The Fredholm representation theory is well adapted to the construction of homotopy invariants of non-simply-connected manifolds by means of the generalized Hirzebruch formula [(M)] = hL(M) chA f, [M]i ∈ K0 A(pt) ⊗ Q, where A = C is the C-algebra of the group , = 1(M). The bundle ∈ K0 A(B) is the canonical A-bundle generated by the natural representation −→ A. Recently, the first author constructed a natural family of Fredholm representations that lead to a symmetric vector bundle on the completion of the fundamental group with a modification of the Higson–Roe corona, provided that the completion is a closed manifold. In the present paper, a homology version of symmetry is discussed for the case in which the completion, with a modification of the Higson–Roe corona, is a manifold with boundary. The results were developed during the visit of the first author to Ancona on March, 2007. The last version is supplemented by details considering the case of manifolds with boundary.
Construction of Fedholm Representations and a Modification of the Higson-Roe Corona. In "Russian Journal of Mathematical Physics" / Alexander, Mischenko; Teleman, NECULAI SINEL. - In: RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS. - ISSN 1061-9208. - 16:(2009), pp. 446-449.