We apply an equivariant version of Perelman's Ricci flow with surgery to study smooth actions by finite groups on closed 3-manifolds. Our main result is that such actions on elliptic and hyperbolic 3-manifolds are conjugate to isometric actions. Combining our results with results by Meeks and Scott [17], it follows that such actions on geometric 3-manifolds (in the sense of Thurston) are always geometric, i.e. there exist invariant locally homogeneous Riemannian metrics. This answers a question posed by Thurston in [32].
We decribe and announce some results (joint with G. Besson, L. Bessieres, M. Boileau and J.Porti) about the geometry and topology of 3-manifolds. Most of the article is primarily intended as an introduction for nonexperts to geometrization of 3-manifolds, Ricci flow with surgery, and the simplicial volume approach to collapsing theorems. In the last section, Ricci flow with surgery on open 3-manifolds and obstructions to positive scalar curvature are discussed.
Given an equivariant knot $K$ of order $2$, we study the induced action of the symmetry on the knot Floer homology. We relate this action with the induced action of the symmetry on the Heegaard Floer homology of large surgeries on $K$. This surgery formula can be thought of as an equivariant analog of the involutive large surgery formula proved by Hendricks and Manolescu. As a consequence, we obtain that for certain double branched covers of $S^{3}$ and corks, the induced action of the involution on Heegaard Floer homology can be identified with an action on the knot Floer homology. As an appl…
This is the first of a series of papers on the long-time behavior of 3 dimensional Ricci flows with surgery. In this paper we first fix a notion of Ricci flows with surgery, which will be used in this and the following three papers. Then we review Perelman's long-time estimates and generalize them to the case in which the underlying manifold is allowed to have a boundary. Eventually, making use of Perelman's techniques, we prove new long-time estimates, which hold whenever the metric is sufficiently collapsed.
The first author's geometric Hopf invariant of a stable map $F:Σ^{\infty}X \to Σ^{\infty}Y$ is a stable ${\mathbb Z}_2$-equivariant map $h(F):Σ^{\infty}X \to Σ^{\infty}(Y \wedge Y)$ constructed by an explicit difference construction applied to $(F \wedge F)Δ_X - Δ_Y F$. The stable ${\mathbb Z}_2$-equivariant homotopy class of $h(F)$ is the primary obstruction to desuspending $F$ up to homotopy. The explicit nature of the construction allows for a $π$-equivariant version of $h(F)$ in the case of a $π$-equivariant $F$, with $π$ a discrete group. In earlier joint work we applied the $π_1(N)$-equi…
In the third part of this series of papers, we establish several topological results that will become important for studying the long-time behavior of Ricci flows with surgery. In the first part of this paper we recall some elementary observations in the topology of 3-manifolds. The main part is devoted to the construction of certain simplicial complexes in a given 3-manifold that exhibit useful intersection properties with embedded, incompressible solid tori. This paper is purely topological in nature and Ricci flows will not be used.
Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the space of ordered pairs of distinct points of M. We show how to define the equivariant cube Q(K) of this Blanchfield pairing with respect to a framed knot K that generates H_1(M)/Torsion. This article is devoted to the study of the invariant Q. We prove many properties for this invariant including two surgery formulae. Via surgery, the invariant Q is equivalent to an invariant of null-homologous kno…
In the following series of papers we analyze the long-time behavior of 3 dimensional Ricci flows with surgery. Our main result will be that if the surgeries are performed correctly, then only finitely many surgeries occur and after some time the curvature is bounded by $C t^{-1}$. This result confirms a conjecture of Perelman. In the course of the proof, we also obtain a qualitative description of the geometry as $t \to \infty$.
This is the fourth and last part of a series of papers on the long-time behavior of 3 dimensional Ricci flows with surgery. In this paper, we prove our main two results. The first result states that if the surgeries are performed correctly, then the flow becomes non-singular eventually and the curvature is bounded by $C t^{-1}$. The second result provides a qualitative description of the geometry as $t \to \infty$.
In this paper we analyze the long-time behaviour of 3 dimensional Ricci flow with surgery. We prove that under the topological condition that the initial manifold only has non-aspherical or hyperbolic components in its geometric decomposition, there are only finitely many surgeries and the curvature is bounded by $C t^{-1}$ for large $t$. This answers an open question in Perelman's work, which was made more precise by Lott and Tian, for this class of initial topologies. More general classes will be discussed in subsequent papers using similar methods.
