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Thread: first homolgy group of a disk with n holes

  1. #1
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    first homolgy group of a disk with n holes

    Hi;

    Let D^2 be a 2-dim disk with n holes, i.e D^2\(S^0 * D^2$)^n. Then is it true that the first homology group of this space is Z^n.

    Thank you in advance
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  2. #2
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    Re: first homolgy group of a disk with n holes

    The "homology group" can be thought of as the group of non-equivalent curves in the set where two curves are equivalent if and only if one can be continuously deformed into the other. First, of course, all closed curves (loops) that do not enclose any hole are equivalent (and that equivalence class corresponds to the group identity), every curve that encloses one and only one hole is equivalent to other curves that surround only that particular hole- so there are n of them. Every curve that encloses exactly two holes forms an equivalence class- and there are \begin{pmatrix}n \\ 2 \end{pmatrix} of those, etc. to the single class of curves that enclose all holes. Think "binomial theorem".
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