%t\documentclass[trans]{beamer}
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\usepackage{beamerthemesplit}
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\usepackage{verbatim}

\usepackage[all,color]{xy}

\useoutertheme{infolines} 

\input{nielsenmacros}

\newcommand{\fpause}{\pause\vfill}

\title[Nielsen equalizer theory]{
Nielsen equalizer theory}
\author[Staecker]{P. Christopher Staecker}
\institute[Fairfield U.]{Fairfield University, Fairfield CT}
\date[]{Capitol Normal University, Beijing China, June 24, 2011}

\newcommand{\aside}{\textcolor{red}}

\begin{document}
\frame{\titlepage}

\frame{
Given a set of maps: $f_1,\dots, f_k\colon X \to Y$, the equalizer set is
\[ \Eq(f_1,\dots,f_k) = \{ x\in X \mid f_1(x) = \dots = f_k(x) \} \]
The points where all the functions agree.

\fpause

For 2 maps, this is the coincidence set.

\fpause
In fact,
\[ \Eq(f_1,\dots,f_k) = \bigcap_{i,j} \Coin(f_i, f_j) \pause = \bigcap_{i}\Coin(f_1,f_i). \]
}

\frame{
$X$ and $Y$ will be closed manifolds.

\fpause

When $X$ and $Y$ are the same dimension, any two maps can be made to have finite coincidence set.

\fpause

\begin{prop}
When $X$ and $Y$ have the same dimension, given $f_1,\dots,f_k\colon X \to Y$ with $k>2$, we can change the maps by homotopy to be equalizer free.
\end{prop}

\fpause
Make $\Coin(f_1,f_i)$ finite for each $i$, then arrange for these sets to be distinct.
}

\frame{
So codimension-zero equalizer theory is uninteresting up to homotopy.
\fpause
To get something interesting we'll increase the dimension in the domain.
\fpause
Then the coincidence sets $\Coin(f_1,f_i)$ will be submanifolds of $X$, and it's possible that their intersections would be essentially nonempty.
}

\frame{
An example: three maps $f,g,h: T^2 \to S^2$ given by $(1\times 2)$ matrices:
\pause
\[ f = (3\, 1), \quad g = (0\, 2)\quad h=(-1\,-1) \]
\fpause
We can compute the coincidence sets:
\fpause
$\Coin(f,g)$ is points $(x,y)$ with $3x+y=0x+2y$ mod $\Z^2$, which is the ``line'' $y = 3x$ mod $\Z^2$. 
}

\frame{
\[ f = (3\, 1), \quad g = (0\, 2)\quad h=(-1\,-1) \]

We can draw the coincidence sets: 
\only<1>{
\[ 
\newcommand{\xwidth}{40}
\newcommand{\axline}{-}
\begin{xy}
% 4 corners
(0,0)="ll";
(0,\xwidth) = "ul";
(\xwidth,0) = "lr";
(\xwidth,\xwidth) = "ur";
% axes
{\ar@{-} "ll";"ul"};
{\ar@{-} "ul";"ur"};
{\ar@{-} "ur";"lr"};
{\ar@{-} "lr";"ll"};
% C_fg
"ll"; "lr"; **\dir{} ?(.333) = "ltt"; ?(.666) = "lt";
"ul"; "ur"; **\dir{} ?(.333) = "utt"; ?(.666) = "ut";
{\color{red}\ar@{-} "ll";"ut"};
{\ar@{-} "lt";"utt"};
{\ar@{-} "ltt";"ur"\color{black}};
(50,30)*{\color{red}\Coin(f,g)};
\end{xy}
\quad 
\]
}
\color{black}
%
\only<2>{
\[ 
\newcommand{\xwidth}{40}
\newcommand{\axline}{-}
\begin{xy}
% 4 corners
(0,0)="ll";
(0,\xwidth) = "ul";
(\xwidth,0) = "lr";
(\xwidth,\xwidth) = "ur";
% axes
{\ar@{-} "ll";"ul"};
{\ar@{-} "ul";"ur"};
{\ar@{-} "ur";"lr"};
{\ar@{-} "lr";"ll"};
% C_fg
"ll"; "lr"; **\dir{} ?(.333) = "ltt"; ?(.666) = "lt";
"ul"; "ur"; **\dir{} ?(.333) = "utt"; ?(.666) = "ut";
{\color{red}\ar@{-} "ll";"ut"};
{\ar@{-} "lt";"utt"};
{\ar@{-} "ltt";"ur"};
% C_gh
"ll"; "ul"; **\dir{} ?(.333)*{} = "ltt"; ?(.666)*{} = "lt";
"lr"; "ur"; **\dir{} ?(.333)*{} = "rtt"; ?(.666)*{} = "rt";
{\color{blue}\ar@{-} "ul";"rtt"};
{\ar@{-} "ltt";"rt"};
{\ar@{-} "lt";"lr"\color{black}};
% C_fh
"ll";"ul"; **\dir{} ?(.5) = "lh";
"lr";"ur"; **\dir{} ?(.5) = "rh";
"ul";"ur"; **\dir{} ?(.25) = "ufff"; ?(.5)="uff"; ?(.75)="uf";
"ll";"lr"; **\dir{} ?(.25) = "lfff"; ?(.5)="lff"; ?(.75)="lf";
{\color{black}\ar@{-} "lh";"lf"};
{\ar@{-} "ul";"lff"};
{\ar@{-} "uf";"lfff"};
{\ar@{-} "uff";"lr"};
{\ar@{-} "ufff";"rh"\color{black}};
(50,30)*{\color{red}\Coin(f,g)};
(50,20)*{\color{blue}\Coin(g,h)};
(50,10)*{\color{black}\Coin(f,h)\color{black}};
\end{xy}
\quad 
\]
}
\color{black}
\pause

