arXiv research

The paper removes a condition for the Nielsen realisation problem using equivariant Poincaré duality.

Classifies $C_p$-surfaces using equivariant surgery methods.

We study the eta invariants of compact flat spin manifolds of dimension n with holonomy group cyclic of odd prime order p. We find explicit expressions for the twisted and relative eta invariants and show that the reduced eta invariant is always an integer, except in a single case, when p=n=3. We use the expressions ob…

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

Let G be the fundamental group of a connected, closed, orientable 3-manifold. We explicitly compute its virtually cyclic geometric dimension. Among the tools we use are the prime and JSJ decompositions of M, several push-out type constructions, as well as some Bredon cohomology computations.

We prove that a prime knot K is not determined by its p-fold cyclic branched cover for at most two odd primes p. Moreover, we show that for a given odd prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold cyclic branched cover of at most one more knot K' non equivalent to K. To prove the main theor…

We use assembly maps to study $\mathbf{TC}(\mathbb{A}[G];p)$, the topological cyclic homology at a prime $p$ of the group algebra of a discrete group $G$ with coefficients in a connective ring spectrum $\mathbb{A}$. For any finite group, we prove that the assembly map for the family of cyclic subgroups is an isomorphis…

Study numerical invariants for groups, computing for cyclic groups and surfaces.

We prove that for n>2 there exists a quandle of cyclic type of size n if and only if n is a power of a prime number. This establishes a conjecture of S. Kamada, H. Tamaru and K. Wada. As a corollary, every finite quandle of cyclic type is an Alexander quandle. We also prove that finite doubly transitive quandles are of…

In a recent paper Y. Hu has given a sufficient condition for the fundamental group of the r-th cyclic branched covering of S^3 along a prime knot to be left-orderable in terms of representations of the knot group. Applying her criterion to a large class of two-bridge knots, we determine a range of the integer r>1 for w…

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 show that a hyperbolic $3$-manifold can be the cyclic branched cover of at most fifteen knots in $\mathbf{S}^3$. This is a consequence of a general result about finite groups of orientation preserving diffeomorphisms acting on $3$-manifolds. A similar, although weaker, result holds for arbitrary irreducible $3$-mani…

It is known that if any prime power branched cyclic cover of a knot in the 3-sphere is a homology sphere, then the knot has vanishing Casson-Gordon invariants. We construct infinitely many examples of (topologically) non-slice knots in the 3-sphere whose prime power branched cyclic covers are homology spheres. We show …

Cyclic covers of knots uniquely determine the original knot.

We provide an alternative proof of a sufficient condition for the fundamental group of the $n^{th}$ cyclic branched cover of $S^3$ along a prime knot $K$ to be left-orderable, which is originally due to Boyer-Gordon-Watson. As an application of this sufficient condition, we show that for any $(p,q)$ two-bridge knot, wi…

In this paper, a vanishing theorem is stated and proved. If a 4-manifold $M$ admits a smooth action by a cyclic group $\mathbb{Z}_r$, then given an $\mathbb{Z}_r$-equivariant $Spin^c$-structure $\mathcal{C}$ on $M$, the Seiberg-Witten invariant $SW\mathcal{(C)}$ is zero modulo $r$ under some slight assumptions. Here $r…

In this paper, we prove the Farrell-Jones Conjecture for the solvable Baumslag-Solitar groups with coefficients in an additive category. We also extend our results to groups of the form, Z[1/p] semidirect product with any virtually cyclic group, where p is a prime number.

Arithmetic study of knots connects homology and SL2 representations.

We give a construction of quandle cocycles from group cocycles, especially, for any integer p \geq 3, quandle cocycles of the dihedral quandle R_p from group cocycles of the cyclic group Z/p. We will show that a group 3-cocycle of Z/p gives rise to a non-trivial quandle 3-cocycle of R_p. When p is an odd prime, since d…

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.

