some inkspace pictures
This commit is contained in:
Maria Marchwicka
parent
b2a61ac843
commit
0583159f09
Binary file not shown.
|
Before Width: | Height: | Size: 3.8 KiB After Width: | Height: | Size: 44 KiB |
Binary file not shown.
|
Before Width: | Height: | Size: 12 KiB After Width: | Height: | Size: 16 KiB |
BIN
images/Hopf.png
BIN
images/Hopf.png
Binary file not shown.
|
Before Width: | Height: | Size: 4.9 KiB After Width: | Height: | Size: 34 KiB |
Binary file not shown.
|
Before Width: | Height: | Size: 12 KiB After Width: | Height: | Size: 57 KiB |
Binary file not shown.
|
|
@ -0,0 +1,63 @@
|
|||
%% Creator: Inkscape inkscape 0.91, www.inkscape.org
|
||||
%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
|
||||
%% Accompanies image file 'ball_4.pdf' (pdf, eps, ps)
|
||||
%%
|
||||
%% To include the image in your LaTeX document, write
|
||||
%% \input{<filename>.pdf_tex}
|
||||
%% instead of
|
||||
%% \includegraphics{<filename>.pdf}
|
||||
%% To scale the image, write
|
||||
%% \def\svgwidth{<desired width>}
|
||||
%% \input{<filename>.pdf_tex}
|
||||
%% instead of
|
||||
%% \includegraphics[width=<desired width>]{<filename>.pdf}
|
||||
%%
|
||||
%% Images with a different path to the parent latex file can
|
||||
%% be accessed with the `import' package (which may need to be
|
||||
%% installed) using
|
||||
%% \usepackage{import}
|
||||
%% in the preamble, and then including the image with
|
||||
%% \import{<path to file>}{<filename>.pdf_tex}
|
||||
%% Alternatively, one can specify
|
||||
%% \graphicspath{{<path to file>/}}
|
||||
%%
|
||||
%% For more information, please see info/svg-inkscape on CTAN:
|
||||
%% http://tug.ctan.org/tex-archive/info/svg-inkscape
|
||||
%%
|
||||
\begingroup%
|
||||
\makeatletter%
|
||||
\providecommand\color[2][]{%
|
||||
\errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%
|
||||
\renewcommand\color[2][]{}%
|
||||
}%
|
||||
\providecommand\transparent[1]{%
|
||||
\errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%
|
||||
\renewcommand\transparent[1]{}%
|
||||
}%
|
||||
\providecommand\rotatebox[2]{#2}%
|
||||
\ifx\svgwidth\undefined%
|
||||
\setlength{\unitlength}{538.34058867bp}%
|
||||
\ifx\svgscale\undefined%
|
||||
\relax%
|
||||
\else%
|
||||
\setlength{\unitlength}{\unitlength * \real{\svgscale}}%
|
||||
\fi%
|
||||
\else%
|
||||
\setlength{\unitlength}{\svgwidth}%
|
||||
\fi%
|
||||
\global\let\svgwidth\undefined%
|
||||
\global\let\svgscale\undefined%
|
||||
\makeatother%
|
||||
\begin{picture}(1,0.36148366)%
|
||||
\put(0,0){\includegraphics[width=\unitlength,page=1]{ball_4.pdf}}%
|
||||
\put(0.62288514,0.24829785){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.62747874\unitlength}\raggedright $B^4$\\ \end{minipage}}}%
|
||||
\put(1.24285327,1.51222577){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.88153562\unitlength}\raggedright \end{minipage}}}%
|
||||
\put(0,0){\includegraphics[width=\unitlength,page=2]{ball_4.pdf}}%
|
||||
\put(0.08156744,0.26027166){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.62747874\unitlength}\raggedright $B^4$\\ \end{minipage}}}%
|
||||
\put(0,0){\includegraphics[width=\unitlength,page=3]{ball_4.pdf}}%
|
||||
\put(0.72054873,0.78436255){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.17856087\unitlength}\raggedright \end{minipage}}}%
|
||||
\put(0.34790192,0.21447686){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.17197199\unitlength}\raggedright $\Sigma$\end{minipage}}}%
|
||||
\put(0.86664402,0.76488318){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{}}}%
|
||||
\put(0.88802542,0.20006421){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.17197199\unitlength}\raggedright $\widetilde{\Sigma}$\end{minipage}}}%
|
||||
\end{picture}%
|
||||
\endgroup%
|
||||
Binary file not shown.
