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\title[Axioms for $L(f)$ as a valuation]{Axioms for the Lefschetz number as a lattice valuation}
\author[Staecker]{P. Christopher Staecker}
\institute[Fairfield U.]{Fairfield University, Fairfield CT}
\date[]{Nielsen Theory and Related Topics 2013}

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\begin{document}

\frame{\titlepage}

\frame{
Axioms for the Lefschetz number and fixed point index have been around for a while. \pause\ A few major axiomatizations:

\fpause

{\bf O'Neill (1953)} Fixed point index for continuous maps on compact polyhedra.

\fpause

{\bf Furi, Pera, \& Spadini (2004)} Fixed point index for continuous maps on differentiable ($C^1$) manifolds.

\fpause

{\bf  Arkowitz \& Brown  (2004)} Lefschetz number for continuous maps on compact polyhedra.
\pause

Based on axioms for $\chi(X)$ by Watts.
}

\frame{
{\bf Arkowitz \& Brown (2004)} for continuous maps on compact polyhedra:
\begin{thm}
The ``reduced Lefschetz number'' is the unique $\Z$-valued function satisfying:\pause
\begin{itemize}
\item (Homotopy) If $f \htp g$, then $\bar L(f) = \bar L(g)$\pause
\item (Cofibration) If $A \subset X$ is a subpolyhedron and $f$ induces maps on $A$ and $X/A$, then 
\[ \bar L(f) = \bar L(f_A) + \bar L(f_{X/A}) \] \par\pause\noindent
\item (Wedge of circles) If $f$ is a map on a wedge of $k$ circles, then
\[ \bar L(f) = -(\deg(f_1) + \dots + \deg(f_k)) \]  \par\pause\noindent
\item (Commutativity) $\bar L(f\circ g) = \bar L(g\circ f)$
\end{itemize}
\end{thm}
}

\frame{
{\bf Furi, Pera, \& Spadini (2004)} for continuous maps on $C^1$ manifolds:
\begin{thm}
The fixed point index is the unique $\R$-valued function satisfying:\pause
\begin{itemize}
\item (Homotopy) If $f \htp g$, then $\ind(f,U) = \ind(g,U)$\pause
\item (Disjoint additivity) If $\Fix(f)\cap U \subset A \sqcup B$, then
\[ \ind(f,U) = \ind(f,A) + \ind(f,B) \]  \par\pause\noindent
\item (Constant map) If $c$ is a constant map, then
\[ \ind(c,X) = 1 \]
\end{itemize}
\end{thm}

}

\begin{comment}
\frame{ 
{\bf O'Neill (1953)} for continuous maps on compact polyhedra:
\begin{thm} The fixed point index is the unique function satsifying:\pause
\begin{itemize}
\item (Small homotopy) There is a neighborhood $\mathcal A$ of $f$ in $X^X$ such that $\ind(f,U) = \ind(g,U)$ for $g \in \mathcal A$\pause
\item (Valuation) $\ind(f,U\cup V) = \ind(f,U) + \ind(f,V) - \ind(f,U\cap V)$\pause
\item (Localization) If $U \subset X$ is a subpolyhedron and $f(U)\subset U$ then $\ind(f,U) = \ind(f|_U, U)$\pause
\item (Commutativity) $\ind(f\circ g, U) = \ind(g \circ f, U)$\pause
\item (Small simplex) If $x$ is a simplex with $x \cup f(x)$ contained in a Euclidean open set, $\ind(f,x) = (-1)^{\dim x} \deg(f,\partial x)$.
\end{itemize}
\end{thm}

\fpause

(O'Neill omits the last one, but I'm not sure about that)
}

\end{comment}

\frame{
Each scheme has:
\begin{itemize}
\item Homotopy \pause
\item Addition (``cofibration'', ``additivity'') \pause
\item A basic computation (``wedge-of-circles'', ``constant map'') 
\end{itemize}

\fpause

Some also have the commutativity property.
}

\frame{
{Extensions}

The A\&B and FPS systems have been recently generalized in various ways:

\vfill

FP\&S:
\begin{itemize}
\only<1>\item \only<2->{\item S. 2007: Coincidence }\only<3->{(Taleshian \& Mirghasemi 2009) }\only<4->{(plagiarized)}
\only<1-5>\item \only<6->{\item S. 2009: $\RT$ for fixed points and coincidences}
\only<1-6>\item \only<7->{\item Gon\c{c}alves \& S. 2012: Coincidence on nonorientable, $C^0$ manifolds}
\end{itemize}

