Mazur, Kapranov, Reznikov, and others developed ``Arithmetic Topology,'' a theory describing some surprising analogies between 3-dimensional topology and number theory, which can be summarized by saying that knots are like prime numbers. We extend their work by proving several formulas concerning branched coverings of 3-manifolds and extensions of number fields and observe that these formulas are almost identical, via the dictionary of arithmetic topology. Until now there is no satisfactory explanation for the coincidences between our formulas. The proofs of topological results use equivariant cohomology and the Leray-Serre spectral sequence. The number theoretic proofs are based on an approach to class field theory via idele groups.
This is an expository article of our work on analogies between knot theory and algebraic number theory. We shall discuss foundational analogies between knots and primes, 3-manifolds and number rings mainly from the group-theoretic point of view.
Following the analogies between 3-dimensional topology and number theory, we study an idèlic form of class field theory for 3-manifolds. For a certain set $\mathcal{K}$ of knots in a 3-manifold $M$, we first present a local theory for each knot in $\mathcal{K}$, which is analogous to local class field theory, and then, getteing together over all knots in $\mathcal{K}$, we give an analogue of idèlic global class field theory for an integral homology sphere $M$.
Quasifuchsian hyperbolic manifolds, or more generally convex co-compact hyperbolic manifolds, have infinite volume, but they have a well-defined ``renormalized'' volume. We outline some relations between this renormalized volume and the volume, or more precisely the ``dual volume'', of the convex core. On one hand, there are striking similarities between them, for instance in their variational formulas. On the other, object related to them tend to be within bounded distance. Those analogies and proximities lead to several questions. Both the renormalized volume and the dual volume can be used …
There are known infinite families of Brieskorn homology 3-spheres which can be realized as boundaries of smooth contractible 4-manifolds. In this paper we show that free periodic actions on these Brieskorn spheres do not extend smoothly over a contractible 4-manifold. We give a new infinite family of examples in which the actions extend locally linearly but not smoothly.
In this note we derive an upper bound on the number of 2-spheres in the fixed point set of a smooth and homologically trivial cyclic group action of prime order on a simply-connected 4-manifold. This improves the a priori bound which is given by one half of the Euler characteristic of the 4-manifold. The result also shows that in some cases the 4-manifold does not admit such actions of a certain order at all or that any such action has to be pseudofree.
We use the equivariant Yang-Mills moduli space to investigate the relation between the singular set, isotropy representations at fixed points, and permutation modules realized by the induced action on homology for smooth group actions on certain 4-manifolds.
Let G be either a finite cyclic group of prime order or S^1. We find new relations between cohomology of a manifold (or a Poincare duality space) M with a G-action on it and cohomology of the fixed point set, M^G. Our main tool is the notion of Poincare duality on the Leray spectral sequence of the map M_G -> BG. We apply our results to study group actions on 3-manifolds.
This article announces joint work with Frank Connolly and Jim Davis. We generalize our classification of pseudo-free involutions on the n-torus, by studying the action of the associated infinite group with torsion in the universal cover. Included is a non-Riemannian example obtained from the restriction of the action of a Coxeter group on a CAT(0) manifold.
We show that the simply-connected four-manifolds which admit locally linear, homologically trivial actions by rank two finite abelian groups are homeomorphic to connected sums of CP^2, -CP^2, and S^2 x S^2 (with one exception: pseudofree Z_3 x Z_3 actions on the Chern manifold), and also establish an equivariant decompostion theorem. This generalizes results from a 1970 paper by Orlik and Raymond, and complements more recent work of Fintushel, Yoshida, and Huck on S^1 actions. In each case, the simply-connected four-manifolds which support such actions are essentially the same.
We consider a purely algebraic result. Then given a circle or cyclic group of prime order action on a manifold, we will use it to estimate the lower bound of the number of fixed points. We also give an obstruction to the existence of $\mathbb{Z}_p$ action on manifolds with isolated fixed points when $p$ is a prime.
We introduce an elementary way of constructing principal (Z_2)^m-bundles over compact smooth manifolds. In addition, we will define a general notion of locally standard (Z_2)^m-actions on closed manifolds for all m>0, and then give a general way to construct all such (Z_2)^m-actions from the orbit space. Some related topology problems are also studied.
Let M be a closed, connected, orientable topological 4-manifold, and G be a finite group acting topologically and locally linearly on M. In this paper we investigate the Borel spectral sequence for the G-equivariant cohomology of M, and establish new bounds on the rank of G for homologically trivial actions with discrete singular set.
In the paper we describe obstructions for the existence of symplectic and Hamiltonian symplectic circle actions on closed compact manifolds in terms of Hirzebruch genera and relations between differential and homotopic invariants of such manifolds.
