\documentclass{slides}
\usepackage{amsmath, amsfonts, amssymb, hyperref}

\include{nielsenmacros}

\newcommand{\adm}{\mathcal{C}}

\newenvironment{head}
        {
        \begin{center}\bf
        }
        {
        \end{center}
        }

\newtheorem*{htpax}{Homotopy Axiom}
\newtheorem*{addax}{Additivity Axiom}
\newtheorem*{normax}{Normalization Axiom}
\newtheorem*{lem*}{Lemma}

\begin{document}
\begin{slide}
\begin{head}
The uniqueness of the coincidence index on orientable differentiable manifolds
\end{head}
\begin{center}P. Christopher Staecker

\url{http://www.messiah.edu/~cstaecker} %
% Given at Joint Meetings, 1/6/07

1/5/07
\end{center}
\end{slide}

\begin{slide}
\begin{head}
Setting
\end{head}

Throughout, we study continuous maps $f,g: X \to Y$, where $X$ and $Y$
are connected, orientable, differentiable manifolds of the same
dimension $n$.

For some $U\subset X$, 
\[ \Coin(f,g,U) = \{ x \in U \mid f(x) = g(x) \}. \]

The \emph{coincidence index} assigns an integer value
\[ \ind(f,g,U) \in \Z, \]
a sort of multiplicity count for coincidence points which sums to the
Lefschetz number and aids in the computation of the Nielsen number.

\end{slide}

\begin{slide}
\begin{head} 
Axioms in fixed point theory
\end{head}
The fixed point index $\ind(f,U)$ for selfmaps is defined on ANRs. A
uniqueness result for polyhedra using 5 axioms was given by O'Neill, 1953.

In 2004, a uniqueness result for (possibly nonorientable) manifolds: Let $\adm$ be the set of
\emph{admissable} pairs $(f,U)$ where $f:X \to X$ is a selfmap of a
differential manifold, and $U \subset X$ is open with $\Fix(f,U)$ compact.
\end{slide}

\begin{slide}
\begin{thm*}[Furi, Pera, Spadini, 2004] There is a unique function 
\[ \ind: \adm \to \R \]
satisfying the following three axioms:
\begin{itemize}
\item {\bf (Normalization)} If $c:X \to X$ is constant, then
\[ \ind(c, U) = 1. \]
\item {\bf (Homotopy)} If $f$ is admissably homotopic to $f'$, then
\[ \ind(f,U) = \ind(f', U). \]
\item {\bf (Additivity)} If $\Fix(f,U) \subset A \sqcup B$, then
\[ \ind(f,U) = \ind(f,A) + \ind(f,B). \]
\end{itemize}
\end{thm*}

\end{slide}

\begin{slide}
\begin{head}
Axioms for the coincidence index
\end{head}

For $X,Y$ orientable differential manifolds of dimension $n$, let
$\adm(X,Y)$ be all \emph{admissable triples} $(f,g,U)$ for $f,g:X \to Y$.

\vspace{1cm}

\begin{thm*} There is a unique function 
\[ \ind: \adm(X,Y) \to \R \]
satisfying three axioms:
\end{thm*}
\vfill
\begin{htpax}
If $(f,g)$ is admissably homotopic to $(f',g')$, then
\[ \ind(f,g,U) = \ind(f',g', U). \]
\end{htpax}
\vfill
\begin{addax} If $\Coin(f,g,U) \subset A \sqcup B$, then
\[ \ind(f,g,U) = \ind(f,g,A) + \ind(f,g,B). \]
\end{addax}

\end{slide}

\begin{slide}
\begin{head}
The Normalization axiom
\end{head}
Recall the fixed-point normalization axiom:
\[ \ind_{\Fix}(c,X) = 1 \]

This is difficult to generalize to coincidence theory. A na\"ive attempt
will try: 
\[ \ind(c, \id, X) = 1 \]
This isn't quite strong enough (though it will work if we allow
only selfmaps $f,g: X \to X$).

