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%% Translation via Omnimark script a2l, March 16, 1998 (all in one day!)
\controldates{30-MAR-1998,30-MAR-1998,30-MAR-1998,30-MAR-1998}
 
\documentclass{era-l}
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\theoremstyle{plain}
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\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
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\newcommand{\cmax}{C^{\ast }_{\max }\Gamma }
\newcommand{\erf}{\varphi }


\begin{document}

\title[Homotopy invariance of relative $\eta$-invariants]{Homotopy 
invariance of relative eta-invariants 
and $C^{*}$-algebra $K$-theory}
\author{Navin Keswani}
\address{Department of Mathematics, The Pennsylvania State University,
University Park, PA 16802}

\email{navin@math.psu.edu}
\keywords{Eta-invariants, $K$-theory}
\subjclass{Primary 19K56}
\thanks{The author would like to thank Nigel Higson for his guidance with this 
project.}

\issueinfo{4}{04}{}{1998}
\dateposted{April 1, 1998}
\pagespan{18}{26}
\PII{S 1079-6762(98)00042-0}
\def\copyrightyear{1998}
\copyrightinfo{1998}{American Mathematical Society}
\commby{Masamichi Takesaki}
\date{January 28, 1998}

\begin{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
$\pi _{1}(M)$.
\end{abstract}
\maketitle

\section{Introduction}

Eta-invariants arose in the work of Atiyah, Patodi, and Singer
\cite{APS1} as the contribution from the boundary in their formula
for the signature of a manifold with boundary.

\begin{dfn}  Let $N$ be a smooth, compact,
Riemannian manifold, and let $D$ be a self-adjoint, first-order elliptic
operator on $N$.  Define 
\begin{equation*}\eta _{D}(s) = \sum _{\lambda \in sp(D), \lambda \ne 0}
\text{sign}(\lambda ) |\lambda |^{-s}, \end{equation*}
where $sp(D)$ is the spectrum of $D$.
\end{dfn}


The sum converges for $\hbox {Re} \,s \gg 0$. It is a deep result that 
$\eta _{D}(s)$ has an analytic continuation that is regular at $s=0$ --- 
see 
\cite{G}, for instance.   The quantity $\eta _{D}(0)$ measures the
``spectral asymmetry'' of
$D$, in the sense that if $D$ is an operator whose spectrum is
symmetric with respect to 0, then $\eta _{D}(0) = 0$.  

Let $N$ be a $4k$-dimensional, oriented, Riemannian manifold with
boundary $M$.   If $N$ is locally a (Riemannian)
product near the boundary, the Atiyah-Patodi-Singer signature
theorem \cite{APS1} states that 
\begin{equation*}Sign(N,M) = \int _{N} \mathcal{L}(TN) + \eta _{D}(0), \end{equation*}
where $D$ is the signature operator on $M$,  and $\mathcal{L}$ is the
Hirzebruch $\mathcal{L}$-class of $N$.

In \cite{APS2}, 
$\eta _{D}(0)$ is examined for signature operators on
manifolds $M$ which are not necessarily boundaries.  Suppose $M$
has fundamental group
$\Gamma $.  Let
$\alpha , \beta :\Gamma \to U(n)$ be unitary, finite-dimensional
representations of $\Gamma $ of the same dimension. Let $L_{\alpha }$ be 
the 
flat vector bundle over $M$ associated to $\alpha $.  Let
$D_{\alpha }$ denote the signature operator with coefficients in
$L_{\alpha }$.

Define
\begin{equation*}\eta _{\alpha }(s) = \eta _{D_{\alpha }}(s), \qquad \eta _{\beta }(s) = 
\eta _{D_{\beta }}(s), \end{equation*}
and set 
\begin{equation*}\rho _{\alpha , \beta }(s) = \eta _{\alpha }(s) - \eta _{\beta }(s).
\end{equation*}
In \cite{APS2} it is proved
that $\rho _{\alpha ,\beta }(0)$ is a {\em differential invariant} of $M$ 
(it
does not depend on the Riemannian structure on $M$).  We denote
$\rho _{\alpha ,\beta }(0)$ by $\rho _{\alpha ,\beta }(M)$; it is called the 
{\em relative
eta-invariant} of $M$ associated to the representations $\alpha , \beta $ 
\cite{APS2}.

