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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?

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    $\begingroup$ There are lots of such invariants - google for combinatorial characteristic classes $\endgroup$ – მამუკა ჯიბლაძე Jul 10 '18 at 5:30
  • $\begingroup$ Thanks, I'll have a look! But are those really of the form above, i.e. only depending on the number of different kinds of objects in a cell/simplicial complex? $\endgroup$ – Andi Bauer Jul 10 '18 at 8:46
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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.

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  • $\begingroup$ So this answers the question in the case where the kinds of objects can only be $d$-cells/simplices for different $d$ - already a good starting point! $\endgroup$ – Andi Bauer Jul 10 '18 at 8:54
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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."

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