Let $\mathscr{F}$ be a presheaf of abelian groups on some topological space $X$. We say that $\mathscr{F}$ is locally constant if there exists an open cover $\mathcal{U}$ of $X$ (i.e. $X=\bigcup_{U\in\mathcal{U}}\ U$ and the elements of $\mathcal{U}$ are open) such that, for all $U\in\mathcal{U}$ and for all $P\in U$, we have $\mathscr{F}(U)=\mathscr{F}_P$.

The task is to prove that for every $U\in\mathcal{U}$ and every connected subset $V\subseteq U$, the composite

$\mathscr{F}(U) \xrightarrow{\ \text{restriction}\ } \mathscr{F}(V) \xrightarrow{\ \text{sheafification}\ } \mathscr{F}^+(V)$

is an isomorphism. This is an exercise in the Book on Algebraic Topology by Spanier. A fellow student asked me about it because he needs the result, but we were unable to figure it out.


closed as too localized by S. Carnahan May 10 '11 at 22:48

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    $\begingroup$ I don't think that your recent questions are appropriate for MO. Or at least you should put some effort to solve these questions by your own and show your ideas ... $\endgroup$ – Martin Brandenburg May 10 '11 at 22:44
  • $\begingroup$ Perhaps you should try asking on math.stackexchange.com $\endgroup$ – S. Carnahan May 10 '11 at 22:48
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    $\begingroup$ I'm sorry, I had read the FAQ and I am not quite sure how my questions are inappropriate. In particular, how is this particular question any less demanding or interesting than, say, that one: mathoverflow.net/questions/24361/… $\endgroup$ – Jesko Hüttenhain May 10 '11 at 23:19
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    $\begingroup$ The question you linked, while easy, is interesting because it is about local-versus-global phenomena. Your question is a diagram chase. $\endgroup$ – S. Carnahan May 11 '11 at 4:49
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    $\begingroup$ Note also that saying "$\mathcal{F}(U) = \mathcal{F}_P$" seems slightly sloppy to me: what you mean is that the canonical map from the guy on the left to the guy on the right is an isomorphism. $\endgroup$ – Pete L. Clark May 11 '11 at 5:02