The following question may be posed: Is $\rho _{\alpha ,\beta }(M)$ a
homotopy invariant of $M$?  In other words, if $h:M \to M'$ is an
orientation-preserving homotopy equivalence between oriented
compact manifolds without boundary, is $\rho _{\alpha , \beta }(M) =
\rho _{\alpha \circ h_{*}, \beta \circ h_{*}}(M')$? 

The answer to this question is no in general, as is illustrated by the
example of the lens spaces $L(7,1)$ and $L(7,2)$.  These are
homotopy equivalent manifolds (\cite{deR}, Theorem 1, p. 97)  which 
have different relative 
eta-invariants \cite{Kes}.  

On the other hand, Neumann has shown in \cite{N} that
$\rho _{\alpha , \beta }(M)$ is a homotopy invariant for manifolds $M$ with 
free abelian fundamental group.  Mathai reproved this result using
index theory \cite{M}.  In 1988, Shmuel Weinberger \cite{W}
noticed a connection between the homotopy invariance problem for 
eta-invariants and the assembly map in surgery theory.  He extended 
Neumann's result, proving the homotopy invariance of
$\rho _{\alpha , \beta }(M)$ for a larger class of manifolds $M$, those for
which the Borel conjecture is known for the fundamental group
$\Gamma $.  

Motivated by the close analogy (see \cite{KM} and \cite{HR}) between 
the assembly map in surgery theory and the index map of Kasparov 
\cite{K2} and Baum-Connes \cite{BCH}, our goal is to give an analytic 
proof of a result parallel to Weinberger's, with 
the Borel conjecture replaced by its operator-theoretic analogue, the 
Baum-Connes conjecture.

Let $\cmax $ and $C^{*}_{\text{red}} \Gamma $ denote the full and
reduced group $C^{*}$-algebras of $\Gamma $, respectively. 
Denote by $B \Gamma $ the classifying space for principal
$\Gamma $-bundles.   For
torsion-free discrete groups $\Gamma $, the Baum-Connes
conjecture (see \cite{BCH}) states that a certain index map
$\mu _{\text{red}} : K_{*}(B \Gamma ) \to K_{*}(C^{*}_{\text{red}} \Gamma )$ is an
isomorphism.  At the present time the  groups $\Gamma $ for which this 
conjecture is known to be true have the property that
$K_{*}(\cmax ) \cong K_{*}(C^{*}_{\text{red}} \Gamma )$.  Thus, for the torsion-free
discrete groups for which the Baum-Connes conjecture is currently known,
there is an index isomorphism $\mu _{\text{max}} : K_{*}(B\Gamma ) \to K_{*}(\cmax )$.  In our work we will need to use the maximal group 
$C^{*}$-algebra
since we require that finite-dimensional unitary representations
of $\Gamma $ induce a representation of the $C^{*}$-algebra
involved. 

\begin{thm}\label{theorem1}  Let $M$ be a closed, smooth, oriented,
odd-dimensional, Riemannian manifold.  Suppose that $\pi _{1}(M) =
\Gamma $ is torsion-free and the Baum-Connes index map
$\mu _{\textup{max}}$ is an isomorphism for $\Gamma $.  Let
$\alpha , \beta $ be finite-dimensional unitary representations of
$\Gamma $ of the same dimension.  Then the relative eta-invariant
$\rho _{\alpha , \beta }(M)$ is an oriented homotopy invariant of $M$.
\end{thm}


