The study explores stable diffeomorphism groups in 4-manifolds using localisation and invariants.
problem Understanding stable diffeomorphism groups in 4-manifolds.
method Localisation of n-manifolds, inverting connected sum construction, using Bauer--Furuta invariants.
result K3-stable Bauer--Furuta invariants determine S^2xS^2-stable invariants.
Study stable equivalence relations on 4-manifolds, proving homotopy equivalent manifolds with abelian fundamental group are stably diffeomorphic.
problem Classifying stable equivalence relations on 4-manifolds.
method Combination of modified and classical surgery, focusing on homotopy equivalence up to stabilisation.
result Closed oriented homotopy equivalent 4-manifolds with abelian fundamental group are stably diffeomorphic.
Classifies stable diffeomorphism of spin 4-manifolds with specific fundamental groups.
problem Classifying stable diffeomorphism of spin 4-manifolds with given fundamental groups.
method Formulated conjectural relationships between algebraic invariants and obstructions, proved for specific groups.
result Proved conjectures for specific fundamental groups, providing complete algebraic stable classification.
New classification for some unorientable 4-manifolds using modified surgery theory.
problem Classifying stable diffeomorphism classes of unorientable 4-manifolds.
method Modified surgery theory applied to unorientable 4-manifolds with specific fundamental groups.
result Found nine stable diffeomorphism classes for pin+ manifolds, one for pin−, and four for neither, under certain conditions. We answer affirmatively a question posed by Morita on homological stability of surface diffeomorphisms made discrete. In particular, we prove that C∞-diffeomorphisms and volume preserving diffeomorphisms of surfaces as family of discrete groups exhibit homological stability. We show that the stable homology o…
The paper introduces a new method to create stable ergodic actions on higher-dimensional manifolds.
problem Stable ergodicity of group actions on smooth manifolds restricted to one-dimensional cases.
method Geometric method using quasi-conformal blender for constructing stable local dynamics.
result Every closed manifold admits stably ergodic finitely generated group actions by diffeomorphisms of class C1+α. Computes homotopy groups of diffeomorphism spaces for high-dimensional manifolds.
problem Determining homotopy groups of diffeomorphism spaces for high-dimensional manifolds.
method Computes rational homotopy groups of classifying spaces of diffeomorphisms.
result Determines rational pseudoisotopy stable range for compact spin manifolds.
We prove that group homology of the diffeomorphism group of #gSn×Sn as a discrete group is independent of g in a range, provided that n>2. This answers the high dimensional version of a question posed by Morita about surface diffeomorphism groups made discrete. The stable homology is isomorphic to the…
Study calculates homotopy groups and derivatives for disc diffeomorphisms.
problem Understanding the homotopy groups of diffeomorphisms of discs.
method Computes rational homotopy groups and uses Weiss' orthogonal calculus.
result Determines optimal rational concordance stable range for high-dimensional discs.
In this paper we prove a stability theorem for block diffeomorphisms of 2d-dimensional manifolds that are connected sums of S^d x S^d. Combining this with a recent theorem of S. Galatius and O. Randal-Williams and Morlet's lemma of disjunction, we determine the homology of the classifying space of their diffeomorphism …
Extended surgery theory proves diffeomorphism for simply-connected 4k-manifolds.
problem Proving diffeomorphism for simply-connected 4k-manifolds.
method Construction of an extended surgery obstruction associated to a normal bordism.
result Identifies the inertia group of a (2k-1)-connected 4k-manifold.
Anosov diffeomorphisms with integrable subbundles have coherent dynamics and spectral rigidity.
problem Characterizing Anosov diffeomorphisms with integrable subbundles.
method Joint integrability of strong stable and unstable subbundles leads to coherent dynamics and spectral rigidity.
result Anosov diffeomorphisms with integrable subbundles are dynamically coherent and have spectral rigidity.
This paper is concerned with the problem of stable diffeomorphism classification of 4-manifolds obtained using the surgery on loops. The main theorem states that under the assumption that the normal 1-type of two 4-manifolds in question is the same, the only classifying invariant is the signature. In particular, in som…
Authors prove existence of exotic surfaces and invariants not detecting self-diffeomorphisms.
problem Existence and properties of exotic surfaces and diffeomorphisms.
method Vanishing theorem of family Bauer--Furuta invariant for diffeomorphisms on spin 4-manifolds.
result Family Bauer--Furuta invariants do not detect exotic self-diffeomorphisms on S4 or S2imesS2. Two 4-manifolds are stably diffeomorphic if they become diffeomorphic after connected sum with S^2 x S^2's. This paper shows that two closed, orientable, homotopy equivalent, smooth 4-manifolds are stably diffeomorphic, provided a certain map from the second homology of the fundamental group with coefficients in Z/2 to…
For a germ of a smooth map f and a subgroup G_V of any of the Mather groups G for which the source or target diffeomorphisms preserve some given volume form V in the source or in the target we study the G_V-moduli space of f that parameterizes the G_V-orbits inside the G-orbit of f. We find, for example, that this modu…
Proves transitivity of real Anosov diffeomorphisms with specific properties.
problem Transitivity of real Anosov diffeomorphisms with specific properties.
method Proves transitivity using specific properties of real Anosov diffeomorphisms.
result Proves transitivity of real Anosov diffeomorphisms.
