TITLE: Mayer-Vietoris Type Sequence For Pushouts
QUESTION [7 upvotes]: 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.

REPLY [7 votes]: 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.