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\usepackage{uwbgeom}
\usepackage[latin2]{inputenc}
\usepackage[mathscr]{euscript}
\usepackage{eepic, float}

\title[The Bundle Theorem...]{The Bundle Theorem in fragments of projective Grassmann spaces}
\author[Mariusz Żynel]{Mariusz Żynel\\ \texttt{mariusz@math.uwb.edu.pl}}
\institute{%
    University of Białystok\\
    Institute of Mathematics}
\date[June 1-6, 2014]{Combinatorics 2014\\Gaeta 2014}

%% Private notation

\DeclareMathOperator{\AfColin}{L^\tau}

\def\StarOf{\mathrm{S}}
\def\TopOf{\mathrm{T}}
\def\fixout{{\mathfrak D}}
\def\fixoutf(#1,#2){\mathord{\fixout_{#1}(#2)}}
\def\bfixoutf(#1,#2){\mathord{\fixout_{#1}\bigl(#2\bigr)}}
\def\linesout{{\Lines_{\W}}}
\def\pointsout{{S_{\W}}}
\def\copl{\mathrel{\pi}}
\def\piclique(#1,#2){\K_\pi^{#2}(#1)}
\def\uzg{\mathrel{\sigma}}

\let\adjac\sim

\DeclareMathOperator{\AP}{AP}
\DeclareMathOperator{\SP}{SP}
\DeclareMathOperator{\PP}{PP}

\def\PPAS(#1,#2,#3){\ensuremath{\PP^{\alpha,\cal S}_{#1,#2}(#3)}}
\def\PPAT(#1,#2,#3){\ensuremath{\PP^{\alpha,\cal T}_{#1,#2}(#3)}}
\def\PPOS(#1,#2,#3){\ensuremath{\PP^{\omega,\cal S}_{#1,#2}(#3)}}
\def\PPOT(#1,#2,#3){\ensuremath{\PP^{\omega,\cal T}_{#1,#2}(#3)}}

\def\SPAS(#1,#2,#3){\ensuremath{\SP^{\alpha,\cal S}_{#1,#2}(#3)}}
\def\SPAT(#1,#2,#3){\ensuremath{\SP^{\alpha,\cal T}_{#1,#2}(#3)}}
\def\SPOS(#1,#2,#3){\ensuremath{\SP^{\omega,\cal S}_{#1,#2}(#3)}}
\def\SPOT(#1,#2,#3){\ensuremath{\SP^{\omega,\cal T}_{#1,#2}(#3)}}

\def\APAS(#1,#2,#3){\ensuremath{\AP^{\alpha,\cal S}_{#1,#2}(#3)}}
\def\APAT(#1,#2,#3){\ensuremath{\AP^{\alpha,\cal T}_{#1,#2}(#3)}}
\def\APOS(#1,#2,#3){\ensuremath{\AP^{\omega,\cal S}_{#1,#2}(#3)}}
\def\APOT(#1,#2,#3){\ensuremath{\AP^{\omega,\cal T}_{#1,#2}(#3)}}

\def\W{{\cal W}}
\def\K{{\cal K}}
\def\X{{\cal X}}
\def\Y{{\cal Y}}
\def\A{{\mathfrak A}}
\def\M{{\mathfrak M}}
\def\B{{\mathscr B}}
\def\BL{{\mathfrak L}}
\def\BS{{\mathfrak B}}
\def\hipa{\mathscr{H}}


\newenvironment{cmath}{%
  \par
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\newenvironment{ctext}{%
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\setlength{\leftmargini}{0pt}

\begin{document}


\frame[plain]{\titlepage}

\begin{frame}
  \frametitle{Motivations and references}

  \begin{thebibliography}{9}
  \bibitem{Wilcox}
    L. R. Wilcox
    \newblock {\it Modularity in the theory of lattices}
    \newblock Ann, of Math. {\bf 40} (1939), 490--505
  
  \bibitem{Kahn}
    J. Kahn
    \newblock {\it Locally projective-planar lattices which satisfy the Bundle Theorem}
    \newblock Math. Z. {\bf 175} (1980), 219--247

