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\title[Fixed points]{85 years of Nielsen theory: Fixed Points}
\author[Staecker]{P. Christopher Staecker}
\institute[Fairfield U.]{Fairfield University, Fairfield CT}
\date[]{Nielsen Theory and Related Topics 2013}

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\frame{
Thanks
\fpause
Who my talk is for.
\fpause
Please ask questions.
\fpause
Videos will be on YouTube.\pause\ (tell your friends)
}

\frame{{Some good books}
There have been a few books about Nielsen theory:
\begin{itemize}
\item $\vcenter{\hbox{\includegraphics[width=50px]{brownbook.jpg}}}$ Bob Brown, \emph{The Lefschetz Fixed Point Theorem}, 1977. \pause
\item $\vcenter{\hbox{\includegraphics[width=50px]{jiangbook}}}$ Boju Jiang, \emph{Lectures on Nielsen Fixed Point Theory}, 1981. 
\end{itemize}

}

\frame{{Some more good books}
\begin{itemize}
\item $\vcenter{\hbox{\includegraphics[width=50px]{kiangbook.jpg}}}$ Tsai-han Kiang, \emph{The Theory of Fixed Point Classes}, 1989. \pause
\item $\vcenter{\hbox{\includegraphics[width=50px]{jmbook.jpg}}}$ Jerzy Jeziersky, Wac\l{}aw Marzantowicz, \emph{Homotopy Methods in Topological Fixed and Periodic Points Theory}, 2006
\end{itemize}
}

\frame{
We'll start longer than 85 years ago: 
\begin{thm}{\bf Brouwer Fixed Point Theorem, before 1912}
Any selfmap of the disc has a ``fixed point'': some $x$ with $f(x)=x$.
\end{thm}

\fpause
What about spaces other than the disc?
\fpause
A more general result was obtained by Lefschetz for any selfmap of a compact polyhedron $X$.
\fpause
First define the \emph{Lefschetz number}: 
\[ L(f) = \sum_{q=0}^{\dim X} (-1)^q \tr(f_{*q}:H_q(X) \to H_q(X)) \]
\pause
This is a homotopy invariant, and it turns out is always an integer.
}

\frame{
Now, (about) 85 years ago:\pause
\begin{thm}{\bf Lefschetz Fixed Point Theorem, 1926}
For a selfmap, if $L(f)\neq 0$, then $f$ has a fixed point.
\end{thm}

\fpause

Lefschetz proved it for compact manifolds, Hopf for compact polyhedra soon after.
\fpause

If our space $X$ is the disc, then $L(f)$ is easy to compute:\pause
\[ L(f) = \sum_{q=0}^{\dim X} (-1)^q \tr(f_{*q}:H_q(X) \to H_q(X)) \]\pause
All $f_{*q}$ are zero except $f_{*0}$ which is identity. So $L(f)=1$, so Lefschetz's theorem implies Brouwer's.
}

\frame{
Why the Lefschetz number?
\[ L(f) = \sum_{q=0}^{\dim X} (-1)^q \tr(f_{*q}:H_q(X) \to H_q(X)) \]
\pause
Let's assume we have a simplicial map on a compact polyhedron, with $L(f)\neq 0$. \fpause
Then there is a nonzero trace $f_{*q}$\pause, and so there is a simplex $s$ with $f_q(s) = s$. \fpause
But $s$ is topologically a $q$-disc, and so there is a fixed point in $s$ by Brouwer.
}

\frame{
This more or less proves Lefschetz's theorem for simplicial maps.
\fpause
For nonsimplicial maps, use the Simplicial Approximation Theorem. 
\fpause
Why the alternating sign in $L(f)$? 
\fpause
I was using simplices instead of homology classes- actually my argument was for the chain sum:
\[ \sum_{q=0}^{\dim X} (-1)^q \tr(f_{*q}:C_q(X) \to C_q(X)) \]
\pause
But this equals $L(f)$ by the Hopf Trace Theorem- \pause\ the alternating sign is necessary to make this work.
}

\frame{
$L(f)$ is a homotopy invariant ``algebraic'' count of the fixed points of $f$. \pause Like counting fixed points ``with multiplicity''.
\fpause
Not a perfect count of the actual number of fixed points:
\fpause
It's possible to have $L(f)=2$ with only one ``double'' fixed point. 
\fpause
Also possible to have $L(f)=0$ even though there are two fixed points with ``opposite signs''. \pause (So generally the converse of Lefschez FPT is not true).
}

