Riemannian \(L^p\) structures and \(K\)-homology. (Structures riemanniennes \(L^p\) et \(K\)-homologie.) (French) Zbl 0942.58016

Hilsum, Michel

Ann. Math. (2) 149, No. 3, 1007-1022 (1999).

Generalizing previous works of N. Teleman for Lipschitz manifolds and of A. Connes, N. Teleman and D. Sullivan for quasi-conformal manifolds of even dimension, the author constructs analytically the signature operator for a new family of topological manifolds. This family contains the quasi-conformal manifolds and the topological manifolds modeled on germs of homeomorphisms of \(\mathbb{R}^n\) possessing a derivative which is in \(L^p\) with \(p>{1\over 2} n(n+1)\).
So, he obtains an unbounded Fredholm module which defines a class in the \(K\)-homology of the manifold, the Chern character of which is the Hirzebruch polynomial in the Pontrjagin classes of the manifold.

Reviewer: Corina Mohorianu (Iaşi)

MSC:

58B34	Noncommutative geometry (à la Connes)
58B20	Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
57N65	Algebraic topology of manifolds
58B15	Fredholm structures on infinite-dimensional manifolds

Keywords:

\(K\)-homology of a manifold; Pontrjagin class; Chern character; quasi-conformal manifolds; topological manifolds