|
|
@ -0,0 +1,58 @@
|
|||
%% Creator: Inkscape inkscape 0.91, www.inkscape.org
|
||||
%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
|
||||
%% Accompanies image file 'ink_diag.pdf' (pdf, eps, ps)
|
||||
%%
|
||||
%% To include the image in your LaTeX document, write
|
||||
%% \input{<filename>.pdf_tex}
|
||||
%% instead of
|
||||
%% \includegraphics{<filename>.pdf}
|
||||
%% To scale the image, write
|
||||
%% \def\svgwidth{<desired width>}
|
||||
%% \input{<filename>.pdf_tex}
|
||||
%% instead of
|
||||
%% \includegraphics[width=<desired width>]{<filename>.pdf}
|
||||
%%
|
||||
%% Images with a different path to the parent latex file can
|
||||
%% be accessed with the `import' package (which may need to be
|
||||
%% installed) using
|
||||
%% \usepackage{import}
|
||||
%% in the preamble, and then including the image with
|
||||
%% \import{<path to file>}{<filename>.pdf_tex}
|
||||
%% Alternatively, one can specify
|
||||
%% \graphicspath{{<path to file>/}}
|
||||
%%
|
||||
%% For more information, please see info/svg-inkscape on CTAN:
|
||||
%% http://tug.ctan.org/tex-archive/info/svg-inkscape
|
||||
%%
|
||||
\begingroup%
|
||||
\makeatletter%
|
||||
\providecommand\color[2][]{%
|
||||
\errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%
|
||||
\renewcommand\color[2][]{}%
|
||||
}%
|
||||
\providecommand\transparent[1]{%
|
||||
\errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%
|
||||
\renewcommand\transparent[1]{}%
|
||||
}%
|
||||
\providecommand\rotatebox[2]{#2}%
|
||||
\ifx\svgwidth\undefined%
|
||||
\setlength{\unitlength}{595.27559055bp}%
|
||||
\ifx\svgscale\undefined%
|
||||
\relax%
|
||||
\else%
|
||||
\setlength{\unitlength}{\unitlength * \real{\svgscale}}%
|
||||
\fi%
|
||||
\else%
|
||||
\setlength{\unitlength}{\svgwidth}%
|
||||
\fi%
|
||||
\global\let\svgwidth\undefined%
|
||||
\global\let\svgscale\undefined%
|
||||
\makeatother%
|
||||
\begin{picture}(1,1.41428571)%
|
||||
\put(0,0){\includegraphics[width=\unitlength,page=1]{ink_diag.pdf}}%
|
||||
\put(0.63682602,0.48349309){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{$B^4$}}}%
|
||||
\put(1.19167986,1.92661837){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.79722134\unitlength}\raggedright \end{minipage}}}%
|
||||
\put(0.35293654,0.78061261){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.46919801\unitlength}\raggedright \end{minipage}}}%
|
||||
\put(0.26158827,0.80552578){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.48165458\unitlength}\raggedright \end{minipage}}}%
|
||||
\end{picture}%
|
||||
\endgroup%
|
||||
Binary file not shown.
