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?

Please see https://math.meta.stackexchange.com/questions/5020/ — Angina Seng, Jul 04 '19 at 15:46
Instead of just stating the problem, specify what you've thought about so far and where exactly you are stuck. — Travis Willse, Jul 04 '19 at 15:49
So assuming $X\subseteq\mathbb{R}^3$ then your $X$ deformation retracts onto the singleton ${(0,0,0)}$ via $(t,v)\mapsto tv$. All positive homologies are trivial. — freakish, Jul 04 '19 at 15:50
Oh gosh! I guess my brain has been occupied with so many strange ideas, not noticing that the space can be retract to origin! — Dick. Y, Jul 04 '19 at 15:58

freakish · Accepted Answer · 2019-07-04T16:42:54.093

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.

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

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

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

The calculation of a homology group

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