389 research outputs found
Higher homotopy of groups definable in o-minimal structures
It is known that a definably compact group G is an extension of a compact Lie
group L by a divisible torsion-free normal subgroup. We show that the o-minimal
higher homotopy groups of G are isomorphic to the corresponding higher homotopy
groups of L. As a consequence, we obtain that all abelian definably compact
groups of a given dimension are definably homotopy equivalent, and that their
universal cover are contractible.Comment: 13 pages, to be published in the Israel Journal of Mathematic
A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets
Results of Smale (1957) and Dugundji (1969) allow to compare the homotopy
groups of two topological spaces and whenever a map with
strong connectivity conditions on the fibers is given. We apply similar
techniques in o-minimal expansions of fields to compare the o-minimal homotopy
of a definable set with the homotopy of some of its bounded hyperdefinable
quotients . Under suitable assumption, we show that and . As a special case,
given a definably compact group, we obtain a new proof of Pillay's group
conjecture ")" largely independent of the
group structure of . We also obtain different proofs of various comparison
results between classical and o-minimal homotopy.Comment: 24 page
Combinatorial complexity in o-minimal geometry
In this paper we prove tight bounds on the combinatorial and topological
complexity of sets defined in terms of definable sets belonging to some
fixed definable family of sets in an o-minimal structure. This generalizes the
combinatorial parts of similar bounds known in the case of semi-algebraic and
semi-Pfaffian sets, and as a result vastly increases the applicability of
results on combinatorial and topological complexity of arrangements studied in
discrete and computational geometry. As a sample application, we extend a
Ramsey-type theorem due to Alon et al., originally proved for semi-algebraic
sets of fixed description complexity to this more general setting.Comment: 25 pages. Revised version. To appear in the Proc. London Math. So
O-minimal cohomology: finiteness and invariance results
We prove that the cohomology groups of a definably compact set over an
o-minimal expansion of a group are finitely generated and invariant under
elementary extensions and expansions of the language. We also study the
cohomology of the intersection of a definable decreas-ing family of definably
compact sets, under the additional assumption that the o-minimal structure
expands a field.Comment: 28 pages, 7 figures and diagrams Added the hypothesis that singletons
are construcible to section 3. Corrected misprint
Discrete subgroups of locally definable groups
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.Comment: Final version. 17 pages. To appear in Selecta Mathematic
Non-archimedean tame topology and stably dominated types
Let be a quasi-projective algebraic variety over a non-archimedean valued
field. We introduce topological methods into the model theory of valued fields,
define an analogue of the Berkovich analytification of ,
and deduce several new results on Berkovich spaces from it. In particular we
show that retracts to a finite simplicial complex and is locally
contractible, without any smoothness assumption on . When varies in an
algebraic family, we show that the homotopy type of takes only a
finite number of values. The space is obtained by defining a
topology on the pro-definable set of stably dominated types on . The key
result is the construction of a pro-definable strong retraction of
to an o-minimal subspace, the skeleton, definably homeomorphic to a space
definable over the value group with its piecewise linear structure.Comment: Final versio
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