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Math Help - A problem from "Introduction to Topological Manifolds"

  1. #1
    Aki
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    A problem from "Introduction to Topological Manifolds"

    Could anyone help me solve the following problem ?

    Show that a covering map is proper if and only if it is finite-sheeted.

    This is the problem 11-10 from "Introduction to Topological Manifolds" by John M. Lee. The definition of a covering map in this book is the following.

    A covering map is a continuous surjective map q : E \rightarrow X such that E is connected and locally path-connected, and every point of X has an evenly covered
    neighborhood.

    My first attempt was to show that the image of any sequence in E that diverges to infinity diverges to infinity in X. But, I noticed that for this to be a sufficient condition for properness of q, E should be a second countable Hausdorff space, and that no additional condition is assumed for E other than connectedness and local path-connectedness.

    I don't have a clue any more.
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    Re: A problem from "Introduction to Topological Manifolds"

    Try to prove that e^{2\pi i(\cdot)} : \mathbb{R} \to \mathbb{S}^1 is not proper using as little special properties of \mathbb{R} as possible, the general idea should follow from there.
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    Aki
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    Re: A problem from "Introduction to Topological Manifolds"

    Thank you for your help.

    Your mapping is not proper since the preimage of a point in \mathbb{S}^1 is an infinite discrete set in \mathbb{R}. I understand that to prove a continuous map being proper, it is necessary to show that it is a closed map in addition to showing that it has compact fibers. The former is harder for me.
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    Re: A problem from "Introduction to Topological Manifolds"

    Quote Originally Posted by Aki View Post
    Thank you for your help.

    Your mapping is not proper since the preimage of a point in \mathbb{S}^1 is an infinite discrete set in \mathbb{R}. I understand that to prove a continuous map being proper, it is necessary to show that it is a closed map in addition to showing that it has compact fibers. The former is harder for me.
    According to the first edition of the book a proper map is one such that the preimage of compact sets are compact.
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    Aki
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    Re: A problem from "Introduction to Topological Manifolds"

    The definition of proper maps in the second edition is the same.
    I referred to Proposition 4.93 (c).

    Proposition 4.93 (Sufficient Conditions for Properness)
    Suppose X and Y are topological spaces, and F : X \rightarrow Yis a continuous map.
    (c) If  F is a closed map with compact fibers, then F is proper.
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    Re: A problem from "Introduction to Topological Manifolds"

    You don't need that proposition: The idea is that you associate with a covering of the preimage of a compact set in X a finite refinement (here's where you use the finite-sheeted-ness) using that the set in X is compact and the special open sets one can define associated with points in X.

    Edit: I'm assuming you have trouble with finite-sheeted implies proper.
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    Aki
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    Re: A problem from "Introduction to Topological Manifolds"

    Yes.
    But, I'm not sure about the converse too.

    What troubles me is the following.
    Given a compact set K in X and an open covering \{U_\lambda\}_{\lambda\in\Lambda} of q^{-1}(K),
    since \{q(U_\lambda\)\}_{\lambda\in\Lambda} is an open covering of K, one can choose from it a finite covering \{q(U_\lambda_i)\}_{i=1,\cdots,N} of K. But, \{U_\lambda_i\}_{i=1,\cdots,N} is not necessarily a covering of q^{-1}(K).
    Last edited by Aki; August 8th 2011 at 12:52 AM. Reason: I corrected a confusion of the notations
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    Re: A problem from "Introduction to Topological Manifolds"

    Quote Originally Posted by Aki View Post
    Yes.
    But, I'm not sure about the converse too.

    What troubles me is the following.
    Given a compact set K in X and an open covering \{U_\lambda\}_{\lambda\in\Lambda} of q^{-1}(K),
    since \{q(U_\lambda\)\}_{\lambda\in\Lambda} is an open covering of K, one can choose from it a finite covering \{q(U_\lambda_i)\}_{i=1,\cdots,N} of K. But, \{U_\lambda_i\}_{i=1,\cdots,N} is not necessarily a covering of q^{-1}(K).
    Where are you using the fact that q is a covering?

