There are a collection of definitions of "combinatorial Euler characteristic", which is different from the "homotopy Euler characteristic". I will describe a few of them and give some references, and then ask how far they can be generalized.
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A good place to start is Hadwiger's theorem. Define a "Hadwiger measure" m on Rn to be a thing that assigns (possibly negative) real numbers to (nice?) subsets of Rn in such a way that the assignment is invariant under rigid transformations (i.e. isometries) and satisfies the "inclusion-exclusion" principle that m(_A_ ∪ B) = m(_A_) + m(_B_) – m(_A_ ∩ B); Hadwidger measures are also required to satisfy some analytic properties. Then Hadwidger proves that the space of measures on Rn is precisely (n+1)-dimensional, and has a basis mi with mi([0,1]i) = 1 and mi(λ A) = λi mi(A), where λ A is the set rescaled by a factor of λ in every direction. In particular, m0 of a finite set counts the number of points, and agrees with Euler characteristic for compact regions; the function m0 is the "combinatorial Euler characteristic". It is not homotopy-invariant: m0([0,1]) = 1 whereas m0(R) = -1. It is multiplicative.
Incidentally, Hadwiger's paper is in German and so I cannot read it. Apparently all this material is in Rota's book "Introduction to Geometric Probability", but I have been away from a library and haven't read it yet. Thus I don't know the precise statement of "nice".
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Schanuel in MR1173024 various "geometric categories". Namely, say that a subset of Rn is a "polyhedron" if it is the positive locus finitely many affine maps to R; close the collection of polyhedra under union, intersection, and complement, and thus recover the notion of "polyhedral set" (so that a polyhedral set is actually a pair (n,_S_) where S is a subset of Rn satisfying certain properties). Then morphism of polyhedral sets is a set-theoretic function whose graph (as a subset of Rn x Rm) is polyhedral. Then it's straightforward to check that a morphism is an isomorphism if it is a set-theoretic bijection — morphisms allow gluing and cutting.
Or replace the word "affine" with "polynomial" and thus recover the notion of "semi-algebraic set". Or restrict your attention to bounded polyhedral sets. Anyway, each of these geometric categories has well-behaved product and coproduct, and so a "Burnside Rig" (ring without negation) whose elements are isomorphism classes of objects. Schanuel computes each of these Burnside rigs, and shows that the universal cancelative quotient of each is the integers; this map to Z is the combinatorial Euler characteristic.
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Apparently there are also more analytic definitions. Schanuel in MR842922 (wonderful but only trying to develop intuition and motivation) suggests that each of the Hadwiger measures can be defined in terms of curvatures and whatnot, but the formulas he gives only make sense for compact manifolds (with boundaries, corners…).
Chen (MR1215324) describes the combinatorial Euler characteristic with the following fun integral: let f: R → R be continuous except for finitely many jump and/or removable discontinuities, and define ∫Eulerf = Σx∈R [ f(x) – (1/2) (f(x+) + f(x–)) ]; then try to compute Euler integrals of characteristic functions. The problem is that he then defines the multi-dimensional version via the Fubini theorem, but suggests that his integrals depend on a choice of basis.
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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.
Anyway, so first, are there references I've missed?
Second and more importantly, all the references consider only subsets of Euclidean space (well, Schanuel briefly mentions the Burnside rig of varieties/C, but only computes a quotient). Why? Why isn't there an intrinsic topological description, or perhaps manifold-theoretic description?
In particular, a "measure-theoretic" version that does not rely on embeddings in Euclidean space would be great, as it would presumably give "measures" against which we could integrate smooth functions. Any ideas?
Best Answer
the best approach to the geometric euler characteristic comes from the theory of o-minimal structures.
the best reference in this area is the book "tame topology and o-minimal structures" by lou van den dries. requires very little background to understand.
in brief: an o-minimal structure is collection of boolean algebras of subsets of $R^n$ which satisfies a short list of axioms. (the name comes from model theory, but you don't need to know any model theory to understand the results)
examples of o-minimal structures include the semialgebraic sets, the globally subanalytic sets, and (if you tweak the definitions a bit) the piecewise-linear sets.
elements of an o-minimal structure are "tame" or "definable" sets. mappings between tame sets are tame iff their graph is a tame set.
basic relevant results:
every tame set has a well-defined euler characteristic.
two tame sets are "definably homeomorphic" (there is a tame bijection between them --- not necessarily continuous!) iff they have the same dimension and euler characteristic.
(yes, i wrote iff - this is the first surprise in this subject)
one can so this for more general manifolds as well.
concerning integration with respect to euler characteristic:
1) in the o-minimal framework, one can integrate all constructible functions, as noted by viro and schapira in the 1980s, based on works of macpherson and kashiwara in the 1970s. these results follow from sheaf theory. though more difficult than the combinatorial approach, all these proofs are "natural" and don't rely on "luck".
2) if you want to integrate non-constructible (e.g., smooth) integrands, the theory of chen (really due to rota) will fail -- that integral vanishes on all continuous integrands.
3) baryshnikov and ghrist have extensions of the integral to definable integrands (see 2009 arxiv paper). there are two such extensions, and they are dual. there are deep connections with morse theory, but the integral operators are unfortunately non-linear, and the fubini theorem does not hold in full generality.