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.
In "On o-minimal homotopy groups", 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 V-definable groups -- which are examples of locally definable spaces. We show that the various concepts of connectedness associated to these groups, which have appeared in the literature, are non-equivalent.
Given a definably compact group G in a saturated o-minimal structure, there is a canonical homomorphism from G to a compact real Lie group F(G). We establish a similar result for the (o-mininimal) universal cover of a definably compact group. We also show that F(G) determines the definable homotopy type of G. A crucial step is to show that the fundamental group of an open subset of F(G) is isomorphic to the definable fundamental group of its preimage in G. Our results depend on the study of the o-minimal fundamental groupoid of G.
Consider an o-minimal structure on the real field. Let $M$ be a definable $C^r$ manifold, where $r$ is a nonnegative integer. We first demonstrate an equivalence of the category of definable $C^r$ vector bundles over $M$ with the category of finitely generated projective modules over the ring $C_{\text{df}}^r(M)$. Here, the notation $C_{\text{df}}^r(M)$ denotes the ring of definable $C^r$ functions on $M$. We also show an equivalence of the category of definable $C^r$ bilinear spaces over $M$ with the category of bilinear spaces over the ring $C_{\text{df}}^r(M)$. The main theorems of this pap…
We work over an o-minimal expansion of a real closed field. The o-minimal homotopy groups of a definable set are defined naturally using definable continuous maps. We prove that any two semialgebraic maps which are definably homotopic are also semialgebraically homotopic. This result together with the study of semialgebraic homotopy done by H. Delfs and M. Knebusch allows us to develop an o-minimal homotopy theory. In particular, we obtain o-minimal versions of the Hurewicz theorems and the Whitehead theorem.
We introduce the o-minimal LS-category of definable sets in o-minimal expansions of ordered fields and we establish a relation with the semialgebraic and the classical one. We also study the o-minimal LS-category of definable groups. Along the way, we show that two definably connected definably compact definable groups G and H are definable homotopy equivalent if and only if L(G) and L(H) are homotopy equivalent, where L is the functor which associates to each definable group its corresponding Lie group via Pillay's conjecture.
We prove new upper bounds on homotopy and homology groups of o-minimal sets in terms of their approximations by compact o-minimal sets. In particular, we improve the known upper bounds on Betti numbers of semialgebraic sets defined by quantifier-free formulae, and obtain for the first time a singly exponential bound on Betti numbers of sub-Pfaffian sets.
We study groups and rings definable in d-minimal expansions of ordered fields. We generalize to such objects some known results from o-minimality. In particular, we prove that we can endow a definable group with a definable topology making it a topological group, and that a definable ring of dimension at least 1 and without zero divisors is a skew field.
In this paper we work in an arbitrary o-minimal structure with definable Skolem functions and we prove that definably connected, locally definable manifolds are uniformly definably path connected, have an admissible cover by definably simply connected, open definable subsets and, definable paths and definable homotopies on such locally definable manifolds can be lifted to locally definable covering maps. These properties allows us to obtain the main properties of the general o-minimal fundamental group, including: invariance and comparison results; existence of universal locally definable cove…
We study the topology of metric spaces which are definable in o-minimal expansions of ordered fields. We show that a definable metric space either contains an infinite definable discrete set or is definably homeomorphic to a definable set equipped with its euclidean topology. This implies that a separable metric space which is definable in an o-minimal expansion of the real field is definably homeomorphic to a definable set equipped with its euclidean topology. We show that almost every point in a definable metric space has a neighborhood which is definably homeomorphic to an open definable su…
We prove that all known examples of weakly o-minimal non-valuational structures have no definable Skolem functions. We show, however, that such structures eliminate imaginaries up to (definable families of) definable cuts. Along the way we give some new examples of weakly o-minimal non-valuational structures.
Let N be an o-minimal structure. In this paper we develop group extension and group cohomology theory over N and use it to describe the N-definable solvable groups. We prove an o-minimal analogue of the Lie-Kolchin-Mal'cev theorem and we describe the N-definable G-modules and the N-definable rings.
We give a necessary and sufficient condition for a one-dimensional regular and Hausdorff topological space definable in a definably complete uniformly locally o-minimal structure of the second kind having definable bounded multiplication being affine.
We show that an infinite group $G$ definable in a $1$-h-minimal field admits a strictly $K$-differentiable structure with respect to which $G$ is a (weak) Lie group, and show that definable local subgroups sharing the same Lie algebra have the same germ at the identity. We conclude that infinite fields definable in $K$ are definably isomorphic to finite extensions of $K$ and that $1$-dimensional groups definable in $K$ are finite-by-abelian-by-finite. Along the way we develop the basic theory of definable weak $K$-manifolds and definable morphisms between them.
