Intersection theory for o-minimal manifolds

Abstract

We develop an intersection theory for definable C^p -manifolds in an o-minimal expansion of a real closed field and we prove the invariance of the intersection numbers under definable C^p -homotopies (p > 2). In particular we define the intersection number of two definable submanifolds of complementary dimensions, the Brouwer degree and the winding numbers. We illustrate the theory by deriving in the o-minimal context the Brouwer fixed point theorem, the Jordan-Brouwer separation theorem and the invariance of the Lefschetz numbers under definable C^p -homotopies. A. Pillay has shown that any definable group admits an abstract manifold structure. We apply the intersection theory to definable groups after proving an embedding theorem for abstract definably compact C^p -manifolds. In particular using the Lefschetz fixed point theorem we show that the Lefschetz number of the identity map on a definably compact group, which in the classical case coincides with the Euler characteristic, is zero

A survey on groups definable in o-minimal structures

“…This conjecture has recently been proved by Y.Peterzil and S.Starchenko in [39]; they also give there -Corollary 4.6 -another proof of Theorem 6.5. They extend further the results in [3] introducing definable Morse functions and prove that E(G) is the degree of a map, a degree that they have shown is 0. To finish with the different proofs of Theorem 6.5, let me note that we can also get one via the structure of the torsion -Theorem 5.9 -which gives E(G) = 0, for G abelian, and then make use of either Theorem 6.3 and Theorem 5.6, or the existence of an infinite abelian subgroup -Corollary 2.4(ii).…”

“…In [5] A.Berarducci and myself proved the result (see the proof of Corollary 3.4 there) via a Lefschetz fixed point theorem for o-minimal expansions of real closed fields, which in turn is proved by transfer from the reals; the transfering can be done after we prove that the top homology group of a definably compact definably connected group is Z (see Theorem 5.2 in [5]). Using a differential topology approach, in [3], we define the Lefschetz number of id G , Ξ(G), as a self intersection number of the diagonal in G × G (see Definition 9.11, there) and we prove in Theorem 11.4 also there, that Ξ(G) = 0; our hope was -as in the classical case and as we conjectured -that Ξ(G) = E(G). This conjecture has recently been proved by Y.Peterzil and S.Starchenko in [39]; they also give there -Corollary 4.6 -another proof of Theorem 6.5.…”

Compact domination for groups definable in linear o-minimal structures

“…Hence, to prove property (1) for G it suffices to show that G(N ) remains definably compact with respect to definability in N , and work in N . To see that definable compactness is preserved under taking expansions, observe that G can be written as G = i∈J G i , where each G i is a closed subset of G contained in one the charts of G see [BO1,Lemmas 10.4,10.5], for example, where the authors work over a real closed field but their arguments go word-byword through in any o-minimal expansion of an ordered group . If φ i : G i → M n denotes the corresponding chart map, then G is definably compact if and only if each φ i (G i ) is closed and bounded.…”

Computing o-minimal topological invariants using differential topology

“…In the last few years there has been a significant progress in applying ideas and methods from algebraic topology to sets definable in R (see, for example, [1], [2], [11], [6], [20]). Probably one of the most important results in this direction was a positive answer by M. Edmundo to the following problem posted by the first author and C. Steinhorn (see [15]).…”