A Morse theoretic description of string topology
نویسندگان
چکیده
Let M be a closed, oriented, n-dimensional manifold. In this paper we give a Morse theoretic description of the string topology operations introduced by Chas and Sullivan, and extended by the first author, Jones, Godin, and others. We do this by studying maps from surfaces with cylindrical ends to M , such that on the cylinders, they satisfy the gradient flow equation of a Morse function on the loop space, LM . We then give Morse theoretic descriptions of related constructions, such as the Thom and Euler classes of a vector bundle, as well as the shriek, or umkehr homomorphism.
منابع مشابه
Morse theory, graphs, and string topology
In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph flows, using homotopy theoretic methods to construct a virtual fundamental class, and evaluating cohomology classes on this fundamental class. By using similar c...
متن کاملA new virtual leader-following consensus protocol to internal and string stability analysis of longitudinal platoon of vehicles with generic network topology under communication and parasitic delays
In this paper, a new virtual leader following consensus protocol is introduced to perform the internal and string stability analysis of longitudinal platoon of vehicles under generic network topology. In all previous studies on multi-agent systems with generic network topology, the control parameters are strictly dependent on eigenvalues of network matrices (adjacency or Laplacian). Since some ...
متن کاملDiscrete Morse Theory for Computing Cellular Sheaf Cohomology
Sheaves and sheaf cohomology are powerful tools in computational topology, greatly generalizing persistent homology. We develop an algorithm for simplifying the computation of cellular sheaf cohomology via (discrete) Morse-theoretic techniques. As a consequence, we derive efficient techniques for distributed computation of (ordinary) cohomology of a cell complex.
متن کاملOn the Normal Cycles of Subanalytic Sets
We present a short proof of the existence of the normal cycle of a compact subanalytic set. The approach is inspired by some old ides of Joseph Fu, uses Morse theoretic techniques and o-minimal topology.
متن کاملThird-order Decentralized Safe Consensus Protocol for Inter-connected Heterogeneous Vehicular Platoons
In this paper, the stability analysis and control design of heterogeneous traffic flow is considered. It is assumed that the traffic flow consists of infinite number of cooperative non-identical vehicular platoons. Two different networks are investigated in stability analysis of heterogeneous traffic flow: 1) inter-platoon network which deals with the communication topology of lead vehicles and...
متن کامل