It is a well-known fact that for every compact oriented odd-dimensional manifold $\mathcal{M}$ with boundary it holds that


$$\chi(\mathcal{M})=\frac{1}{2}\chi(\partial\mathcal{M}).$$


In particular, if you take a $3$-dimensional manifold with boundary given by a genus $g$ surface, then its Euler characteristic is $\chi=1-g$.


Is there any relation known between the Euler characteristic of a pseudomanifold with boundary and its boundary? Maybe some explicit formula as above, or at least some inequality relating the two Euler characteristics? I am mainly interested in the $3$-dimensional case. Maybe pseudomanifolds are too general and something like the statement above is only true in the case of "normal" pseudomanifolds, in which all links are themselve pseudomanifolds.




---


Let me briefly define, what I mean when taking about "pseudomanifolds with boundary":


**Simplicial Complex:**



> 
> Let $\mathcal{V}$ be a finite set. Then a collection of non-empty
> finite subsets of $\mathcal{V}$, denoted by
> $\Delta\subset\mathcal{P}(\mathcal{V})$, is called "(abstract)
> simplicial complex", if it satisfies the following two properties:
> 
> 
> 1. $\Delta$ contains all singletons, i.e. $\{v\}\in\Delta$ for all $v\in\mathcal{V}$.
> 2. For any non-empty $\tau\subset\sigma$ for some $\sigma\in\Delta$ it holds that $\tau\in\Delta$.
> 
> 
> 


**Pseudomanifolds:**



> 
> Let $\Delta$ be a finite abstract $d$-dimensional simplicial complex.
> We call the corresponding geometric realization $\vert\Delta\vert$ a
> "$d$-dimensional pseudomanifold", if the following conditions are
> fulfilled:
> 
> 
> 1. $\Delta$ is "pure", i.e. every simplex $\sigma\in\Delta$
> is the face of some $d$-simplex.
> 2. $\Delta$ is "non-branching", i.e. every $(d-1)$-simplex
> is face of exactly one or two $d$-simplices.
> 3. $\Delta$ is "strongly-connected", i.e. for every pair of
> $d$-simplices $\sigma,\tau\in\Delta\_{d}$, there is a sequence of
> $d$-simplices $\sigma=\sigma\_{1},\sigma\_{2},\dots,\sigma\_{k}=\tau$
> such that the intersection $\sigma\_{l}\cap\sigma\_{l+1}$ is a
> $(d-1)$-simplex for every $l\in\{1,\dots,k-1\}$.
> 
> 
> The $(d-1)$-simplices from condition (2), which are the face of only
> one $d$-simplices are called "boundary simplices". The (geometrical
> realization of the) subcomplex of all these simplices is called
> "boundary of the pseudomanifold" and we denote this subcomplex by
> $\partial\Delta$. If $\partial\Delta\neq\emptyset$, then we call
> $\vert\Delta\vert$ "pseudomanifold with boundary", otherwise just "pseudomanifold."
> 
> 
> 



Euler characteristic of pseudomanifolds with boundary
Ok, to convert my comment to an answer. Let $S$ be a closed orientable triangulated surface of genus $\ge 1$. Let $M$ be the cone over $S$. Then $M$ has a natural orientable pseudomanifold structure. However, $\chi(M)=1$, while $\chi(S)$ can be any nonpositive even number.


The moral is that there are way too many pseudomanifolds (even "normal" ones, in your sense). If you want to have the standard relation $\chi(\partial M)=\chi(M)/2$, consider working with (say, rational) "homology manifolds" (with boundary). But in the 3-dimensional setting, all homology manifolds are manifolds.