\frame{
In fact this can be made a bit more formal: \pause\ $L(f)$ can be ``localized'' to a specific integer ``multiplicity'' for each fixed point.
\fpause
This is called the \emph{fixed point index}.
\fpause
In dimension 1, it's easy to define the index:
\fpause
Fixed points are intersections of the graph of $f$ and the diagonal $\Delta$.
}

\frame{
\[ 
\only<1>{\includegraphics[width=200px]{fixpt1}}%
\only<2>{\includegraphics[width=200px]{fixpt2}}%
\only<3->{\includegraphics[width=200px]{fixpt3}}%
\]
\pause\pause\pause
The index depends on the \emph{slope} as $f$ passes through $\Delta$.
}

\frame{
Specifically, when this is nonzero, 
\[ \ind(f,x) = \sign(1 - df_x) \]
\pause
Turns out, a similar definition works in higher dimensions.
\fpause
If $f$ is differentiable, and has isolated fixed points which are ``transverse'', then \pause
\[ \ind(f,x) = \sign \det(I - df_x) \]
where $I$ is the identity matrix.
\fpause
There is a much more general homological definition of the index for nonsmooth maps, and nonisolated fixed points. 
\fpause
Axiomatic definitions exist too.
}

\frame{
When $x$ is an isolated fixed point, $\ind(f,x) \in \Z$ satisfies:
\[ L(f) = \sum_{x\in \Fix(f)} \ind(f,x). \]
\pause
This is the Lefschetz-Hopf theorem (Hopf, 1929).
\fpause
So $\ind(f,x)$ sums up to $L(f)$ which is a homotopy invariant. 
\fpause
How does a homotopy affect the individual fixed point indices?
\fpause
When we change $f$ by a small homotopy, the fixed points move around by a small amount\pause, and the indices are preserved.
}

\frame{
When we change $f$ by a small homotopy, the fixed points move around by a small amount, and the indices are preserved.
\fpause
\[ 
\vcenter{\hbox{\includegraphics[width=100px]{graphhtp1}}} \pause \to 
\vcenter{\hbox{\includegraphics[width=100px]{graphhtp2}}} \pause \to
\vcenter{\hbox{\includegraphics[width=100px]{graphhtp3}}} \] \fpause
When fixed points combine, the indices add. \fpause
When can fixed points be combined?
}

\frame{
Let's talk about the actual number of fixed points.
\fpause
Specifically: How many fixed points can be achieved by changing the map by homotopy?
\fpause
Easy: we can always change by homotopy to \emph{increase} the number of fixed points
\fpause
\[ 
\vcenter{\hbox{\includegraphics[width=100px]{maxfp1}}} \pause \to
\vcenter{\hbox{\includegraphics[width=100px]{maxfp2}}}  \]
}

\frame{
What about minimizing the number of fixed points by homotopy?
\pause \[ MF(f) = \min\{ \#\Fix(f') \mid f' \htp f \} \]
\pause
This is much harder, and this is what Nielsen Theory is about.
\fpause
Nielsen's idea (for torus homeomorphisms in 1913, surfaces in 1927, about 85 years ago): group the fixed points into classes.
\fpause
The classes are meant to group those fixed points which can be combined by homotopies. \pause The number of such classes will be a lower bound for the minimal number of fixed points.
}

\frame{
The basic theory of fixed point classes is from Nielsen (1927)\pause, much formalization and basic properties proved by Reidemeister \& Wecken (1930s \& 1940s). 
\fpause
Let $\lift X$ be the universal covering space with projection $p:\lift X \to X$, and consider the fixed point sets of the liftings of $f$. 
\fpause
If we choose a ``reference lift'' $\lift f$, then any other lift is $\gamma \lift f$ for various $\gamma \in \pi = \pi_1(X)$. 
\fpause
It's easy to show that 
\[ \Fix(f) = \bigcup_{\gamma \in \pi} p( \Fix(\gamma \lift f)) \]
\pause
These sets in the union are the fixed point classes.
}

\frame{
So $x, y \in \Fix(f)$ are in the same fixed point class (or \emph{Nielsen class}) when they both come from fixed points of the same lifting.
\fpause 
Nielsen saw that this is a necessary condition for fixed points to be combined by a homotopy.
}

\frame{
An equivalent definition of the fixed point classes, also by Nielsen: \fpause $x, y$ are in the same Nielsen class if and only if there is a path $\alpha$ from $x$ to $y$ with $\alpha \htp f(\alpha)$.\pause
\[ 
\only<3>{\includegraphics[width=150px]{pathhtp1}}
\only<4->{\includegraphics[width=150px]{pathhtp2}}
\]
\pause
\pause
Pretty clear that this is necessary for $x$ and $y$ to be combined.
}

