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Cup product structure of the n-torus

  1. Nov 17, 2008 #1
    I have some questions about an example in Hatcher's "Algebraic Topology". This book is freely available online at http://www.math.cornell.edu/~hatcher/AT/ATchapters.html and I have attached the relevant part. My questions are about Example 3.11 where Hatcher computes the cup product structure of the n-torus.

    1) In the second paragraph of this example, Hatcher seems to take the cross product of a relative homology class [itex]\alpha[/itex] and an absolute homology class [itex]\beta[/itex]. That is, he seems to use

    [tex]H^1(I,\partial I;R) \times H^n(Y;R) \to H^{n+1}(I \times Y, \partial I \times Y; R)[/tex]

    But above this example he only defines cross products between absolute homology groups or between relative homology groups. So, what does [itex]\alpha \cup \beta[/itex] mean?

    2) I don't understand at all why Hatcher says at the top of page 211 that [itex]\delta[/itex] is an isomorphism when restricted to the copy of [itex]H^n(Y;R)[/itex] corresponding to [itex]{0} \times Y [/itex]. I thought that [itex] \delta [/itex], the connecting homomorphism, was a rather complicated object and I don't see why that doesn't require justification...
     

    Attached Files:

    Last edited: Nov 17, 2008
  2. jcsd
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