  \bibitem{Kr96}
    A. Kreuzer
    \newblock {\it Locally projective spaces which satisfy the {B}undle {T}heorem}
    \newblock J.~Geom. {\bf 56} (1996), 87--98

  \bibitem{wielohol}
    K. Petelczyc, M. {\.Z}ynel
    \newblock {\it The complement of a point subset in a projective space and a Grassmann space}
    \newblock J. Appl. Logic (to appear)
  \end{thebibliography}
\end{frame}


\begin{frame}
  \frametitle{Objectives}
  
  \LARGE
  Our goal is to recover a deleted hyperplane\\[6pt]
  from a projective Grassmann space.
\end{frame}


\begin{frame}
  \frametitle{Pencils}
  
  \begin{itemize}
  \item
    $V$ -- a vector space of dimension $n$ with $3\le n<\infty$
    
  \item
    $\Sub_k(V)$ -- the set of all $k$-dimensional subspaces of $V$
    
  \item
    Assume that $0 < k < n$.
  \end{itemize}
  
  \begin{block}{}
    For $H\in\Sub_{k-1}(V)$, $B\in\Sub_{k+1}(V)$ with $H\subset B$
    a \structure{$k$-pencil} is a set of the form
    \begin{cmath}
      \Pencil(H, B) := \bset{ U\in\Sub_k(V)\colon H\subset U\subset B}.
    \end{cmath}
  \end{block}
  \begin{center}
    \scriptsize
    \setlength{\unitlength}{0.0005in}
    \begin{picture}(3624,2139)(0,-10)
      \put(1562,512){\circle*{50}}
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	      (2112,2112)(12,612)
      \path(1337,362)(2902,1527)
      \path(1852,407)(652,912)
      \path(1452,222)(2022,1807)
      \put(3222,1402){$B$}
      \put(1540,265){$H$}
      \put(2717,1227){$U_1$}
      \put(2017,1587){$U_2$}
      \put(607,660){$U_3$}
    \end{picture}
  \end{center}
\end{frame}


\begin{frame}
  \frametitle{Grassmann spaces}
  
  \begin{block}{}
    The point-line structure
    \begin{cmath}
      \M = \SpPencils(k, V) = \bstruct{\Sub_k(V), \SpPencilsLines(k, V)},
    \end{cmath}
    where $\SpPencilsLines(V, k)$ is the family of all $k$-pencils, is a \structure{Grassmann space}.
  \end{block}
  
  \bigskip
  
  \begin{itemize}
  \item
    If $0 < k < n$, then $\M$ is a \emph{partial linear space}.
    
  \item
    If $k=1$ or $k=n-1$, then $\M$ is a \emph{projective space}.
  \end{itemize}
\end{frame}


\begin{frame}
  \frametitle{Basic properties and facts}
  
  \begin{itemize}
  \item
    Every strong subspace of\/ $\M$ is a projective space.
    
  \pause\item
    There are two disjoint classes of maximal strong subspaces in $\M$.
  \end{itemize}  
    
  \begin{block}{Stars}
    For $H\in\Sub_{k-1}(V)$ a \structure{star} is a set of the form
    \begin{cmath}
      \StarOf(H) = [H)_k = \{U\in\Sub_k(V)\colon H\subset U\}.
    \end{cmath}
  \end{block}
  \begin{block}{Tops}
   For $B\in\Sub_{k+1}(V)$ a \structure{top} is a set of the form
   \begin{cmath}
     \TopOf(B) = (B]_k = \{U\in\Sub_k(V)\colon U\subset B\}.
    \end{cmath}
  \end{block}
  
