Andrei Agrachev, Marco Caponigro
Given a compact manifold M, we prove that any bracket generating and invariant under multiplication on smooth functions family of vector fields on M generates the connected component of unit of the group Diff(M).
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This paper is devoted to the specific class of pseudoconformal mappings of quaternion and octonion variables. Normal families of functions are defined and investigated. Four criteria of a family being normal are proven. Then groups of pseudoconformal diffeomorphisms of quaternion and octonion manifolds are investigated. It is proven, that they are finite dimensional Lie groups for compact manifolds. Their examples are given. Many charactersitic features are found in comparison with commutative geometry over $\bf R$ or $\bf C$.
Diffeomorphism groups and gauge theory for families
本文综述了族的规范理论，重点探讨了2021-2025年间发展出的用于4维流形微分同胚群的应用方法。
An average intersection estimate for families of diffeomorphisms
本文证明了闭Riemannian流形上充分丰富的紧致$C^1$微分同胚族中，互补维数子流形的平均几何相交数近似于其体积乘积，并给出了此类族的存在性构造。
Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory
本文通过Seiberg-Witten理论构造4维流形微分同胚群的高阶同伦和同调中的无限秩直和项，并应用于嵌入空间和正标量曲率度量空间的无限生成性。
We introduce an invariant of tuples of commutative diffeomorphisms on a 4-manifold using families of Seiberg-Witten equations. This is a generalization of Ruberman's invariant of diffeomorphisms defined using 1-parameter families of Seiberg-Witten equations. Our invariant yields an application to the homotopy groups of the space of positive scalar curvature metrics on a 4-manifold. We also study the extension problem for families of 4-manifolds using our invariant.
Let $X$ be a smooth, compact, oriented $4$-manifold. Building upon work of Li-Liu, Ruberman, Nakamura and Konno, we consider a families version of Seiberg-Witten theory and obtain obstructions to the existence of certain group actions on $X$ by diffeomorphisms. The obstructions show that certain group actions on $H^2(X , \mathbb{Z})$ preserving the intersection form can not be lifted an action of the same group on $X$ by diffeomorphisms. Using our obstructions, we construct numerous examples of group actions which can be realised continuously but can not be realised smoothly for any differenti…
We give constraints on smooth families of 4-manifolds with boundary using Manolescu's Seiberg-Witten Floer stable homotopy type, provided that the fiberwise restrictions of the families to the boundaries are trivial families of 3-manifolds. As an application, we show that, for a simply-connected oriented compact smooth 4-manifold $X$ with boundary with an assumption on the Frøyshov invariant or the Manolescu invariants $α, β, γ$ of $\partial X$, the inclusion map $\mathrm{Diff}(X,\partial) \hookrightarrow \mathrm{Homeo}(X,\partial)$ between the groups of diffeomorphisms and homeomorphisms whic…
Exotic diffeomorphisms of reducible $4$-manifolds with odd $b_+$
利用族Bauer-Furuta不变量，在具有奇数$b_+$的可约$4$流形上构造奇异微分同胚，并发现映射类群非有限生成的单连通$4$流形新例子。
Exotic Dehn twists on 4-manifolds
该研究率先探索4-流形内沿非$S^3$ 3-流形的奇异Dehn扭，借助$\text{RP}^2$上的2参数族Seiberg-Witten理论，构造出多类新的奇异微分同胚例子，还证明了相关微分同胚群基本群同态的非满射性。
Controllability and diffeomorphism groups on manifolds with boundary
本文研究具有光滑边界的流形的微分同胚群，证明其与边界微分同胚群构成具有局部截面的短正合序列，并建立向量场族的可控性结果，推广了Agrachev和Caponigro的结论，表明该群由指数映射的像生成。
Gauge theory for families
本文综述了族的规范理论及其在4维流形的微分同胚群与同胚群比较中的应用，直至2021年。
We consider the following problem: given two parallel and identically oriented bundles of light rays in n-dimensional Euclidean space and given a diffeomorphism between the rays of the former bundle and the rays of the latter one, is it possible to realize this diffeomorphism by means of several mirror reflections? We prove that a 2-mirror realization is possible if and only if the diffeomorphism is the gradient of a function. We further prove that any orientation reversing diffeomorphism of domains in the plane is locally the composition of two gradient diffeomorphisms, and therefore can be r…
On localizing groups of exotic diffeomorphisms of 4-manifolds
该研究围绕4-流形外徽微分同胚群的无限生成能否局部化到更小嵌入子流形的问题展开，通过Seiberg-Witten理论等得到多项相关结论，并指出4-流形映射类群也有类似结果。
Exotic diffeomorphisms of 4-manifolds with b_+ = 2
本文针对b_+=2的4维流形，利用单参数族Seiberg-Witten不变量及连通和族粘合公式，证明了特定n值下2ℂP²#nℂP²̅的Torelli群满射至ℤ^∞、n=10时其映射类群非有限生成。
Superconformal Gravity And The Topology Of Diffeomorphism Groups
本文推广扭曲四维超对称杨-米尔斯理论，通过路径积分构造光滑四流形族的唐纳森不变量推广，并证明其可解释为耦合共形超引力背景场的拓扑扭曲路径积分，从而揭示微分同胚群的拓扑信息。
We construct infinite families of topologically isotopic but smoothly distinct knotted spheres in many simply connected 4-manifolds that become smoothly isotopic after stabilizing by connected summing with $S^2 \times S^2$, and as a consequence, analogous families of diffeomorphisms and metrics of positive scalar curvature for such 4-manifolds. We also construct families of smoothly distinct links, all of whose corresponding proper sublinks are smoothly isotopic, that become smoothly isotopic after stabilizing.