In this expository article, we introduce the topological ideas and context central to the Poincare Conjecture. Our account is intended for a general audience, providing intuitive definitions and spatial intuition whenever possible. We define surfaces and their natural generalizations, manifolds. We then discuss the classification of surfaces as it relates to the Poincare and Thurston Geometrization conjectures. Finally, we survey Perelman's results on Ricci flows with surgery.
Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the configuration space of ordered pairs of distinct points of M. We show how to define the equivariant cube Q(M,K) of this Blanchfield pairing with respect to a framed knot K that generates H_1(M;Z)/Torsion. We present the invariant Q(M,K) and some of its properties including a surgery formula. Via surgery, the invariant Q is equivalent to an invariant Q' of null-homologous knots in rational homology …
In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of N. Higson and J. Roe. This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topological manifolds. Crucial to our treatment is the Lipschitz signature operator of Teleman. We also give a generalization to the equivariant setting of the product defined by Siegel in his Ph.D. thesis. Geometric applications are given to stability results for rho classes. We also o…
We derive a cut-and-paste surgery formula of Seiberg--Witten invariants for negative definite plumbed rational homology 3-spheres. It is similar to (and motivated by) Okuma's recursion formula [arXiv:math.AG/0610464, 4.5] targeting analytic invariants of splice quotient singularities. The two formulas combined provide automatically a proof of the equivariant version [arXiv:math.AG/0310084, 5.2(b)] of the `Seiberg--Witten invariant conjecture' [arXiv:math.AG/0111298] for these singularities.
In this paper we analyze the long-time behavior of 3 dimensional Ricci flows with surgery. Our main result is that if the surgeries are performed correctly, then only finitely many surgeries occur and after some time the curvature is bounded by $C t^{-1}$. This result confirms a conjecture of Perelman. In the course of the proof, we also obtain a qualitative description of the geometry as $t \to \infty$. This paper is the third part of a series. Previously, we had to impose a certain topological condition $\mathcal{T}_2$ to establish the finiteness of the surgeries and the curvature control. T…
We show an equivariant bordism principle for constructing metrics of positive scalar curvature that are invariant under a given group action. Furthermore, we develop a new codimension-2 surgery technique which removes singular strata from fixed point free $S^1$-manifolds while preserving equivariant positive scalar curvature. These results are applied to derive the following generalization of a result of Gromov and Lawson: Each closed fixed point free $S^1$-manifold of dimension at least 6 whose isotropy groups have odd order and whose union of maximal orbits is simply connected and not spin c…
This article is a sequel to the book `Ricci Flow and the Poincare Conjecture' by the same authors. Using the main results of that book we establish the Geometrization Conjecture for all compact, orientable three-manifolds following the approach indicated by Perelman in his preprints on the subject. This approach is to study the collapsed part of the manifold as time goes to infinity in a Ricci flow with surgery. The main technique for this study is the theory of Alexandrov spaces. This theory gives local models for the collapsed part of the manifold. These local models can be glued together to…
In a previous article, we constructed an invariant Z for null-homologous knots in rational homology spheres, from equivariant intersections in configuration spaces. Here we present an equivalent definition of Z in terms of configuration space integrals, we prove that Z is multiplicative under connected sum, and we prove null Lagrangian-preserving surgery formulae for Z. Our formulae generalize similar formulae that are satisfied by the Kricker rational lift of the Kontsevich integral for null Borromean surgeries. They imply that Z is universal with respect to a natural filtration. According to…
This is the second part of a series of papers analyzing the long-time behaviour of 3 dimensional Ricci flows with surgery. We generalize the methods developed in the first part and use them to treat cases in which the initial manifold satisfies a certain purely topological condition which is far more general than the one that we previously had to impose. Amongst others, we are able to treat initial topologies such as the 3-torus or $Σ\times S^1$ where $Σ$ is any surface of genus $\geq 1$. We prove that under this condition, only finitely many surgeries occur and that after some time the curvat…
We apply Lescop's construction of $\mathbb{Z}$-equivariant perturbative invariant of knots and 3-manifolds to the explicit equivariant propagator of "AL-paths" given in arXiv:1403.8030. We obtain an invariant $\hat{Z}_n$ of certain equivalence classes of fiberwise Morse functions on a 3-manifold fibered over $S^1$, which can be considered as a higher loop analogue of the Lefschetz zeta function and whose construction will be applied to that of finite type invariants of knots in such a 3-manifold. We also give a combinatorial formula for Lescop's equivariant invariant $\mathscr{Q}$ for 3-manifo…
