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From diffeomorphism groups to loop spaces via cyclic homology. (English) Zbl 0928.19001
Connes, A. (ed.) et al., Quantum symmetries/ Symétries quantiques. Proceedings of the Les Houches summer school, Session LXIV, Les Houches, France, August 1 - September 8, 1995. Amsterdam: North-Holland. 727-755 (1998).
In this survey paper the author describes the relations between diffeomorphism groups of manifolds, i.e., pseudoisotopy theory, the algebraic \(K\)-theory of topological spaces of Waldhausen, and the cyclic homology of Connes and Tsygan. After recalling some classical results of pseudoisotopy theory the relation between the stable pseudoisotopy space, the Whitehead space \(Wh(X)\), and Waldhausen’s space \(A(X)\) is explained. After this, the construction of algebraic \(K\)-theory á la Quillen, respectively Waldhausen is outlined. Then the cyclic homology of Connes and Tsygan is introduced and identified with the primitive Lie algebra homology of stable matrices. Finally, it is explained how the Chern character on \(K\)-theory with values in cyclic homology can be used to calculate the homotopy groups of \(A(\Sigma X)\) rationally for any suspension \(\Sigma X\).
For the entire collection see [Zbl 0902.00046].

19D10 Algebraic \(K\)-theory of spaces
57R50 Differential topological aspects of diffeomorphisms
55P35 Loop spaces
17B55 Homological methods in Lie (super)algebras
19D55 \(K\)-theory and homology; cyclic homology and cohomology
58B34 Noncommutative geometry (à la Connes)