# Interesting topological spaces to calculate the homology groups.

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.

• You might try looking at the exercises in an algebraic topology textbook. 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.