\vfill

A\&B:
\begin{itemize}
\only<1-4>{\item}
\only<5->{\item Gon\c{c}alves \& Weber, 2008: Equivariant $L(f)$ and $\RT(f)$}
\end{itemize}

\fpause\pause\pause\pause\pause\pause\pause
A\&B seems hard to extend to coincidence theory, \pause

FP\&S seems hard to do for nonmanifolds. (without commutativity)
}

\frame{
{Hadwiger's Theorem}
Our scheme for $L(f)$ is based on Hadwiger's Theorem (1950s), for subcomplexes of an abstract simplicial complex:
\begin{thm} (Hadwiger)
The Euler characteristic is the unique $\R$-valued function on subcomplexes of a simplicial complex satisfying:\pause
\begin{itemize}
\item (Valuation axiom) $\chi(\emptyset)=0$ and if $A$, $B$ are subcomplexes of $X$, then 
\[ \chi(A \cup B) = \chi(A) + \chi(B) - \chi(A\cap B) \]\pause
\item (Simplex axiom) If $x$ is a simplex, then $\chi(x) = 1$.
\end{itemize}
\end{thm}
}

\frame{
Hadwiger's Theorem is an approach to the Euler characteristic ``without algebraic topology''.
\fpause
Comes from a well-developed theory of lattice valuations.
\fpause
If we consider subcomplexes of a complex, this forms a ``distributive lattice''
\fpause
Two operations $\cap$ and $\cup$ which are commutative, associative, distributive, with a few more properties.
}

\frame{
Hadwiger's result is obvious if you believe the following Lemma:
\begin{lem}
Any valuation on a complex is determined uniquely by its values on simplices, which may be assigned arbitrarily. 
\end{lem}

\fpause
To prove the Lemma:\pause\ just check that, when you assign values to the simplices, the valuation property gives a unique well-defined extension to the whole complex.

\fpause
From the lemma, there must be a unique valuation which is 1 on simplices\pause, and we know it is the Euler characteristic.

\fpause
We'll use the same lemma for our result.
}

\frame{
$L(f)$ is slightly more complicated than $\chi$: \pause we need maps too.

\fpause

For a complex $X$, let $M(X)$ be the set of pairs $(f,A)$ where $f:X\to X$ is a simplicial selfmap and $A \subset X$ is a subcomplex.

\fpause

\begin{thm}
There is a unique function $L:M(X) \to \R$ satisfying:
\begin{itemize}
\item (Valuation axiom) $L(f,\emptyset)=0$ and
if $A$, $B$ are subcomplexes of $X$, then \[ L(f,A \cup B) = L(f,A) + L(f,B) - L(f,A\cap B) \]\pause
\item (Simplex axiom) If $x$ is a simplex, then  and 
\[ L(f,x) = (-1)^{\dim x} c(f,x) + L(f,\partial x).\]
\end{itemize}
\end{thm}
}

\frame{
\[ L(f,x) = (-1)^{\dim x} c(f,x) + L(f,\partial x).\]
Here $\partial x$ is the boundary of $x$.

\fpause
$c(f,x) \in \{-1, 0, 1\}$ is the orientation of how $x$ maps onto itself:
\fpause

If $f(x) \neq x$ then $c(f,x) = 0$. \pause

If $f(x)=x$ then $c(f,x) = \pm 1$ depending on the orientation. 
}

\frame{
This $c(f,x)$ should look familiar -- \pause it's the coefficient on $x$ in $f_q(x)$, the chain map. 
\fpause
Adding these up, it's easy to verify that 
\[ L(f,X) = \sum_q (-1)^q \tr(f_q:C_q(X) \to C_q(X)), \] \pause
and so 
\[ L(f,X) = \sum_q (-1)^q \tr(f_{*q}:H_q(X) \to H_q(X)) \]
as expected.
\fpause
Note: we obtain this trace formula even without assuming a homotopy invariance axiom.
}

\frame{
We want to extend this to continuous maps on polyhedra.
\fpause
The usual approach is to use a simplicial approximation to the map.
\fpause
But our setting above is simplicial maps $X \to X$, which is too restrictive.
\fpause
To use simplicial approximations we need to subdivide the domain.
\fpause
So we need a ``subdivision'' version of the theorem.
}

