Let us consider the quotient of the disjoint union of the spaces; $D^{2}$, the two dimensional disc, and $M$, the Mobius strip with the equivalence relation that identifies the boundary of $D^{2}$ and $M$.In other words $D^{2} \bigsqcup M / \sim $.
I would like to compute the singular homology of this quotient.
I know that homology is homotopic invariant and hence it follows that it is invariant under homeomorphism, and so I need to find a space which is homeomorphic to this of which I known the homology.
How can I do this?