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Math Help - Topology

  1. #1
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    Topology

    Why is it obvious that attaching an (m+n)-cell to SmVSn is SmxSn?

    Also, if you have a space X that is the real line, with two sphere's (S2) wedged at each integer, to determine its homology group, I figured I should use the Mayer–Vietoris sequence, so I split it such that U = X - {odd integers}, V = X - {even integers}. Do you think that's what I should do?
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  2. #2
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    Re: Topology

    For your first question, note that S^n \cong R^n \cup \{\infty\}, that is, S^n is the one point compactification of R^n.
    Now S^m \times S^n \cong (R^m \cup \{\infty\}) \times (R^n \cup \{\infty\})
     = R^{m+n} \cup (R^m \times \{\infty\}) \cup (R^n \times \{\infty\}) \cup  (\{\infty\} \times \{\infty\})
     = R^{m+n} \sqcup R^m \sqcup R^n \sqcup \{\infty\}(disjoint union)
    First glue R^m \sqcup R^n \sqcup \{\infty\} we get S^m \vee S^n, then glue R^{m+n} by identifying it with D^{m+n} the unit disk, then glue it to the S^m \vee S^n frame. It's not easy to express but you can always take m=n=1 as an example to study the procedure to glue (1 square)+(2 segments)+(1 point) together to get a torus.
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