|
|
@ -0,0 +1,62 @@
|
|||
%% Creator: Inkscape inkscape 0.91, www.inkscape.org
|
||||
%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
|
||||
%% Accompanies image file 'link_diagram.pdf' (pdf, eps, ps)
|
||||
%%
|
||||
%% To include the image in your LaTeX document, write
|
||||
%% \input{<filename>.pdf_tex}
|
||||
%% instead of
|
||||
%% \includegraphics{<filename>.pdf}
|
||||
%% To scale the image, write
|
||||
%% \def\svgwidth{<desired width>}
|
||||
%% \input{<filename>.pdf_tex}
|
||||
%% instead of
|
||||
%% \includegraphics[width=<desired width>]{<filename>.pdf}
|
||||
%%
|
||||
%% Images with a different path to the parent latex file can
|
||||
%% be accessed with the `import' package (which may need to be
|
||||
%% installed) using
|
||||
%% \usepackage{import}
|
||||
%% in the preamble, and then including the image with
|
||||
%% \import{<path to file>}{<filename>.pdf_tex}
|
||||
%% Alternatively, one can specify
|
||||
%% \graphicspath{{<path to file>/}}
|
||||
%%
|
||||
%% For more information, please see info/svg-inkscape on CTAN:
|
||||
%% http://tug.ctan.org/tex-archive/info/svg-inkscape
|
||||
%%
|
||||
\begingroup%
|
||||
\makeatletter%
|
||||
\providecommand\color[2][]{%
|
||||
\errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%
|
||||
\renewcommand\color[2][]{}%
|
||||
}%
|
||||
\providecommand\transparent[1]{%
|
||||
\errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%
|
||||
\renewcommand\transparent[1]{}%
|
||||
}%
|
||||
\providecommand\rotatebox[2]{#2}%
|
||||
\ifx\svgwidth\undefined%
|
||||
\setlength{\unitlength}{595.27559055bp}%
|
||||
\ifx\svgscale\undefined%
|
||||
\relax%
|
||||
\else%
|
||||
\setlength{\unitlength}{\unitlength * \real{\svgscale}}%
|
||||
\fi%
|
||||
\else%
|
||||
\setlength{\unitlength}{\svgwidth}%
|
||||
\fi%
|
||||
\global\let\svgwidth\undefined%
|
||||
\global\let\svgscale\undefined%
|
||||
\makeatother%
|
||||
\begin{picture}(1,1.41428571)%
|
||||
\put(0,0){\includegraphics[width=\unitlength,page=1]{link_diagram.pdf}}%
|
||||
\put(0.31120017,0.7509725){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.77132343\unitlength}\raggedright $B^4$\\ \end{minipage}}}%
|
||||
\put(1.19167986,1.92661837){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.79722134\unitlength}\raggedright \end{minipage}}}%
|
||||
\put(0,0){\includegraphics[width=\unitlength,page=2]{link_diagram.pdf}}%
|
||||
\put(0.30826412,1.20312347){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.77132343\unitlength}\raggedright $B^4$\\ \end{minipage}}}%
|
||||
\put(0,0){\includegraphics[width=\unitlength,page=3]{link_diagram.pdf}}%
|
||||
\put(0.71933102,1.26837147){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.16148245\unitlength}\raggedright \end{minipage}}}%
|
||||
\put(0.63565374,1.14799364){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.21139524\unitlength}\raggedright $\Sigma$\end{minipage}}}%
|
||||
\put(0.85145305,1.2507552){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{}}}%
|
||||
\end{picture}%
|
||||
\endgroup%
|
||||
|
|
@ -1,4 +1,5 @@
|
|||
\documentclass[12pt, twoside]{article}
|
||||
\usepackage[pdf]{pstricks}
|
||||
|
||||
\usepackage{amssymb}
|
||||
\usepackage{amsmath}
|
||||
|
|
@ -14,7 +15,8 @@
|
|||
\usepackage{advdate}
|
||||
\usepackage{amsthm}
|
||||
\usepackage[useregional]{datetime2}
|
||||
