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Help with gluing together surfaces of infinite genus

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1

$begingroup$

Let $S=bigwedge_minmathbbZmathbbS^1$ denote the wedge sum of countably infinitely many circles—that is, the topological space obtained by removing countably infinitely many distinct points from, say, $mathbbC$: $S=mathbbCbackslashleft z_m:minmathbbZright$. Let $nu$ be an integer $geq2$, and suppose we have $nu$ copies of $S$ (written $S_1,ldots,S_nu$) to be glued together along the “holes” (ex: for each $m$, the surfaces $S_1,ldots,S_nu$ get glued together at the hole at $z_m$). I'm pretty sure that the resultant topological space (call it $T$) is then: $$T=prod_n=1^nuS=prod_n=1^nubigwedge_minmathbbZmathbbS^1$$ that is, $T$ is the direct product of $nu$ copies of $S$. Is there a simpler description of $T$ (i.e., is there a way to interchange $prod$ and $bigwedge$; is there a way to write things in terms of $mathbbT^k$ for some $k$s (where $mathbbT^k=prod_i=1^kmathbbS^1$), etc.)?

general-topology riemann-surfaces

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asked Mar 25 at 21:18

MCSMCS

984313

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Be careful. The wedge sum of circles is not the same topological space as $mathbbC$ without some points. They are homotopy equivalent, but not topologically the same. It is not clear to me how you would glue something together along a hole. Perhaps along the boundary of a hole?
  $endgroup$
  – Strichcoder
  Mar 25 at 22:15
  

  
  
  
  

add a comment | 

1

1

$begingroup$

Let $S=bigwedge_minmathbbZmathbbS^1$ denote the wedge sum of countably infinitely many circles—that is, the topological space obtained by removing countably infinitely many distinct points from, say, $mathbbC$: $S=mathbbCbackslashleft z_m:minmathbbZright$. Let $nu$ be an integer $geq2$, and suppose we have $nu$ copies of $S$ (written $S_1,ldots,S_nu$) to be glued together along the “holes” (ex: for each $m$, the surfaces $S_1,ldots,S_nu$ get glued together at the hole at $z_m$). I'm pretty sure that the resultant topological space (call it $T$) is then: $$T=prod_n=1^nuS=prod_n=1^nubigwedge_minmathbbZmathbbS^1$$ that is, $T$ is the direct product of $nu$ copies of $S$. Is there a simpler description of $T$ (i.e., is there a way to interchange $prod$ and $bigwedge$; is there a way to write things in terms of $mathbbT^k$ for some $k$s (where $mathbbT^k=prod_i=1^kmathbbS^1$), etc.)?

general-topology riemann-surfaces

share|cite|improve this question

asked Mar 25 at 21:18

MCSMCS

984313

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Be careful. The wedge sum of circles is not the same topological space as $mathbbC$ without some points. They are homotopy equivalent, but not topologically the same. It is not clear to me how you would glue something together along a hole. Perhaps along the boundary of a hole?
  $endgroup$
  – Strichcoder
  Mar 25 at 22:15
  

  
  
  
  

add a comment | 

$begingroup$

Let $S=bigwedge_minmathbbZmathbbS^1$ denote the wedge sum of countably infinitely many circles—that is, the topological space obtained by removing countably infinitely many distinct points from, say, $mathbbC$: $S=mathbbCbackslashleft z_m:minmathbbZright$. Let $nu$ be an integer $geq2$, and suppose we have $nu$ copies of $S$ (written $S_1,ldots,S_nu$) to be glued together along the “holes” (ex: for each $m$, the surfaces $S_1,ldots,S_nu$ get glued together at the hole at $z_m$). I'm pretty sure that the resultant topological space (call it $T$) is then: $$T=prod_n=1^nuS=prod_n=1^nubigwedge_minmathbbZmathbbS^1$$ that is, $T$ is the direct product of $nu$ copies of $S$. Is there a simpler description of $T$ (i.e., is there a way to interchange $prod$ and $bigwedge$; is there a way to write things in terms of $mathbbT^k$ for some $k$s (where $mathbbT^k=prod_i=1^kmathbbS^1$), etc.)?

general-topology riemann-surfaces

share|cite|improve this question

asked Mar 25 at 21:18

MCSMCS

984313

$endgroup$

share|cite|improve this question

asked Mar 25 at 21:18

MCSMCS

984313

share|cite|improve this question

asked Mar 25 at 21:18

MCSMCS

984313

asked Mar 25 at 21:18

MCSMCS

984313

asked Mar 25 at 21:18

MCSMCS

984313