scholarly journals The topological spectrum of a smooth closed manifold

1995 ◽  
Vol 44 (2) ◽  
pp. 0-0 ◽  
Author(s):  
Jyh-Wang Wu
Keyword(s):  
Closed Manifold ◽  
Smooth Closed Manifold

scholarly journals On the Level Set of a Function with Degenerate Minimum Point

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Yasuhiko Kamiyama
Keyword(s):  
Critical Point ◽  
Level Set ◽  
Smooth Function ◽  
Closed Manifold ◽  
Minimum Point ◽  
Unique Point ◽  
Degenerate Minimum ◽  
Nondegenerate Critical Point ◽  
Smooth Closed Manifold

Forn≥2, letMbe ann-dimensional smooth closed manifold andf:M→Ra smooth function. We setminf(M)=mand assume thatmis attained by unique pointp∈Msuch thatpis a nondegenerate critical point. Then the Morse lemma tells us that ifais slightly bigger thanm,f-1(a)is diffeomorphic toSn-1. In this paper, we relax the condition onpfrom being nondegenerate to being an isolated critical point and obtain the same consequence. Some application to the topology of polygon spaces is also included.

Algebraic Structures Associated to Orbifold Wreath Products

2010 ◽  
Vol 8 (2) ◽  
pp. 323-338
Author(s):  
Carla Farsi ◽  
Christopher Seaton
Keyword(s):  
Hopf Algebra ◽  
Wreath Product ◽  
Finite Groups ◽  
Lie Group ◽  
Wreath Products ◽  
Closed Manifold ◽  
Algebraic Structures ◽  
Present Structure ◽  
Smooth Closed Manifold ◽  
Connected Lie Group

AbstractWe present structure theorems in terms of inertial decompositions for the wreath product ring of an orbifold presented as the quotient of a smooth, closed manifold by a compact, connected Lie group acting almost freely. In particular we show that this ring admits λ-ring and Hopf algebra structures both abstractly and directly. This generalizes results known for global quotient orbifolds by finite groups.

scholarly journals Morse–Conley–Floer homology

2014 ◽  
Vol 06 (03) ◽  
pp. 305-338 ◽  
Author(s):  
T. O. Rot ◽  
R. C. A. M. Vandervorst
Keyword(s):  
Morse Function ◽  
Floer Homology ◽  
Gradient Flow ◽  
Homology Theory ◽  
Closed Manifold ◽  
Invariant Sets ◽  
Invariance Properties ◽  
Singular Homology ◽  
Morse Homology ◽  
Smooth Closed Manifold

The gradient flow of a Morse function on a smooth closed manifold generates, under suitable transversality assumptions, the Morse–Smale–Witten complex. The associated Morse homology is an invariant for the manifold, and equals the singular homology, which yields the classical Morse relations. In this paper we define Morse–Conley–Floer homology, which is an analogous homology theory for isolated invariant sets of smooth, not necessarily gradient-like, flows. We prove invariance properties of the Morse–Conley–Floer homology, and show how it gives rise to the Morse–Conley relations.

scholarly journals Higher spectral flow and an entire bivariant JLO cocycle

2012 ◽  
Vol 11 (1) ◽  
pp. 183-232 ◽  
Author(s):  
Moulay-Tahar Benameur ◽  
Alan L. Carey
Keyword(s):  
Dirac Operator ◽  
Chern Character ◽  
Dirac Operators ◽  
Cyclic Cohomology ◽  
Closed Manifold ◽  
Spectral Flow ◽  
Smooth Functions ◽  
Base Manifold ◽  
K Theory ◽  
Higher Spectral Flow

AbstractFor a single Dirac operator on a closed manifold the cocycle introduced by Jaffe-Lesniewski-Osterwalder [19] (abbreviated here to JLO), is a representative of Connes' Chern character map from the K-theory of the algebra of smooth functions on the manifold to its entire cyclic cohomology. Given a smooth fibration of closed manifolds and a family of generalized Dirac operators along the fibers, we define in this paper an associated bivariant JLO cocycle. We then prove that, for any l ≥ 0, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the Cl+1 topology and functions on the base manifold with the Cl topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fréchet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow.

scholarly journals The Lusternik–Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles

2016 ◽  
Vol 08 (03) ◽  
pp. 545-570 ◽  
Author(s):  
Luca Asselle ◽  
Gabriele Benedetti
Keyword(s):  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Symplectic Form ◽  
Critical Value ◽  
Closed Manifold ◽  
Hamiltonian Flow ◽  
Cotangent Bundles ◽  
Almost All

Let [Formula: see text] be a closed manifold and consider the Hamiltonian flow associated to an autonomous Tonelli Hamiltonian [Formula: see text] and a twisted symplectic form. In this paper we study the existence of contractible periodic orbits for such a flow. Our main result asserts that if [Formula: see text] is not aspherical, then contractible periodic orbits exist for almost all energies above the maximum critical value of [Formula: see text].

A new invariant and decompositions of manifolds

2018 ◽  
Vol 27 (02) ◽  
pp. 1850019
Author(s):  
Eiji Ogasa
Keyword(s):  
Topological Invariant ◽  
The Body ◽  
Precise Definition ◽  
Closed Manifold ◽  
Compact Manifolds ◽  
Connected Components ◽  
Fixed Base

We introduce a new topological invariant [Formula: see text] of compact manifolds-with-boundaries [Formula: see text] which is much connected with boundary-unions. A boundary-union is a kind of decomposition of compact manifolds-with-boundaries. See the body of the paper for the precise definition. Let [Formula: see text] and [Formula: see text] be [Formula: see text]-dimensional compact manifolds-with-boundaries. Let [Formula: see text] be a boundary-union of [Formula: see text] and [Formula: see text]. Then we have [Formula: see text] We define [Formula: see text] as follows: First, define an invariant of [Formula: see text]-closed manifolds. Take the maximum of the invariant of all connected-components of the boundary of each handle-body of an ordered-handle-decomposition with a fixed base [Formula: see text], where we impose the condition that the base [Formula: see text] is a (not necessarily connected) closed manifold. Take the minimum of the maximum for all ordered-handle-decompositions with the base [Formula: see text]. It is our another invariant [Formula: see text]. Take the maximum of the minimum, [Formula: see text], for all basis to satisfy the above condition. It is [Formula: see text]. See the body of the paper for the precise definition.

scholarly journals On realization of splitting obstructions in Browder-Livesay groups for closed manifold pairs

2000 ◽  
Vol 43 (1) ◽  
pp. 15-25 ◽  
Author(s):  
P. M. Akhmetiev ◽  
A. Cavicchioli ◽  
D. Repovš
Keyword(s):  
Closed Manifold ◽  
Realization Problem

AbstractWe consider some new types of realization problem for obstructions in the Browder-Livesay groups by homotopy equivalences of closed manifold pairs. We give several examples of calculations. We also consider relations with classical surgery problems.

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