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\newtheorem*{htpax}{Homotopy Axiom}
\newtheorem*{addax}{Additivity Axiom}
\newtheorem*{normax}{Normalization Axiom}
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\begin{document}
\begin{slide}
\begin{head}
Axioms for a local Reidemeister trace in fixed point and coincidence theory
\end{head}
\begin{center}P. Christopher Staecker

\url{http://www.messiah.edu/~cstaecker} %

July 2007
\end{center}
\end{slide}

\begin{slide}
\begin{head}
The Reidemeister trace
\end{head}

Given $f:X \to X$ and $\lift f:\lift X \to \lift X$, choose a
$\Z\pi$-basis for $C_q(\lift X)$, and we have
\[ T = \sum_q (-1)^q \tr(\lift f_q) \in \Z \pi \]
The projection into the Reidemeister (twisted conjugacy) classes gives
the \emph{Reidemeister trace}: 
\[ \RT(f,\lift f) = \rho(T) \in \Z\Reid(f) \]

$\RT$ does not depend on the choice of $\Z\pi$ basis or the simplicial
structure, but does depend (in a predictable way) on the choice of
lift $\lift f$.

Wecken's Trace Theorem:
\[ \RT(f,\lift f) = \sum_{[\alpha] \in \Reid(f)} \ind([\alpha])
   [\alpha], \]
where 
\[ \ind([\alpha]) = \ind(f, p(\Fix(\alpha \lift f))) \]
\end{slide}

\begin{slide}
\begin{head}
Properties of the Reidemeister trace
\end{head}
\begin{itemize}
\item {\bf (Homotopy)} If $f \htp f'$ and this homotopy lifts to
  $\lift f \htp \lift f'$, then
\[ \RT(f,\lift f) = \RT(f', \lift f'). \]
\item {\bf (Lift invariance)} If 
\[ \RT(f,\lift f) = \sum_{[\sigma]} k_{[\sigma]} [\sigma], \]
then 
\[ \RT(f, \alpha \lift f) = \sum_{[\sigma]} k_{[\sigma]} [\sigma
  \alpha^{-1}] \]
\item {\bf (Fixed point of lift)} If $[\alpha]$ appears in
  $\RT(f,\lift f)$, then $\alpha \lift f$ has a fixed point.
\end{itemize}
\end{slide}

\begin{slide}
\begin{head}Roadmap \end{head}

We will use the above (sometimes weakened) as axioms, and obtain a
uniqueness theorem for the Reidmeister trace in coincidence theory
using a similar result (S. 2007) for the local coincidence index.
\end{slide}

\begin{slide}
\begin{head} The uniqueness of the local coincidence index \end{head}
\begin{thm}[S. 2007]
For maps $f,g: X \to Y$ of orientable differentiable manifolds of the
same dimension with $U \subset X$ open with compact closure, there is a unique
integer-valued function 
\[ \ind(f,g,U) \]
satisfying:
\begin{itemize}
\item {\bf (Normalization)} $\ind(f,g,X) = L(f,g)$.
\item {\bf (Homotopy)} If $f,g$ are admissably homotopic to $f', g'$,
  then $\ind(f,g,U) = \ind(f', g', U)$.
\item {\bf (Additivity)} If $\Coin(f,g,U) \subset A \sqcup B$, then
\[ \ind(f,g,U) = \ind(f,g,A) + \ind(f,g,B). \]
\end{itemize}
\end{thm}

In the case $X=Y$, orientability is not needed, and the normalization
axiom can be weakened to: $\ind(\text{const},\id,X) = 1$.

