Are there invariants of cell complexes similar to the Euler characteristic?

Question

The Euler characteristic is an invariant (under homeomorphism) of manifolds that can be computed from a cellulation by (weighted) counting of different kinds of objects, namely \begin{equation} \chi=\textrm{#vertices}-\textrm{#edges}+\textrm{#faces}-\ldots \end{equation}

Are there any other (independent) invariants that are computed in a similar way? I would allow for arbitrary weights (not only $\pm 1$) and arbitrary kinds of "objects" (not only $d$-cells, but also things like "corners", "cycles consisting of $3$ edges", "pairs of a $3$-cell and a $1$-cell that is part of its boundary").

In particular, in odd dimensions where the Euler characteristic is trivial by Poincare duality, is there any sort of replacement for it?

Answer 1

In a certain restricted setting the answer is that the Euler characteristic is the only such invariant. This is not a complete answer to the question since more general things are allowed. However, it may still be of interest to to OP and tells us where not to look for such invariants.

In A property that characterizes Euler characteristic among invariants of combinatorial manifolds by Li Yu the following is shown (quoting the abstract):

If a real valued invariant of compact combinatorial manifolds (with or without boundary) depends only on the number of simplices in each dimension in the manifold, then the invariant is completely determined by the Euler characteristic of the manifold and its boundary. So essentially, the Euler characteristic is the unique invariant of this type.

Answer 2

Meanwhile it seems to me that the discrete analogues to all Stiefel-Whitney numbers of $d$-dimensional manifolds are invariants of this type:

First, for every $n$ there is a rule to color every $d-n$-simplex of a simplicial complex (with branching structure) describing a $d$-manifold by a $Z_2$ element such that

1. the color or each simplex only depends on how the simplicial complex looks in a small patch around it.
2. the coloring forms a $Z_2$ $d-n$-cycle in the simplicial complex.
3. this cycle is a combinatorial representant of the $n$th Stiefel-Whitney class of the $d$-manifold.

Second, there is another rule to $Z_2$ color every $d-n-m$-simplex of a simpicial complex given a $n$-cycle and a $m$-cycle in the simplicial complex such that

1. the color or each simplex only depends on how the simplicial complex and the two cycles look in a small patch around it.
2. the rule is such that the coloring is a $Z_2$ $d-m-n$-cycle.
3. this cycle is a combinatorial representant of the cup product of the two cycles.

Such rules are for example given, at least partly, in https://arxiv.org/abs/1505.05856.

Now a Stiefel-Whitney number is an integral over a cup product of different Stiefel-Whitney classes. So combinatorially we can combine the two rules above to get a $0$-cycle and then count $Z_2$-colorings of the vertices mod $2$. So this corresponds to an invariant of the above form if we choose as "kind of object":

"Every vertex that is colored by $1\in Z_2$ via the combination of the two rules above."

Answer 3

As pointed out in the comments, every characteristic class in $H^d(BO(d), G)$ provides a $G$-valued locally computable invariant of $d$-manifolds, by pulling back via the classifying map of the tangent bundle. As argued by Levitt and Rourke, there are local combinatorial formulas on trianglulations for dual simplicial $d-i$-cycles representing any degree-$i$ characteristic classes, whose values on a $d-i$-simplex only depend on the star of that simplex. To be precise, for this to work we need to add some decorations to the triangulation, namely a local ordering.

So every degree-$d$ characteristic class yields a local combinatorial formula for a simplicial $0$-cycle, whose summation over vertices is an invariant of the type in the question. The "objects" are the different stars of vertices, to which we associate elements of $G$ according to the local formula. Summing up all the $G$-values on the vertices corresponds to evaluating the invariant.

Concrete local formulas are in fact known for many cases. For $G=\mathbb{Z}_2$, the relevant characteristic classes are generated by degree-$d$ polynomials (with $\mathbb{Z}_2$ sum and cup product) of Stiefel-Whitney classes. For the latter, combinatorial formulas are given by Goldstein and Turner, and a combinatorial cup product for cycles can be easily obtained via its geometric interpretation as intersection. For $G=\mathbb{Z}$ on oriented manifolds, we additionally need the Bockstein homomorphism for the short exact sequence $$\mathbb{Z}\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}_2$$ and the Pontryagin classes. The former only involves simplex-wise operations and the co-boundary operator. The Pontryagin classes are the only case for which no satisfactory local formulas exist to date. $\mathbb{Q}$-valued formulas have been described by Gaifullin, but not very explicitly.

I'm not aware of any converse argument that every local formula for a $0$-cycle invariant corresponds to a characteristic class, however I don't think there are any known examples for invariants which don't.