Transfer methods for o-minimal topology

Abstract

LetMbe an o-minimal expansion of an ordered field. Letφbe a formula in the language of ordered domains. In this note we establish some topological properties which are transferred fromφMtoφRand vice versa. Then, we apply these transfer results to give a new proof of a result ofM. Edmundo—based on the work of A. Strzebonski—showing the existence of torsion points in any definably compact group defined in an o-minimal expansion of an ordered field.

Definably Compact Abelian Groups

“…Given X an oriented definable manifold of dimension n with orientation map s, and a definably compact definable set N ⊂ X. Then, by (the proof of) Theorem 5.2 in [2], there is a unique relative homology class ζ N ∈ H def n (X, X \ N ) (the fundamental class around N ) such that for each p ∈ N the homomorphism H…”

“…When our definable set is a definably connected definable group the o-minimal cohomology Q-algebra becomes, in analogy with the Lie case, a free commutative algebra over Q with odd degree generators, i.e., an exterior algebra over Q (Corollaries 3.5 and 3.6). In Section 4 we consider definable manifolds and use the existence of the o-minimal fundamental class of an orientable definable manifold, proved by Beraducci and Otero in [2], to define the degree of a definable map between oriented definable manifolds of the same dimension and with the target manifold definably compact. We then prove properties of degree of the relevant maps which will be specially useful later for the definable groups we are interested on (Proposition 4.6).…”

Poincaré - Verdier duality in o-minimal structures

“…O-minimal singular homology theory can be used to obtain an orientation theory for definable manifolds ( [2], [1]). (In the papers [2] and [1], orientation is defined by taking homology with coefficients in Z but replacing Z by k and considering homology groups as k-vector spaces one gets the theory of k-orientations.)…”

“…(In the papers [2] and [1], orientation is defined by taking homology with coefficients in Z but replacing Z by k and considering homology groups as k-vector spaces one gets the theory of k-orientations.) Our goal here is to show an Alexander duality for homology and to conclude that the two orientation theories agree.…”

A fixed point theorem in o-minimal structures

“…Also, since on definably compact definable sets the Euler-Poincaré characteristic coincides with the o-minimal Euler characteristic we get the following solution to a problem from [1]: Corollary 3.6. -Suppose that f : X −→ X is a definable continuous map definably homotopic to the identity, where X is a definably compact, definable set of o-minimal Euler characteristic E(X) different from zero.…”