We will prove this theorem by means of the following association
of ideas:
\begin{equation*}\text{Eta-invariants} \longleftrightarrow \text{Winding
numbers}  \longleftrightarrow \text{$K$-theory}. \end{equation*}
In Section \ref{sec2} we establish a link between eta-invariants and
the winding numbers of (open) paths of unitary
operators.  Using these paths and a homotopy equivalence $h:M \to M'$ we construct a {\em loop} of unitary operators describing an
element of the $K$-theory of $\cmax $.  At this point we will 
bring the Baum-Connes machinery to bear upon the problem of
understanding the invariance of the winding number of this loop 
under twistings by finite-dimensional unitary representations of
$\Gamma $.

Most of the known groups covered by our theorem are covered
by Weinberger's and vice versa.  There are however, two examples
of groups for which we are able to extend Weinberger's result, 
namely,  amenable groups and other groups which act metrically
properly on Hilbert spaces \cite{HK}.

Some of the techniques developed in the proof may be of interest
for other purposes.  In particular, this applies to our observation 
that $K$-homological equivalences
between elliptic operators on manifolds  are
realized through paths of operators with certain controlled analytic
properties.  

\section{Winding numbers}\label{sec2}

First we review an integral formula for the eta-invariant.  Notice
that 
\begin{equation*}\int _{0}^{\infty } t^{\frac{s-1}{2}} \lambda e^{-\lambda ^{2}t} dt =
\Gamma (\tfrac{s+1}{2}) \text{sign}(\lambda ) |\lambda |^{-s} \quad\text{ for  
} \hbox {Re} \,s > -1. \end{equation*}
Here $\Gamma (s) = \int _{0}^{\infty } t^{s-1} e^{-t} dt$ is the gamma
function. Let $D$ be the signature operator on a smooth, closed 
manifold $M$.  Summing over
all $\lambda \in sp(D)$, $\lambda \ne 0$, we get for $\hbox {Re}\,s \gg 0$,
\begin{equation*}\int _{0}^{\infty } t^{\frac{s-1}{2}} \text{trace}(D e^{-tD^{2}})
dt = \Gamma (\tfrac{s+1}{2}) \eta _{D}(s). \end{equation*}
Analytically continuing to $s=0$ and noting that $\Gamma (1/2) =
\sqrt {\pi }$, we obtain
\begin{equation*}\frac{1}{\sqrt {\pi }} \int _{0}^{\infty } t^{-\frac{1}{2}}
\text{trace}(De^{-tD^{2}}) dt = \eta _{D}(0). \end{equation*}
From the substitution $t \to t^{2}$ we get 
\begin{equation*}\frac{2}{\sqrt {\pi }} \int _{0}^{\infty } \text{trace}(D
e^{-t^{2}D^{2}}) dt = \eta _{D}(0). \tag{{2.1}} \end{equation*} 
For a detailed derivation see \cite{G}.  The convergence of this
integral is a delicate matter which is also treated in \cite{G}.  Notice
though that according to this formula,
\begin{equation*}\rho _{\alpha , \beta }(0) = \frac{2}{\sqrt {\pi }} \int _{0}^{\infty }
\text{trace}(D_{\alpha } e^{-t^{2}D_{\alpha }^{2}}) -
\text{ trace}(D_{\beta } e^{-t^{2} D_{\beta }^{2}}) dt, \end{equation*}
where $\alpha , \beta : \Gamma \to U(n)$ are finite-dimensional unitary 
representations of $\Gamma $ of the same dimension.
The right-hand side of the above equation exists for much easier
reasons than the integral appearing in (2.1) since it involves a
{\em difference} of integrands.  For small $t$, the Schwartz kernel of
$D_{\alpha } e^{-t^{2} D_{\alpha }^{2}}$  is concentrated near the diagonal
and (up to a small error which vanishes as $t \to 0$) depends only on 
the local geometry of the manifold and the bundle $L_{\alpha }$.  Since 
$L_{\alpha }$ is {\em flat}, it has no local geometry.  It follows that 
as $t \to 0$ the difference of local traces
\begin{equation*}\text{trace}_{x}(D_{\alpha } e^{-t^{2} D_{\alpha }^{2}}) -
\text{ trace}_{x}(D_{\beta } e^{-t^{2} D_{\beta }^{2}})\end{equation*} (where $x \in M$) 
converges uniformly to 0.

