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

Higson, Nigel¹ ; Schick, Thomas² ; Xie, Zhizhang³

¹ Department of Mathematics, Pennsylvania State University, University Park, PA, United States
² Mathematisches Institut, Georg-August-Universität Göttingen, Göttingen, Germany
³ Department of Mathematics, Texas A&M University, College Station, TX, United States

Geometry & topology, Tome 22 (2018) no. 6, pp. 3671-3699.

Résumé

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).

Détail

DOI : 10.2140/gt.2018.22.3671

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

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