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Covering maps between Seifert fibered manifolds

2

$begingroup$

Let $M$ and $widetildeM$ be two Seifert fibered three manifolds. Suppose that there exists a covering projection $p: widetildeM to M$ preserving the Seifert structure. What is the relation between the Euler numbers $e left( widetildeM right)$ and $e(M)$ of the two Seifert manifolds?

geometric-topology low-dimensional-topology

share|cite|improve this question

asked Aug 23 '15 at 10:01

Antonio AlfieriAntonio Alfieri

1,222412

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  This depends on two parameters: degree $d_1$ of the covering of the generic fiber and the degree $d_2$ of the covering between the bases (understood in the orbifold sense). Then $e(tilde M)= e(M)d_2/d_1$, I think.
  $endgroup$
  – Moishe Kohan
  Aug 25 '15 at 21:04
  

  
  
  
  

add a comment | 

2

$begingroup$

Let $M$ and $widetildeM$ be two Seifert fibered three manifolds. Suppose that there exists a covering projection $p: widetildeM to M$ preserving the Seifert structure. What is the relation between the Euler numbers $e left( widetildeM right)$ and $e(M)$ of the two Seifert manifolds?

geometric-topology low-dimensional-topology

share|cite|improve this question

asked Aug 23 '15 at 10:01

Antonio AlfieriAntonio Alfieri

1,222412

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  This depends on two parameters: degree $d_1$ of the covering of the generic fiber and the degree $d_2$ of the covering between the bases (understood in the orbifold sense). Then $e(tilde M)= e(M)d_2/d_1$, I think.
  $endgroup$
  – Moishe Kohan
  Aug 25 '15 at 21:04
  

  
  
  
  

add a comment | 

$begingroup$

Let $M$ and $widetildeM$ be two Seifert fibered three manifolds. Suppose that there exists a covering projection $p: widetildeM to M$ preserving the Seifert structure. What is the relation between the Euler numbers $e left( widetildeM right)$ and $e(M)$ of the two Seifert manifolds?

geometric-topology low-dimensional-topology

share|cite|improve this question

asked Aug 23 '15 at 10:01

Antonio AlfieriAntonio Alfieri

1,222412

$endgroup$

share|cite|improve this question

asked Aug 23 '15 at 10:01

Antonio AlfieriAntonio Alfieri

1,222412

share|cite|improve this question

asked Aug 23 '15 at 10:01

Antonio AlfieriAntonio Alfieri

1,222412

asked Aug 23 '15 at 10:01

Antonio AlfieriAntonio Alfieri

1,222412

asked Aug 23 '15 at 10:01

Antonio AlfieriAntonio Alfieri

1,222412

1 Answer
1

active

oldest

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0

$begingroup$

Let's say that a regular fiber of $M$ is covered by $u$ regular fibers of $widetildeM$ and restriction of $p$ to each regular fiber has order $v$. Then $e(widetildeM) = e(M) u / v$.

I don't have any reference for that fact from top of my head but it is visible just by thinking of the geometric role of numbers $(alpha_i, beta_i)$ for each singular fiber. I'm sure it's written down somewhere in Neumann-Raymond's book.

share|cite|improve this answer

edited Mar 25 at 20:35

answered Mar 25 at 20:17

Stephen DedalusStephen Dedalus

1,0091022

$endgroup$

add a comment | 

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0

$begingroup$

Let's say that a regular fiber of $M$ is covered by $u$ regular fibers of $widetildeM$ and restriction of $p$ to each regular fiber has order $v$. Then $e(widetildeM) = e(M) u / v$.

I don't have any reference for that fact from top of my head but it is visible just by thinking of the geometric role of numbers $(alpha_i, beta_i)$ for each singular fiber. I'm sure it's written down somewhere in Neumann-Raymond's book.

share|cite|improve this answer

edited Mar 25 at 20:35

answered Mar 25 at 20:17

Stephen DedalusStephen Dedalus

1,0091022

$endgroup$

add a comment | 

0

$begingroup$

Let's say that a regular fiber of $M$ is covered by $u$ regular fibers of $widetildeM$ and restriction of $p$ to each regular fiber has order $v$. Then $e(widetildeM) = e(M) u / v$.

I don't have any reference for that fact from top of my head but it is visible just by thinking of the geometric role of numbers $(alpha_i, beta_i)$ for each singular fiber. I'm sure it's written down somewhere in Neumann-Raymond's book.

share|cite|improve this answer

edited Mar 25 at 20:35

answered Mar 25 at 20:17

Stephen DedalusStephen Dedalus

1,0091022

$endgroup$

add a comment | 

$begingroup$

Let's say that a regular fiber of $M$ is covered by $u$ regular fibers of $widetildeM$ and restriction of $p$ to each regular fiber has order $v$. Then $e(widetildeM) = e(M) u / v$.

I don't have any reference for that fact from top of my head but it is visible just by thinking of the geometric role of numbers $(alpha_i, beta_i)$ for each singular fiber. I'm sure it's written down somewhere in Neumann-Raymond's book.

share|cite|improve this answer

edited Mar 25 at 20:35

answered Mar 25 at 20:17

Stephen DedalusStephen Dedalus

1,0091022

$endgroup$

share|cite|improve this answer

edited Mar 25 at 20:35

answered Mar 25 at 20:17

Stephen DedalusStephen Dedalus

1,0091022

answered Mar 25 at 20:17

Stephen DedalusStephen Dedalus

1,0091022

answered Mar 25 at 20:17

Stephen DedalusStephen Dedalus

1,0091022