Study K-theory of Etesi C∗-algebras to understand smooth manifolds.
problem Understanding smooth manifolds through K-theory of Etesi C∗-algebras. method Calculate topological and smooth invariants of manifolds using K-theory of Etesi C∗-algebras. result Smoothings of a manifold form a torsion abelian group isomorphic to the Brauer group of a number field.
Paper proves every stable 4-sphere has a unique diffeomorphism class.
problem Identifying stable 4-spheres and their diffeomorphisms.
method Using Wall's result and properties of surface-knot spaces.
result Every stable 4-sphere has a unique orientation-preserving diffeomorphism class.
We study closed, oriented 4-manifolds whose fundamental group is that of a closed, oriented, aspherical 3-manifold. We show that two such 4-manifolds are stably diffeomorphic if and only if they have the same w_2-type and their equivariant intersection forms are stably isometric. We also find explicit algebraic invaria…
Study cohomotopy classes for 4-manifolds using complex spin structures.
problem Understanding cohomotopy classes for families of 4-manifolds with complex spin structures.
method Using Bauer--Furuta invariants in parametrised stable homotopy theory.
result Definition of characteristic cohomotopy classes on Thom spectra.
Study the topology of stable vector fields and Lyapunov functions on R^n.
problem Topology of stable vector fields and Lyapunov functions on R^n.
method Differential topology, Lyapunov theory, and results on diffeomorphism groups of discs.
result Path-connected and simply connected spaces of stable vector fields for n≠4,5 and weakly contractible for n≤3.
Let G be a compact Lie group and X be a compact smooth G-manifold with finitely many G-fixed points. We show that if X admits a G-equivariant hyperbolic diffeomorphism having a certain convergence property, there exists an open covering of X indexed by the G-fixed points so that each open set is G-stable and G-equivari…
Study constraints on diffeomorphisms and homeomorphisms of 4-manifolds with boundary.
problem Constraints on smooth families of 4-manifolds with boundary.
method Use Manolescu's Seiberg-Witten Floer stable homotopy type.
result Inclusion map between diffeomorphisms and homeomorphisms is not a weak homotopy equivalence.
The paper studies pseudo-isotopies of spherical 3-manifolds and computes ranks of certain groups.
problem Computing ranks of abelian groups related to spherical 3-manifolds.
method Surgery on theta-graphs embedded in spherical 3-manifolds, study of pseudo-isotopy behavior under suspension.
result Lower bounds of ranks of abelian groups π0Diff(X,∂) are computed. The paper computes smooth structures on a specific product manifold.
problem Computing the number of smooth structures on a product manifold.
method Using known low-dimensional computations of stable homotopy groups of spheres, the paper determines the inertia group of the product manifold.
result The paper establishes a diffeomorphism classification of all smooth manifolds homeomorphic to CP3imesSk for 1≤k≤7. New stable exotic 4-manifolds found with specific topological properties.
problem Identifying conditions for the existence of stable exotic 4-manifolds.
method Investigated fundamental group, Stiefel-Whitney classes, and H_5(π;Z) to determine stable exotic pairs.
result Produced new stable exotica and settings where they do not arise.
Study shows almost all Arnold stable solutions have no conjugate points.
problem Existence of conjugate points in Arnold stable solutions.
method Analysis of Misiołek curvature for Arnold stable solutions.
result Almost all Misiołek curvature is nonpositive for Arnold stable solutions.
The paper constructs multiple manifolds with similar properties.
problem Finding multiple distinct manifolds with stable diffeomorphism.
method Constructing n pairwise homotopically inequivalent manifolds. result Each constructed manifold is stably diffeomorphic to one another.
Characterizes the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.
problem Characterizing the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.
method Introduce the diffeomorphic logarithm of special orthogonal matrices and an efficient algorithm.
result The region containing the principal logarithm has a special multiplicity structure.
