scholarly journals On the iterated Hamiltonian Floer homology

2020 ◽  
pp. 2050026
Author(s):  
Erman Çı̇nelı̇ ◽  
Viktor L. Ginzburg
Keyword(s):  
Fixed Point ◽  
Euler Characteristic ◽  
Group Action ◽  
Floer Homology ◽  
Local Homology ◽  
Lefschetz Index ◽  
Cyclic Group Action ◽  
Symplectically Aspherical Manifold ◽  
Aspherical Manifold ◽  
Symplectically Aspherical

The focus of the paper is the behavior under iterations of the filtered and local Floer homology of a Hamiltonian on a symplectically aspherical manifold. The Floer homology of an iterated Hamiltonian comes with a natural cyclic group action. In the filtered case, we show that the supertrace of a generator of this action is equal to the Euler characteristic of the homology of the un-iterated Hamiltonian. For the local homology the supertrace is the Lefschetz index of the fixed point. We also prove an analog of the classical Smith inequality for the iterated local homology and the equivariant versions of these results.

Uniqueness of Free Actions on S3 Respecting a Knot

1987 ◽  
Vol 39 (4) ◽  
pp. 969-982 ◽  
Author(s):  
Michel Boileau ◽  
Erica Flapan
Keyword(s):  
Fixed Point ◽  
Cyclic Group ◽  
Group Action ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Cyclic Group Action ◽  
Periodic Diffeomorphisms ◽  
Free Actions

In this paper we consider free actions of finite cyclic groups on the pair (S3, K), where K is a knot in S3. That is, we look at periodic diffeo-morphisms f of (S3, K) such that fn is fixed point free, for all n less than the order of f. Note that such actions are always orientation preserving. We will show that if K is a non-trivial prime knot then, up to conjugacy, (S3, K) has at most one free finite cyclic group action of a given order. In addition, if all of the companions of K are prime, then all of the free periodic diffeo-morphisms of (S3, K) are conjugate to elements of one cyclic group which acts freely on (S3, K). More specifically, we prove the following two theorems.THEOREM 1. Let K be a non-trivial prime knot. If f and g are free periodic diffeomorphisms of (S3, K) of the same order, then f is conjugate to a power of g.

The Topology of a Group Action

2020 ◽  
pp. 205-212
Author(s):  
Loring W. Tu
Keyword(s):  
Fixed Point ◽  
Vector Bundle ◽  
Group Action ◽  
Lie Group ◽  
Tubular Neighborhood ◽  
Compact Lie Group ◽  
Fixed Point Set ◽  
Smooth Action ◽  
Point Set ◽  
Neighborhood Theorem

This chapter describes the topology of a group action. It proves some topological facts about the fixed point set and the stabilizers of a continuous or a smooth action. The chapter also introduces the equivariant tubular neighborhood theorem and the equivariant Mayer–Vietoris sequence. A tubular neighborhood of a submanifold S in a manifold M is a neighborhood that has the structure of a vector bundle over S. Because the total space of a vector bundle has the same homotopy type as the base space, in calculating cohomology one may replace a submanifold by a tubular neighborhood. The tubular neighborhood theorem guarantees the existence of a tubular neighborhood for a compact regular submanifold. The Mayer–Vietoris sequence is a powerful tool for calculating the cohomology of a union of two open subsets. Both the tubular neighborhood theorem and the Mayer–Vietoris sequence have equivariant counterparts for a G-manifold where G is a compact Lie group.

scholarly journals Lescop's invariant and gauge theory

2015 ◽  
Vol 24 (09) ◽  
pp. 1550050 ◽  
Author(s):  
Prayat Poudel
Keyword(s):  
Gauge Theory ◽  
Euler Characteristic ◽  
Betti Number ◽  
Floer Homology ◽  
Integral Homology ◽  
Similar Relationship

Taubes proved that the Casson invariant of an integral homology 3-sphere equals half the Euler characteristic of its instanton Floer homology. We extend this result to all closed oriented 3-manifolds with positive first Betti number by establishing a similar relationship between the Lescop invariant of the manifold and its instanton Floer homology. The proof uses surgery techniques.

scholarly journals Fixed Point Theory for Digital k-Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1244 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Fixed Point ◽  
Euler Characteristic ◽  
Fixed Point Theory ◽  
Geometric Realization ◽  
Point Theory ◽  
Fixed Point Property ◽  
Point Property ◽  
Euler Characteristics ◽  
Surface Theory ◽  
Closed Surfaces

The present paper studies the fixed point property (FPP) for closed k-surfaces. We also intensively study Euler characteristics of a closed k-surface and a connected sum of closed k-surfaces. Furthermore, we explore some relationships between the FPP and Euler characteristics of closed k-surfaces. After explaining how to define the Euler characteristic of a closed k-surface more precisely, we confirm a certain consistency of the Euler characteristic of a closed k-surface and a continuous analog of it. In proceeding with this work, for a simple closed k-surface in Z 3 , say S k , we can see that both the minimal 26-adjacency neighborhood of a point x ∈ S k , denoted by M k ( x ) , and the geometric realization of it in R 3 , denoted by D k ( x ) , play important roles in both digital surface theory and fixed point theory. Moreover, we prove that the simple closed 18-surfaces M S S 18 and M S S 18 ′ do not have the almost fixed point property (AFPP). Consequently, we conclude that the triviality or the non-triviality of the Euler characteristics of simple closed k-surfaces have no relationships with the FPP in digital topology. Using this fact, we correct many errors in many papers written by L. Boxer et al.

