
Topology Topology Math Forum 
 LinkBack  Thread Tools  Display Modes 
April 1st, 2018, 11:06 AM  #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, 08:03 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.


Tags 
analytic, connected, distinct, genus, germ, manifold, manifolds, seeking, single, topologies 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Show that two topologies are homeomorph  wuspengret  Real Analysis  4  November 12th, 2012 12:10 PM 
Ordered distribution of distinct objects into distinct conta  metamath101  Algebra  2  June 22nd, 2012 04:59 PM 
A Question in Manifolds  Hooman  Real Analysis  6  April 9th, 2012 10:01 PM 
Open or Closed Topologies.  shmounal  Real Analysis  0  November 2nd, 2011 08:38 AM 
Help with coarser/finer topologies  pruzchett  Real Analysis  1  August 6th, 2008 05:34 AM 