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When formally defining Betti number, we often use homology group - but I am not sure how we can use that definition to prove the informal definition of Betti number - that talks about "unconnected and connected" ones.

from Wikipedia (http://en.wikipedia.org/wiki/Betti_number )

Informally, the $k$th Betti number refers to the number of unconnected $k$-dimennsional surfaces.

Actually, I also feel confused about what this actually means. (Example of torus would be appreciated for an explanation. I am having a hard time understanding what "one-dimensional" holes of a torus exactly refers to, and same with two-dimensional holes.)

addition: But what exactly is an outside hole of a torus? I only see one hole inside.... (I know I am stupid for not figuring out, but I just cannot find it...)

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Differentio
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  • i think this thread should help you out: http://math.stackexchange.com/questions/40149/soft-question-intuition-of-the-meaning-of-homology-groups – citedcorpse Jun 15 '13 at 08:50
  • in Carlson's reference (page 7), he says "independant" k-dim. surfaces (doesn't change much). I don't know either how to interpret it as anything related to homology, so it is unhelpful (to me). Your best bet would be to ask him what he meant then. – mercio Jun 15 '13 at 09:06
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    The "inside hole" of the the torus is the hole you had to be in if you would have wanted to know what the torus looks like from the inside. The "outside hole" of the torus is the hole you can stick your arm through and "wear" the torus like a bracelet. I never thought I'd use those terms in a mathematical context. – Idan Jun 15 '13 at 10:23
  • Yes, but then second Betti number also seems to refer to the inside hole.. So let's say that we fill the inside hole of a torus. Then, the second Betti number would be zero, while the first Betti number changes to one? That's my current understanding. – Differentio Jun 16 '13 at 02:19

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