We settle some open problems in the special case of groups in o-minimal structures, such as the equality of G^00 and G^000 and the equivalence of definable amenability and existence of a type with bounded orbit. We prove almost exactness of the G to G^00 functor. We ask further questions about types with bounded orbits in NIP theories.
We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,τ\right)$ is definably homeomorphic to an affine definable space (namely, a definable subset of $M^{n}$ with the induced subspace topology). One of the main results says that it is sufficient for $X$ to be regular and decompose into finitely many definably connected components.
We completely characterize definable linear orders in o-minimal structures expanding groups. For example, let (P,<_p) be a linear order definable in the real field R. Then (P,<_p) embeds definably in (R^{n+1},<_l), where <_l is the lexicographic order and n is the o-minimal dimension of P. This improves a result of Onshuus and Steinhorn in the o-minimal group context.
Let $R$ be an o-minimal expansion of the real field. We show that the Hausdorff dimension of an $R$-definable metric space is an $R$-definable function of the parameters defining the metric space. We also show that the Hausdorff dimension of an $R$-definable metric space is an element of the field of powers of $R$. The proof uses a basic topological dichotomy for definable metric spaces due to the second author, and the work of the first author and Shiota on measure theory over nonarchimedean o-minimal structures.
We consider a global semianalytic set defined by real analytic functions definable in an o-minimal structure. When the o-minimal structure is polynomially bounded, we show that the closure of this set is a global semianalytic set defined by definable real analytic functions. We also demonstrate that a connected component of a planar global semianalytic set defined by real analytic functions definable in a substructure of the restricted analytic field is a global semianalytic set defined by definable real analytic functions.
We develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable sets, definable functions are almost everywhere continuous, and definable sets are finite unions of graphs of definable continuous "multi-valued functions". This generalizes known statements about weakly o-minimal, C-minimal and P-minimal theories.
We analyze definably compact groups in o-minimal expansions of ordered groups as a combination of semi-linear groups and groups definable in o-minimal expansions of real closed fields. The analysis involves structure theorems about their locally definable covers. As a corollary, we prove the Compact Domination Conjecture in o-minimal expansions of ordered groups.
We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o-minimal structure. This fact together with the results in a previous paper implies tame dimension theory and decomposition theorem into good-shaped definable subsets called quasi-special submanifolds. Using this fact, in the latter part of this paper, we investigate definably complete locally o-minimal expansions of ordered groups when the restriction of multiplication to an arbitrary bounded open box is definable. Similarly to o-minimal expansions of ordered fiel…
In this paper we prove a single exponential upper bound on the number of possible homotopy types of the fibres of a Pfaffian map, in terms of the format of its graph. In particular we show that if a semi-algebraic set $S \subset {\R}^{m+n}$, where $\R$ is a real closed field, is defined by a Boolean formula with $s$ polynomials of degrees less than $d$, and $π: {\R}^{m+n} \to {\R}^n$ is the projection on a subspace, then the number of different homotopy types of fibres of $π$ does not exceed $s^{2(m+1)n}(2^m nd)^{O(nm)}$. As applications of our main results we prove single exponential bounds o…
We study the definable topological dynamics $(G(M), S_G(M))$ of a definable group acting on its type space, where $M$ is either an $o$-minimal structure or a $p$-adically closed field, and $G$ a definable amenable group. We focus on the problem raised by Neweslki of whether weakly generic types coincide with almost periodic types, showing that the answer is positive when $G$ has boundedly many global weakly generic types. We also give two "minimal counterexamples" where $G$ has unboundedly many global weakly generic types, extending the main results of "On minimal flows, definably amenable gro…
In this note we show that every definably connected, definably compact abelian definable group in an o-minimal expansion of a real closed field of dimension not 4 is definably homeomorphic to a torus of the same dimension. Moreover, in the semialgebraic case the result holds for all dimensions.
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 ${\mathcal M}'$ is valuational. As a corollary, we get an elementary proof that the theory of any such ${\mathcal M}'$ does not satisfy definable choice.
We introduce a non real-valued measure on the definable sets contained in the finite part of a cartesian power of an o-minimal field $R$. The measure takes values in an ordered semiring, the Dedekind completion of a quotient of $R$. We show that every measurable subset of $R^n$ with non-empty interior has positive measure, and that the measure is preserved by definable $C^1$-diffeomorphisms with Jacobian determinant equal to $\pm 1$.