\frame{
Let $M'(X)$ be the set of pairs $(f,A)$, where $A$ is a subcomplex of some subdivision $X'$ of $X$, and $f:X'\to X$ is simplicial.
\fpause


\begin{thm}
There is a unique function $L:M'(X) \to \R$ satisfying:
\begin{itemize}
\item (Valuation axiom) $L(f,\emptyset) = 0$  and if $A$, $B$ are subcomplexes of a common subdivision of $X$, then 
\[ L(f,A \cup B) = L(f,A) + L(f,B) - L(f,A\cap B) \]
\item (Simplex axiom) If $x$ is a simplex, then 
\[ L(f,x) = (-1)^{\dim x} c(f,x) + L(f,\partial x).\]
\end{itemize}
\end{thm}
}

\begin{comment}
\frame{
\[ L(f,x) = (-1)^{\dim x} c(f,x) + L(f,\partial x).\]
We need to say a bit more clearly what $c(f,x)$ means when $f:X'\to X$ is simplicial.
\fpause
If $x$ is not a subdivision of $f(x)$, then $c(f,x)=0$. 
\fpause
If $x$ is a subdivision of $f(x)$, then $c(f,x) = \pm 1$ according to orientation. 
}
\end{comment}

\frame{
This setting now allows for subdivisions, we can do simplicial approximations to continuous maps.

\fpause

Our final setting is continuous maps on compact polyhedra:
\fpause
Let $X$ be a compact polyhedron, and let $N(X)$ be the set of pairs $(f,A)$ where $f:X\to X$ is continuous and $A$ is a subpolyhedron of some subdivision of $X$. 

\fpause
Then our previous arguments suffice in this setting, using a homotopy property to get simplicial approximations.
}

\frame{

\begin{thm}
There is a unique function $\Lambda:N(X) \to \R$ satisfying:
\begin{itemize}
\item (Homotopy axiom) If $f\htp g$, then $\Lambda(f,A) = \Lambda(g,A)$.\pause
\item (Valuation axiom) $\Lambda(f,\emptyset) = 0$ and 
if $A$, $B$ are subpolyhedra of a common subdivision of $X$, then \[ \Lambda(f,A \cup B) = \Lambda(f,A) + \Lambda(f,B) - \Lambda(f,A\cap B) \]\pause
\item (Simplicial map axiom) If $f$ is simplicial and $x$ is a simplex, then
\[ \Lambda(f,x) = (-1)^{\dim x} c(f, x) + \Lambda(f, \pd x). \] 
\end{itemize}

\end{thm}
}

\frame{
Idea: 
\fpause
Replace $f$ by a simplicial approximation using the homotopy axiom
\fpause
Previous theorem gets the uniqueness
\fpause
Need to check that alternative homotopies don't change the value,\pause\ but we already have the trace formula which is homotopy invariant.
}

\frame{
Actually we can do better- \pause use the Hopf construction and you can put all fixed points in the interior of maximal simplicies. 
\fpause
Call such a map a \emph{Hopf simplicial map}.
\fpause
A Hopf simplicial map has no fixed points on the boundaries\pause,
so in 
\[ \Lambda(f,x) = (-1)^{\dim x} c(f, x) + \Lambda(f, \pd x), \]
we'll always have $\Lambda(f,\pd x) = 0$.
}


\frame{
So we get a weaker simplicial map axiom:
\begin{thm}
There is a unique function $\Lambda:N(X) \to \R$ satisfying:
\begin{itemize}
\item (Homotopy axiom) If $f\htp g$, then $\Lambda(f,A) = \Lambda(g,A)$.
\item (Valuation axiom) If $A$, $B$ are subpolyhedra of a common subdivision of $X$, then $\Lambda(f,\emptyset) = 0$ and 
\[ \Lambda(f,A \cup B) = \Lambda(f,A) + \Lambda(f,B) - \Lambda(f,A\cap B) \]
\item (Hopf simplicial map axiom) Let $f$ be Hopf simplicial. If $x$ is a nonmaximal simplex then $L(f,x) = 0$. If $x$ is a maximal simplex, then
\[ \Lambda(f,x) = (-1)^{\dim X} c(f, x). \] 
\end{itemize}

\end{thm}
}

\frame{
Can we weaken the homotopy axiom?
\fpause
We can't just remove it, since $L(f,A)$ would be undefined when $f$ is not simplicial.
\fpause
But the simplicial approximation theorem and  Hopf construction require only small homotopies. 
\fpause
Actually the set of Hopf simplicial maps with fixed points  in maximal simplices is a dense set in $X^X$, the space of selfmaps.
}

\frame{

Since the valuation and simplicial map axioms determine $\Lambda$ on a dense set, we need only assume \emph{continuity} of $\Lambda$ to have uniqueness on all of $X^X$.
\fpause
``Homotopy invariance'' means that $\Lambda$ is constant on path components of $X^X$.

