Notes on string topology
arXiv:math/0503625
Abstract
This paper is an exposition of the new subject of String Topology. We present an introduction to this exciting new area, as well as a survey of some of the latest developments, and our views about future directions of research. We begin with reviewing the seminal paper of Chas and Sullivan, which started String Topology by introducing a BV-algebra structure on the homology of a loop space of a manifold, then discuss the homotopy theoretic approach to String Topology, using the Thom-Pontrjagin construction, the cacti operad, and fat graphs. We review quantum field theories and indicate how string topology fits into the general picture. Other topics include an open-closed version of string topology, a Morse theoretic interpretation, relation to Gromov-Witten invariants, and "brane'' topology, which deals with sphere spaces. The paper is a joint account of the lecture series given by each of us at the 2003 Summer School on String Topology and Hochschild Homology in Almeria, Spain.
95 pages
References in corpus (9)
- Batalin-Vilkovisky algebras and two-dimensional topological field theories
- Floer homology and the heat flow
- Perturbed closed geodesics are periodic orbits: Index and Transversality
- Deformation complex of a d-algebra is a (d+1)-algebra
- De Rham model for string topology
- A bordism approach to string topology
- Homology of moduli spaces of curves and commutative homotopy algebras
- Duality in Gerstenhaber algebras
- Combinatorial Lie bialgebras of curves on surfaces
Cited by in corpus (14)
- On homotopy invariance for algebras over colored PROPs
- The string topology BV algebra, Hochschild cohomology and the Goldman bracket on surfaces
- The topological chiral homology of the spherical category
- A Batalin-Vilkovisky Algebra structure on the Hochschild Cohomology of Truncated Polynomials
- String Bracket and Flat Connections
- Calabi-Yau Frobenius algebras
- Spectral Sequences in String Topology
- Homological Quantum Field Theory
- Higher dimensional algebras via colored PROPs
- The topological cyclic Deligne conjecture
- Chen-Ruan Cohomology of cotangent orbifolds and Chas-Sullivan String Topology
- Graded Poisson algebras on bordism groups of garlands
- Orbifold String Topology
- Groups, cacti and framed little discs