Now, let $U_{t}$, $a \leq t \leq b$, be a norm continuous path of
unitary operators on a Hilbert space $H$ satisfying:
\begin{enumerate}
\item $U_{t} = I + \text{Trace Class}$, for $a \leq t \leq b$; and
\item the path $U_{t}$ is smooth in the trace norm.
\end{enumerate}Define the {\em winding number} of $\mathcal{U} = \{ U_{t} \}_{a \leq t \leq b}$:
\begin{equation*}w(\mathcal{U}) = \frac{1}{2\pi i} \int _{a}^{b} \text{trace}(U_{t}^{-1}
\frac{dU_{t}}{dt}) dt \end{equation*}
(compare \cite{HS}).
 Let $ \erf (x) =
\frac{2}{\sqrt {\pi }}
\int _{0}^{x} e^{-t^{2}} dt $.
Define a path of unitary operators  by
\begin{equation*}V_{t} = -\exp (i \pi \erf (tD)). \end{equation*}
Since $1 + \exp (i \pi \erf (x))$ and the derivative of $\exp (i \pi \erf (x))$ are Schwartz class functions, the path
$V_{t}$ satisfies conditions
%\therosteritem 
1 and
%\therosteritem 
2 above (except that it is defined on the open interval 
$0 < t < \infty $).  Formally calculating the winding
number of $\mathcal{V} = \{ V_{t} \}_{t > 0}$, we get
\begin{align*}
w(\mathcal{V})  &= \frac{1}{2 \pi i} \int _{0}^{\infty }
\text{trace}\bigl ( V_{t}^{-1} \frac{d V_{t}}{dt} \bigr ) dt \\
   &= \frac{i \pi }{2 \pi i} \int _{0}^{\infty } \text{trace} \bigl ( 
\exp (-i \pi \erf (tD)) \exp (i\pi \erf (tD)) \erf '(tD) D \bigr ) dt \\
   &= \frac{1}{\sqrt {\pi }} \int _{0}^{\infty } \text{trace} \bigl ( D
e^{-t^{2}D^{2}} \bigr ) dt
\\
   &=  \frac{1}{2} \eta _{D}(0).
\end{align*}

Twist $V_{t}$ by $\alpha $ to obtain the path $V_{t}^{\alpha }$ as
follows:
\begin{equation*}V_{t}^{\alpha } = - \exp ( i \pi \erf (tD_{\alpha })). \end{equation*}
Then
\begin{align*}
2 \lim _{\epsilon \to 0} (w(\mathcal{V}_{\epsilon }^{\alpha }) -
w(\mathcal{V}_{\epsilon }^{\beta })) &= \lim _{\epsilon \to 0}
\frac{2}{\sqrt {\pi }}
\int _{\epsilon }^{1/\epsilon } \text{trace}(D_{\alpha } e^{-t^{2}
D_{\alpha }^{2}}) - \text{trace}(D_{\beta } e^{t^{2} D_{\beta }^{2}}) dt
\\
  &= \rho _{\alpha ,\beta }(M).
\end{align*}