\end{slide}

\begin{slide}
\begin{head} A local Reidemeister trace \end{head}
Fares and Hart [1994] gave a local Reidemeister trace $\RT(f,\lift f,
\liftinc, U) \in \Z\Reid(f)$, and showed that it satisfies:
\begin{itemize}
\item {\bf (Homotopy)} If $f \htp f'$, and
  this homotopy lifts to $\lift f \htp \lift f'$, then
\[ \RT(f,\lift f, \liftinc, U) = \RT(f', \lift f', \liftinc, U). \]
\item {\bf (Additivity)} If $\Fix(f,U) \subset A \sqcup B$, then
\[ \RT(f,\lift f, \liftinc, U) = \RT(f, \lift f, \liftinc, A) + \RT(f,
\lift f, \liftinc, B). \]
\item {\bf (Lift Invariance)}
If $\RT(f,\lift f, \liftinc, U) = \sum_{[\sigma]} i_\sigma [\sigma],$ then 
\[ \RT(f, \alpha \lift f, \liftinc, U) = \sum_{[\sigma]} i_\sigma [\sigma \alpha^{-1}] \]
\end{itemize}
\end{slide}

\begin{slide}
\begin{head}Our uniqueness result\end{head}
We use the above as axioms to show that the local Reidemeister trace
is unique in coincidence theory (and fixed point theory). 

Letting $\pi = \pi_1(Y)$, the Reidemeister classes $\Reid(f,g)$ are
the classes with respect to doubly twisted conjugacy:
\[ [\alpha] = [\beta] \iff \alpha = \psi(\sigma)^{-1} \beta
\phi(\sigma) \]
where $\phi = \pi_1(f)$ and $\psi = \pi_1(g)$.

Our theorem is:

\begin{thm} If $X, Y$ are orientable differentiable manifolds of the
same dimension, there is a unique function
\[ \RT(f,\lift f, g, \lift g, U) \in \Z\Reid(f,g) \]
satisfying five axioms:
\end{thm}
\end{slide}

\begin{slide}
\begin{itemize}
\item {\bf (Normalization)}
\[ c(\RT(f,\lift f, g, \lift g, X)) = L(f,g) \]
\item {\bf (Additivity)} If $\Coin(f,g,U) \subset A \sqcup B$, then
\begin{align*} \RT(f,\lift f, &g, \lift g, U) \\
&= \RT(f, \lift f, g, \lift g, A) +
\RT(f, \lift f, g, \lift g, B). 
\end{align*}
\item {\bf (Homotopy)} If $f \htp f'$ and $g \htp g'$ and these
  homotopies lift to $\lift f \htp \lift f'$ and $\lift g \htp \lift
  g'$, then
\[ \RT(f,\lift f, g, \lift g, U) = \RT(f', \lift f', g', \lift g',
  U). \]
\item {\bf (Lift invariance)}
\[ c(\RT(f,\lift f, g, \lift g, U)) = c(\RT(f,\alpha \lift f, g,
\beta \lift g, U)). \]
\item {\bf (Coincidence of lifts)} If $[\alpha]$ appears in
  $\RT(f,\lift f, g, \lift g, U)$, then $\alpha \lift f$ and $\lift g$
  have a coincidence on $\lift U$.
\end{itemize}
\end{slide}

\begin{slide}
\begin{head}Connections to the index\end{head}
\begin{lem}If $c: \Z\Reid(f,g) \to \Z$ is the sum of the coefficients,
  then
\[ c(\RT(f,\lift f, g, \lift g, U)) = \ind(f,g,U). \]
\end{lem}
\begin{proof}
By the lift invariance axiom, $c \circ \RT$ does not depend on $\lift
f$ or $\lift g$. Consider
\[ c \circ \RT(f,g,U) \in \Z \]
The function $c \circ \RT$ is additive, homotopy invariant, and gives the 
Lefschetz number if $U=X$. These are the three axioms for the
coincicdence index, and so $c \circ \RT$ is the coincidence
index.
\end{proof}

(Note: didn't need the Coincidence of Lifts axiom.)
\end{slide}

\begin{slide}
\begin{head} Coincidence classes \end{head}
\begin{lem}
Given a tuple $(f,\lift f, g, \lift g, U)$, there is a homotopic tuple
having isolated coincidence points (on the same $U$).
\end{lem}