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{cd}
|
||||
\hypersetup{
|
||||
colorlinks,
|
||||
citecolor=black,
|
||||
|
|
@ -25,7 +27,6 @@
|
|||
\usepackage{fontspec}
|
||||
\usepackage{mathtools}
|
||||
\usepackage{unicode-math}
|
||||
|
||||
\graphicspath{ {images/} }
|
||||
|
||||
\newtheorem{lemma}{Lemma}
|
||||
|
|
@ -47,6 +48,7 @@
|
|||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\overbar}[1]{\mkern 1.5mu\overline{\mkern-1.5mu#1\mkern-1.5mu}\mkern 1.5mu}
|
||||
\DeclareMathOperator{\Hom}{Hom}
|
||||
\DeclareMathOperator{\rank}{rank}
|
||||
\AtBeginDocument{\renewcommand{\setminus}{\mathbin{\backslash}}}
|
||||
|
||||
\input{knots_macros}
|
||||
|
|
@ -54,17 +56,15 @@
|
|||
\titleformat{\section}{\normalfont \large \bfseries}
|
||||
{Lecture\ \thesection}{2.3ex plus .2ex}{}
|
||||
\titlespacing{\subsection}{2em}{*1}{*1}
|
||||
|
||||
\usepackage{enumitem}
|
||||
\setlist[itemize]{topsep=0pt,before=\leavevmode\vspace{0.5em}}
|
||||
|
||||
\begin{document}
|
||||
%\tableofcontents
|
||||
\tableofcontents
|
||||
%\newpage
|
||||
%\input{myNotes}
|
||||
|
||||
\section{}
|
||||
\begin{flushright}
|
||||
\DTMdate{2019-02-25}
|
||||
\end{flushright}
|
||||
\section{\hfill\DTMdate{2019-02-25}}
|
||||
\begin{definition}
|
||||
A knot $K$ in $S^3$ is a smooth (PL - smooth) embedding of a circle $S^1$ in $S^3$:
|
||||
\begin{align*}
|
||||
|
|
@ -107,17 +107,18 @@ Links:
|
|||
\begin{itemize}
|
||||
\item
|
||||
a trivial link with $3$ components:
|
||||
\includegraphics[width=0.13\textwidth]{3unknots.png},
|
||||
\includegraphics[width=0.2\textwidth]{3unknots.png},
|
||||
\item
|
||||
a hopf link: \includegraphics[width=0.13\textwidth]{Hopf.png},
|
||||
\item
|
||||
a Whitehead link:
|
||||
\includegraphics[width=0.13\textwidth]{WhiteheadLink.png},
|
||||
\item
|
||||
Borromean link
|
||||
Borromean link:
|
||||
\includegraphics[width=0.1\textwidth]{BorromeanRings.png},
|
||||
\end{itemize}
|
||||
\end{example}
|
||||
|
||||
\begin{definition}
|
||||
A link diagram is a picture over projection of a link is $S^3$($\mathbb{R}^3$) such that:
|
||||
\begin{enumerate}
|
||||
|
|
@ -132,7 +133,7 @@ ${D_{\pi}}_{\big|L}$ is non degenerate
|
|||
\end{definition}
|
||||
There are under- and overcrossings (tunnels and bridges) on a link diagrams with an obvious meaning.\\
|
||||
Every link admits a link diagram.
|
||||
\begin{comment}
|
||||
%\begin{comment}
|
||||
|
||||
\subsection{Reidemeister moves}
|
||||
A Reidemeister move is one of the three types of operation on a link diagram as shown in Figure~\ref{fig: reidemeister}.
|
||||
|
|
@ -160,7 +161,7 @@ deformed into each other by a finite sequence of Reidemeister moves (and isotopy
|
|||
\end{align*}
|
||||
Fact (Milnor Singular Points of Complex Hypersurfaces):
|
||||
\end{example}
|
||||
\end{comment}
|
||||
%\end{comment}
|
||||
|
||||
An oriented knot is called negative amphichiral if the mirror image $m(K)$ if $K$ is equivalent the reverse knot of $K$. \\
|
||||
\begin{example}[Problem]
|
||||
|
|
@ -168,10 +169,7 @@ Prove that if $K$ is negative amphichiral, then $K \# K$ in
|
|||
$\mathbf{C}$
|
||||
\end{example}
|
||||
|
||||
\section{}
|
||||
\begin{flushright}
|
||||
\DTMdate{2019-03-04}
|
||||
\end{flushright}
|
||||
\section{\hfill\DTMdate{2019-03-04}}
|
||||
\begin{proof}("joke")\\
|
||||
Let $K \in S^3$ be a knot and $N$ be its tubular neighbourhood.
