arXiv research

Odd primes act on alternating knots via flypes.

Study of manifolds with prime cyclic group actions and curvature properties.

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.

Let $\widetilde{S}$ be a closed (compact without boundary) oriented surface with genus $g$, and $G$ be a group isomorphic to $$, where $p$ is a prime integer. An action of $G$ on $S$ is a pair $(\widetilde{S},f)$, where $f$ is a representation of $G$ in the group of orientation preserving autohomeom…

Smooth and symplectic symmetries of an infinite family of distinct exotic $K3$ surfaces are studied, and comparison with the corresponding symmetries of the standard $K3$ is made. The action on the $K3$ lattice induced by a smooth finite group action is shown to be strongly restricted, and as a result, nonsmoothability…

We show that the Frölicher spectral sequence of a complex parallelizable solvmanifold is degenerate at $E_{2}$-term. For a semi-direct product $G=\C^{n}\ltimes_φN$ of Lie-groups with lattice $Γ=Γ^{\prime}\ltimes Γ^{\prime\prime}$ such that $N$ is a nilpotent Lie-group with a left-invariant complex structure and $φ$ is …

Let $p$ be an odd prime. We construct a non-abelian extension $Γ$ of $S^1$ by $Z/p \times Z/p$, and prove that any finite subgroup of $Γ$ acts freely and smoothly on $S^{2p-1} \times S^{2p-1}$. In particular, for each odd prime $p$ we obtain free smooth actions of infinitely many non-metacyclic rank two $p$-groups on $…

We confirm the Halperin-Carlsson Conjecture for free $Z_p$-torus actions (p is a prime) on 2-dimensional finite CW-complexes and free $Z_2$-torus actions on compact 3-manifolds.

In this paper we find a unique normal form for the symplectic matrix representation of the conjugacy class of a prime order element of the mapping-class group. We find a set of generators for the fundamental group of a surface with a conformal automorphism of prime order which reflects the action the automorphism in an…

We discuss techniques for analysing the structure of the group obtained by reducing the image of the Burau representation of the braid group modulo a prime. The main tools are a certain sesquilinear form first introduced by Squier and consideration of the action of the group on a Euclidean building.

We show that a rank two finite group G admits a finite G-CW-complex X homotopy equivalent to a sphere, with rank one prime power isotropy, if and only if G does not p'-involve Qd(p) for any odd prime p. This follows from a more general theorem which allows us to construct a finite G-CW-complex by gluing together a give…

The aim of this paper is to show the rigidity of homologically trivial actions of prime order on K3 surfaces. To be precise, we show that homotopy K3 surfaces do not admit a periodic diffeomorphism of odd prime order 3 acting trivially on cohomology. Moreover, we give an obstruction in terms of the rationality and sign…

In this paper, we consider a Finsler sphere $(M,F)=(S^n,F)$ with the dimension $n>1$ and the flag curvature $K\equiv 1$. The action of the connected isometry group $G=I_o(M,F)$ on $M$, together with the action of $T=S^1$ shifting the parameter $t\in \mathbb{R}/\mathbb{Z}$ of the closed curve $c(t)$, define an action of…

Classifies $C_p$-surfaces using equivariant surgery methods.

When a closed Finsler manifold admits continuous isometric actions, estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics. To generalize the works of H. Duan, Y. Long, H.B. Rademacher, W. Wang and others on the existence of two…

Study shows how repetition affects learning in bandit settings, providing algorithms with sublinear regret.

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.

Let p be an odd prime and r be relatively prime to p. Let G be a finite p-group. Suppose an oriented 3-manifold M-tilde has a free G-action with orbit space M. We consider certain Witten-Reshetikhin-Turaev SU(2) invariants w_r(M). We will give a fomula for w_r(M) in terms of the defect of M-tilde --> M and the number o…

The union of singular orbits of an effective locally smooth circle action on the 4-sphere consists of two 2-knots, $K$ and $K^{\prime}$, intersecting at two points transversely. Each of $K$ and $K^{\prime}$ is called a branched twist spin. A twist spun knot is an example of a branched twist spin. The Gluck twists along…

New $\mathfrak{sl}_2$ action on link homologies discovered.

The study restricts matrix group actions on CAT(0) spaces and uniquely arcwise connected spaces, proving fixed points are inevitable.

Let M be a closed simply connected n-manifold of positive sectional curvature. We determine its homeomorphism or homotopic type if M also admits an isometric elementary p-group action of large rank. Our main results are: There exists a constant p(n)>0 such that (1) If M^{2n} admits an effective isometric \Bbb Z_p^k-act…

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 ap…

When a cyclic group G of prime order acts on a 4-manifold X, we prove a formula which relates the Seiberg-Witten invariants of X to those of X/G.

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 sh…

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 pr…

The paper constructs non-smoothable actions on spin 4-manifolds.