  \pause

  \begin{itemize}
  \item
    Every line of $\M$ can be uniquely extended to a star and a top.
  \end{itemize}
\end{frame}


\begin{frame}
  \frametitle{Spine spaces}
  
  \begin{itemize}
  \item
    $W$ -- a fixed subspace of $V$
    
  \item
    $m$ -- an integer such that $k-\codim(W)\le m\le k, \dim(W)$
    
  \item
    $\SpineSpPoints(k,m,W) := \set{U\in\Sub_k(V)\colon \dim(U\cap W) = m}$
    
  \item
    $\SpineSpLines(k,m,W) := \set{L\cap\SpineSpPoints(k,m,W)\colon L\in\SpPencilsLines(k, V) \text{ and }
      \abs{L\cap\SpineSpPoints(k,m,W)} \ge 2 }$
  \end{itemize}

  \begin{block}{}
    The point-line structure
    \begin{cmath}
      \SpineSp(k,m,V,W)=\bstruct{\SpineSpPoints(k,m,W),\/ \SpineSpLines(k,m,W)}
    \end{cmath}
    is called a \structure{spine space}.
  \end{block}

  \begin{thebibliography}{9}  
    \bibitem{autspine}
    K. Pra\.zmowski, M. Żynel
    \newblock {\it Automorphisms of spine spaces},
    \newblock Abh. Math. Sem. Univ. Hamb. {\bf 72} (2002), 59--77.
  \end{thebibliography}
\end{frame}


\begin{frame}
  \frametitle{Bundles and planes in partial linear spaces}

  \begin{itemize}
  \item
    The set $[U]$ of all lines through a point $U$ is called a \structure{\emph{bundle}}.
    
  \medskip
  
  \pause\item
    Given three lines $L_1, L_2, L_3$ that form a triangle in a partial linear space,
    the \structure{\emph{plane}} $\Pi(L_1, L_2, L_3)$
    spanned by this triangle is the set of those points which lie on the lines
    that intersect two of those three lines $L_1, L_2, L_3$ in two distinct points.
  
  \medskip  
    
  \pause\item
    In linear spaces where the dimension function can be defined we can say
    that a plane is a subspace of dimension 2.
    
  \medskip  
    
  \pause\item
    Two lines $L_1, L_2\in\Lines$ are said to be \structure{\emph{coplanar}}, which is written
    as $L_1\copl L_2$, iff there is a plane containing them.
  \end{itemize}
\end{frame}


\begin{frame}
  \frametitle{Hyperplane complement in a Grassmann space}
  
  \begin{itemize}
  \item
    $W$ -- a fixed subspace of $V$ with $\codim(W) = k$
    
  \item
    $\hipa(W) := \bset{U\in\Sub_k(V)\colon U\cap W \neq 0}$ -- a hyperplane in $\M$  
    
  \item
    $S_{\hipa(W)} := \bset{\Sub_k(V)\setminus\hipa(W)}$
    
  \item  
    $\Lines_{\hipa(W)} := \bset{L\cap S_{\hipa(W)}\colon L\in\SpPencilsLines(k, V) \text{ and } 
      \abs{L\cap S_{\hipa(W)}} \ge 2}$
  \end{itemize}  
    
  \begin{block}{}
    \begin{cmath}
      \fixout = \bfixoutf(\M,{\hipa(W)}) = \bstruct{S_{\hipa(W)},\ \Lines_{\hipa(W)}}
    \end{cmath}
    is the \structure{complement of the hyperplane $\hipa(W)$ in $\M$}.
  \end{block}

  \pause
  
  \begin{fact}
    In every hyperplane of the form\/ $\hipa(W)$ in\/ $\M$ there are points
    collinear with no point outside $\hipa(W)$.
  \end{fact}

  \pause
  
  \begin{itemize}
  \item
    $\fixout = \bstruct{\SpineSpPoints(k,0,W), \SpineSpLines(k,0,W)} = \SpineSp(k,0,V,W)$
    
