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Simple Topology Question

  1. Apr 18, 2008 #1
    Hi,

    I have a question that I'm not sure about.
    If f:A->C is continuous and B is a subset of C that is simply connected, is f(^-1)(B) necessarily connected or simply connected for that matter? Since the spaces are not necessarily homeomorphic I cannot consider it a topological invariant.

    Thanks
     
  2. jcsd
  3. Apr 18, 2008 #2

    quasar987

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    Consider the old canonical mapping from [0,2pi) to the circle S^1 in the complex plane (f(t)=e^it). Then for r small enough, f^-1(B(1,r)) is clearly not connected (and hence not simply connected either) since it consist of a little neighborhoods of 0 and a little neighborhoods of 2pi.
     
  4. Apr 18, 2008 #3
    Actually, we can be even simpler than that: let f:R->R be given by f(x)=x^2, and let B={1}. Then f^-1(B)={1,-1}.
     
  5. Apr 18, 2008 #4
    Thanks for the help! The last one is a very simple counterexample.
     
  6. Apr 20, 2008 #5

    WWGD

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    Maybe for a more extreme counterexample re connectedness, consider the case of

    IR as a covering space of S^1 . Maybe if you had 1-1 -ness. (tho not in this case,

    since continuous bijection bet. compact and hausdorff is a homeo., which is sufficient,

    tho I don't know if it is necessary).


    For a trivial counterexample re simple-connectedness, consider a constant map

    defined on an annulus.
     
  7. Apr 22, 2008 #6

    mathwonk

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    trivial counterexamples do exist for zero dimensional B, but try higher dimensional B's.

    and try it for algebraic maps.

    i.e. if you map an algebraic variety X to an algebraic variety Y, what does the inverse image of an irreducible curve in Y look like?

    more specifically, project a surface onto P^2, and ask what the inverse image of a general line looks like?

    see the fulton - hansen connectedness theorem, and various versions of bertini's thoerem.
     
    Last edited: Apr 22, 2008
  8. Apr 25, 2008 #7

    mathwonk

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    here is a nice theorem of fulton and collaborators:

    if L is a linear subspace of a projective space P, having codimension d,

    and if X-->P is any morphism from a projective variety X,

    having image in P of dimension larger than d,

    then the inverse image in X of L is connected.
     
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