We investigate continuous functions definable in a definably complete uniformly locally o-minimal expansion of the second kind of a densely linearly ordered abelian group (DCULOAS structure). We prove a variant of the Arzela-Ascoli theorem for uniformly continuous definable functions and the following assertion: Consider the parameterized function $f:C \times P \rightarrow M$ which is equi-continuous with respect to $P$. The projection image of the set at which $f$ is discontinuous to the parameter space $P$ is of dimension smaller than $\dim P$ when $C$ is closed and bounded. In addition, we …
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 ordered. As part of the proof we show: Theorem 1: If the M-dimenson of N is 1 then any 1-N-type is either strongly stable or finite by o-minimal. Theorem 2: If N is N-minimal then it is 1-M-dimensional.
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 subgroups, and for every such a K there is a maximal definable torsion-free subgroup H such that G/N can be decomposed as G/N = KH, and the intersection between K and H is trivial. Thus G is definably hom…
We modify the definable ultrapower construction of Kanovei and Shelah (2004) to develop a ZF-definable extension of the continuum with transfer provable using countable choice only, with an additional mild hypothesis on well-ordering implying properness. Under the same assumptions, we also prove the existence of a definable, proper elementary extension of the standard superstructure over the reals. Keywords: definability; hyperreal; superstructure; elementary embedding.
Let $\mathcal M$ be a weakly o-minimal non-valuational structure, and $\mathcal N$ its canonical o-minimal extension (by Wencel). We prove that every group $G$ definable in $\mathcal M$ is a subgroup of a group $K$ definable in $\mathcal N$, which is canonical in the sense that it is the smallest such group. As an application, we obtain that $G^{00}= G\cap K^{00}$, and establish Pillay's Conjecture in this setting: $G/G^{00}$, equipped with the logic topology, is a compact Lie group, and if $G$ has finitely satisfiable generics, then $\dim_{Lie}(G/G^{00})= \dim(G)$.
We prove that in an arbitrary o-minimal structure, every interpretable group is definably isomorphic to a definable one. We also prove that every definable group lives in a cartesian product of one-dimensional definable group-intervals (or one-dimensional definable groups). We discuss the general open question of elimination of imaginaries in an o-minimal structure.
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 bijection with any definable set. This gives a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Additionally, we show that interpretable sets in dense o-minimal structures admit definable topologies which are "tame" in several ways: (a) they are Hausdorff, (b) every point has a neighborhood which is definably homeomorphic to a definabl…
We study sets and groups definable in tame expansions of o-minimal structures. Let $\mathcal {\widetilde M}= \langle \mathcal M, P\rangle$ be an expansion of an o-minimal $\mathcal L$-structure $\cal M$ by a dense set $P$, such that three tameness conditions hold. We prove a structure theorem for definable sets and functions in analogy with the influential cell decomposition theorem known for o-minimal structures. The structure theorem advances the state-of-the-art in all known examples of $\mathcal {\widetilde M}$, as it achieves a decomposition of definable sets into \emph{unions} of `cones'…
This article explains and extends semialgebraic homotopy theory (developed by H. Delfs and M. Knebusch) to o-minimal homotopy theory (over a field). The homotopy category of definable CW-complexes is equivalent to the homotopy category of topological CW-complexes (with continuous mappings). If the theory of the o-minimal expansion of a field is bounded, then these categories are equivalent to the homotopy category of weakly definable spaces. Similar facts hold for decreasing systems of spaces. As a result, generalized homology and cohomology theories on pointed weak polytopes uniquely correspo…
We state conditions for which a definable local homomorphism between two locally definable groups $\mathcal{G}$, $\mathcal{G^{\prime}}$ can be uniquely extended when $\mathcal{G}$ is simply connected (Theorem 2.1). As an application of this result we obtain an easy proof of [3, Thm. 9.1] (see Corollary 2.2). We also prove that Theorem 10.2 in [3] also holds for any definably connected definably compact semialgebraic group $G$ not necessarily abelian over a sufficiently saturated real closed field $R$; namely, that the o-minimal universal covering group $\widetilde{G}$ of $G$ is an open locally…
We demonstrate the following uniform local definable cell decomposition theorem in this paper. Consider a structure $\mathcal M = (M, <,0,+, \ldots)$ elementarily equivalent to a locally o-minimal expansion of the group of reals $(\mathbb R, <,0,+)$. Let $\{A_λ\}_{λ\inΛ}$ be a finite family of definable subsets of $M^{m+n}$. There exist an open box $B$ in $M^n$ containing the origin and a finite partition of definable sets $M^m \times B = X_1 \cup \ldots \cup X_k$ such that $B=(X_1)_b \cup \ldots \cup (X_k)_b$ is a definable cell decomposition of $B$ for any $b \in M^m$ and $X_i \cap A_λ= \emp…
Using a non-Laver modification of Uri Abraham's minimal $\varDelta^1_3$ collapse function, we define a generic extension $L[a]$ by a real $a$, in which, for a given $n\ge3$, $\{a\}$ is a lightface $\varPi^1_n$ singleton, $a$ effectively codes a cofinal map $ω\toω_1^L$ minimal over $L$, while every $\varSigma^1_n$ set $X\subseteqω$ is still constructible.