Surface symmetry and homology

1986 ◽  
Vol 99 (1) ◽  
pp. 73-77 ◽  
Author(s):  
Allan L. Edmonds ◽  
John H. Ewing
Keyword(s):  
Euler Characteristic ◽  
Group Action ◽  
Intersection Numbers ◽  
Oriented Surface ◽  
Effective Group

In his study of the structure of periodic homeomorphisms of surfaces, Jakob Nielsen [7] asked, in effect, how much about an effective group action on a surface is determined by its induced action on homology. (It is well known that the induced action on homology is faithful, provided the surface has negative Euler characteristic. See Farkas and Kra[3], v·3, for example.) Two periodic maps T1 and T2 on an oriented surface M are conjugate if there is an orientation preserving homeomorphism h: M → M such that hT1h–1 = T2; the two maps T1 and T2 are symplectically equivalent if there is an orientation preserving homeomorphism h: M → M so that hT1h–1 and T2 induce the same automorphisms on H1(M). By standard results this is the same as T1* and T2* being conjugate by an automorphism of H1(M) which preserves intersection numbers.

scholarly journals Examples of Calabi-Yau threefolds with small Hodge numbers

2021 ◽  
Vol 2081 (1) ◽  
pp. 012030
Author(s):  
A O Shishanin
Keyword(s):  
Fixed Point ◽  
Fundamental Group ◽  
Gauge Group ◽  
Complete Intersection ◽  
Euler Characteristic ◽  
Discrete Group ◽  
Free Action ◽  
Discrete Groups ◽  
Hodge Numbers ◽  
Fixed Point Sets

Abstract We observe some suitable examples of Calabi-Yau threefolds for heterotic superstring compactifications. It is reasonable to seek CY threefolds with Euler characteristic equals ±6 because of generation’s number. Hosotani mechanism for violations of the gauge group by the Wilson loops requires such CY space has a non-trivial fundamental group. These spaces can be obtained by factoring the complete intersection Calabi-Yau spaces by the free action of some discrete group. Also we shortly discuss cases when discrete groups act with fixed point sets.

scholarly journals The Combinatorics of Hyperbolized Manifolds

2015 ◽  
Vol 117 (1) ◽  
pp. 31
Author(s):  
Allan L. Edmonds ◽  
Steven Klee
Keyword(s):  
Euler Characteristic ◽  
Generating Functions ◽  
Characteristic Sign ◽  
Combinatorial Properties ◽  
Topological Version ◽  
Aspherical Manifold

A topological version of a longstanding conjecture of H. Hopf, originally proposed by W. Thurston, states that the sign of the Euler characteristic of a closed aspherical manifold of dimension $d=2m$ depends only on the parity of $m$. Gromov defined several hyperbolization functors which produce an aspherical manifold from a given simplicial or cubical manifold. We investigate the combinatorics of several of these hyperbolizations and verify the Euler Characteristic Sign Conjecture for each of them. In addition, we explore further combinatorial properties of these hyperbolizations as they relate to several well-studied generating functions.

scholarly journals Link Floer Homology Categorifies the Conway Function

2016 ◽  
Vol 59 (4) ◽  
pp. 813-836
Author(s):  
Mounir Benheddi ◽  
David Cimasoni
Keyword(s):  
Euler Characteristic ◽  
Floer Homology ◽  
Alexander Polynomial ◽  
Oriented Link

AbstractGiven an oriented link in the 3-sphere, the Euler characteristic of its link Floer homology is known to coincide with its multi-variable Alexander polynomial, an invariant only defined up to a sign and powers of the variables. In this paper we remove this ambiguity by proving that this Euler characteristic is equal to the so-called Conway function, the representative of the multi-variable Alexander polynomial introduced by Conway in 1970 and explicitly constructed by Hartley in 1983. This is achieved by creating a model of the Conway function adapted to rectangular diagrams, which is then compared to the Euler characteristic of the combinatorial version of link Floer homology.

scholarly journals Fixed points of Lie group actions on surfaces

1986 ◽  
Vol 6 (1) ◽  
pp. 149-161 ◽  
Author(s):  
J. F. Plante
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Lie Groups ◽  
Euler Characteristic ◽  
Lie Group ◽  
Stationary Point ◽  
Compact Surface ◽  
The Other ◽  
Continuous Action ◽  
Finite Dimensional

AbstractLetGbe a connected finite-dimensional Lie group andMa compact surface. We investigate whether, for a givenGandM, every continuous action ofGonMmust have a fixed (stationary) point. It is shown that whenGis nilpotent andMhas non-zero Euler characteristic that every action ofGonMmust have a fixed point. On the other hand, it is shown that the non-abelian 2-dimensional Lie group (affine group of the line) acts without fixed points on every compact surface. These results make it possible to complete this investigation for Lie groups of dimension at most 3.

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