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I find lots of comment about the connection between of n dimension hole and homology group, but it is hard for me to find a "standard" textbook discussing about the definition and the construction, can someone recommend some textbook about the fact. My level about algebraic topology in about the textbook, An introduction to algebraic topology,by Joseph J. Rotman, Thanks a lot.

J. W. Tanner
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lee
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  • I'm not sure what exactly you're asking about (excision?), but Hatcher is another canonical intro algebraic topology text. – anomaly May 02 '24 at 15:33
  • I mean I want to know about the why homology group will express the n dimensional hole of the space, I want to know this fact systematically. – lee May 02 '24 at 15:36
  • You're not going to get a better answer than you got from reading this answer where you posted a comment, because the very concept of "hole" is too informal, too broad, and too vague. It's only purpose is to serve in a pedagical discussion preceding the formal development of homotopy groups and homology groups. – Lee Mosher May 02 '24 at 15:58
  • You can find some further discussion of this informal concept here, here, here, here, and perhaps other places that you can find by searching the site. – Lee Mosher May 02 '24 at 16:02
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    It's really the other way around: "Hole" is an attempt to stick an intuitive term on a nontrivial (co)homology element, by analogy with the circle and torus. But it's not even true that every homology class can be represented by an embedded submanifold (and there are corresponding results for rational multiples of classes). – anomaly May 02 '24 at 16:09

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