Interesting topological spaces to calculate the homology groups.

I am calculating homology groups of several topological spaces to learn and I have already calculated the homology groups of $\mathbb{S}^m$, $\mathbb{R}P^2$, the Klein bottle, $\mathbb{R}^2$ minus a finite number of points and I am going to calculate the homology groups to the moebius band and I was wondering what other interesting topological spaces I can easily calculate the homology groups, thank you very much.

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    $\begingroup$ You might try looking at the exercises in an algebraic topology textbook. $\endgroup$ Jul 1 '19 at 18:27

Not sure whether you consider them to be interesting, but these are some spaces whose homology groups I once computed in the past when I was studying algebraic topology:

1) The torus $T^2 = S^1 \times S^1$ or more generally $T^n = S^1 \times \dots \times S^1$

2) The space you get when you take $S^2$ and identify the north and south poles

3) Take $T^2 = S^1 \times S^1$ and quotient out the circle $S^1 \times \lbrace x \rbrace$ for some point $x \in S^1$

4) Take $T^2 = S^1 \times S^1$ and quotient out two different circles $S^1 \times \lbrace x \rbrace$ and $S^1 \times \lbrace y \rbrace$

5) The space $X$ one gets by glueing two solid tori $S^1 \times D^2$ along their boundaries via the identity $S^1 \times S^1 \rightarrow S^1 \times S^1$

6) The dunce hat/cap (Take a solid triangle and identify the sides by the edge word $a^3$ (or $a^2a^{-1}$ - definition varies a bit in the literature)

7) If you managed to compute your examples plus these and you still want more, then I suggest that you just construct some examples and try to compute the homology groups. Sometimes it will work out and otherwise you can still ask questions here.

I will not be able to remember all the homology groups for you to compare though.


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