A compact imbedding result on Lipschitz manifolds

Abstract

In [14], Teleman developed a theory of signature operators on Lipschitz manifolds. The theory includes both the classical index operator _D + and "twisted" operators with values in vector bundles. The main result of 1,-14] is Theorem 13.1 which states, in part, that for any closed, oriented, Lipschitz Riemannian manifold M" and for any Lipschitz complex vector bundle ~ over M, there is a natural signature operator _D[; the index of this operator is a Lipschitz invariant of the pair (M, ~). Using this result, Sullivan and Teleman 1-13] were able to show that (except in dimension 4) the index of _D~is actually a topological invariant of(M, 4). The computation of the index of _D~requires three tools: Hodge theory, a compact imbedding result and an excision theorem. The proof of the imbedding result in particular is quite complicated, using ideas from the theory of pseudodifferential operators. The main purpose of the present paper is to provide a simpler proof of the compact imbedding, based on ideas in 1-11], and to show that Hodge theory may be derived easily as a consequence of the imbedding result. (In the smooth case, this approach to Hodge theory may be traced back to Gaffney 15] and Friedrichs I-4].) Our techniques use only elementary facts from functional analysis. (For a concise summary of these see Appendix A of 1,8].) Before proceeding with the rigorous development, we would like to explain briefly the essential new idea in our approach. (The precise definitions of all spaces and operators may be found in the next section.) We denote the Hilbert space of square-integrable differential forms on M by L2(M), and define