We have 10 isolated equalizer points.
}

\frame{
\color{black}
In fact we'll show that in this example any maps homotopic to $f,g,h$ must 
have at least 10 equalizers.
\fpause
We'll define a Nielsen number (easy matrix formula for tori), and in this case
\[ N(f,g,h) = 10. \]
}

\frame{
\frametitle{Oops!}
On Tuesday, Peter Wong suggested I have a look at: 
\fpause

Dobre\'nko, Kucharski, \emph{On the generalization of the Nielsen number}, \emph{Fundamenta Mathematicae} {\bf 134}:1--14, 1990.

\fpause
They give a very general theory for maps $f:X \to Y$ and a subset $B\subset Y$, and a Nielsen theory for counting $\#f^{-1}(B)$. 
\fpause
In various special cases, in appropriate codimensional settings, this gives:
\begin{itemize}
\item Nielsen fixed point theory ($B=\Delta$) \pause
\item root theory ($B=pt$) \pause
\item coincidence theory of $k$ maps ($B = \Delta \subset Y^k$)
\end{itemize}
}

\frame{
The DK theory treats only the smooth orientable case, \pause and they don't discuss positive codimension coincidence theory. \fpause But the theory is otherwise the same as ours, though developed completely differently.

\fpause
My lesson learned:
\fpause
\begin{center} \it talk to Peter more often \end{center}
\fpause
or\dots
\fpause
\begin{center} \it never talk to Peter \end{center}
\fpause
With apologies to Dobre\'nko and Kucharski, let's continue.
}

\frame{
Our theory is based on a simple trick:
\fpause
Let $f_1,\dots f_k:X \to Y$ with $\dim X = (k-1)n$ and $\dim Y = n$. 
\fpause
Let $F,G:X\to Y^{k-1}$ be maps (codimension 0) given by
\[ F(x) = (f_1(x), \dots, f_1(x)), \quad G(x) = (f_2(x), \dots, f_k(x)). \]

\fpause
Then $F,G$ are maps of manifolds of the same dimension, and 
\[ \Eq(f_1,\dots,f_k) = \Coin(F,G). \]
}

\frame{
\[ F(x) = (f_1(x), \dots, f_1(x)), \quad G(x) = (f_2(x), \dots, f_k(x)) \]
\[ \Eq(f_1,\dots,f_k) = \Coin(F,G) \]

This connection is deep enough to build our whole theory, in the case where $\dim X = (k-1)n$ and $\dim Y = n$.
\fpause
\begin{thm}
With these dimensions, the maps can be changed by homotopy so that $\Eq(f_1,\dots,f_k)$ is finite.
\end{thm}
\fpause
Since $F,G$ are codimension zero maps, we can change them to have finite coincidence set. \pause Actually we need only change $G$. \pause Thus we obtain
\[ F = (f_1,\dots, f_1) \quad G' = (f_2', \dots, f_k') \]
\pause with $\Coin(F,G') = \Eq(f_1,f'_2,\dots,f'_k)$ finite.
}

\frame{
We get an equalizer index (or semi-index, or $\Z \oplus \Z_2$ index): 
\fpause
Define $\ind(f_1,\dots,f_k,U) = \ind(F,G,U)$. 
\fpause
Easy to show that this is homotopy invariant, and has appropriate other properties.
}