\section{The Baum-Connes conjecture}\label{sec3}

Let $h:M \to M'$ be a homotopy equivalence between manifolds
$M,M'$. Let $\Gamma $ be the fundamental group of $M$ and let $D$ and $D'$
denote the signature operators on $M$ and $M'$ respectively.   Let
$L_{\text{max}}$ denote the Miscenko bundle \cite{Ros1}; this is
a flat bundle over $B \Gamma $ whose fibers are projective
$\cmax $-modules.  Let
$D_{\text{max}}, D_{\text{max}}'$ denote the signature operators
with coefficients in the pullback of $L_{\text{max}}$ over $M,M'$
respectively.  Let
\begin{equation*}V_{t} = -\exp (i \pi \erf (tD_{\text{max}})), 
\qquad V_{t}^{'} =
-\exp (i \pi \erf (tD_{\text{max}}')).\end{equation*} 
If $\epsilon > 0$, then we shall obtain, from the homotopy equivalence 
$h: M \to M'$, a loop $[h]$ of operators on Hilbert $\cmax $-modules 
which is comprised of  four segments, two of which are the closed paths 
$\{V_{t}\}_{t=\epsilon }^{1/\epsilon }$ and $\{V_{t}
^{'} \}_{t=\epsilon }^{1/\epsilon }$.  The remaining two segments are 
constructed as follows:  we use the hypothesis of {\em injectivity} of 
$\mu _{\text{max}}$ to connect $V_{\epsilon }$ to $V_{\epsilon }^{'}$ 
through what we shall call the small time path 
(denoted $ST_{\epsilon }$) of unitary operators.  This will be done in 
Section \ref{sec5}.  In Section \ref{sec4} 
we shall connect $V_{1/\epsilon }$ to 
$V_{1/\epsilon }^{'}$ through a large time path (denoted 
$LT_{1/\epsilon }$) of unitary operators (see Figure \ref{fg1}).  

\begin{figure}
\includegraphics[scale=2.20]{era42e-fig-1}
\caption{{} The loop $[h]$.}\label{fg1}
\end{figure}


Note that the large and small time paths actually depend on $\epsilon > 
0$.  However, the homotopy class of $[h]$ is independent of $\epsilon $. 
 Putting the constructions of these paths aside for a moment, we shall 
outline in this section our proof of 
Theorem \ref{theorem1}.

Notice that $[h]$ is a loop of unitary operators, each of which is a 
compact perturbation of the identity.  So
$[h]$ corresponds to an element of
$\pi _{1}(GL(\cmax ))$, which by Bott periodicity is the same as
$K_{0}(\cmax )$; see \cite{B}. 

Now every finite-dimensional unitary representation $\alpha : 
\Gamma \to U(n)$ determines a trace map 
$\text{trace}_{\alpha }: K_{0}(\cmax ) \to \mathbb{C}$.

\begin{lem}\label{theorem2} Let the twisted index map
$\textup{Index}_{\alpha }: K_{0}(B \Gamma ) \to \mathbb{Z}$ be given by
\begin{equation*}\textup{Index}_{\alpha }([\mathcal{H},F]) = \textup{Index}(F_{\alpha }).\end{equation*}
{\rm (}Here $[\mathcal{H},F]$ is an abstract elliptic operator on $B \Gamma $; 
these form the basic cycles of Kasparov's model for K-homology 
\cite{K1}.{\rm )} Then, the following diagram commutes:
\begin{equation*}\begin{CD}
K_{0}(B\Gamma )   @>\mu _{\textup{max}}>>   K_{0}(\cmax ) \\
@V\textup{Index}_{\alpha }VV          @VV\textup{trace}_{\alpha }V \\
\mathbb{Z}             @>>>     \mathbb{C}
\end{CD}\end{equation*}
\end{lem}


\begin{prop}\label{theorem3} If the Baum-Connes assembly map 
\begin{equation*}\mu _{\textup{max}}: K_{0}(B \Gamma ) \to K_{0}(\cmax )\end{equation*}
is surjective, and if $\alpha , \beta : \Gamma \to U(n)$ are 
finite-dimensional unitary representations of $\Gamma $ of the same 
dimension, then $\textup{trace}_{\alpha } = \textup{trace}_{\beta }$ on 
$K_{0}(\cmax )$.  In particular,
\begin{equation*}\textup{trace}_{\alpha }[h] - \textup{trace}_{\beta }[h] = 0.\end{equation*}
\end{prop}
\begin{proof} Let $[P] \in K_{0}(\cmax )$. Since $\mu _{\text{max}}$ is 
surjective, there is a preimage $[\mathcal{H},F] \in K_{0}(B\Gamma )$ to 
$[P]$.   By Lemma \ref{theorem2}, all we need to check 
is that Index$(F_{\alpha }) = $ Index$(F_{\beta })$. 

