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- A note on generic subsets of definable groupsPublication . Edmundo, Mário; Terzo, G.We generalize the theory of generic subsets of definably compact de-finable groups to arbitrary o-minimal structures. This theory is a crucial part of the solution to Pillay's conjecture connecting definably compact definable groups with Lie groups.
- Definably compact abelian groupsPublication . Edmundo, Mário; Otero, MargaritaLet M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n-dimensional group definable in M. We show the following: the o-minimal fundamental group of G is isomorphic to ℤ^n; for each k>0, the k-torsion subgroup of G is isomorphic to (ℤ/kℤ)^n, and the o-minimal cohomology algebra over ℚ of G is isomorphic to the exterior algebra over ℚ with n generators of degree one.
- Locally definable groups in o-minimal structuresPublication . Edmundo, MárioIn this paper we develop the theory of locally definable groups in o-minimal structures generalizing in this way the theory of definable groups.
- Definable group extensions in semi-bounded o-minimal structuresPublication . Edmundo, Mário; Eleftheriou, P. E.In this note we show: Let ℛ = 〈 R, <, +, 0,...〉 be a semi-bounded (respectively, linear) o-minimal expansion of an ordered group, and G a group definable in R of linear dimension m ([2]). Then G is a definable extension of a bounded (respectively, definably compact) definable group B by 〈 Rm, +〉.
- A fixed point theorem in o-minimal structuresPublication . Edmundo, MárioHere we prove an o-minimal fixed point theorem for definable continuous maps on definably compact definable sets, generalizing Brumfiel's version of the Hopf fixed point theorem for semi-algebraic maps.
- Comparison theorems for o-minimal singular (co)homologyPublication . Edmundo, Mário; Woerheide, A.Here we show the existence of the o-minimal simplicial and singular (co)homology in o-minimal expansions of real closed fields and prove several comparison theorems for o-minimal (co)homology theories
- Invariance results for definable extensions of groupsPublication . Edmundo, Mário; Jones, G. O.; Peatfield, N. J.We show that in an o-minimal expansion of an ordered group finite definable extensions of a definable group which is defined in a reduct are already defined in the reduct. A similar result is proved for finite topological extensions of definable groups defined in o-minimal expansions of the ordered set of real numbers.
- Integration of positive constructible functions against Euler characteristic and dimensionPublication . Edmundo, Mário; Cluckers, R.Following recent work of R. Cluckers and F. Loeser [Fonctions constructible et intégration motivic I, C. R. Math. Acad. Sci. Paris 339 (2004) 411–416] on motivic integration, we develop a direct image formalism for positive constructible functions in the globally subanalytic context. This formalism is generalized to arbitrary first-order logic models and is illustrated by several examples on the p-adics, on the Presburger structure and on o-minimal expansions of groups. Furthermore, within this formalism, we define the Radon transform and prove the corresponding inversion formula.
- On freely generated E-subringsPublication . Edmundo, Mário; Terzo, G.In this paper we prove, without assuming Schanuel's conjecture, that the E-subring generated by a real number not definable without parameters in the real exponential field is freely generated. We also obtain a similar result for the complex exponential field.
- Discrete subgroups of locally definable groupsPublication . Berarducci, A.; Edmundo, Mário; Mamino, M.We work in the category of locally definable groups in an o-minimal expansion of a field. Eleftheriou and Peterzil conjectured that every definably generated abelian connected group G in this category is a cover of a definable group. We prove that this is the case under a natural convexity assumption inspired by the same authors, which in fact gives a necessary and sufficient condition. The proof is based on the study of the zero-dimensional compatible subgroups of G. Given a locally definable connected group G (not necessarily definably generated), we prove that the n-torsion subgroup of G is finite and that every zero-dimensional compatible subgroup of G has finite rank. Under a convexity hypothesis, we show that every zero-dimensional compatible subgroup of G is finitely generated