  \item
    $\hipa(W) = \SpineSpPoints(k,1,W)\cup\dots\cup\SpineSpPoints(k,{\min(k,n-k)},W)$
  \end{itemize}
\end{frame}


\begin{frame}
  \frametitle{Key facts about the complement}

  \begin{lem}
    \begin{sentences}
    \item
      If\/ $L\in\AfLines(k,1,W)$, then\/ $\StarOf(L), \TopOf(L)\subseteq\hipa(W)$.
    \item
      If\/ $L\in\aLines(k,1,W)$, then\/ $\TopOf(L)\nsubseteq\hipa(W)$ and\/ $\StarOf(L)\subseteq\hipa(W)$.
    \item
      If\/ $L\in\oLines(k,1,W)$, then\/ $\StarOf(L)\nsubseteq\hipa(W)$ and\/ $\TopOf(L)\subseteq\hipa(W)$.
    \end{sentences}
  \end{lem}
  
  \pause\medskip

  \begin{block}{}
    \centering
    In what follows we assume that $2 < k < n - 2$.
  \end{block}
  
  \medskip

  \begin{cor}
    For every point\/ $U\in\SpineSpPoints(k,1,W)$ there are
    $U_1, U_2, U_3\in\SpineSpPoints(k,0,W)$ such that $U, U_1, U_2, U_3$
    form a tetrahedron in $\M$.
  \end{cor}
\end{frame}


\begin{frame}
  \frametitle{Bundle Theorem}

  \begin{itemize}
  \item
    Both in the Grassmann space $\M$ and in the hyperplane complement $\fixout$ 
    the following is true.
  \end{itemize}

  \begin{thm}[Bundle Theorem]
    Let\/ $L_1, L_2, L_3, L_4$ be lines such that no three of them are coplanar.
    If five of the six pairs\/ $\set{L_i, L_j}, 1\le i<j\le 4$ are coplanar, then
    so is the sixth pair.
  \end{thm}
    
  \begin{itemize}
  \item
    In locally-projective linear spaces (cf.\ [Kreuzer, 1996]) this theorem
    is used to prove that two lines determine a bundle uniquely.
  \end{itemize}
\end{frame}


\begin{frame}
  \frametitle{Maximal $\copl$-clique and bundles}

  \begin{itemize}
  \item
    Let $\K$ be a maximal $\copl$-clique in $\fixout$. There are two possibilities:
    \begin{sentences}
    \item[(i)]
      there is a point $U$ in $\M$ such that $U\in\overline{L}$ for all $L\in\K$, or
    \item[(ii)]
      there is a plane $\Pi$ in $\fixout$ such that $L\subset\Pi$ for all $L\in\K$.
    \end{sentences}
    
  \medskip
    
  \pause\item
    They can be distinguished by requirement that:
    \begin{sentences}
    \item[$(\ast)$]
      there are three non-coplanar lines in $\K$.
    \end{sentences}
    The $\copl$-clique $\K$ is of type (i) iff it satisfies $(\ast)$.
    
  \medskip
    
  \pause\item
    There is a strong subspace in $\fixout$ containing $\K$.
  
  \medskip
    
  \pause\item
    Every bundle $[U]$ breaks up into $\copl$-cliques.
  \end{itemize}
\end{frame}


\begin{frame}
  \frametitle{Gluing $\copl$-cliques together}

  \begin{itemize}
  \item
    $X_1, X_2$ -- strong subspaces of different types in $\fixout$
    \begin{multline*}
      \piclique(U_1, X_1)\uzg \piclique(U_2, X_2) :\iff
        U_1, U_2 \in X_1\cap X_2 \Land \\
        \bigl(\exists L_1\in\piclique(U_1, X_1)\bigr)\bigl(\exists L_2\in\piclique(U_2, X_2)\bigr)
        \bigl[\, L_1, L_2\neq X_1\cap X_2 \Land  \\
          \TopOf(L_1)\cap\StarOf(L_2) \text{ or } \StarOf(L_1)\cap\TopOf(L_2)
            \text{ is a proper line distinct from } X_1\cap X_2\,\bigr]
    \end{multline*}
    
  \item
    $X_1, X_2$ -- strong subspaces of the same type in $\fixout$
    \begin{multline*}
      \piclique(U_1, X_1)\uzg \piclique(U_2, X_2) :\iff
        (\exists U\in\SpineSpPoints(k,0,W)\cup\SpineSpPoints(k,1,W)) \\
        (\exists \text{a maximal strong subspace } X  \text{ of the other type than } X_1,X_2 ) \\
          \bigl[\, U\in X \Land \piclique(U_1, X_1)\uzg \piclique(U, X) \uzg \piclique(U_2, X_2) \,\bigr]
    \end{multline*}

  \pause\item
    $\uzg$ is an equivalence relation; its equivalence classes are bundles.
    
