Mayer-Vietoris for a triple

Question

Let $X$ be a topological space with $X = A_1 \cup A_2 \cup A_3$. In the case I am interested all of these spaces are manifolds and submanifolds. Is there something like an exact sequence relating $H_\ast(X)$ with $H_\ast(A_i), H_\ast(A_i \cap A_j), H_\ast(A_1 \cap A_2 \cap A_3)$, and $H_\ast(A_i \cup A_j)$. In the application that I have in mind I would like to understand $H_\ast(X)$ an dI understand the homology of all of all of the smaller pieces.

Answer 1

This is not an exact answer to your question, but an indication of a related question: what should be "homotopical excision" for a space which is the union of more than two open sets?

An answer is given in Section $1$ of this paper

R. Brown and J.-L. Loday, "Homotopical excision and Hurewicz theorems for $n$-cubes of spaces" Proc. London Math. Soc. (3) 54 (1987) 176-192.

There are two ideas explained there.

Let $\square(n)$ be the category determined by the partial order $\{0,1\}^n$, where $0 < 1$. An $n$-cube of spaces is defined to be a functor $X: \square(n) \to Top$. Such a functor determines an $n$-cube $X^2$ of $n$-cubes of spaces by the rule $$X^2(\alpha)(\beta)= X(\alpha \wedge \beta), \quad \alpha, \beta \in \square(n).$$

For example, a square of spaces $$\begin{matrix} C & \to & B \\ \downarrow && \downarrow\\ A& \to & Y \end{matrix}$$ determines a square of squares of spaces $$ \begin{matrix} \begin{matrix}C&C\\ C&C \end{matrix}& \to & \begin{matrix}C&C\\A&A\end{matrix}\\ \downarrow & & \downarrow \\ \begin{matrix}C&B\\C&B\end{matrix}& \to & \begin{matrix} C&B\\A&Y\end{matrix} \end{matrix}$$ Further, an $n$-cube of spaces $X$ can be regarded as a map of $(n-1)$-cubes of spaces $$e(X): \partial^-_n X \to \partial^+_n X. $$ The usual excision is where a square of spaces is regarded as a map of maps of spaces.

Now a $3$-cube $X$ of spaces determines a map $e(X)$ of squares of spaces, and so a map $e(X)^2$ of squares of squares of spaces, which then determines a $3$-cube $Y$ of squares of spaces!

I leave you to write out this $3$-cube in the case say of $X= U \cup V \cup W$.

The advantage of this approach in the cited paper was that we used functors on $n$-cubes of spaces, and a van Kampen theorem for these, giving new connectivity and algebraic homotopical results in Theorems $4.3$ and $6.1$.

This suggests that for a homological result in the case $n=3$ you need to use $H_3(X;A,B)$ .

I'd be interested to hear if this sketch proves useful.