C∗–algebraic higher signatures and an invariance theorem in codimension two

Abstract

We revisit the construction of signature classes in C * -algebra K-theory, and develop a variation that allows us to prove equality of signature classes in some situations involving homotopy equivalences of noncompact manifolds that are only defined outside of a compact set. As an application, we prove a counterpart for signature classes of a codimension two vanishing theorem for the index of the Dirac operator on spin manifolds (the latter is due to Hanke, Pape and Schick, and is a development of well-known work of Gromov and Lawson).

The Gromov-Lawson codimension 2 obstruction to positive scalar curvature and the C*-index

“…In this note, we give an account of the proof in [6], together with a slight simplification avoiding relative higher index theory. Moreover, we observe that this method also gives a simpler proof of the main result of [5], even of the following strengthening. Here we write Sgn(M ; V M ) for the C * -algebraic higher signatures, i.e.…”

“…Higson, Schick, and Xie proved in [5] a companion result: the unexpected homotopy invariance of higher signatures of submanifolds of codimension 2 in the situation of Theorem 1.1, with the spin condition replaced by orientability.…”

Codimension 2 transfer of higher index invariants

“…According to [HPS15, Theorem 1.1], if M and N satisfies some assumptions on homotopy groups (listed in Proposition 2.8), and if the Rosenberg index α π (N ) vanishes, then M does not admit any psc metric. The corresponding result in higher signature is established by Higson-Schick-Xie [HSX18].…”