  \pause\item
    To every point $U\in\SpineSpPoints(k,0,W)\cup\SpineSpPoints(k,1,W)$
    we can associate a unique bundle $[U]_\fixout$ of lines of $\fixout$.
  \end{itemize}
\end{frame}


\begin{frame}
  \frametitle{Adjacency of bundles}

  \begin{itemize}
  \item
    We will write $\B(\fixout)$ for the set of all bundles in $\fixout$.
  
  \item
    If $a\in\B(\fixout)$ we write $\bar{a}$ for the bundle of $\M$ with $a\subseteq\bar{a}$.

  \item
    For a point $U$ of $\fixout$ and a bundle $a\in\B(\fixout)$ we write $U\adjac a$ if there
    is a line $L\in a$ such that $U\in L$, we write $L = \LineOn(U,a)$.

  \item
    Frankly, two bundles $a,b\in\B(\fixout)$ are adjacent iff they share a line.
    \begin{cmath}
      a\adjac b :\iff
        \bigl(\exists U\in\SpineSpPoints(k,0,W)\bigr)
          \bigl[\, U\adjac a \Land U\adjac b \Land \LineOn(U,a) \copl \LineOn(U,b) \,\bigr]
    \end{cmath}
    
    \begin{center}
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        \put(1500,90){$U$}
        \put(1770,820){$b$}
        \put(330,170){$a$}
      \end{picture}
    \end{center}

    \begin{lem}
      For two distinct bundles $a, b\in\B(\fixout)$ we have $a\adjac b$ iff
        there is a proper line $L$ or $L\in\aLines(k,1,W)\cup\oLines(k,1,W)$ such that
        $L\in\bar{a}\cap\bar{b}$.
    \end{lem}
  \end{itemize}
\end{frame}


\begin{frame}
  \frametitle{Lines of bundles}

  \begin{itemize}
  \item
    Given two distinct adjacent bundles $a, b\in\B(\fixout)$ we can define the new
    line  through $a, b$:
    \begin{multline*}
      [a,b] := \bigl\{c\in\B(\fixout)\colon \text{for all points $U$ of $\fixout$ with $[U]_\fixout\neq a, b, c$ } \\
          \text{if $U\adjac a, b$, then $U\adjac c$
            and $\LineOn(U,a), \LineOn(U,b), \LineOn(U,c)$ are coplanar} \bigr\}.
    \end{multline*}
    
    \begin{center}
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        \path(1100,680)(782,1527)
        \path(1100,680)(932,1602)
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        \path(1100,680)(1507,302)
        \put(1500,90){$U$}
        \put(330,170){$a$}
        \put(1770,820){$b$}
        \put(1157,627){$c$}
      \end{picture}
    \end{center}
  \end{itemize}
  
  \begin{lem}
    If\/ $a, b\in\B(\fixout)$, $a\neq b$, $a\adjac b$, and $L\in\bar{a}\cap\bar{b}$,
    then
    \begin{cmath}
      [a,b] = \bset{c\in\B(\fixout)\colon L\in\bar{c}}.
    \end{cmath}
  \end{lem}
\end{frame}


\begin{frame}
  \frametitle{Bundle space}

  \begin{itemize}
  \item
    The set of all new lines will be written as
    \begin{cmath}
      \BL(\fixout) := \bset{[a,b]\colon a,b\in\B(\fixout),  a\adjac b \text{ and } a\neq b}.
    \end{cmath}

  \item
    The structure
    \begin{equation*}
      \BS(\fixout) := \bstruct{\B(\fixout), \BL(\fixout)}
    \end{equation*}
    is called the \structure{\emph{bundle space}} over $\fixout$. It is a partial linear space.
  \end{itemize}
    