\fpause
A ``continuity axiom'' is weaker.
\vfill
}

\frame{
So our final result is:
\begin{thm}
There is a unique function $\Lambda:N(X) \to \R$ satisfying:
\begin{itemize}
\item (Continuity axiom) The value $\Lambda(f,A)$ depends continuously on $f \in X^X$.
\item (Valuation axiom) If $A$, $B$ are subpolyhedra of a common subdivision of $X$, then $\Lambda(f,\emptyset) = 0$ and 
\[ \Lambda(f,A \cup B) = \Lambda(f,A) + \Lambda(f,B) - \Lambda(f,A\cap B) \]
\item (Hopf simplicial map axiom) Let $f$ be Hopf simplicial. If $x$ is a nonmaximal simplex then $L(f,x) = 0$. If $x$ is a maximal simplex, then
\[ \Lambda(f,x) = (-1)^{\dim X} c(f, x). \] 
\end{itemize}
\end{thm}
}

\begin{comment}

\frame{
There is a unique function $\Lambda:N(X) \to \R$ satisfying:
\begin{itemize}
\item (Valuation axiom) If $A$, $B$ are subpolyhedra of a common subdivision of $X$, then $\Lambda(f,\emptyset)=0$ and 
\[ \Lambda(f,A \cup B) = \Lambda(f,A) + \Lambda(f,B) - \Lambda(f,A\cap B) \]
\item (Simplex axiom) If $x$ and $f(x)$ are both simplices, then
\[ \Lambda(f,x) = (-1)^{\dim x} c(f, x) + \Lambda(f, \pd x). \] 
\end{itemize}
\fpause
This can't be true though.
\fpause
If $f$ is continuous but not simplicial, then the simplex axiom will never apply.
\fpause
So the function's actual values would never be proscribed, \pause\ 
so the zero valuation would satisfy both the axioms.
}

\frame{
But the homotopy axiom is only used to get a simplicial approximation. \pause\ We don't actually need to allow ``big'' homotopies.
\fpause
We could change it to an $\epsilon$-homotopy axiom\pause, but this isn't any weaker on a compact space.
\fpause
Any homotopy is a concatenation of finitely many $\epsilon$-homotopies.
\fpause
Our weak version allows only a \emph{single} $\epsilon$-homotopy.
}

\frame{
\begin{thm}
For any $\epsilon>0$, there is a unique function $\Lambda:N(X)\to \R$ satisfying the following axioms:
\begin{itemize}
\item (Valuation axiom) Let $A,B$ be subcomplexes of some common subdivision $X'$ of $X$. Then $\Lambda(f,\emptyset)=0$, and 
\[ \Lambda(f,A\cup B) = \Lambda(f,A) + \Lambda(f,B) - \Lambda(f,A\cap B). \]
\item (Simplicial approximation axiom) Let $x$ be a simplex of some subdivision of $X$, and let $f'$ be a map $\epsilon$-homotopic to $f$ such that $f'(x)$ is a simplex of $X$. Then
\[ \Lambda(f,x) = (-1)^{\dim x} c(f', x) + \Lambda(f', \pd x), \]
\end{itemize}
\end{thm}
}
\end{comment}

\frame{
By the way, a similar weakening may be possible in the FPS approach.
\fpause
\begin{conj}
The fixed point index is the unique $\R$-valued function satisfying the following axioms:
\begin{itemize}
\item (Continuity) $\ind(f,U)$ depends continuously on $f\in X^X$
\item (Additivity) If $\Fix(f)\cap U \subset U_1 \sqcup U_2$, then
\[ \ind(f,U) = \ind(f,U_1) + \ind(f,U_2)
\] 
\item (Constant map) If $c$ is a constant map, then
\[ \ind(c,U) = 1 \]
\end{itemize}
\end{conj}

\fpause
Not sure if this will work for the A\&B approach.
}

\frame{
Thanks!
}
\end{document}