Finding the singular homology of the quotient of the disjoint union of two spaces

Asked May 01 '20 at 18:05

Active May 01 '20 at 18:20

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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?

edited May 01 '20 at 18:20

J. W. Tanner

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asked May 01 '20 at 18:05

Morettin

1

https://math.stackexchange.com/questions/2963241/how-to-see-the-real-projective-plane-is-a-m%C3%B6bius-band-glued-to-a-disk This shows that the space you are looking for is $\mathbb R P^2$. – fish May 01 '20 at 18:18
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I think this should answer your question. – Mnifldz May 01 '20 at 18:18
Thank you! This definitely helps – Morettin May 01 '20 at 18:21

Finding the singular homology of the quotient of the disjoint union of two spaces

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