\frame{
The definition with liftings is a bit easier to work with:
\[ \Fix(f) = \bigcup_{\gamma \in \pi} p( \Fix(\gamma \lift f)) \]
\pause
This union is not disjoint, however. \pause But it's not too hard to decide when the sets on the right intersect.
\fpause
We say $\gamma, \sigma \in \pi_1(X)$ are in the same \emph{Reidemeister class} or \emph{twisted-conjugacy class} when:
\[ \exists z \in \pi \text{ such that } \gamma = z^{-1} \sigma f_\#(z) \]
where $f_\#$ is the induced map in $\pi_1$. 
\fpause
In this case write $[\gamma] = [\sigma]$.
}

\frame{
Not hard to prove: in the union
\[ \Fix(f) = \bigcup_{\gamma \in \pi} p( \Fix(\gamma \lift f)) \]
\pause
$p\Fix(\gamma \lift f) = p\Fix(\sigma \lift f)$ iff $[\gamma]=[ \sigma]$, \pause and when $[\gamma]\neq [\sigma]$ we have $p\Fix(\gamma\lift f) \cap p\Fix(\sigma\lift f) = \emptyset$.
\fpause
So the Nielsen classes of fixed points are more or less in correspondence to the Reidemeister classes of $\pi_1$ elements.
\fpause
Actually some sets $\Fix(\gamma \lift f)$ may be empty, so really there's an inclusion:
\[ \{ \text{ Fixed point classes } \} \hookrightarrow \{ \text{ Reidemeister classes } \} \]
}

\frame{
The algebraic decision problem of twisted conjugacy in various groups is hotly studied\pause, even outside of Nielsen theory.

\pause
\begin{itemize}
\item Given $f_\#:G \to G$ and $g, h \in G$, is there an algorithm for deciding whether $[g]=[h]$?\pause\ ``The twisted conjugacy problem''\pause
\item Let $\Reid(f)$ be the set of Reidemeister classes in $G$. \pause Is $\Reid(f)$ finite or infinite?\pause
\item For which $G$ is $\Reid(f)$ always infinite when $f$ is an isomorphism?\fpause
This is called the $R_\infty$ property, lots of work now. (Nasybullov, Fel'shtyn, J. B. Lee)
\end{itemize}
\fpause
Lots of these become easier if we assume $f_\#$ is a group isomorphism.
}

\frame{
Back to $MF(f)$:
\fpause
The smallest possible number of fixed points would be achieved when each fixed point class has only 1 point. \pause Or zero points.
\fpause
How can we know if a class can be totally removed by a homotopy? \pause The fixed point index.
\fpause
A Nielsen class is called \emph{essential} if its total fixed point index sum is nonzero. \pause These ones cannot be made empty by homotopies.
\fpause 
The number of essential fixed point classes is called the \emph{Nielsen number} $N(f)$.
\fpause
Automatically
\[ N(f) \le MF(f). \]
}

\frame{
Let's do some simple examples. \pause Selfmaps on the circle.
\fpause
Since $N(f)$ is homotopy invariant, the only relevant information is the degree of our selfmap.
\fpause
Any degree $d$ map can be changed by homotopy to $f(z) = z^d$, which has $|1-d|$ fixed points.
\fpause
These fixed points each have the same index $\pm 1$, so $L(f) = \pm(1-d)$
}

\frame{
What about the Reidemeister classes?
\fpause
For the circle, $\pi_1 = \Z$. \pause When are two numbers twisted-conjugate?
\fpause
For $x,y \in \Z$, we have $[x]=[y]$ iff there is some $z$ with 
\[ x = -z + y + f_\#(z) \pause = -z + y + dz \pause= y - (1 - d)z. \]
\fpause
So $[x]=[y]$ iff $x = y \mod (1-d)$.
\fpause
So $\Reid(f) = \Z_{|1-d|}$.
}

\frame{
Recall we had $|1-d|$ fixed points of the same index, and it's easy to show that they all have different Reidemeister classes.
\fpause
So we have $N(f) = |1-d|$, and also $MF(f) = |1-d|$ since $f(z)=z^d$ has $|1-d|$ fixed points.
\fpause
So the Nielsen theory of the circle is easy.
}

\frame{
What about tori?
\fpause
\[\includegraphics[width=200px]{torinielsen}\]
\pause
\[\text{Tori Nielsen}\]
}