We obtain constraints on the topology of families of smooth $4$-manifolds arising from a finite dimensional approximation of the families Seiberg-Witten monopole map. Amongst other results these constraints include a families generalisation of Donaldson's diagonalisation theorem and Furuta's $10/8$ theorem. As an application we construct examples of continuous $\mathbb{Z}_p$-actions for any odd prime $p$, which can not be realised smoothly. As a second application we show that the inclusion of the group of diffeomorphisms into the group of homeomorphisms is not a weak homotopy equivalence for …
We use a 1-parameter version of gauge theory to investigate the topology of the diffeomorphism group of 4-manifolds. A polynomial invariant, analogous to the Donaldson polynomial, is defined, and is used to show that the diffeomorphism group of certain simply-connected 4-manifolds has infinitely generated π_0.
Non-trivial smooth families of $K3$ surfaces
该研究针对复$K3$曲面微分同胚群恒等分支的基本群结构问题，利用族Seiberg–Witten不变量与Einstein度量模空间，证明其包含可数无限秩的自由阿贝尔群直和项。
A morph between two Riemannian $n$-manifolds is an isotopy between them together with the set of all intermediate manifolds equipped with Riemannian metrics. We propose measures of the distortion produced by some classes of morphs and diffeomorphisms between two isotopic Riemannian $n$-manifolds and, with respect to these classes, prove the existence of minimal distortion morphs and diffeomorphisms. In particular, we consider the class of time-dependent vector fields (on an open subset $Ω$ of $ \R^{n+1}$ in which the manifolds are embedded) that generate morphs between two manifolds $M$ and $N…
In this paper we study the uniform perfectness, boundedness and uniform simplicity of diffeomorphism groups of compact manifolds with boundary and open manifolds and obtain some upper bounds of their diameters with respect to commutator length, those with support in balls and conjugation-generated norm.
Let (M,J) be an almost complex manifold. We show that the infinite-dimensional space Tau of totally real submanifolds in M carries a natural connection. This induces a canonical notion of geodesics in Tau and a corresponding definition of when a functional, defined on Tau, is convex. Geodesics in Tau can be expressed in terms of families of J-holomorphic curves in M; we prove a uniqueness result and study their existence. When M is Kähler we define a canonical functional on Tau; it is convex if M has non-positive Ricci curvature. Our construction is formally analogous to the notion of geodesic…
The Cauchy problem for gradient generalized Ricci solitons on a bundle gerbe
研究梯度广义里奇溶子在阿贝尔纤维丛束上的柯西问题，证明分析解的存在性和唯一性，并解决在紧黎曼面上的初值方程。
Boundary values of diffeomorphisms of simple polytopes, and controllability
本文研究简单多胞形上微分同胚群的边界值问题，建立了商空间的典范李群结构，并证明恒等分量由指数映射生成，得出可控性结论。
We examine the existence of one parameter groups of diffeomorphisms whose infinitesimal generators annihilate all scalar polynomial curvature invariants through the application of the Lie derivative, known as $\mathcal{I}$-preserving diffeomorphisms. Such mappings are a generalization of isometries and appear to be related to nil-Killing vector fields, for which the associated Lie derivative of the metric yields a nilpotent rank two tensor. We show that the set of nil-Killing vector fields contains Lie algebras, although the Lie algebras may be infinite and can contain elements which are not $…
Smooth structures on four-manifolds with finite cyclic fundamental groups
本文研究有限循环基本群四维流形的光滑结构分类，证明除七个可能例外外，Z_{4m+2} 且奇相交形式的流形或无光滑结构或有无穷多个，并构造多类非复不可约伪射影平面。
We develop a powerful new analytic method to construct complete non-compact G2-manifolds, i.e. Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction starts with a complete non-compact asymptotically conical Calabi-Yau 3-fold B and a circle bundle M over B satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete G2-metrics on M that collapses to the original Calabi-Yau metric on the base B as the parameter converges to 0. The G2-metrics we construct have controlled asymptotic ge…
Generalized Positive Energy Representations of the Group of Compactly Supported Diffeomorphisms
受广义相对论启发，本文研究了光滑流形上紧支撑微分同胚群的广义正能射影酉表示，证明了在流形连通且维数大于1时该表示在单位分量上必然平凡，并确定了紧支撑向量场李代数的连续第二上同调。
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