We deduce an index jump formula for first order elliptic complexes over end-periodic manifolds, which generalizes the corresponding result for the DeRham complex. In the case of the anti-self-dual DeRham complex, we define the periodic rho invariant for a class of $4$-manifolds, and identify it with the periodic spectral flow of this complex. As an application, we prove the equivalence (under a mild homological assumption) of two signatures invariants defined by means of Yang-Mills theory and geometric topology respectively for essentially embedded tori in homology $S^1 \times S^3$. We also pr…
In this paper we prove several related results concerning smooth $\Z_p$ or $\s^1$ actions on 4-manifolds. We show that there exists an infinite sequence of smooth 4-manifolds $X_n$, $n\geq 2$, which have the same integral homology and intersection form and the same Seiberg-Witten invariant, such that each $X_n$ supports no smooth $\s^1$-actions but admits a smooth $\Z_n$-action. In order to construct such manifolds, we devise a method for annihilating smooth $\s^1$-actions on 4-manifolds using Fintushel-Stern knot surgery, and apply it to the Kodaira-Thurston manifold in an equivariant setting…
Geometric Meta-Learning via Coupled Ricci Flow: Unifying Knowledge Representation and Quantum Entanglement
School of Aeronautics and Astronautics, Shanghai Jiao Tong University（上海交通大学航空航天学院）；Météo-France（法国气象局）；CNRS（法国国家科学研究中心）；Mathematical Institute of Toulouse（图卢兹数学研究所）
We use equivariant surgery to classify all involutions on closed surfaces, up to isomorphism. Work on this problem is classical, dating back to the nineteenth century, but some questions seem to have been left unanswered. We give a modern treatment that leads to a complete classification.
We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. We then describe Floer stable homotopy types, the related Pin(2)-equivariant Seiberg-Witten Floer homology, and its application to the triangulation conjecture.
We prove a sharp area estimate for catenoids that allows us to rule out the phenomenon of multiplicity in min-max theory in several settings. We apply it to prove that i) the width of a three-manifold with positive Ricci curvature is realized by an orientable minimal surface ii) minimal genus Heegaard surfaces in such manifolds can be isotoped to be minimal and iii) the "doublings" of the Clifford torus by Kapouleas-Yang can be constructed variationally by an equivariant min-max procedure. In higher dimensions we also prove that the width of manifolds with positive Ricci curvature is achieved …
A long-standing conjecture due to Michael Freedman asserts that the 4-dimensional topological surgery conjecture fails for non-abelian free groups, or equivalently that a family of canonical examples of links (the generalized Borromean rings) are not A-B slice. A stronger version of the conjecture, that the Borromean rings are not even weakly A-B slice, where one drops the equivariant aspect of the problem, has been the main focus in search for an obstruction to surgery. We show that the Borromean rings, and more generally all links with trivial linking numbers, are in fact weakly A-B slice. T…
Let $n$ be a positive integer, and let $\ell>1$ be square-free odd. We classify the set of equivariant homeomorphism classes of free $C_\ell$-actions on the product $S^1 \times S^n$ of spheres, up to indeterminacy bounded in $\ell$. The description is expressed in terms of number theory. The techniques are various applications of surgery theory and homotopy theory, and we perform a careful study of $h$-cobordisms. The $\ell=2$ case was completed by B Jahren and S Kwasik (2011). The new issues for the case of $\ell$ odd are the presence of nontrivial ideal class groups and a group of equivarian…
The problem of equivariant rigidity is the $Γ$-homeomorphism classification of $Γ$-actions on manifolds with compact quotient and with contractible fixed sets for all finite subgroups of $Γ$. In other words, this is the classification of cocompact $E_{fin}Γ$-manifolds. We use surgery theory, algebraic $K$-theory, and the Farrell--Jones Conjecture to give this classification for a family of groups which satisfy the property that the normalizers of nontrivial finite subgroups are themselves finite. More generally, we study cocompact proper actions of these groups on contractible manifolds and pr…
We study finite type invariants of nullhomologous knots in a closed 3-manifold $M$ defined in terms of certain descending filtration $\{\mathscr{K}_n(M)\}_{n\geq 0}$ of the vector space $\mathscr{K}(M)$ spanned by isotopy classes of nullhomologous knots in $M$. The filtration $\{\mathscr{K}_n(M)\}_{n \geq 0}$ is defined by surgeries on special kinds of claspers in $M$ having one special leaf. More precisely, when $M$ is fibered over $S^1$ and $H_1(M)=\mathbb{Z}$, we study how far the natural surgery map from the space of $\mathbb{Q}[t^{\pm 1}]$-colored Jacobi diagrams on $S^1$ of degree $n$ to…
