# Invariance of combinatorial/geometric euler characteristic

I am trying to read and understand the paper:

TARGET ENUMERATION VIA EULER CHARACTERISTIC INTEGRALS

by YULIY BARYSHNIKOV AND ROBERT GHRIST.

And I am having trouble with a statement. First of all, definitions:

First remark The definition of $k$-simplex should be with $t_i \in (0,1]$.

Second remark The above definition differs from the common one.

Now we continue and we introduce my problem:

Remark It is clear it is not an homotopic invariant.

MY QUESTION(S):

• Is there another easier (which needs less machinery) way to prove the topological invariance of this Euler Characteristic?
• I mean, I have seen that Borel-Moore Homology is defined using sheafs or sheaves. I am still a masters student and I have never heard about that. Is there a proof of the statement using machinery from a first course in Algebraic Topology (Some homotopy theory, a bit of homology and cohomology)?
• In the case some heavy machinery is needed could you provide me some references to look at?

WHAT I HAVE TRIED:

• I have tried using Cohomology with compact supports since in Massey's book titled Singular Homology Theory he uses it to deal with non compact manifolds in order to prove Poincarè Duality. He refers to

H. Cartan, Seminaire Henri Cartan 1948/49: Topologie Algebrique

which I have partially read. However, since the spaces we are dealing with are not locally-compact, I can't use that stuff.

• I have read in the paper: T. Beke, “Topological invariance of the combinatorial euler characteristic of tame spaces,” an idea about using one-point compactifications ... but I didn't achieve my goal yet.
• I have had a look at Stack Exchange. There is a related this question. However, I haven't found an answer there. I quote the part I find most related.

The definition of combinatorial Euler characteristic is great for "finite polyhedral complexes", I think. By a "finite polyhedral complexes" I mean glue together finitely many polyhedra, but you're allowed to leave some faces open, so that unlike a CW complex not every cell must have complex closure. Then you can calculate Euler characteristic with the usual formula: (number of cells of even dimension) - (number of cells of odd dimension). I think this is a topological (but not homotopy!) invariant.

So thanks in advance and any help will be appreciated.

1. I do not see the point of introducing Borel-Moore homology here, but maybe it is used elsewhere in the paper, I do not know. For simplicial complexes, singular homology suffices. My suggestion is to stick to closed simplicial complexes.

2. For closed finite simplicial complexes, $\chi$ is a homotopy-invariant. In particular, if you have two closed finite simplicial complexes $X, Y$ whose geometric realizations are homeomorphic then $\chi(X)=\chi(Y)$. I do not think there is any easier proof of this fact than identifying $\chi$ with $$\sum_{i} (-1)^i b_i(X)$$ (using simplicial homology) and then proving that simplicial homology is isomorphic to the singular homology. Proving these facts is not hard but a bit tedious. You can find proofs in any algebraic topology textbook, for instance in Hatcher's "Algebraic Topology".

3. For the definition of a closed simplex, it is indeed $t_i\in [0,1]$, not $t_i\in (0,1)$. However, for some reason, they also want to work with open simplices, in which case indeed $t_i\in (0,1)$. Why is it needed in their paper I do not know. Their definition of a (finite) simplicial complex seems sloppy, I think, they are assuming that it is (linearly) embedded in $R^n$ for some $n$ in which case it is OK. For simplicial complexes in their sense, $\chi$ is not a homotopy invariant. One can see this as follows. Take $X=(0,1)$, an open 1-simplex. It is homotopy-equivalent to $[0,1)$. But their Euler characteristics (as defined in their paper) are different.