We give a congruence for the quantum invariant of a Z_p-quotient of a 3$-manifold with a Z_{p^2} action. We show the congruence does not hold for quotients of 3--manifolds with a Z_{5}xZ_{5} action.
We produce infinite families of exotic actions of finite cyclic groups on simply connected smooth 4-manifolds with nontrivial Seiberg-Witten invariants.
In two previous papers the author presented a general construction of finite, fiber- and orientation-preserving group actions on orientable Seifert manifolds. In this paper we restrict our attention to elliptic 3-manifolds. A proof is given that orientation-reversing and fiber-preserving diffeomorphisms of Seifert manifolds do not exist for nonzero Euler class, in particular elliptic 3-manifolds. Each type of elliptic 3-manifold is then considered and the possible group actions that fit the given construction. This is shown to be all but a few cases that have been considered elsewhere. Finally…
It is proved that an arbitrary finite group acting locally linearly, homologically trivially, and pseudofreely on a closed, simply connected 4-manifold must in fact be cyclic and act semifreely, provided the second betti number of the manifold is at least three.
In this paper, we study the action of $\text{Homeo}_0(M)$, the identity component of the group of homeomorphisms of an $n$-dimensional manifold $M$ with an $\mathbb{F}_p$-free action, on another manifold $N$ of dimension $n+k<2n$. We prove that if $M$ is not an $\mathbb{F}_p$-homology sphere, then $N\cong M\times K$ for a homology manifold $K$ such that the action of $\text{Homeo}_0(M)$ on $M$ is standard and on $K$ is trivial. In particular, for $M=S^n$ a sphere, any nontrivial action is a generalization of the "coning-off" construction.
We explore transformation groups of manifolds of the form $M\times S^n$, where $M$ is an asymmetric manifold, i.e. a manifold which does not admit any non-trivial action of a finite group. In particular, we prove that for $n=2$ there exists an infinite family of distinct non-diagonal effective circle actions on such products. A similar result holds for actions of cyclic groups of prime order. We also discuss free circle actions on $M \times S^1$, where $M$ belongs to the class of "almost asymmetric" manifolds considered previously by V. Puppe and M. Kreck.
Over 50 years of work on group actions on $4$-manifolds, from the 1960's to the present, from knotted fixed point sets to Seiberg-Witten invariants, is surveyed. Locally linear actions are emphasized, but differentiable and purely topological actions are also discussed. The presentation is organized around some of the fundamental general questions that have driven the subject of compact transformation groups over the years and their interpretations in the case of $4$-manifolds. Many open problems are formulated. Updates to previous problems sets are given. A substantial bibliography is include…
We consider finite group-actions on closed, orientable and nonorientable 3-manifolds M which preserve the two handlebodies of a Heegaard splitting of M of some genus g > 1 (maybe interchanging the two handlebodies). The maximal possible order of a finite group-action on a handlebody of genus g>1 is 12(g-1) in the orientation-preserving case and 24(g-1) in general, and the maximal order of a finite group preserving the Heegaard surface of a Heegaard splitting of genus g is 48(g-1). This defines a hierarchy for finite group-actions on 3-manifolds which we discuss in the present paper; we present…
We show that recent results of Friedl-Vidussi and Chen imply that a symplectic manifold admits a fixed point free circle action if and only if it admits a symplectic circle action and we give a complete description of the symplectic cone in this case. This then completes the characterisation of symplectic 4-manifolds that admit non-trivial circle actions.
We show that every closed, simply connected, spin topological 4-manifold except $S^4$ and $S^2\times S^2$ admits a homologically trivial, pseudofree, locally linear action of $\mathbb{Z}_p$ for any sufficiently large prime number $p$ which is nonsmoothable for any possible smooth structure.
We give an affirmative answer to the Halperin-Carlsson conjecture for the homologically injective torus actions on closed manifolds. This class contains holomorphic torus actions on compact Kahler manifolds, torus actions on compact Riemannian flat manifolds.
We prove rigidity and vanishing theorems for several holomorphic Euler characteristics on complex contact manifolds admitting holomorphic circle actions preserving the contact structure. Such vanishings are reminiscent of those of LeBrun and Salamon on Fano contact manifolds but under a symmetry assumption instead of a curvature condition.
In this paper, we study smooth, semi-free actions on closed, smooth, simply connected manifolds, such that the orbit space is a smoothable manifold. We show that the only simply connected $5$-manifolds admitting a smooth, semi-free circle action with fixed-point components of codimension $4$ are connected sums of $S^3$-bundles over $S^2$. Furthermore, the Betti numbers of the $5$-manifolds and of the quotient $4$-manifolds are related by a simple formula involving the number of fixed-point components. We also investigate semi-free $S^3$ actions on simply connected $8$-manifolds with quotient a…
The aim of this paper is to study compact 5--manifolds which admit fixed point free circle actions. The first result implies that the torsion in the second homology and the second Stiefel--Whitney class have to satisfy strong restrictions. We then show that for simply connected 5--manifolds these restrictions are necessary and sufficient.
