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Locally definable homotopy

Elías Baro & Marg\ Otero

Annals of Pure and Applied Logic 161 (4):488-503 (2010)

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Definable homomorphisms of abelian groups in o-minimal structures.Ya'acov Peterzil & Sergei Starchenko - 1999 - Annals of Pure and Applied Logic 101 (1):1-27.details We investigate the group of definable homomorphisms between two definable abelian groups A and B, in an o-minimal structure . We prove the existence of a “large”, definable subgroup of . If contains an infinite definable set of homomorphisms then some definable subgroup of B admits a definable multiplication, making it into a field. As we show, all of this can be carried out not only in the underlying structure but also in any structure definable in. Download Export citation Bookmark 16 citations
G-linear sets and torsion points in definably compact groups.Margarita Otero & Ya’Acov Peterzil - 2009 - Archive for Mathematical Logic 48 (5):387-402.details Let G be a definably compact group in an o-minimal expansion of a real closed field. We prove that if dim(G\X) < dim G for some definable ${X \subseteq G}$ then X contains a torsion point of G. Along the way we develop a general theory for the so-called G-linear sets, and investigate definable sets which contain abstract subgroups of G. Download Export citation Bookmark 3 citations
A semi-linear group which is not affine.Pantelis E. Eleftheriou - 2008 - Annals of Pure and Applied Logic 156 (2):287-289.details In this short note we provide an example of a semi-linear group G which does not admit a semi-linear affine embedding; in other words, there is no semi-linear isomorphism between topological groups f:G→G′Mm, such that the group topology on G′ coincides with the subspace topology induced by Mm. Download Export citation Bookmark 3 citations
The universal covering homomorphism in o‐minimal expansions of groups.Mário J. Edmundo & Pantelis E. Eleftheriou - 2007 - Mathematical Logic Quarterly 53 (6):571-582.details Suppose G is a definably connected, definable group in an o-minimal expansion of an ordered group. We show that the o-minimal universal covering homomorphism equation image: equation image→ G is a locally definable covering homomorphism and π1 is isomorphic to the o-minimal fundamental group π of G defined using locally definable covering homomorphisms. Download Export citation Bookmark 6 citations
Definably compact Abelian groups.Mário J. Edmundo & Margarita Otero - 2004 - Journal of Mathematical Logic 4 (02):163-180.details Let 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 ℤn; for each k>0, the k-torsion subgroup of G is isomorphic to 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. Download Export citation Bookmark 19 citations

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