|
||||
\begin{align*}
|
||||
|
|
@ -182,10 +180,7 @@ For a pair $(S^3, S^3 \setminus N)$ we have:
|
|||
H^0(S^3)
|
||||
\end{align*}
|
||||
\end{proof}
|
||||
\section{}
|
||||
\begin{flushright}
|
||||
\DTMdate{2019-03-18}
|
||||
\end{flushright}
|
||||
\section{\hfill\DTMdate{2019-03-18}}
|
||||
\begin{definition}
|
||||
A knot $K$ is called (smoothly) slice if $K$ is smoothly concordant to an unknot. \\
|
||||
A knot $K$ is smoothly slice if and only if $K$ bounds a smoothly embedded disk in $B^4$.
|
||||
|
|
@ -216,10 +211,7 @@ Are there in concordance group torsion elements that are not $2$ torsion element
|
|||
\end{example}
|
||||
\noindent
|
||||
Remark: $K \sim K^{\prime} \Leftrightarrow K \# -K^{\prime}$ is slice.
|
||||
\section{}
|
||||
\begin{flushright}
|
||||
\DTMdate{2019-04-08}
|
||||
\end{flushright}
|
||||
\section{\hfill\DTMdate{2019-04-08}}
|
||||
$X$ is a closed orientable four-manifold. Assume $\pi_1(X) = 0$ (it is not needed to define the intersection form). In particular $H_1(X) = 0$.
|
||||
$H_2$ is free (exercise).
|
||||
\begin{align*}
|
||||
|
|
@ -230,10 +222,114 @@ $H_2(X, \mathbb{Z}) \times
|
|||
H_2(X, \mathbb{Z}) \longrightarrow \mathbb{Z}$ - symmetric, non singular.
|
||||
\\
|
||||
Let $A$ and $B$ be closed, oriented surfaces in $X$.
|
||||
\section{}
|
||||
\begin{flushright}
|
||||
\DTMdate{2019-05-20}
|
||||
\end{flushright}
|
||||
|
||||
|
||||
\section{\hfill\DTMdate{2019-04-15}}
|
||||
In other words:\\
|
||||
Choose a basis $(b_1, ..., b_i)$ \\
|
||||
???\\
|
||||
of $H_2(Y, \mathbb{Z}$, then $A = (b_i, b_y)$ \\??\\ is a matrix of intersection form:
|
||||
\begin{align*}
|
||||
\quot{\mathbb{Z}^n}{A\mathbb{Z}^n} \cong H_1(Y, \mathbb{Z}).
|
||||
\end{align*}
|
||||
In particular $\mid \det A\mid = \# H_1(Y, \mathbb{Z}$.\\
|
||||
That means - what is happening on boundary is a measure of degeneracy.
|
||||
\\
|
||||
\vspace{1cm}
|
||||
\begin{center}
|
||||
\begin{tikzcd}
|
||||
[
|
||||
column sep=tiny,
|
||||
row sep=small,
|
||||
ar symbol/.style = {draw=none,"\textstyle#1" description,sloped},
|
||||
isomorphic/.style = {ar symbol={\cong}},
|
||||
]
|
||||
H_1(Y, \mathbb{Z})&
|
||||
\times \quad H_1(Y, \mathbb{Z})&
|
||||
\longrightarrow &
|
||||
\quot{\mathbb{Q}}{\mathbb{Z}}
|
||||
\text{ - a linking form}
|
||||
\\
|
||||
\quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] &
|
||||
\quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] &\\
|
||||
\end{tikzcd}
|
||||
$(a, b) \mapsto aA^{-1}b^T$
|
||||
\end{center}
|
||||
The intersection form on a four-manifold determines the linking on the boundary. \\
|
||||
|
||||
\noindent
|
||||
Let $K \in S^1$ be a knot, $\Sigma(K)$ its double branched cover. If $V$ is a Seifert matrix for $K$, then
|
||||
$H_1(\Sigma(K), \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}$ where
|
||||
$A = V \times V^T$, where $n = \rank V$.
