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April 1st, 2018, 12:06 PM  #1 
Newbie Joined: Apr 2018 From: Frederick, Maryland, USA Posts: 2 Thanks: 0  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 proofbycounterexample, 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 4dimensional 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. 
April 2nd, 2018, 09:03 AM  #2 
Newbie Joined: Apr 2018 From: Frederick, Maryland, USA Posts: 2 Thanks: 0 
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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analytic, connected, distinct, genus, germ, manifold, manifolds, seeking, single, topologies 
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