To simplify the argument slightly we will
assume for now that $B\Gamma $ is a manifold.
By using Chern-Weil theory and the Chern isomorphism, one can
show  that in topological $K$-theory, flat bundles are of the following
form \cite{Ros2}:
\begin{equation*}[L_{\alpha }] = \text{dim}(\alpha ).1 + \text{torsion} \in K^{0}(B\Gamma ).\end{equation*}
Thus, since $\text{dim}(\alpha ) = \text{dim}(\beta )$, we have that 
\begin{equation*}[L_{\alpha }] - [L_{\beta }] = \text{torsion} \in K^{0}(B\Gamma ). \end{equation*}
Now we know that there is a canonical pairing 
\begin{equation*}K^{0}(X) \otimes K_{0}(X) \to \mathbb{Z} \end{equation*}
given by 
\begin{equation*}[E] \otimes [F] \to \text{Index}(F_{E}). \end{equation*}
And thus, 
\begin{align*}
\text{Index}(F_{\alpha }) - \text{Index}(F_{\beta }) &= ([L_{\alpha }] -
[L_{\beta }]) \otimes [F] \\
	&= \text{torsion} \otimes [F] \\
	&= 0. \qquad \end{align*} \end{proof}


To calculate $\text{trace}_{\alpha }[h]$ we note that tensor product 
with $\alpha $ constructs from a loop of unitary operators on a Hilbert 
$\cmax $-module a loop of Hilbert space unitary operators.  Let us 
denote by $[h]_{\alpha }$ the loop so obtained from
 $[h]$.  Like $[h]$ it is composed of four segments: the paths 
$\{V_{\alpha ,t}\}_{t=\epsilon }^{1/\epsilon }$ and 
$\{V_{\alpha ,t}^{'}\}_{t=\epsilon }^{1/\epsilon }$ together with paths 
$ST_{\epsilon , \alpha }$ and $LT_{1/\epsilon , \alpha }$.  We shall prove 
in Sections \ref{sec4} and \ref{sec5} that:

\begin{thm}\label{theorem4}  If $\alpha , \beta : \Gamma \to U(n)$ are 
finite-dimensional unitary representations of $\Gamma $ of the same 
dimension, then in the limit as $\epsilon \to 0$, the winding numbers of 
$ST_{\epsilon , \alpha }$ and $ST_{\epsilon , \beta }$ 
are equal, whereas the winding numbers of $LT_{1/\epsilon ,\alpha }$ and 
$LT_{1/\epsilon , \beta }$ are zero.
\end{thm}


Now,
\begin{lem}\label{theorem5} Let $\alpha $ be a finite-dimensional,
unitary representation of $\Gamma $.
Then \begin{equation*}\textup{trace}_{\alpha }([h]) =
w([h]_{\alpha }).\end{equation*}
\end{lem}


From Theorem \ref{theorem4} and Lemma \ref{theorem5} we obtain:
\begin{thm}\label{theorem6} For $\alpha ,\beta : \Gamma \to U(n)$ 
finite-dimensional unitary representations of $\Gamma $ of the same 
dimension and $[h]$ as above,
\begin{equation*}\textup{trace}_{\alpha }[h] - \textup{trace}_{\beta }[h] = 
\rho _{\alpha ,\beta }(M) - \rho _{\alpha ,\beta }(M').\end{equation*}
\end{thm}
\begin{proof} By Lemma \ref{theorem5} the traces of $[h]$ are winding numbers. 
Decomposing the winding numbers into contributions from the four 
segments of $[h]$, we see from Theorem \ref{theorem4} that the $\alpha $ and 
$\beta $ contributions from the large and small time paths 
cancel one another as $\epsilon \to 0$, whereas the remaining 
contributions converge to $\rho _{\alpha ,\beta }(M) - 
\rho _{\alpha ,\beta }(M')$. \quad \end{proof}


\begin{proof}[Proof of Theorem \ref{theorem1}]  
This is immediate from Proposition \ref{theorem3} and 
Theorem \ref{theorem6}. \quad \end{proof}