\frame{
There is a similar formula for maps on tori by Brooks, Brown, Pak, Taylor (1975).
\fpause
We view the $n$-torus as $\R^n / \Z^n$.
\fpause
A map on the $n$-torus can be ``linearized'' by homotopy into a $n\times n$ matrix $A$ with entries in $\Z$.
\fpause
They showed that this linear map has $|\det(I - A)|$ fixed points.
\fpause
Further, these are all in different classes, and they all have the same index $\pm 1$.
}

\frame{
So on tori, we have $L(f) = \pm \det(I-A)$ and $N(f) = |\det(I-A)|$.
\fpause
Some similar results are possible on nilmanifolds.
\fpause
These are quotients of a nilpotent Lie group by a discrete set. \pause (so tori are nilmanifolds)
\fpause
Nilmanifolds allow a similar linearization of maps, and good formulas for Nielsen theory result. (Anosov, Fadell \& Husseini 1985)
}

\frame{
The results on nilmanifolds and solvmanifolds use some general properties of Nielsen theory on fibrations.
\fpause
Consider a fibration $F \to E \to B$ and a fiber map $f:E\to E$ with
\[ \begin{CD}
F @>>> E @>>> B \\
@V\bar fVV @VfVV @Vf_bVV \\
F @>>> E @>>> B
\end{CD} \]
\fpause
Brown (1967) looked at this setting. \pause When is there a product formula like
\[ N(f) \stackrel?= N(\bar f)N(f_b) \]
}

\frame{
For cartesian products, this ``naive product formula'' was already known for a long time for $L(f)$ and $\ind(f,x)$. \pause Easy to do it for $N(f)$.
\fpause
For general fibrations, the product formula is not always satisfied. 
\fpause
In 1981 You gave necessary and sufficient conditions for the formula to hold. 
\fpause
The conditions are a bit complicated\pause, but fibrations over tori behave very nicely.
\fpause
See Heath's talk for more on fiber (fibre) methods.
}

\frame{
A major theme in Nielsen theory has been: \pause Choose a category of spaces and selfmaps, and try to compute the Nielsen number.
\fpause
Surfaces have been a major topic. \pause \talkbox{Hart mini-lecture, Gon\c{c}alves later today}
\fpause
The geometrization theorem has allowed new techniques on 3-manifolds according to their geometries. \pause \talkbox{Wong, later today}
}

\frame{
Methods for computation are reckoned to be successful when $N(f)$ can be reduced to calculations of $L(f)$ or algebraic calculations of the Reidemeister classes. 
\fpause
In many cases $N(f)$ can be reduced to $L(f)$ and $R(f) = \#\Reid(f)$. \pause\ (The Reidemeister number)
\fpause
This is true for a large class of spaces called Jiang spaces, which include:\pause
\begin{itemize} 
\item Lie groups\pause, topological groups\pause, H-spaces\pause
\item generalized lens spaces\pause
\item simply connected spaces\pause
\item quotients of Lie groups by finite subgroups\fpause
\end{itemize}
Unfortunately Jiang spaces all have $\pi_1$ abelian.
}

\frame{
Some other spaces are still ``weakly Jiang'', which means that when $L(f)=0$ we have $N(f)=0$, and otherwise $N(f) = R(f)$.
\fpause
In these cases, the geometry of $\Fix(f)$ is very closely tied to the algebra of $\Reid(f)$. 
\fpause
For some spaces this is known to be impossible. 
\fpause
Any space such that $\pi_1$ has $R_\infty$ property cannot be a weakly Jiang space. \pause (This isn't quite true)
}

\frame{
So far we have $L(f)$ from 1926, and $N(f)$ from 1927, the index and Lefschetz-Hopf theorem in 1929. \pause This is the beginning of the ``85 years''.
\fpause
These two invariants were combined in a clever way by Reidemeister and Wecken:
\fpause
Let's do the Lefschetz trace:
\[ \sum_{q} (-1)^q \tr(f_q:C_q(X) \to C_q(X)), \]
\pause
but do it in $\lift X$ instead of $X$.
}
\frame{
$\lift X$ has the same simplicial structure as $X$, only every simplex has copies parameterized by $\pi_1$.
\fpause
So we can consider $C_q(\lift X)$ as the same as $C_q(X)$, only allowing coefficients from $\Z\pi$ instead of $\Z$.
\fpause
Then we can write $\lift f_q:C_q(\lift X) \to C_q(\lift X)$ as a matrix with entries in $\Z\pi$\pause, and we can do
\[ \tr(\lift f_q:C_q(\lift X) \to C_q(\lift X)) \in \Z\pi. \]
}

