# Euler characteristic of pseudomanifolds with boundary

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."

• Your question is be more appropriate for Math Stack Exchange. Think of the cone over a closed, connected, orientable triangulated surface of genus $\ge 1$. Sep 23, 2021 at 15:26
• @moishe Even if the answer winds up being (relatively) easy, I still think this is a reasonable MO question. Pseudomanifolds are research-level gadgets. Sep 26, 2021 at 22:35
• @GregFriedman: Maybe you are right. I definitely discussed them in my MSE answers in the past though. Sep 26, 2021 at 22:48

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.