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String topology and cyclic homology. (English) Zbl 1089.57002

This book is based upon a series of lectures given by the three authors at the summer school organized by D. Chataur, J.-L. Rodrígues and J. Scherer in Almería (Spain) from September 16 to 20, 2003. The first part of this book focuses on string topology (R. Cohen and S. Voronov). String topology is the study of algebraic and differential topological properties of paths and loops in manifolds. It gives both an introduction and a status report to this new and deep field. In Chapter 1, basic intersection theory for compact manifolds is reviewed and the Chas-Sullivan constructions are redefined. In Chapter 2, the concepts of operads and PROPS are studied in details (cacti operad). In Chapter 3, the field theoretic properties of string topology are discussed (fat graphs). In Chapter 4, the Morse theoretic interpretation of the string theory is given. In Chapter 5 the authors study the similar structure on spaces of maps of higher-dimensional spheres to manifolds (brane theory). The second part of this book, due to K. Hess, is devoted to the construction of a cochain complex \(tc^* (X)\) which computes the mod 2 topological cyclic homology of a space \(X\). In Chapter 1, basic algebraic and topological constructions are recalled. In Chapter 2, a simplicial model of the free loop space \(LX\) is constructed. This allows to define a cochain complex computing the cohomology of \(LX\) using a refinement of the Dupont-Hess methods. Chapter 3 is devoted to the study of the homotopy orbit space of the canonical circle action. In the last chapter a model for the Frobenius map \(LX\to LX\), \((z\to\gamma(z))\mapsto(z\to\gamma (z^2))\) is constructed. Thus, using a convenient definition of the mod 2 topological cyclic homology of \(X\), the author completes the construction of \(tc^*(X)\).

MSC:

57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57R19 Algebraic topology on manifolds and differential topology
55P35 Loop spaces
55N91 Equivariant homology and cohomology in algebraic topology
55P42 Stable homotopy theory, spectra
19D55 \(K\)-theory and homology; cyclic homology and cohomology
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