Let X be a definable C manifold, Y1, Y2 definably compact definable C r submanifolds of X such that dimY1 + dimY2 < dimX and Y1 has a trivial normal bundle. We prove that there exists a definable isotopy {hp}p∈J such that h0 = idX and h1(Y1) ∩ Y2 = ∅. 2010 Mathematics Subject Classification. 14P10, 14P20, 58A05, 03C64.

Given a Dedekind incomplete ordered field, a pair of convergent nets of gaps which are respectively increasing or decreasing to the same point is used to obtain a further equivalent criterion for Dedekind completeness of ordered fields: Every continuous one-to-one function defined on a closed bounded interval maps interior of that interval to the interior of the image. Next, it is shown that ov...

zw Let G be a finite abelian group and M an o-minimal expansion of Rexp = (R, +, ·, <, e) admitting the C∞ cell decomposition. Everything is considered in M. We prove that every definable C∞G manifold is affine. Moreover we prove that if X1, . . . , Xn (resp. Y1, . . . , Yn) are definable C ∞G submanifolds of a definable C∞G manifold X (resp. Y ) in general position, then every definable CG map...

In [2] o-minimal homotopy was developed for the definable category, proving o-minimal versions of the Hurewicz theorems and the Whitehead theorem. Here, we extend these results to the category of locally definable spaces, for which we introduce homology and homotopy functors. We also study the concept of connectedness in ∨ -definable groups – which are examples of locally definable spaces. We s...

Theorem 1.1 (Hrushovski’s Theorem 3.5). Let G be a definable group, X a definable subset, and G̃ be the ∨-definable group generated by X. Suppose all this data is defined over a model M . Suppose μ is an M-invariant translation-invariant ideal on subsets of G̃. Let q be a wide type over M , contained in G̃. Identify q with its ∧-definable set of realizations. Suppose (F) there exist two realizatio...

X ′ is the canonical example of a set which is definable from X but not recursive in X. The Turing degree of X ′ depends only on the Turing degree of X, so the jump induces an increasing function on the Turing degrees D. In this paper, we will discuss two aspects of the jump and its iterations. First, we will show that they are implicitly characterized by general properties of relative definabi...

Cost monadic logic has been introduced recently as a quantitative extension to monadic second-order logic. A sentence in the logic defines a function from a set of structures to N∪{∞}, modulo an equivalence relation which ignores exact values but preserves boundedness properties. The rich theory associated with these functions has already been studied over finite words and trees. We extend the ...

Let N = (R,+, ·, <, . . . ) be an o-minimal expansion of the standard structure of a real closed field R. A definably compact definable group G is a definable Borsuk-Ulam group if there exists an isovariant definable map f : V → W between representations of G, then dimV − dimV G ≤ dimW − dimW. We prove that if a finite group G satisfies the prime condition, then G is a definable Borsuk-Ulam gro...

Extending the work of [7] on groups definable in compact complex manifolds and of [1] on strongly minimal groups definable in nonstandard compact complex manifolds, we classify all groups definable in nonstandard compact complex manifolds showing that if G is such a group then there are a linear algebraic group L, a definably compact group T , and definable exact sequence 1→ L → G → T → 1.

There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study definable groups G which are not definably compact showing that they have a unique maximal normal definable torsion-free subgroup N ; the quotient G/N always has maximal definably compact ...