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I realize that this question might seem ambiguous, is there a topological notion for what a Hole is? I think it has something to do with the fundamental groups of the topological space but I don't really know, could someone please tell me the basics and what is a basic tool for detecting when a topological space has "holes"?

Background: I am currently taking topology 1 and I am halfway through my second semester of abstract algebra.

Thank you kindly, regards.

Asinomás
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  • I also heard somebody mentioned we should check if the homology vanishes. – Asinomás Apr 07 '15 at 03:02
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    The fundamental group and the homology of a topological space can help you detect this. Any course in algebraic topology should cover this, and it looks like you'll have the prerequisites for that course very soon! – Jack Davies Apr 07 '15 at 03:06
  • Thank you very much! I am very much looking forward to it, do you think you could help me make this joke mathematically rigorous? – Asinomás Apr 07 '15 at 03:09
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    Very nice, you could say "How does a topologist check his proof is correct? He makes sure the fundamental group vanishes", or maybe you should say algebraic topologist haha – Jack Davies Apr 07 '15 at 03:11

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My professor exposed me to L.S. Pontryagin's "Foundations of Combinatorial Topolology," just recently, and I would say that I had about the same background as you did when I started it. I struggled at some point, but in general I was able to get through it pretty okay. The main interest of the book is simplicial homology, and this will address your question. The text is short, and relatively dense, but it was possible when I had taken a course in abstract algebra and point-set topology.

To be fair, this is probably overkill just to understand a joke about topological holes, but it is a thorough treatment of homology theory.

A. Thomas Yerger
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Homology counts the number of "holes".

Math
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