This paper initiated an investigation on the following question: Suppose a smooth 4-manifold does not admit any smooth circle actions. Does there exist a constant $C>0$ such that the manifold support no smooth $\Z_p$-actions of prime order for $p>C$? We gave affirmative results to this question for the case of holomorp…

We give an alternative proof of the mod $p$ vanishing theorem by F.Fang of Seiberg-Witten invariants under a cyclic group action of prime order, and generalize it to the case when $b_1>0$. Although we also use the finite dimensional approximation of the monopole map as well as Fang, our method is rather geometric. Furt…

We answer a weaker version of the classification problem for the homotopy types of $(n-2)$-connected closed orientable $(2n-1)$-manifolds. Let $n\geq 6$ be an even integer, and $X$ be a $(n-2)$-connected finite orientable Poincaré $(2n-1)$-complex such that $H^{n-1}(X;\mathbb{Q})=0$ and $H^{n-1}(X;\mathbb{Z}_2)=0$. The…

We construct finitely generated groups with strong fixed point properties. Let $\mathcal{X}_{ac}$ be the class of Hausdorff spaces of finite covering dimension which are mod-$p$ acyclic for at least one prime $p$. We produce the first examples of infinite finitely generated groups $Q$ with the property that for any act…

THEOREM. For every prime $p$ and each $n=2, 3, ... \infty$, there is an action of $G=\prod_{i=1}^{\infty}(Z/ pZ)$ on a two-dimensional compact metric space $X$ with $n$-dimensional orbit space. This theorem was proved in [DW: A.N. Dranishnikov and J.E. West, Compact group actions that raise dimension to infinity, Topol…

Let $p$ be an odd prime and $G$ the finite cyclic group of order $p$. We use the Casson-Walker-Lescop invariant to find a necessary condition for a three-manifold to have an action of $G$ with a circle as the set of fixed points.

Study cyclic group actions on specific high-dimensional manifolds.

Let $p$ be an odd regular prime, and let $G_p$ denote the extraspecial $p$--group of order $p^{3}$ and exponent $p$. We show that $G_p$ acts freely and smoothly on $S^{2p-1} \times S^{2p-1}$. For $p=3$ we explicitly construct a free smooth action of a Lie group $\widetilde{G}_3$ containing $G_3$ on $S^{5} \times S^{5}$…

The paper builds complex hyperbolic 2-manifolds with isolated singularities.

The paper proves new applications of knot invariants and smooth group actions on 3-spheres.

We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to $S^3.$ Such involutions are called hyperelliptic as the manifolds admitting such an action. We consider finite groups acting on 3-manifolds and containing hyperelliptic involutions whose fixed-point set has $r…

Let $X_0$ denote a compact, simply-connected smooth $4$-manifold with boundary the Poincaré homology $3$-sphere $Σ(2,3,5)$ and with even negative definite intersection form $Q_{X_0}=E_8$. We show that free $\mathbb{Z}/p$ actions on $Σ(2,3,5)$ do not extend to smooth actions on $X_0$ with isolated fixed points for any p…

Let $n \geq 3$. In this paper, we study the problem of whether a given finite group $G$ embeds in a quotient of the form $B_n/Γ_k(P_n)$, where $B_n$ is the $n$-string Artin braid group, $k \in \{2, 3\}$, and $\{Γ_l(P_n)\}_{l\in \mathbb{N}}$ is the lower central series of the $n$-string pure braid group $P_n$. Previous …

Fix a prime number ell. In this paper we develop the theory of relative pro-ell completion of discrete and profinite groups -- a natural generalization of the classical notion of pro-ell completion -- and show that the pro-ell completion of the Torelli group does not inject into the relative pro-ell completion of the c…

Researchers classify quotients of lens spaces using linear actions and topological tools.

A study of symplectic actions of a finite group $G$ on smooth 4-manifolds is initiated. The central new idea is the use of $G$-equivariant Seiberg-Witten-Taubes theory in studying the structure of the fixed-point set of these symmetries. The main result in this paper is a complete description of the fixed-point set str…

Let Gamma be a semidirect product of the form Z^n rtimes Z/p where p is prime and the Z/p-action on Z^n is free away from the origin. We will compute the topological K-theory of the real and complex group C*-algebra of Gamma and show that Gamma satisfies the unstable Gromov-Lawson-Rosenberg Conjecture. On the way we wi…

New algebraic structure for 2-string links and long knots.

Classifies symmetries of knots using group actions and orthogonal representation theory.

We show that a closed orientable 3-manifold M admits an action of Z_p with fixed point set S^1 iff M can be obtained as the result of surgery on a p-periodic framed link L and Z_p acts freely on the components of L. We prove a similar theorem for free Z_p-actions. As an interesting application, we prove the following, …

The equivalence (or weak equivalence) classes of orientation-preserving free actions of a finite group G on an orientable 3-dimensional handlebody of genus g can be enumerated in terms of sets of generators of G. They correspond to the equivalence classes of generating n-vectors of elements of G, where n=1+(g-1)/|G|, u…