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Homotopy invariance of relative eta-invariants and $C^{*}$-algebra $K$-theory

Author: Navin Keswani
Journal: Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 18-26
MSC (1991): Primary 19K56
Published electronically: April 1, 1998
MathSciNet review: 1613055
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Abstract: We prove a close cousin of a theorem of Weinberger about the homotopy invariance of certain relative eta-invariants by placing the problem in operator $K$-theory. The main idea is to use a homotopy equivalence $h:M \to M'$ to construct a loop of invertible operators whose ``winding number" is related to eta-invariants. The Baum-Connes conjecture and a technique motivated by the Atiyah-Singer index theorem provides us with the invariance of this winding number under twistings by finite-dimensional unitary representations of $\pi _{1}(M)$.

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Navin Keswani
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, PA 16802

Keywords: Eta-invariants, $K$-theory
Received by editor(s): January 28, 1998
Published electronically: April 1, 1998
Additional Notes: The author would like to thank Nigel Higson for his guidance with this project.
Communicated by: Masamichi Takesaki
Article copyright: © Copyright 1998 American Mathematical Society