\section{The large time path}\label{sec4}

In Theorem 3.18 in \cite{HR}, Higson and Roe write down an explicit
path that realizes the equality in $K_{*}(\cmax )$ of the indices of the 
signature operators of homotopy equivalent manifolds having
fundamental group $\Gamma $ (compare also \cite{KM}).  We use this path 
to obtain the large time path.  One checks by an explicit calculation 
that for any finite-dimensional unitary representation $\alpha $ of 
$\Gamma $,
\begin{lem}\label{theorem7}   As $t \to \infty $, the winding number of
$LT_{t,\alpha }$ converges to $0$. 
\end{lem}


\section{The small time path}\label{sec5}

The small time path is obtained from the equivalence relation defined in
$K$-homology and a result of Kasparov.

\begin{lem}[\cite{KM}]\label{theorem8} If the Baum-Connes assembly
map is injective, then $[D]=[D']$ in $K_{1}(B\Gamma )$.
\end{lem}
\begin{proof}[Sketch of Proof]  It is first established that due to the
homotopy equivalence $h:M \to M'$, $D$ and $D'$ have the same
index in $K_{1}(\cmax )$; see Theorem 3.18 in \cite{HR}.  The result
then follows from the injectivity of $\mu _{\textup{max}}$. \quad \end{proof}


\begin{dfn} Let $\epsilon > 0$.  An
{\em $\epsilon $-compression} of a bounded operator $F$ is an operator
$F_{\epsilon }$ satisfying the following conditions:
\begin{enumerate}
\item $F_{\epsilon }$ is a trace class perturbation of $F$; and
\item the propagation of $F_{\epsilon }$ is no more than $\epsilon $.
\end{enumerate}\end{dfn}


\begin{dfn}  An operator $F$ 
is said to have {\em polynomial growth} if there is a polynomial $p$
such that for each $\epsilon > 0$, there is an
$\epsilon $-compression of $F$, $F_{\epsilon }$, satisfying 
\begin{equation*}\|F-F_{\epsilon }\|_{1} < p \left ( \frac{1}{\epsilon } \right ). \end{equation*}  
\end{dfn}


\begin{dfn} Let $Y$ be a metric space.  A
path $F_{t}$ of bounded operators on a Hilbert space $\mathcal{H}$
equipped with an action of $C(Y)$ is called a {\em controlled path}
provided the following are true:
\begin{enumerate}
\item the path $F_{t}$ has polynomial growth; and
\item the paths $F_{t}^{2} - 1$ and $F_{t}(F_{t}^{2} - 1)$
are paths made up of trace class operators and are trace-norm
continuous and piecewise continuously differentiable in the trace
norm.
\end{enumerate}\end{dfn}

Note that this definition is modelled on the equivalence relation in 
Kasparov's 
realization of $K$-homology \cite{K1}.  In particular, a controlled 
path is a homotopy of abstract elliptic operators in the sense of 
\cite{K1}.
 
\begin{dfn} A chopping function is a
continuous function
$f$ on
$\mathbb{R}$ which satisfies the following:
\begin{enumerate}
\item $|f| \leq 1$;
\item $\lim _{x \to \pm \infty } f(x) = \pm 1.$
\end{enumerate}\end{dfn}


