Mayer-Vietoris Type Sequence For Pushouts

Pushouts in the category $\mathsf{Top}$ of topological spaces exist and under certain conditions are known as adjunction spaces. Rigorously, if

is a diagram in $\mathsf{Top}$, then there exists a universal commutative diagram

Question 1. Is there a long exact sequence (similar to the Mayer-Vietoris sequence) relating the homology groups of $A$, $X$, $Y$, and $P$?

If there is no known sequence in this general situation, then I'm curious about the special case of adjunction spaces.

Question 2. If $g$ is the inclusion of a subspace $A$ into $Y$, then $P$ is the so-called adjunction space $X\cup_f Y$. Is there a long exact sequence relating the homology groups of $A$, $X$, $Y$, and $X\cup_f Y$?

Finally, since intersections are just special cases of pullbacks in $\mathsf{Top}$, I'm curious if this line of questioning provides a generalized Mayer-Vietoris sequence.

Question 3. Let $P$ be the pullback of a $\mathsf{Top}$ diagram $X\rightarrow A\leftarrow Y$. Is there a long exact sequence relating the homology groups of $A$, $X$, $Y$, and $P$?

Obviously I won't be too upset if anyone knows of a nice answer that relies on replacing $\mathsf{Top}$ with a "nicer" subcategory such as cw-complexes or whatever.

As an answer to your first two questions, the answer is yes if the pushout computes the so-called homotopy pushout.

In elementary terms this means that the map $C(f, g) \rightarrow P$ from the double mapping cylinder to the pushout is a homotopy equivalence. It is for example sufficient for one of $f, g$ to be a cofibration.

If this is the case, then we get a long exact sequence as Mayer Vietoris for the covering $U, V \subseteq C(f, g)$, where $U = X \cup A \times [0, \frac{2}{3})$, $V = Y \cup A \times (\frac{1}{3}, 1]$. Indeed, it is not hard to see that $U \simeq X, V \simeq A$ and $U \cap V \simeq A$.

If the pushout of your diagram is not a homotopy pushout, then I believe there is little hope for any long exact sequence. Suppose such thing would exist and would behave functorially with respect to map of diagrams. Then some form of five-lemma would imply that the obvious map from $D^{n} \leftarrow S^{n-1} \rightarrow D^{n}$ to $\{ * \} \leftarrow S^{n-1} \rightarrow \{ * \}$ must give a homology isomorphism on the pushouts, as level-wise this map of diagrams is a homotopy equivalence. But the pushout of the first diagram is $S^{n}$ and it is $\{ * \}$ for the second one, so their homology is different.

A nice exotic example is to take $A=X=Y=S^1$ and $f,g$ maps of degree $2$ and $3$ respectively, e.g. $z \mapsto z^n$ for $n=2,3$. Then the pushout $P$ is not even Hausdorff. hence we quickly replace the pushout by the double mapping cylinder, i.e. homotopy pushout, and get a nice CW-complex, whose fundamental group is is the trefoil group, $T$ with presentation $\{x,y\mid x^2=y^3\}$. A variation is to take the fundamental groupoid based on two points $0,1$ at each end of the mapping cylinder and get the fundamental groupoid on these two points having generators $x$ at $0$, $y$ at $1$ and $\iota :0 \to 1$, with the relation $\iota x^2= y^3 \iota$.

Nov 14,2020There is more on this in terms of exact sequences (not of homology) at (R, Brown, P.R. HEATH and H. KAMPS), ``Groupoids and the Mayer-Vietoris sequence'', — J. Pure Appl. Alg. 30 (1983) 109-129. This is useful for looking at pullbacks of covering maps.