FREE LOOP SPACES IN GEOMETRY AND TOPOLOGY. INCLUDING THE MONOGRAPH SYMPLECTIC COHOMOLOGY AND VITERBO’S THEOREM BY MOHAMMED ABOUZAID

Introduction . . . 1
I A panorama of topology, geometry and algebra 19
1 Basics on free loop spaces . 21
David Chataur and Alexandru Oancea
2 Morse theory, closed geodesics, and the homology of free loop spaces . . 67
Alexandru Oancea
3 Rational homotopy – Sullivan models . . . . . . . . . 111
Luc Menichi
4 Free loop space and homology . . . 137
Jean-Louis Loday
5 Appendix to the chapter by J.-L. Loday . . . . . . . . 157
Janko Latschev
6 On algebraic structures of the Hochschild complex . . 165
Hossein Abbaspour
7 Basic rational string topology . . . . 223
Yves Felix ´
8 Fukaya’s work on Lagrangian embeddings . . . . . . 243
Janko Latschev
II Symplectic cohomology and Viterbo’s theorem 271
Mohammed Abouzaid
9 Symplectic cohomology of cotangent bundles . . . . . 279
10 Operations in symplectic cohomology . . . . . . . . . 323
11 String topology using piecewise geodesics . . . . . . . 355
12 From symplectic cohomology to loop homology . . . . 377
13 Viterbo’s theorem: surjectivity . . . 405
14 Viterbo’s theorem: isomorphism . . . . . 455
Bibliography to Part II . . . . 481
List of contributors . . . . . . 487
Index . .

In the late 1990s two initially unrelated developments brought free loop spaces into renewed focus. In 1999, Chas and Sullivan introduced a wealth of new algebraic operations on the homology of these spaces under the name of string topology, the full scope of which is still not completely understood. A few years earlier, Viterbo had discovered a first deep link between the symplectic topology of cotangent bundles and the topology of their free loop space. In the past 15 years, many exciting connections between these two viewpoints have been found. Still, researchers working on one side of the story often know quite little about the other.
One of the main purposes of this book is to facilitate communication between topologists and symplectic geometers thinking about free loop spaces. It was written by active researchers coming to the topic from both perspectives and provides a concise overview of many of the classical results, while also beginning to explore the new directions of research that have emerged recently. As one highlight, it contains a research monograph by M. Abouzaid which proves a strengthened version of Viterbo’s isomorphism between the homology of the free loop space of a manifold and the symplectic cohomology of its cotangent bundle, following a new strategy.
The book grew out of a learning seminar on free loop spaces held at Strasbourg University in 2008–2009, and should be accessible to a graduate student with a general interest in the topic. It focuses on introducing and explaining the most important aspects rather than offering encyclopedic coverage, while providing the interested reader with a broad basis for further studies and research.
Keywords: Loop space, symplectic geometry, symplectic topology, string topology, Morse theory, Hochschild and cyclic homology, operations on Hochschild and cyclic homology, rational homotopy theory, minimal models, Lagrangian embeddings, pseudo-holomorphic curves

Authors
• Janko Latschev (Universität Hamburg, Germany)
• Alexandru Oancea (Sorbonne Universités, Paris, France)