In this paper we consider the finite groups that act fiber- and orientation-preservingly on closed, compact, and orientable Seifert manifolds that fiber over an orientable base space. We establish a method of constructing such group actions and then show that if an action satisfies a condition on the obstruction class of the Seifert manifold, it can be derived from the given construction. The obstruction condition is refined and the general structure of the finite groups that act via the construction is provided.
The main results of this paper describes a formula for the Seiberg-Witten invariant of a 4-manifold which admits a nontrivial free S^1-action. We use this theorem to produce a nonsymplectic 4-manifold with a free circle action whose orbit space fibers over S^1. We also describe a 3-manifold which is not the orbit space of any symplectic 4-manifold with a free circle action. A corollary of the main theorem is a formula for the 3-dimensional Seiberg-Witten invariants of the total space of a circle bundle over a surface.
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…
An upper bound is obtained on the rank of a torus which can act smoothly and effectively on a smooth, closed, simply connected, rationally elliptic manifold. In the maximal-rank case, the manifolds admitting such actions are classified up to equivariant rational homotopy type.
We construct a nonsmoothable Z\times Z-action on the connected sum of an Enriques surface and S^2\times S^2, such that each of generators is smoothable. We also construct a nonsmoothable self-homeomorphism on an Enriques surface.
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 prove that closed symplectic four-manifolds do not admit any smooth free circle actions with contractible orbits, without assuming that the actions preserve the symplectic forms. In higher dimensions such actions by symplectomorphisms do exist, and we give explicit examples based on a construction of Fernandez, Gray and Morgan. We also prove that the only compact complex surfaces admitting free circle actions with contractible orbits are primary Hopf surfaces.
The content of this unpublished paper is subsumed in successive work of the authors, in particular arXiv:1102.0820, arXiv:1102.0821 and arXiv:1205.2434.
We study low-dimensional representations of matrix groups over general rings, by considering group actions on CAT(0) spaces, spheres and acyclic manifolds.
Every lattice H in a connected semi-simple Lie group G acts properly discontinuously by isometries on the contractible manifold G/K (K a maximal compact subgroup of G). We prove that if H acts on a contractible manifold W and if either 1) the action is properly discontinuous, or 2) W is equipped with a complete Riemannian metric, the action is by isometries and with unbounded orbits, G is simple with finite center and rank >1, then dim W is not less than dim G/K.
A survey of finite group actions on symplectic 4-manifolds is given with a special emphasis on results and questions concerning smooth or symplectic classification of group actions, group actions and exotic smooth structures, and homological rigidity and boundedness of group actions. We also take this opportunity to include several results and questions which did not appear elsewhere.
Let $M$ be a closed 4-manifold with a free circle action. If the orbit manifold $N^3$ satisfies an appropriate fibering condition, then we show how to represent a cone in $H^2(M;\R)$ by symplectic forms. This generalizes earlier constructions by Thurston, Bouyakoub and Fernández-Gray-Morgan. In the case that $M$ is the product 4-manifold $S^1\times N$ our construction complements the results of \cite{FV08} (arXiv:0805:1234 [math.GT]) and allows us to completely determine the symplectic cone of such 4-manifolds. The content of this paper partly overlaps with the content of the unpublished prepr…
This is the first of a series of papers devoted to the topology of symplectic Calabi-Yau $4$-manifolds endowed with certain symplectic finite group actions. We completely determine the fixed-point set structure of a finite cyclic action on a symplectic Calabi-Yau $4$-manifold with $b_1>0$. As an outcome of this fixed-point set analysis, the $4$-manifold is shown to be a $T^2$-bundle over $T^2$ in some circumstances, e.g., in the case where the group action is an involution which fixes a $2$-dimensional surface in the $4$-manifold. Our project on symplectic Calabi-Yau $4$-manifolds is based on …
This paper is devoted to a systematic study of the geometry of nondegenerate $\bbR^n$-actions on $n$-manifolds. The motivations for this study come from both dynamics, where these actions form a special class of integrable dynamical systems and the understanding of their nature is important for the study of other Hamiltonian and non-Hamiltonian integrable systems, and geometry, where these actions are related to a lot of other geometric objects, including reflection groups, singular affine structures, toric and quasi-toric manifolds, monodromy phenomena, topological invariants, etc. We constru…
