close but not quite that simple - you need a version that involves more homological algebra. Specifically, there is a commutative diagram of cochain algebras $$\begin{array}{ccc} C^*(A)&\leftarrow &C^*(B)\\ \uparrow & & \uparrow\\ C^*(C)&\leftarrow& C^*(D). \end{array}$$ and Eilenberg-Moore used this to construct a map out of the derived tensor product $$C^*(B) \mathop{\otimes}^{\mathbb L}_{C^*(D)} C^*(C) \to C^*(A).$$ In the simply-connected case, if $A$ is a homotopy pullback then this map is a quasi-isomorphism.

As one consequence, if $H^*(B)$ or $H^*(C)$ is a projective module over $H^*(D)$, then $A$ being a homotopy pullback implies that a homotopy pullback it satisfies $H^*(A) \cong H^*(B) \otimes_{H^*(D)} H^*(C)$.

So far as the rest of your questions:

• You might be able to combine this with the universal coefficient theorem and with the homology Whitehead theorem to get a result. Specifically, if you check it with field coefficients for an arbitrary field then you get your desired result.

• Unfortunately there's not a purely algebraic criterion, because being a homotopy pullback won't always be detected by the maps on cohomology; sometimes it will involve delicate analysis of secondary products.

• I don't have a nice, complete answer for you about the non-simply-connected case other than carefully looking at what's going on with the fundamental group and universal covers. The Eilenberg-Moore result wlll have issues with "exotic" convergence.

• Finally, I'm afraid that I can't really address your desire for references without "weird" spaces, since most of my friends fall into that category. Maybe someone else can help there.

1

I think that part of what you are looking for is from work of Eilenberg-Moore.

Suppose that all four spaces are simply-connected. On taking cohomology, you get a commutative diagram of graded-commutative rings $$\begin{array}{ccc} H^*(A)&\leftarrow &H^*(B)\\ \uparrow & & \uparrow\\ H^*(C)&\leftarrow& H^*(D). \end{array}$$ The simplest possible criterion would be that $H^*(A)$ is a pushout (of graded-commutative rings) in this diagram, or equivalently that the map $H^*(B) \otimes_{H^*(D)} H^*(C) \to H^*(A)$ is an isomorphism.

Unfortunately, the answer is close but not quite that simple - you need a version that involves more homological algebra. Specifically, there is a commutative diagram of cochain algebras $$\begin{array}{ccc} C^*(A)&\leftarrow &C^*(B)\\ \uparrow & & \uparrow\\ C^*(C)&\leftarrow& C^*(D). \end{array}$$ and Eilenberg-Moore used this to construct a map out of the derived tensor product $$C^*(B) \mathop{\otimes}^{\mathbb L}_{C^*(D)} C^*(C) \to C^*(A).$$ In the simply-connected case, if $A$ is a homotopy pullback then this map is a quasi-isomorphism.

As one consequence, if $H^*(B)$ or $H^*(C)$ is a projective module over $H^*(D)$, then $A$ being a homotopy pullback implies that a homotopy pullback satisfies $H^*(A) \cong H^*(B) \otimes_{H^*(D)} H^*(C)$.

So far as the rest of your questions:

• You might be able to combine this with the universal coefficient theorem and with the homology Whitehead theorem to get a result. Specifically, if you check it with field coefficients for an arbitrary field then you get your desired result.

• Unfortunately there's not a purely algebraic criterion, because being a homotopy pullback won't always be detected by the maps on cohomology; sometimes it will involve delicate analysis of secondary products.

• I don't have a nice, complete answer for you about the non-simply-connected case other than carefully looking at what's going on with the fundamental group and universal covers. The Eilenberg-Moore result wlll have issues with "exotic" convergence.

• Finally, I'm afraid that I can't really address your desire for references without "weird" spaces, since most of my friends fall into that category. Maybe someone else can help there.