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April 1st, 2018, 11:06 AM   #1
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seeking example of single germ in manifolds with distinct topologies

What would be an example of a single germ that could inhabit two analytic, connected manifolds $M_1$ and $M_2$, such that the genus of the first manifold does not equal the genus of the second?

My question is motivated by the claim that I believe to be true, but I have not seen proven, that the germ of an analytic function $f$ at the point $x \in M$ is insufficient to determine the global topology of $M$, where $M$ is a connected analytic manifold, and $f$ is everywhere analytic in $M$. Therefore, there should be a proof-by-counterexample, but I have been unable to come up with one. Any ideas?

David

Addendum: I am particularly interested in the case where the function $f$ is the metric $g_{ij}$ of a 4-dimensional manifold. However, I'm not sure if that changes the essence of the question or not, which is why I didn't formulate it that way.
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April 2nd, 2018, 08:03 AM   #2
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I think perhaps that I should pose the question such that the analytic function in question is the metric. And perhaps I should also add "nontrivial" i.e. assume that $f$ is not constant.
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