In the following theorem we use the $(M,E,\phi )$ description of 
$K$-homology due to P. Baum \cite{BD}.  We also use the notion of a 
degenerate operator as defined by Kasparov in his formulation of 
$K$-homology \cite{K1}.  
\begin{dfn} Let $Y$ be a compact Riemannian manifold 
(possibly with boundary). Let $H$ be a Hilbert space equipped with an 
action of $C(Y)$.  A {\em degenerate operator} is a bounded 
self-adjoint operator $F$ on $H$ satisfying:
\begin{enumerate}
\item $F^{2} - I = 0$;
\item $Ff-fF = 0$ for all $f \in C(Y)$.
\end{enumerate}\end{dfn}

    
\begin{thm}\label{theorem9}  Let $Y$ be a compact
Riemannian manifold with boundary.  Let $(M,E,\phi )$ and
$(M',E',\phi ')$ be two Baum $K$-cycles on $Y$ and suppose that
$\phi : M \to Y$ and $\phi ' : M' \to Y$ are
Lipschitz maps.  Let $\chi (x)$ be a chopping function such that: 
\begin{enumerate}
\item the derivative of $\chi $ is Schwartz class;
\item the Fourier transform of $\chi $ is smooth and is supported in 
$[-1,1]$; and
\item the functions $\chi ^{2} - 1$ and $\chi (\chi ^{2} - 1)$ are
Schwartz class and their Fourier transforms are supported in
$[-1,1]$.
\end{enumerate}Let
$D_{E}, D_{E'}'$ be the Dirac operators on $M,M'$ respectively, with
coefficients in $E,E'$ respectively.  
If $\big [(M,E,\phi )\big ] = \big [(M',E',\phi ')\big ] \in K_{*}(Y)$,
then there are degenerate operators
$A,A'$ such that 
\begin{enumerate}
\item $\chi (D_{E}) \oplus A$ and $A' \oplus \chi (D_{E'}')$ are
defined on the same Hilbert space $\mathcal{H}$;
\item the Hilbert space $\mathcal{H}$ has an action of $C(Y)$;
\item $\chi (D_{E}) \oplus A$ is connected to $A' \oplus \chi (D_{E'}')$
by a controlled path.
\end{enumerate}\end{thm}


A consequence of Lemma \ref{theorem8} and Theorem \ref{theorem9} is
\begin{cor}\label{theorem10}  There is a controlled path 
$F_{s,\epsilon }$ 
connecting $\erf (\epsilon D) \oplus I$ and $\erf (\epsilon D') \oplus I$. 
\end{cor}


Let $ST_{\epsilon } = \{-\exp (i \pi F_{s,\epsilon }) | 0 \leq s \leq 1\}$.

If $\alpha $ and $\beta $ are finite-dimensional unitary
representations of $\Gamma $ of the same dimension, then
\begin{lem}\label{theorem11}   As $\epsilon \to 0$, the difference
of the winding numbers of $ST_{\epsilon ,\alpha }$ and
$ST_{\epsilon ,\beta }$ converges to zero.   
\end{lem}
\begin{proof}[Sketch of Proof] Using standard techniques (see \cite{R},
Proposition 5.11), one writes 
\begin{equation*}ST_{\epsilon ,\alpha }(s) = F_{s,\epsilon } + G_{s,\epsilon } \end{equation*}
where $F_{s,\epsilon }$ has small propagation and
$tr(G_{s,\epsilon }) \to 0$ as $\epsilon \to 0$.  Now
the Schwartz kernels of $F_{s,\epsilon }$ are localized near the
diagonal and hence depend only on the local geometry of the
manifold and the bundle $L_{\alpha }$.  Since $L_{\alpha }$ is flat,
it has no local geometry and thus the kernel of $F_{s,\epsilon }$
does not detect it.  Thus $w(F_{s,\epsilon })$ is independent of
$L_{\alpha }$ and $w(G_{s,\epsilon }) \to 0$ as $\epsilon \to 0$.   
\end{proof}


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\end{document}