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?



Are there invariants of cell complexes similar to the Euler characteristic?
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](https://www.ams.org/journals/tran/1978-239-00/S0002-9947-1978-0494134-6/S0002-9947-1978-0494134-6.pdf), 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](https://www.ams.org/journals/proc/1976-058-01/S0002-9939-1976-0415643-5/S0002-9939-1976-0415643-5.pdf), 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](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/gaifullin2.pdf), 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.