\frame{
Reidemeister defined:
\[ RT(\lift f) = \rho(\sum_{q} (-1)^q \tr(\lift f_q:C_q(\lift X) \to C_q(\lift X))) \]
now called the \emph{Reidemeister trace} or \emph{generalized Lefschetz number}.
\fpause
Here $\rho: \Z\pi \to \Z\Reid(f)$ puts group elements into Reidemeister classes.
}

\frame{
In an example, this $RT(\lift f)$ would look something like:
\[ RT(\lift f) = 2[\gamma] - 3[\sigma] + 1[e], \]
\pause
Which indicates the fixed point class with Reidemeister class $[\gamma]$ has index sum 2, \pause the one with Reidemeister class $[\sigma]$ has index sum $-3$, \pause the one with class $[e]$ has index sum 1, \pause and all others have index 0. 
\fpause
Thus $L(f) = 2-3+1=0$, and $N(f) = 3$. 
\fpause
In general, the sum of the coefficients in $R(\lift f)$ is $L(f)$, and the number of nonzero terms is $N(f)$.
\fpause
The trace formula often makes this easily computable. \pause\ (except for the $\rho$ part)
}

\frame{
Let's talk about 
\[ N(f) \le MF(f) \]
\pause
When are they equal?
\fpause
Nielsen's original setting (1920s) was surfaces homeomorphisms, in which it's not clear if they are always equal\pause, though Nielsen seems to have believed that they were.
\fpause
Wecken showed (1940s) that $N(f) = MF(f)$ for compact manifolds of dimension $\neq 2$.
\fpause
This is called the Wecken Theorem.
\fpause
Dimension 1 is easy, \pause for dimension $\ge 3$ there is enough ``room'' to deform $f(X)$ so that it intersects the diagonal $\Delta$ once for each essential class.
}

\frame{
What about for polyhedra? 
\fpause
Shi (1966) proved that $N(f) = MF(f)$ for polyhedra with dimension $\ge 3$ and no local separating points.
\fpause
Jiang (1979) proved that $N(f)= MF(f)$ for any polyhedron without local separating points which is not a surface.
\fpause
What about surfaces?
}

\frame{
The Wecken issue for surfaces was also resolved by Jiang in early 1980s. 
\fpause
Until this time there was no known example with $N(f) \neq MF(f)$ on a surface.
\fpause
Jiang constructed a map on the pants surface with $N(f) = 0$ and $MF(f) = 2$. 
\fpause
In this example there are 2 fixed points in the same class of index $+1$ and $-1$, so the class is not essential\pause, but Jiang showed that $MF(f) = 2$.
\fpause
The paper is \emph{Fixed points and braids} (1984 \& 1985).
}

\frame{
A very vague idea of why braids are important:
\fpause
Consider a map with two fixed points, and we change the map by homotopy.
\fpause
Let's use the pants surface $P$, and the homotopy itself is a map on $P\times [0,1]$. 
}

\frame{
This looks like:
\vfill
\[ 
\only<1>{\includegraphics[width=200px]{braid1} }
\only<2>{\includegraphics[width=200px]{braid2} }
\]
\pause
}

\frame{
\[ \includegraphics[width=100px]{braid2}\]
\vfill
This thing is called a ``two strand braid on $P$''.
\fpause
There is an algebraic theory for surface braids\pause, using the ``surface braid groups''.
}

\frame{
\[ \includegraphics[width=100px]{braid2}\]
\vfill
Surface braid groups have finite presentations with relators like in the classical braid groups\pause, plus some relators depending on the topology of the surface.
\fpause
Jiang shows that in his example, removing the two fixed points would require an algebraic formula to hold in the surface braid group. \fpause Then he proves using the relations that this would be impossible.
}

\frame{
Braid groups now play a big role in Nielsen theory (Ferrario's talk)
\fpause
Jiang showed that his example can be embedded to make non-Wecken maps on any surface of negative Euler characteristic.
\fpause
Several people asked whether $N(f)$ can be arbitrarily distant from $MF(f)$. \pause Kelly showed that the difference can be arbitrarily large for any hyperbolic surface.
}

\frame{
Some related questions:\pause
\begin{itemize}
\item If we choose a surface selfmap ``at random'', is it likely that $N(f) = MF(f)$? (S.W. Kim's talk)\pause
\item If $X$ is a smooth manifold, can $N(f) = MF(f)$ be realized by a smooth map? (Jezierski's talk)
\end{itemize}

\fpause

By the way, the role of smoothness is another theme in several people's work.

\fpause
Does it matter when we restrict to smooth maps? \pause (for the original map, or the intermediate maps in a homotopy, etc)

\fpause
Sometimes it does, sometimes it doesn't. (Khamsemanan's talk)
}

\frame{
That's all for now!
}

\end{document}