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Math Help - Mapping cone, Homology groups and universal cover

  1. #1
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    Mapping cone, Homology groups and universal cover

    Let W be the mapping cone of the map f: S^1 \rightarrow S^1 defined by f(z)= z^p.

    How do you compute the homology groups of W? What about the homology groups of the Universal covering of W?

    I know that the mapping cone C_f of f: X \rightarrow Y, is defined to be the quotient of the mapping cylinder of f with X.
    Or we can say,
    Given a map f: X \rightarrow Y, the mapping cone C_f is defined to be the quotient of the topological space of (X \times I) \coprod Y with respect to the equivalence relation (x,0) \sim (x',0), (x,1) \sim   f(x) , on X. Here I denotes the unit interval [0,1] with its standard topology.

    But I am not sure how to start using this definition of the mapping cone, to find the homology groups of W and of its universal cover.
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  2. #2
    Senior Member Tinyboss's Avatar
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    The "cone" part is contractible to the "point" X\times\{0\}, so its homology is easy to compute. I'd guess that the Mayer-Vietoris sequence might be the way to proceed, but my algebraic topology class just ended and we didn't do mapping cones.
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  3. #3
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    So, can we say W deformation retracts to S^1 \times \{0 \}, so they are homotopic equivalent, hence H_n(W) = H_n(S^1 \times \{0 \} ) = Z if n=1 or 0 and 0 if n>1 ?

    What would be the universal cover of W?
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