  \pause
    
  \begin{prop}
    The bundle space $\BS(\fixout)$ is definable in terms of\/ $\fixout$.
  \end{prop}

  \pause
  
  \begin{itemize}
  \item
    The bundle space $\BS(\fixout)$ is isomorphic to
    \begin{equation*}
      \fixout' = \bstruct{\SpineSpPoints(k,0,W)\cup\SpineSpPoints(k,1,W),\ \SpineSpLines(k,0,W)\cup\aLines(k,1,W)\cup\oLines(k,1,W)}.
    \end{equation*}

  \item
    This is a new proof of a know fact that $\fixout'$ is definable in $\fixout$.
  \end{itemize}
\end{frame}


\begin{frame}
  \frametitle{Affine lines on the horizon}

  \begin{itemize}
  \item
    The affine lines $\AfLines(k,1,W)$ are not yet recovered.

  \pause\item
    Let $\Pi$ be a plane in $\fixout'$. There are two possibilities:
    \begin{sentences}
    \item
      $\Pi$ is a strong subspace of $\fixout'$, then\\
      $\Pi$ is a proper plane in $\fixout$ or it is a projective plane in $\SpineSp(k,1,V,W)$,
    \item
      $\Pi$ contains collinear and non-collinear points, then\\
      $\Pi$ is a semi-affine plane in $\SpineSp(k,1,V,W)$.
    \end{sentences}

  \pause\item
    Collinearity w.r.t.\ affine lines $\AfLines(k,1,W)$ can be defined as follows:
    \begin{multline*}
      \AfColin(a, b, c) :\iff
        (\exists\text{a semi-affine plane $\Pi$ in $\fixout'$}) \\
          \bigl[\, a, b, c\in\Pi \Land \mathop{\not\adjac}(a,b,c) \,\bigr].
    \end{multline*}


    \begin{prop}[Prażmowski and Żynel, 2002]
      For points $a, b, c$ in $\fixout'$ we have $\AfColin(a, b, c)$ iff
      there is a line $L\in\AfLines(k,1,W)$ such that $a, b, c\in L$.
    \end{prop}
  \end{itemize}
\end{frame}


\begin{frame}
  \frametitle{Recovery of the hyperplane $\hipa(W)$}

  \begin{prop}[Prażmowski and Żynel, 2002]
    $\SpineSp(k,1,V,W)$ is definable in terms of\/ $\SpineSp(k,0,V,W)$.
  \end{prop}
  
  \pause\medskip

  \begin{itemize}
  \item
    Continuing this procedure we can recover $\SpineSp(k,2,V,W)$, then
    $\SpineSp(k,3,V,W)$, and so on. 
    
  \medskip
  
  \pause\item
    Finally, $\SpineSp(k,{\min(k,n-k)},V,W)$ is recoverable in $\fixout = \SpineSp(k,0,V,W)$.
  \end{itemize}
  
  \pause\medskip

  \begin{thm}[Prażmowski and Żynel, 2002]
    The hyperplane $\hipa(W)$ and the ambient space $\M$ are 
    recoverable from the complement\/ $\fixout$.
  \end{thm}
\end{frame}


\begin{frame}
  \frametitle{Summary and final remarks}

  \structure{DONE:}
  \begin{itemize}
  \item
    We have successfully recovered the ambient projective Grassmann space form
    its hyperplane complement.
  \end{itemize}
  
  \vfill\pause
  
  \structure{TO DO:}
  \begin{itemize}
  \item
    Express the relation $\uzg$ in terms of the points, lines and planes 
    of the complement $\fixout$.
  
  \medskip
    
  \item
    Delete two hyperplanes from a Grassmann space and check how to recover 
    the ambient space from their complement.
  \end{itemize}

\end{frame}



\begin{frame}
  \thispagestyle{empty}
  \begin{centering}
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      \usebeamerfont{part title}Thank you for your attention
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