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

### Navin Keswani

**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
$\pi _{1}(M)$.

*Copyright 1998 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**04** (1998), pp. 18-26
- Publisher Identifier: S 1079-6762(98)00042-0
- 1991
*Mathematics Subject Classification*. Primary 19K56
*Key words and phrases*. Eta-invariants, K-theory
- Received by the editors January 28, 1998
- Posted on April 1, 1998
- Communicated by Masamichi Takesaki
- Comments (When Available)

**Navin Keswani**

Department of Mathematics, The Pennsylvania State University,
University Park, PA 16802

*E-mail address:* `navin@math.psu.edu`

The author would like to thank Nigel Higson for his guidance
with this project.

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