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
DOI:
https://doi.org/10.1090/S1079-6762-98-00042-0
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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- N. Higson and G. Kasparov, Operator $K$-theory for groups which act properly and isometrically on Hilbert space, ERA Amer. Math. Soc. 3 (1997), 131–142.
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Additional Information
Navin Keswani
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
Email:
navin@math.psu.edu
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