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The dual homological theory of topological K-theory, called K-homology and denotedKi (X) (i = 0,1) is described (thanks to the work of Atiyah, Brown-Douglas-Fillmore and Kasparov) from the notion of Fredholm module on the algebra A = C (X) of continuous functions on X . The aim of my lecture this year was to study, in the particularly simple case where X = S1, the mutiplicative analogue of the notion of Fredholm module on A. The second quantification with Fermi-Dirac statistics allows us to construct the multiplicative analogue of the fundamental class ofS1 in K homology.
We then show how to construct other multiplicative Fredholm modules on A from the C* algebras of V. Jones and A. Ocneanu (*).

(*) This construction is a collaboration with D. Evans.