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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

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 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).

$K$–theory, $C^*$–algebraic signature, partitioned manifold theorem, eventual homotopy equivalence
Mathematical Subject Classification 2010
Primary: 19K56, 57R19
Received: 12 October 2017
Accepted: 2 April 2018
Published: 23 September 2018
Proposed: Benson Farb
Seconded: Tobias H Colding, Martin Bridson
Nigel Higson
Department of Mathematics
Pennsylvania State University
University Park, PA
United States
Thomas Schick
Mathematisches Institut
Georg-August-Universität Göttingen
Zhizhang Xie
Department of Mathematics
Texas A&M University
College Station, TX
United States