We give necessary and sufficient conditions on a non-oscillatory curve in an o-minimal field such that, for any bounded definable function, the germ of the function on an initial segment of the curve can be continuously extended to a closed definable set. This situation is translated into a question about types: What are the conditions on an $n$-type such that, for any bounded definable function, there is a definable set containing the type on which the function is continuous, and can be extended continuously to the set's closure? All such types are definable, and we give the precise condition…
We show that every definable nested family of closed and bounded subsets of a $P$-minimal field $K$ has non-empty intersection. As an application we answer a question of Darnière and Halupczok showing that $P$-minimal fields satisfy the "extreme value property": for every closed and bounded subset $U\subseteq K$ and every interpretable continuous function $f\colon U \to Γ_K$ (where $Γ_K$ denotes the value group), $f(U)$ admits a maximal value. Two further corollaries are obtained as a consequence of their work. The first one shows that every interpretable subset of $K\timesΓ_K^n$ is already in…
For a dependent theory T, in C_T for every type definable group G, the intersection of type definable subgroups with bounded index is a type definable subgroup with bounded index.
We study the definable topological dynamics $(G,S_G(M))$ of a definable group acting on its type space, where $M$ is a structure and $G$ is a group definable in $M$. In \cite{Newelski-I}, Newelski raised a question of whether weakly generic types coincide with almost periodic types in definable topological dynamics. In \cite{YZ-Sta}, we introduced the notion of stationarity, showing the answer is positive when $G$ is a stationary definably amenable group definable over the field of $p$-adic numbers or an $o$-minimal expansion of real closed field. In this paper, we continue with the work of \c…
We study definably compact definably connected groups definable in a sufficiently saturated real closed field $R$. We introduce the notion of group-generic point for $\bigvee$-definable groups and show the existence of group-generic points for definably compact groups definable in a sufficiently saturated o-minimal expansion of a real closed field. We use this notion along with some properties of generic sets to prove that for every definably compact definably connected group $G$ definable in $R$ there are a connected $R$-algebraic group $H$, a definable injective map $ϕ$ from a generic defina…
Let R be an o-minimal field and V a proper convex subring with residue field k and standard part (residue) map st: V \to k. Let k_{ind} be the expansion of k by the standard parts of the definable relations in R. We investigate the definable sets in k_{ind} and conditions on (R,V) which imply o-minimality of k_{ind}. We also show that if R is omega-saturated and V is the convex hull of the rationals in R, then the sets definable in k_{ind} are exactly the standard parts of the sets definable in (R,V).
We establish the first global results for groups definable in tame expansions of o-minimal structures. Let $\mathcal N$ be an expansion of an o-minimal structure $\mathcal M$ that admits a good dimension theory. The setting includes dense pairs of o-minimal structures, expansions of $\mathcal M$ by a Mann group, or by a subgroup of an elliptic curve, or a dense independent set. We prove: (1) a Weil's group chunk theorem that guarantees a definable group with an o-minimal group chunk is o-minimal, (2) a full characterization of those definable groups that are o-minimal as those groups that have…
We prove that in a $P$-minimal structure, every definable set can be partitioned as a finite union of classical cells and regular clustered cells. This is a generalization of previously known cell decomposition results by Denef and Mourgues, which were dependent on the existence of definable Skolem functions. Clustered cells have the same geometric structure as classical, Denef-type cells, but do not have a definable function as center. Instead, the center is given by a definable set whose fibers are finite unions of balls.
Let K be an expansion of either an ordered field or a valued field. Given a definable set X $\subseteq$ K<sup>m</sup> let C(X) be the ring of continuous definable functions from X to K. Under very mild assumptions on the geometry of X and on the structure K, in particular when K is o-minimal or P-minimal, or an expansion of a local field, we prove that the ring of integers Z is interpretable in C(X). If K is o-minimal and X is definably connected of pure dimension 2, then C(X) defines the subring Z. If K is P-minimal and X has no isolated points, then there is a discrete ring Z contained in K …