    Okay, here's a bigger hint: Let K \subset X compact and a covering (V_a)_{a\in A} of the inverse image. For every x\in K there is an U_x such that W_x=q^{-1}(U_x) =\cup_{j=1}^n U_j where the U_j are open and disjoint and q:U_j\to U is an homeomorphism (this is just the definition of a covering map). Try to shrink the W_j so that they are inside one V_a but in a way that they are still preimages of a set and have analogous properties to W_j. Now use the compacteness of K to finish the argument.
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    Aki
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    Re: A problem from "Introduction to Topological Manifolds"

    Thanks to your guidance, I managed to come up with something resembling a proof of the first half of the problem.

    Let K \subset X compact and \{U_\alpha\}_{\alpha\in A} a covering of q^{-1}(K). One can construct a covering \{V_\lambda\}_{\lambda\in \Lambda} of K satisfying the condition that
    (i) V_\lambda is evenly covered by q for each \lambda \in \Lambda,
    (ii) for each \lambda\in\Lambda there exists \alpha\in A such that V_\lambda \subset q(U_\alpha),
    (iii) \bigcup_{\alpha\in A}q(U_\alpha)=\bigcup_{\lambda\in \Lambda}V_\lambda.
    For the condition (i), one can use the fact that every connected open
    subset of an evenly covered open subset is itself evenly covered.

    It follows from the compactness of K that there exists a finite set \Lambda_K \subset \Lambda such that K\subset\bigcup_{\lambda\in \Lambda_K}V_\lambda. Since V_\lambda (\lambda\in\Lambda_K) are evenly covered by q, we have q^{-1}(V_\lambda)=\bigcup_{j=1}^n W_\lambda^j where W_\lambda^j \subset E (\lambda\in\Lambda_K, j=1,\cdots,n) are disjoint open sets such that q|_{W_\lambda^j} W_\lambda^j \rightarrow V_\lambda is a homeomorphism. Let I_\lambda^j be a subset of Adefined by
    I_\lambda^j:=\{\alpha\in A : W_\lambda^j \subset U_\alpha\}, \lambda\in\Lambda_K, j=1,\cdots,n.

    Finally, the index set {\cal I} can be constructed by choosing one element arbitrarily from I_\lambda^j that is not empty for \lambda\in\Lambda_K, j=1,\cdots,n. Then \{U_\alpha\}_{\alpha \in {\cal I}} is a finite covering of q^{-1}(K).

    I'm looking forward to your comment.

    [I corrected a mistake in the definition of the index set {\cal I}.]
    Last edited by Aki; August 9th 2011 at 03:13 AM.
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    Aki
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    Re: A problem from "Introduction to Topological Manifolds"

    I noticed a serious error in my "proof".
    By the definition of I_\lambda^j, only one necessary open set in \{U_\alpha\}_{\alpha\in A} is guaranteed to be chosen for a finite cover of q^{-1}(K). Other open sets necessary in the finite cover on the other sheets can be lost, if they intersect with W_\lambda^j but don't contain it.
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    Aki
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    Re: A problem from "Introduction to Topological Manifolds"

    I revised my proof.

    Let K \subset X be compact and \{U_\alpha\}_{\alpha\in A} an open cover of q^{-1}(K). For each x \in K there exists an open set V'_x \subset X evenly covered by q. Let y_1,\cdots,y_n be the preimages of x by q. For each j\in \{1, \cdots, n\} there exists \alpha_j \in A such that y_j \in U_{\alpha_j}.
    Define V_x by V_x := V'_x \bigcap \left(\bigcap_{j=1}^n q(U_{\alpha_j})\right). Then, \{V_x\}_{x\in K} is a cover of K. Since K is compact, there exists a finite subcover of it. Let it denoted by \{V_\lambda\}_{\lambda\in \Lambda}. Since V_\lambda (\lambda\in \Lambda) is evenly covered by q, we have q^{-1}(V_\lambda)=\bigcup_{j=1}^n W_\lambda^j where W_\lambda^j \subset E (\lambda\in\Lambda, j=1,\cdots,n) are disjoint open sets such that q|_{W_\lambda^j} W_\lambda^j \rightarrow V_\lambda is a homeomorphism. Let I_\lambda^j be a subset of Adefined by
    I_\lambda^j:=\{\alpha\in A : W_\lambda^j \subset U_\alpha\}, \lambda\in\Lambda, j=1,\cdots,n. Finally, the index set {\cal I} can be constructed by choosing one element arbitrarily from I_\lambda^j for each \lambda\in\Lambda, j=1,\cdots,n. Then \{U_\alpha\}_{\alpha \in {\cal I}} is a finite cover of q^{-1}(K).
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