We prove that for many degrees in a stable range the homotopy groups of the moduli space of metrics of positive scalar curvature on S^n and on other manifolds are non-trivial. This is achieved by further developing and then applying a family version of the surgery construction of Gromov-Lawson to an exotic smooth famil…
Characteristic classes of oriented vector bundles can be identified with cohomology classes of the disjoint union of classifying spaces BSO_n of special orthogonal groups SO_n with n=0,1,... A characteristic class is stable if it extends to a cohomology class of a homotopy colimit BSO of classifying spaces BSO_n. Simil…
Decouples homotopy quotients of generalised configuration spaces on surfaces.
problem Homological stability of generalised configuration spaces on surfaces.
method Analyzes actions of diffeomorphism groups and uses homotopy quotients.
result Decouples theorem for homology of homotopy quotients on surfaces.
In this paper we prove a homological stability theorem for the diffeomorphism groups of high dimensional manifolds with boundary, with respect to forming the boundary connected sum with the product Dp+1×Sq for ∣q−p∣<min{p,q}−2. In a recent joint paper with Boris Botvinnik (see arXiv:1509.03359…
We provide an alternative proof that Crosscaps are diffeomorphically stable.
Homotopy theory for (2n+1)-dimensional manifold triads with fixed boundary.
problem Classifying stable moduli spaces of (2n+1)-dimensional manifold triads. method Homotopy-theoretic description of stable moduli spaces, stabilization by boundary connected sum with SnimesDn+1. result Established homology of stable moduli spaces for (2n+1)-dimensional manifold triads. The paper proves vanishing bounded cohomology for various groups.
problem Vanishing bounded cohomology for specific groups.
method Elementary algebraic criterion called the commuting cyclic conjugates condition.
result Many groups of interest have vanishing bounded cohomology.
Smooth manifolds have equivalent diffeomorphism groups if and only if they are diffeomorphic.
problem Understanding when diffeomorphism groups of smooth manifolds are elementarily equivalent.
method Analyzing the equivalence of Cr and Cs diffeomorphism groups of smooth manifolds. result Equivalent diffeomorphism groups imply diffeomorphic manifolds, strengthening previous results.
Study on group cocycles for volume-preserving diffeomorphisms.
problem Understanding group cocycles on volume-preserving diffeomorphisms.
method Constructed two types of group cocycles on the volume-preserving diffeomorphism group.
result One cocycle yields the Euler class of flat sphere bundles for the sphere.
Extends Palais' result on diffeomorphisms to homeomorphisms and bi-Lipschitz mappings.
problem Extending diffeomorphisms to global mappings of manifolds.
method Elementary argument for diffeomorphisms, deep results for homeomorphisms and bi-Lipschitz mappings.
result Extension of Palais' result to homeomorphisms and bi-Lipschitz mappings.
We show that, up to topological conjugation, the equivalence class of a Morse-Smale diffeomorphism without heteroclinic curves on 3-manifold is completely defined by an em- bedding of two-dimensional stable and unstable heteroclinic laminations to a characteristic space.
New Lie groups found for Poisson diffeomorphisms.
problem Finding Lie group structures on Poisson diffeomorphism groups.
method Using Poisson groupoids, develop Lie group structures.
result Poisson diffeomorphism groups of various Poisson manifolds are regular Lie groups.
We show that in the first sub-Riemannian Heisenberg group there are intrinsic graphs of smooth functions that are both critical and stable points of the sub-Riemannian perimeter under compactly supported variations of contact diffeomorphisms, despite the fact that they are not area-minimizing surfaces. In particular, w…
Study length functions on various groups and prove homomorphisms to finite groups.
problem Understanding length functions on different types of groups.
method Analyzing length functions on Lie groups, Gromov hyperbolic groups, arithmetic subgroups, matrix groups, and Cremona groups.
result Prove that homomorphisms to certain groups must have finite images.
Survey on foliations and diffeomorphism groups.
problem Relationship between algebraic and homotopical properties.
method Survey and analysis of existing literature.
result Explains the connection between diffeomorphism groups and foliations.
Wall's result extended to 4-manifolds with definite intersection forms.
problem Realizing automorphisms of definite intersection forms.
method Using a specific 4-manifold construction and Wall's original result.
result Automorphisms of definite intersection forms are realized by diffeomorphisms of the constructed 4-manifold.
Diffeomorphisms of convex polytopes form a Lie group.
problem Understanding transformations of convex polytopes.
method Forming a Lie group from diffeomorphisms of convex polytopes.
result The group of diffeomorphisms of a convex polytope is a regular Lie group.
Complex surfaces show non-simply connected diffeomorphism groups with non-homotopic loops.
problem Complex surfaces with non-simply connected diffeomorphism groups and non-homotopic loops.
method Exhibited examples of complex surfaces.
result Diffeomorphism groups of complex surfaces are not simply-connected and contain non-homotopic loops.