Thus we can assume without loss of generality (by the homotopy axiom)
that our maps have isolated coincidence points. We also know that these
coincidence points are partitioned into \emph{coincidence classes} of the form
\[ C[\alpha] = p_X \Coin(\alpha \lift f, \lift g, \lift U) \subset X. \]
If $[\alpha] = [\beta]$, then $C[\alpha] = C[\beta]$. If $[\alpha]
\neq [\beta]$, then $C[\alpha]$ and $C[\beta]$ are disjoint.

\end{slide}

\begin{slide}
For a coincidence point $x$, let $[x]$ be the Reidmeister class
$[\alpha]$ for which $x \in C[\alpha]$.

\begin{thm}
For each coincidence point $x \in U$, let $U_x$ be an isolating
neighborhood of $x$. Then
\[ \RT(f,\lift f, g, \lift g, U) = \sum_x \ind(f,g,U_x) [x] \]
\end{thm}
\begin{proof}
Clearly
\[ \RT(f, \lift f, g, \lift g, U) = \sum_x \RT(f, \lift f, g, \lift g,
U_x) \]
by the additivity axiom. 

It remains to show that 
\[ \RT(f,\lift f, g, \lift g, U_x) = \ind(f,g,U_x)[x]. \]

\newpage

It remains to show that 
\[ \RT(f,\lift f, g, \lift g, U_x) = \ind(f,g,U_x)[x]. \]

First observe that $\RT(f,\lift f, g, \lift g, U_x)$ is an integral
multiple of $[x]$: If $\RT$ contains a term $[\beta]$, then by the
coincidence of lifts axiom $\beta \lift f$ and $\lift g$ have a
coincidence above $U_x$. But since $x$ is the only coincidence point
in $U_x$, it must belong to the coincidence class $C[\beta]$, which is
to say that $[\beta] = [x]$. Thus we know that
\[ \RT(f, \lift f, g, \lift g, U_x) = k [x], \]
and we need only show that $k= \ind(f,g,U_x)$.

We already know that the sum of coefficients in $\RT$ equals the
index, so we are done. 
\end{proof}
\end{slide}

\begin{slide}
\begin{head}Uniqueness\end{head}
The above gives a formula for the computation of $\RT$, which means
that it must be unique (and also that it exists). We also have shown a
coincidence version of Wecken's trace theorem:
\[ \RT(f, \lift f, g, \lift g, U) = \sum_{[\alpha]} \ind(C[\alpha])
   [\alpha]. \]

This leads fairly easily to a strong lift invariance property: If
\[ \RT(f,\lift f, g, \lift g, U) = \sum_{[\sigma]} k_{[\sigma]}
[\sigma], \] then 
\[ \RT(f, \alpha \lift f, g, \beta \lift g, U) = \sum_{[\sigma]}
k_{[\sigma]} [\beta \sigma \alpha^{-1}]. \]
\end{slide}

\begin{slide}
\begin{head} Selfmaps \end{head}

In the case $X = Y$, we obtain a uniqueness result without requiring
orientability, using only a weak normalization axiom:

If $f$ is a constant map, then
\[ c \circ \RT(f, \lift f, \id, \lift \id, U) = 1 \]

This result specializes directly to the fixed-point local Reidemeister
trace of Fares and Hart. 
\end{slide}

\begin{slide}
\begin{head} A lingering question \end{head}

Are all five axioms needed to establish uniqueness?

Just the first four do not suffice: defining
\[ \RT(f, \lift f, g, \lift g, U) = \ind(f, g, U) [1] \]
satisfies normalization, homotopy, additivity, and lift invariance,
but is not the ``correct'' Reidemeister trace.

But is the lift invariance axiom necessary? 

\[ ? \]
\end{slide}

\end{document}