Euler characteristic of pseudomanifolds with boundary

Question

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.

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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."

Answer 1

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.