In this article http://en.wikipedia.org/wiki/Mayer%E2%80%93Vietoris_sequence, and in the section about the connecting homomorphism associated to a Mayer-Vietoris sequence, the connecting homomorphhism $\partial_*$ is defined by:
$$H_{n}(X) \xrightarrow{\partial_*}\,H_{n-1} (A\cap B)$$
My problem is the following. Suppose we have a class in $Im(\partial_*)\subset H_{n-1}(A\cap B)$ and we want to determine one of the preimages of this class in $H_{n}(X)$, is there a general way to find this preimage, knowing that the map $\partial:X\rightarrow A\cap B$ from which $\partial_*$ is induced is not known in general or maybe need not exist!!