The calculation of a homology group

Question

This is a question on Massey's A Basic Course of Algebraic Topology. I met some problem in calculating a homology group of a specific space when dealing with a question.

Let $X=\{ (x,y,z) \mid xyz=0 \}$ then how to calculate $H_2(X)$? By the way, are there any widely used important skills in calculating such homology groups?

Answer 1

It seems that you actually mean $X=\{(x,y,z)\in\mathbb{R}^3\ |\ xyz=0\}$ judging from comments (always write down the space you start from). In that situation there's a deformation retraction

$$F:I\times X\to X$$ $$F(t,v)=tv$$

onto the origin $\{(0,0,0)\}$ and so $X$ is contractible.

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As for the skills/tools. Well there are lots of them, to name few important:

1. homotopy invariance (the one I've used above)
2. the long exact sequence of homology
3. Mayer-Vietoris sequence
4. Eilenberg-Steenrod axioms (which generalizes 1 and 2)
5. Kunneth formula
6. Hurewicz theorem (although seems to be more useful for homotopy calculation)
7. Lefschetz duality
8. spectral sequences (heavy artillery)

and many, many more. The research on the topic is still ongoing, probably will never end.