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Induced Maps on Homology

  1. Aug 28, 2011 #1
    Hi, All:

    I am curious to find examples of maps f:X-->X ; X an n-dimensional manifold

    that induce isomorphisms on , say, the first k<n homology groups, but not

    so on the remaining n-k groups. I can see if we had maps g:X-->Y, we could start

    with Y=X, let f be an automorphism, and then cap some boundaries of X, i.e., all j-

    boundaries for j>k , but not so for maps f:X-->X . Any Ideas?


    others, so that the induced maps on H_k(X) are not isomorphsims
  2. jcsd
  3. Aug 28, 2011 #2
    Hint: "skeleton".
  4. Aug 28, 2011 #3

    I guess you're suggesting some obstruction theory issues; spin structure, etc?

    Unfortunately, I haven't been able to find much in this area from a geometric

    perspective; most of the info nowadays seems to be done in terms of abstract

    obstruction theory, spin structures, etc. Still, I ordered Steenrod's book on the

    geometry of bundles from the library recently. Is this what you were referring to?
  5. Aug 29, 2011 #4


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    Take any map of degree higher than one from a sphere to itself. Using Cartesian products of spheres with other manifolds I think you should be able to get all of the examples except iso up to dimension n-1 and not iso in the top dimension. Cartesian product of spheres with tori should do it.
    Last edited: Aug 29, 2011
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