#### Volume 22, issue 6 (2018)

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$C^*$–algebraic higher signatures and an invariance theorem in codimension two

### Nigel Higson, Thomas Schick and Zhizhang Xie

Geometry & Topology 22 (2018) 3671–3699
##### Abstract

We revisit the construction of signature classes in ${C}^{\ast }$–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 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).

##### Keywords
$K$–theory, $C^*$–algebraic signature, partitioned manifold theorem, eventual homotopy equivalence
##### Mathematical Subject Classification 2010
Primary: 19K56, 57R19