How useless can the Mayer-Vietoris sequence be in general?

Question

In an algebraic topology course I'm taking we are often asked to compute the homology groups of a space $X = A \cup B$ using the Mayer-Vietoris sequence, and it happens in all of the examples I've seen so far that it's possible to do this without knowing anything about the connecting homomorphisms $\partial_{\ast}$ (say on the level of chains); we only end up needing $H_{\ast}(A), H_{\ast}(B), H_{\ast}(A \cap B)$ and possibly some of the inclusion maps.

My guess is that this is not a typical situation; is there a relatively simple example of a nice space $X$ and nice subspaces $A, B$ such that knowing $H_{\ast}(A), H_{\ast}(B), H_{\ast}(A \cap B)$ is not enough to compute $H_{\ast}(X)$ without knowing the specific form of the connecting homomorphisms? (For maximal relevance to the course $X, A, B, A \cap B$ should be finite simplicial complexes.)

Answer 1

Given a long exact sequence

$$ \cdots \to C_{i+1} \to A_i \to B_i \to C_i \to A_{i-1} \to \cdots $$

let the map $A_i \to B_i$ be denoted $f_i$. Then you have that $C_i$ is an extension

$$ 0 \to coker(f_i) \to C_i \to ker(f_{i-1}) \to 0$$

so up to that extension problem, the maps $f_i$ always determine the $C_i$ groups. So if you want a situation where the group $C_i$ is ambiguous, you could have $ker(f_{i-1}) = \mathbb Z_2$ and $coker(f_i) = \mathbb Z$, that way $C_i$ could be either $\mathbb Z$ or $\mathbb Z \oplus \mathbb Z_2$.

Regardless, the connecting map $\partial_i : C_i \to A_{i-1}$ is determined by this extension problem, and it's easy enough to cook up examples either-way.

So I'm a little confused as to the nature of your question. I guess what I'm saying is that you are in the typical situation, and Grigory's example is also typical in that it's the inclusion map that makes the differences between his examples.

Regarding how useful/useless the MVS is for a typical problem, it really depends on how easily-expressible your space is as a union of spaces you understand (and their intersections). If your space doesn't fit that profile, you've got potentially a lot of work to do. The Serre Spectral Sequence of a Fibration is in a sense something of a souped-up Mayer-Vietoris sequence, and there are plenty of papers where people are happy just computing the $E_3$-page, or determining which page the SS collapses on, or computing a differential. These extension issues tend to be very thorny and consume much literature.

Answer 2

Knowing only $H(A)$, $H(B)$ and $H(A\cap B)$ is not enough, of course. For example, taking $A=B=S^1\times D^2$ and gluing them by $S^1\times S^1=A\cap B$ one can get either $X_1=S^2\times S^1$ or $X_2=S^3$. This gives two Mayer-Vietoris sequences with identical $H(A)$, $H(B)$ and $H(A\cap B)$ but different H(X).

As for the situation where one also knows inclusion maps, see Ryan Budney's excellent answer.