Related papers: A remark on $\mathscr{C}^\infty$ definable equival…

For a given Riemannian manifold $(M^n, g)$ which is near standard sphere $(S^n, g_{round})$ in the Gromov-Hausdorff topology and satisfies $Rc \geq n-1$, it is known by Cheeger-Colding theory that $M$ is diffeomorphic to $S^n$. A…

Suppose that $M$ is a connected orientable $n$-dimensional manifold and $m>2n$. If $H^i(M,\R)=0$ for $i>0$, it is proved that for each $m$ there is a monomorphism $H^m(W_n,\on{O}(n))\to H^m_{\on{cont}}(\on{Diff}M,\R)$. If $M$ is closed and…

In this paper, we study definably compact semigroups in o-minimal structures, aiming to extend the theory of definable groups to a broader algebraic setting. We show that any definably compact semigroup contains idempotents and admits a…

Marker and Steinhorn shown that given two models $M\prec N$ of an o-minimal theory, if all 1-types over $M$ realized in $N$ are definable, then all types over $M$ realized in $N$ are definable. In this article we characterize pairs of…

The real Calogero--Moser space $\mathcal{C}_n^\mathbb{R}$ is a noncompact, totally real submanifold of the complex Calogero--Moser space $\mathcal{C}_n$. We prove that every symplectic diffeomorphism of $\mathcal{C}_n^\mathbb{R}$ smoothly…

We give an example of a dense o-minimal structure in which there is a definable quotient that cannot be eliminated, even after naming parameters. Equivalently, there is an interpretable set which cannot be put in parametrically definable…

Let $S$ be a generic submanifold of $C^N$ of real codimension m. In this work we continue the study, carried over by various authors, of the set of analytic discs attached to S. Let $M$ be the set of analytic discs attached to $S.$ Given $q…

We present a diagram surveying equivalence or strict implication for properties of different nature (algebraic, model theoretic, topological, etc.) about groups definable in o-minimal structures. All results are well-known and an extensive…

Let $X$ be a compact Gromov-Hausdorff limit space of a collapsing sequence of compact $n$-manifolds, $M_i$, of Ricci curvature $\text{Ric}_{M_i}\ge -(n-1)$ and all points in $M_i$ are $(\delta,\rho)$-local rewinding Reifenberg points, or…

Let $F^{n+p}(c)$ be an $(n+p)$-dimensional simply connected space form with nonnegative constant curvature $c$. We prove that if $M^n(n\geq4)$ is a compact submanifold in $F^{n+p}(c)$, and if $Ric_M>(n-2)(c+H^2),$ where $H$ is the mean…

We construct an injective map from the set of holomorphic equivalence classes of neighborhoods $M$ of a compact complex manifold $C$ into ${\mathbb C}^m$ for some $m<\infty$ when $(TM)|_C$ is fixed and the normal bundle of $C$ in $M$ is…

Let $f: M \to M$ be a diffeomorphism defined on a compact boundaryless $d$-dimensional manifold $M$, $d\geq 2$. C. Morales has proposed the notion of measure expansiveness. In this note we show that diffeomorphisms in a residual subset far…

In this paper we study the relation between the category of real Lie groups and that of groups definable in o-minimal expansions of the real field, which we will refer to as ``definable groups''. With this terminology, it is known…

For A a separable unital C*-algebra and M a separable McDuff II_1-factor, we show that the space Hom_w(A,M) of weak approximate unitary equivalence classes of unital *-homomorphisms A \rightarrow M may be considered as a closed, bounded,…

Let $N$ be a smooth manifold and $f:N\to N$ be a $C^l$, $l\geq 2$ diffeomorphism. Let $M$ be a normally hyperbolic invariant manifold, not necessarily compact. We prove an analogue of the $\lambda$-lemma in this case.

We prove that a compactly supported homeomorphism of a smooth manifold of dimension greater or equal to 5 can be approximated uniformly by compactly supported diffeomorphisms if and only if it is isotopic to a diffeomorphism. If the given…

We show that on any smooth compact connected manifold of dimension $m\geq 2$ admitting a smooth non-trivial circle action $\mathcal{S} = \left\{S_t\right\}_{t \in \mathbb{R}}$, $S_{t+1}=S_t$, the set of weakly mixing…

Let M be an o-minimal structure with elimination of imaginaries, N an unstable structure definable in M. Then there exists X, interpretable in N, such that X with all the structure induced from N is o-minimal. In particular X is linearly…

Given an o-minimal structure ${\mathcal M}$ with a group operation, we show that for a properly convex subset $U$, the theory of the expanded structure ${\mathcal M}'=({\mathcal M},U)$ has definable Skolem functions precisely when…

We determine all the normal subgroups of the group of C^r diffeomorphisms of R^n, r = 1,2,...,infinity, except when r=n+1 or n=4, and also of the group of homeomorphisms of R^n (r=0). We also study the group A_0 of diffeomorphisms of an…