We extend the construction of upsilon-type invariants to null-homologous knots in rational homology three-spheres. By considering $m$-fold cyclic branched covers with $m$ a prime power, this extension provides new knot concordance invariants $Υ_m^C (K)$ of knots in $S^3$. We give computations of these invariants for so…

Algebraic methods prove knot primality using Floer homology.

We study knots in $S^3$ with infinitely many $SU(2)$-cyclic surgeries, which are Dehn surgeries such that every representation of the resulting fundamental group into $SU(2)$ has cyclic image. We show that for every such nontrivial knot $K$, its set of $SU(2)$-cyclic slopes is bounded and has a unique limit point, whic…

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…

Let K be a knot in the 3-sphere with 2-fold branched covering space M. If for some prime p congruent to 3 mod 4 the p-torsion in the first homology of M is cyclic with odd exponent, then K is of infinite order in the knot concordance group. As one application, recall that the n-twisted double of an arbitrary knot has o…

In this paper we study smooth orientation-preserving free actions of the cyclic group $\mathbb Z/m$ on a class of $(n-1)$-connected $2n$-manifolds, $\sharp g (S^n \times S^n)\sharp Σ$, where $Σ$ is a homotopy $2n$-sphere. When $n=2$ we obtain a classification up to topological conjugation. When $n=3$ we obtain a classi…

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.

We prove that an integral homology 3-sphere is S^3 if and only if it admits four periodic diffeomorphisms of odd prime orders whose space of orbits is S^3. As an application we show that an irreducible integral homology sphere which is not S^3 is the cyclic branched cover of odd prime order of at most four knots in S^3…

Detect slopes in toroidal 3-manifolds to prove properties of fundamental groups.

We show that, for any prime p, a knot K in the 3-sphere is determined by its p-fold cyclic unbranched covering. We also investigate when the m-fold cyclic unbranched covering of a knot coincides with the n-fold cyclic unbranched covering of another knot, for different coprime integers m and n.

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…

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

We start by studying the distribution of (cyclically reduced) elements of the free groups Fn with respect to their abelianization (or equivalently, their integer homology class. We derive an explicit generating function, and a limiting distribution, by means of certain results (of independent interest) on Chebyshev pol…

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…

This paper studies connectivity of cyclic-Schottky strata in Schottky space.

Study shows periodic cohomology of non-orientable surface mapping class groups for odd primes.

The paper provides bounds for embedding manifolds into Euclidean spaces with group actions.

3-manifolds study Hasse norm principle, akin to number fields.

Study algebraic K-theory of 3-manifold groups using Farrell-Jones isomorphism and geometrization.

This paper demonstrates a topological meaning of quandle cocycle invariants of links with respect to finite connected quandles $X$, from a perspective of homotopy theory: Specifically, for any prime $\ell$ which does not divide the type of $X$, the $\ell$-torsion of this invariants is equal to a sum of the colouring po…

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…

A 3-manifold $M$ is said to be $p$-periodic ($p\geq 2$ an integer) if and only if the finite cyclic group of order $p$ acts on $M$ with a circle as the set of fixed points. This paper provides a criterion for periodicity of rational homology three-spheres. Namely, we give a necessary condition for a rational homology t…

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 possibilities for new or unusual kinds of topological, locally linear periodic maps of non-prime order on closed, simply connected 4-manifolds with positive definite intersection pairings are explored. On the one hand, certain permutation representations on homology are ruled out under appropriate hypotheses. On th…

Knots generating infinite subgroup bound rational homology balls.

A generalized Baumslag-Solitar group (GBS group) is a finitely generated group $G$ which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that Out(G) either contains non-abelian free groups or is virtually nilpotent of class at most 2. It has torsion only at finitely many primes. One may dec…

The paper studies mapping class groups of cyclic covers and their liftable counterparts.

This paper provides a construction of a quantum statistical mechanical system associated to knots in the 3-sphere and cyclic branched coverings of the 3-sphere, which is an analog, in the sense of arithmetic topology, of the Bost-Connes system, with knots replacing primes, and cyclic branched coverings of the 3-sphere …