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Relation between the homological groups of a set and the ones of its boundary (for pseudo-manifolds)

0

$begingroup$

I heard that when $X$ is a connected compact topological $n$-manifold in $mathbbR^n$ whose boundary $partial X$ is a connected compact (n-1)-manifold, then we can deduce the Betti numbers of $X$ from the Betti numbers of $partial X$. Also, we can deduce the Euler characteristic of $X$ from the one of $partial X$.

However, I do not know if some similar results exist when we work with connected compact topological sets whose boundary is a combinatorial pseudo-manifold (that means that the boundary can own "pinches").

I would be very grateful if someone can help me on this.

Thanks for your help.

manifolds homology-cohomology

share|cite|improve this question

asked Apr 1 at 13:06

Nicolas BoutryNicolas Boutry

287

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Is the fact that $X$ has the same dimension as the ambient space $mathbbR^n$ on purpose or is it an oversight?
  $endgroup$
  – Najib Idrissi
  Apr 1 at 14:52
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It is on purpose: I want my boundary to own separation properties.
  $endgroup$
  – Nicolas Boutry
  Apr 1 at 17:29
  

  
  
  
  

add a comment | 

0

$begingroup$

I heard that when $X$ is a connected compact topological $n$-manifold in $mathbbR^n$ whose boundary $partial X$ is a connected compact (n-1)-manifold, then we can deduce the Betti numbers of $X$ from the Betti numbers of $partial X$. Also, we can deduce the Euler characteristic of $X$ from the one of $partial X$.

However, I do not know if some similar results exist when we work with connected compact topological sets whose boundary is a combinatorial pseudo-manifold (that means that the boundary can own "pinches").

I would be very grateful if someone can help me on this.

Thanks for your help.

manifolds homology-cohomology

share|cite|improve this question

asked Apr 1 at 13:06

Nicolas BoutryNicolas Boutry

287

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Is the fact that $X$ has the same dimension as the ambient space $mathbbR^n$ on purpose or is it an oversight?
  $endgroup$
  – Najib Idrissi
  Apr 1 at 14:52
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It is on purpose: I want my boundary to own separation properties.
  $endgroup$
  – Nicolas Boutry
  Apr 1 at 17:29
  

  
  
  
  

add a comment | 

$begingroup$

I heard that when $X$ is a connected compact topological $n$-manifold in $mathbbR^n$ whose boundary $partial X$ is a connected compact (n-1)-manifold, then we can deduce the Betti numbers of $X$ from the Betti numbers of $partial X$. Also, we can deduce the Euler characteristic of $X$ from the one of $partial X$.

However, I do not know if some similar results exist when we work with connected compact topological sets whose boundary is a combinatorial pseudo-manifold (that means that the boundary can own "pinches").

I would be very grateful if someone can help me on this.

Thanks for your help.

manifolds homology-cohomology

share|cite|improve this question

asked Apr 1 at 13:06

Nicolas BoutryNicolas Boutry

287

$endgroup$

share|cite|improve this question

asked Apr 1 at 13:06

Nicolas BoutryNicolas Boutry

287

share|cite|improve this question

asked Apr 1 at 13:06

Nicolas BoutryNicolas Boutry

287

asked Apr 1 at 13:06

Nicolas BoutryNicolas Boutry

287

asked Apr 1 at 13:06

Nicolas BoutryNicolas Boutry

287