\vspace{1in}

\begin{normax}
If $L(f,g)$ is the coincidence Lefschetz number, then
\[ \ind(f,g,X) = L(f,g) \]
\end{normax}

\end{slide}

\begin{slide}
\begin{head}
Straight from the axioms
\end{head}
\vfill
\begin{prop}[Weak normalization] If $c:X \to X$ is constant and $\id:
  X \to X$ is identity, then
\[ \ind(c,\id,X) = 1. \]
\end{prop}
\vfill
\begin{prop}[Fixed point index] For a selfmap $f:X \to X$,
\[ \ind_{\Fix}(f,U) = \ind_{\Coin}(f,\id,U) \]
\end{prop}
\vfill
\begin{prop}[Solution] If $\ind(f,g,U) \neq 0$, then $f$ and $g$ have
  a coincidence on $U$.
\end{prop}
\end{slide}

\begin{slide}
\begin{head}Proof strategy\end{head}

We prove the uniqueness in steps:
\begin{itemize}
\item Linear maps $\R^n \to \R^n$
\item Nondegenerate maps $\R^n \to \R^n$
\item Nondegenerate maps $X \to Y$
\item All maps $X \to Y$
\end{itemize}
\end{slide}

\begin{slide}
\begin{head}Linear maps $\R^n \to \R^n$\end{head}

For linear maps $A,B: \R^n \to \R^n$, the coincidence set is easy:
\[ \Coin(A,B,\R^n) = \ker(B-A) \]

Triples like $(A,B,\R^n) \in \adm(\R^n, \R^n)$ with $A, B$ linear are
just maps with $B-A \in \Gl_n$.

We know $\Gl_n$ has two path components, and so by the homotopy axiom,
the index is constant on each component. The weak normalization
property gives
\vspace{\parskip}
\begin{lem*}
If $A,B$ are linear, then 
\[ \ind(A,B,\R^n) = \sign \det(B - A). \]
\end{lem*}
\end{slide}

\begin{slide}
\begin{head}Nondegenerate maps $\R^n \to \R^n$\end{head}

We say $(f,g,U) \in \adm(X,Y)$ is \emph{nondegenerate} if $f,g$ are
differentiable and 
\[ dg_x - df_x \in \Gl_n \]
for every coincidence point $x \in U$.

\vspace{\parskip}

\begin{lem*} If $(f,g,U) \in \adm(\R^n, \R^n)$, then each coincidence
  point is isolated, and for $V_x \subset U$ an isolating neighborhood
  of $x$, we have
\begin{align*}
\ind(f,g,V_x) &= \ind(df_x, dg_x, \R^n) \\
&= \sign \det(dg_x - df_x). 
\end{align*}
\end{lem*}

The isolation is an analytical argument. The index is a fancy
application of the homotopy axiom. 
\end{slide}

\begin{slide}
\begin{head} Nondegenerate maps $X \to Y$ \end{head}

For maps $f,g:X \to Y$ and a coincidence point $x$ with $y = f(x) =
g(x)$, take charts $\phi_x: U_x \to \R^n$ and $\psi_y: W_y \to
\R^n$. Then there is a correspondence: 
\[
\Coin(f,g,U_x) \leftrightarrow
\Coin(\psi_y \circ f \circ \phi_x^{-1}, \psi_y \circ g \circ
\phi_x^{-1}, \R^n)
\]
We use this correspondence to leverage our earlier results on $\adm(\R^n,
\R^n)$.

\vspace{\parskip}
\begin{lem*}
If $(f,g,U)$ is nondegenerate, each coincidence point $x$ is isolated,
and 
\[ \ind(f,g,U) = \sum_{x \in \Coin(f,g,U)} \sign \det(dg_x - df_x). \]
\end{lem*}

The key step is to show that the correspondence via the charts
preserves the index. This requires the full normalization axiom.

\end{slide}

\begin{slide}
\begin{head}Maps $X \to Y$\end{head}
\vspace{\parskip}
\begin{lem*} Every triple in $\adm(X,Y)$ is admissably homotopic to a
  nondegenerate triple
\end{lem*}

Now to compute $\ind(f,g,U)$, first replace $(f,g,U)$ by a
nondegenerate triple. Then the index is simply the sum of the signs of
determinants of derivatives.

By the homotopy axiom, the above is independant of our choice of
nondegenerate triple, and so uniquely defines the index.
\end{slide}

\end{document}