\frame{
For differentiable maps we can compute the index at a point using derivative maps: 
\begin{eqnarray*} 
\ind(f_1,\dots,f_k, x) &=& \ind(F,G,x) \\ 
\pause &=& \sign \det(dF - dG) \\ 
\pause &=& \sign \det \left( \begin{bmatrix} df_1 - df_2\\ \vdots \\ df_1 - df_k\end{bmatrix} \right)
\end{eqnarray*}
}

\frame{
The other ingredient to the theory is the Nielsen equalizer classes.
\fpause
Again we can define the equalizer classes to be the coincidence classes of $F$ and $G$. 
\fpause
Equivalently, $x,x'\in \Eq(f_1,\dots,f_k)$ are in the same class when 
\[ x,x' \in p\Eq(\lift f_1,\alpha_2\lift f_2,\dots,\alpha_k\lift f_k) \]
for $\alpha_i \in \pi_1(Y)$. 
}

\frame{
This is equivalent to making Reidemeister classes $\pi_1(Y)^{k-1}/\sim$.
\fpause
We say $(\alpha_2,\dots \alpha_k) \sim (\beta_2,\dots,\beta_k)$ if and only if there is $z \in \pi_1(X)$ with
\[ \beta_i = \phi_1(z)\alpha_i\phi_i(z)^{-1} \text{ for all $i$} \]
}

\frame{
Also equivalent in terms of paths: 
\fpause
$x,x' \in \Eq(f_1,\dots,f_k)$ are in the same class when there is a path $\gamma$ from $x$ to $x'$ with
\[ f_i(\gamma) \htp f_1(\gamma) \text{ for all $i$} \]
}

\frame{
A class is essential when its index (or semi-index) is nonzero, and the number of such classes is $N(f_1,\dots,f_k)$. 
\fpause
We also get $R(f_1,\dots f_k)$ and $L(f_1,\dots,f_k)$ in the usual way.\fpause 
Also we have a ``minimal equalizer number'' with 
\[ \ME(f_1,\dots,f_k) \le N(f_1,\dots,f_k), \]
and these are equal when $(k-1)n \neq 2$. 
\fpause For more than 2 maps, this always holds except 3 maps on dimensions $2\to 1$. 
}

\frame{
We can get all the usual results. 
\fpause
\begin{thm}
If $Y$ is a Jiang space, then all nonempty equalizer classes have the same index.\end{thm}
\fpause
\begin{thm}
If $f_1,\dots,f_k:T^{(k-1)n} \to T^n$ by matrices $A_1,\dots,A_k$, then
\[ N(f_1,\dots,f_k) = \abs \det\begin{bmatrix}A_1 - A_2 \\ \vdots \\ A_1 - A_k \end{bmatrix} \]
\end{thm}
}

\frame{
Our old example: $f,g,h\colon T^2 \to S^1$ by
\[ f = (3\, 1), \quad g = (0\, 2)\quad h=(-1\,-1). \]
\fpause
Then we have 
\[ N(f,g,h) = \abs \det \begin{bmatrix} 3 & -1 \\ 4 & 2 \end{bmatrix} \pause = 10. \]
}

\frame{
Hopefully, this theory is useful for coincidence theory with positive codimensions:
\fpause
The set $\Eq(f_1,\dots,f_k)$ includes a lot of information about the coincidence sets.
\fpause
For each $i,j$ we have
\[ \Eq(\alpha_1 \lift f_1, \alpha_2\lift f_2, \dots \alpha_k\lift f_k) \subset \Coin(\alpha_i\lift f_i, \alpha_j\lift f_j) \]
\pause So every equalizer class is a subset of a coincidence class. 
\fpause
\begin{thm}
Any coincidence class containing an essential equalizer class must be geometrically essential.
\end{thm}
}

