Homology union three subspaces.

Question

Consider the following subspaces of $\mathbb{R}^3$: $$A=\{(x,y,z):x^{2}+y^{2}=1, z\in[-1,1] \}$$ $$B=\{(x,y,z): x^{2}+y^{2}\leq 1, z\in\{-1,0,1\} \}$$ $$C=\{(0,0,z):z\in[-1,1]\}.$$ Let $X=A\cup B\cup C$. How can I compute the singular homology groups $H_{k}(X)$ for $k\geq 0$?

Answer 1

1. Make a picture.
2. Try to see if the set deformation retracts onto a simpler set.
3. Decompose the simpler set into simplices glued to each other along their boundary (exhibit a CW-complex structure).
4. Turn the gluing information from 3. into a cellular chain complex.
5. Compute the (co)homology.