Homology groups, homotopy equivalence and one point compactification

Let and .

Do the spaces X and Y have the same homology groups? Are they homotopy equivalent? Are they Homeomorphic? Do they have homeomorphic one-point compactifications?

First, I was wondering if the space Y is homeomorphic to a cylinder? To me it looks like it's a Torus where one circle has been cut off. Or maybe, it is a Torus, where only point is missing?

I know that if and otherwise. (Where is the i-dimensional reduced homology of the space )

Also I know as an application of the Mayer-Vietoris sequence, that If is a subspace of homeomorphic to for , then if and otherwise.

Also, I know that ) for (Where is the i-dimensional homology group of ) and that

But does any of this help answer the questions? Or is it totally irrelevant in this case? If this doesn't help, how do you go about answering the questions?