\frame{
Our old example:
\[ 
\newcommand{\xwidth}{40}
\newcommand{\axline}{-}
\begin{xy}
% 4 corners
(0,0)="ll";
(0,\xwidth) = "ul";
(\xwidth,0) = "lr";
(\xwidth,\xwidth) = "ur";
% axes
{\ar@{-} "ll";"ul"};
{\ar@{-} "ul";"ur"};
{\ar@{-} "ur";"lr"};
{\ar@{-} "lr";"ll"};
% C_fg
"ll"; "lr"; **\dir{} ?(.333) = "ltt"; ?(.666) = "lt";
"ul"; "ur"; **\dir{} ?(.333) = "utt"; ?(.666) = "ut";
{\color{red}\ar@{-} "ll";"ut"};
{\ar@{-} "lt";"utt"};
{\ar@{-} "ltt";"ur"};
% C_gh
"ll"; "ul"; **\dir{} ?(.333)*{} = "ltt"; ?(.666)*{} = "lt";
"lr"; "ur"; **\dir{} ?(.333)*{} = "rtt"; ?(.666)*{} = "rt";
{\color{blue}\ar@{-} "ul";"rtt"};
{\ar@{-} "ltt";"rt"};
{\ar@{-} "lt";"lr"\color{black}};
% C_fh
"ll";"ul"; **\dir{} ?(.5) = "lh";
"lr";"ur"; **\dir{} ?(.5) = "rh";
"ul";"ur"; **\dir{} ?(.25) = "ufff"; ?(.5)="uff"; ?(.75)="uf";
"ll";"lr"; **\dir{} ?(.25) = "lfff"; ?(.5)="lff"; ?(.75)="lf";
{\color{black}\ar@{-} "lh";"lf"};
{\ar@{-} "ul";"lff"};
{\ar@{-} "uf";"lfff"};
{\ar@{-} "uff";"lr"};
{\ar@{-} "ufff";"rh"\color{black}};
(50,30)*{\color{red}\Coin(f,g)};
(50,20)*{\color{blue}\Coin(g,h)};
(50,10)*{\color{black}\Coin(f,h)\color{black}};
\end{xy}
\quad 
\]

\color{black}
\fpause
$\Coin(f,h)$ has 2 components which are 2 different classes. \pause Are these classes geometrically essential? \pause Each one contains essential equalizer points\dots \pause so {\bf yes} they are essential. 
\fpause
So $N(f,h) = 2$ in this case. \pause Similarly $N(f,g) = N(g,h) = 1$. 
}

\frame{
In positive dimension coincidence theory, one major problem is judging essentiality.
\fpause
There is no coincidence index, but we can use the equalizer index if we create extra maps.
\fpause
Start with two maps $f_1,f_2$, and a coincidence class $C$ that we want to show is essential.
\fpause
We invent a set of maps $f_3,\dots,f_k$ and show that $C$ contains equalizer points of nonzero index.
\fpause
This only works for maps $f_1,f_2:X\to Y$ with $\dim Y = n$ and $\dim X = (k-1)n$ for some $k$. \pause ($\dim X$ must be a multiple of $\dim Y$)
}

\frame{
But even when $\dim X$ isn't a multiple of $\dim Y$, maybe we can still make it work. 
\fpause
Take two maps $f_1,f_2:T^7 \to T^2$ with matrices $A_1, A_2$, and assume that $A_2-A_1$ has rank 2. 
\fpause
Jezierski, \emph{The Nielsen coincidence number of maps into tori}, \emph{Quaestiones Mathematicae}, 2001

gives a method for finding $N(f_1,f_2)$ by observing that they restrict to maps $T^2 \to T^2$, and this restriction respects the Nielsen number.
\fpause
So Jezierski \emph{decreases} the domain dimension to get codimension 0.
\fpause
This only works because $T^2 \subset T^7$.
}

\frame{
Take two maps $f_1,f_2:T^7 \to T^2$ with matrices $A_1, A_2$, and assume that $A_2-A_1$ has rank 2. 
\fpause
We do the opposite:
\fpause
\emph{Increase} the domain dimension: \pause let $\bar f_1,\bar f_2:T^8 \to T^2$ by adding columns of 0s to $A_1,A_2$.
\fpause
Not hard to show that $N(f_1,f_2) = N(\bar f_1,\bar f_2)$.
\fpause
Let $B_1,B_2$ be matrices of $\bar f_1,\bar f_2$, and $B_2-B_1$ still has rank 2.
}
\frame{
So we can invent matrices $B_3,\dots B_5$ with
\[ \begin{bmatrix} B_2 - B_1 \\ \vdots \\ B_5 - B_1 \end{bmatrix} \]
of full rank (8).
\fpause
Thus $(\bar f_1,\bar f_2, g_3,\dots,g_5)$ has essential equalizer classes and so $N(f_1,f_2) = N(\bar f_1,\bar f_2) \neq 0$
\fpause
Hopefully this trick can be used elsewhere when we need to prove that coincidence classes are essential.
}

\frame{
Thank you!
\vfill
Paper at arxiv: ``Nielsen Equalizer Theory'', and in \emph{Topology and its applications}
}
\end{document}