|
||||
%\input{ink_diag}
|
||||
\begin{figure}[h]
|
||||
\fontsize{40}{10}\selectfont
|
||||
\centering{
|
||||
\def\svgwidth{\linewidth}
|
||||
\resizebox{0.5\textwidth}{!}{\input{images/ball_4.pdf_tex}}
|
||||
\caption{Pushing the Seifert surface in 4-ball.}
|
||||
\label{fig:pushSeifert}
|
||||
}
|
||||
\end{figure}
|
||||
\noindent
|
||||
Let $X$ be the four-manifold obtained via the double branched cover of $B^4$ branched along $\widetilde{\Sigma}$.
|
||||
\begin{fact}
|
||||
\begin{itemize}
|
||||
\item $X$ is a smooth four-manifold,
|
||||
\item $H_1(X, \mathbb{Z}) =0$,
|
||||
\item $H_2(X, \mathbb{Z}) \cong \mathbb{Z}^n$
|
||||
\item The intersection form on $X$ is $V + V^T$.
|
||||
\end{itemize}
|
||||
\end{fact}
|
||||
\noindent
|
||||
Let $Y = \Sigma(K)$. Then:
|
||||
\begin{align*}
|
||||
&H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}\\ &(a,b) \mapsto a A^{-1} b^{T},\qquad
|
||||
A = V + V^T\\
|
||||
&H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\\
|
||||
&A \longrightarrow BAC^T \quad \text{Smith normal form}
|
||||
\end{align*}
|
||||
???????????????????????\\
|
||||
In general
|
||||
|
||||
\section{\hfill\DTMdate{2019-05-20}}
|
||||
|
||||
Let $M$ be closed, oriented, compact four-dimensional manifold.\\
|
||||
??????????????????????????????????\\
|
||||
If $H_1(M, \mathbb{Z}) = 0$ then there exists a
|
||||
bilinear form - the intersection form on $M$:
|
||||
|
||||
\begin{center}
|
||||
\begin{tikzcd}
|
||||
[
|
||||
column sep=tiny,
|
||||
row sep=small,
|
||||
ar symbol/.style = {draw=none,"\textstyle#1" description,sloped},
|
||||
isomorphic/.style = {ar symbol={\cong}},
|
||||
]
|
||||
H_2(M, \mathbb{Z})&
|
||||
\times & H_2(M, \mathbb{Z})
|
||||
\longrightarrow &
|
||||
\mathbb{Z}
|
||||
\\
|
||||
\ar[u,isomorphic] \mathbb{Z}^n && &\\
|
||||
\end{tikzcd}
|
||||
\end{center}
|
||||
\noindent
|
||||
Let us consider a specific case: $M$ has a boundary $Y = \partial M$.
|
||||
\\??????\\
|
||||
Betti number $b_1(Y) = 0$, $H_1(Y, \mathbb{Z})$ is finite. \\
|
||||
Then: $H_2(M, \mathbb{Z})
|
||||
\times H_2(M, \mathbb{Z})
|
||||
\longrightarrow
|
||||
\mathbb{Z}$ can be degenerate in the sense that
|
||||
\begin{align*}
|
||||
H_2(M, \mathbb{Z}) \longrightarrow \Hom (H_2(M, \mathbb{Z}), \mathbb{Z})\\
|
||||
(a, b) \mapsto \mathbb{Z}\\
|
||||
a \mapsto (a, \_) H_2(M, \mathbb{Z})
|
||||
\end{align*} has coker
|
||||
|
||||
|
||||
Let $K \subset S^3$ be a knot, \\
|
||||
$X = S^3 \setminus K$ - a knot complement, \\
|
||||
$\widetilde{X} \xrightarrow{\enspace \rho \enspace} X$ - an infinite cyclic cover (universal abelian cover).
|
||||
|
|
@ -249,10 +345,10 @@ H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \longrightarrow \quot{\mathbb{Q}}{\mat
|
|||
|
||||
\begin{fact}
|
||||
\begin{align*}
|
||||
H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \cong
|
||||
\quot{\mathbb{Z}{[t, t^{-1}]}^n}{(tV - V^T)\mathbb{Z}[t, t^{-1}]^n}
|
||||
&H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \cong
|
||||
\quot{\mathbb{Z}{[t, t^{-1}]}^n}{(tV - V^T)\mathbb{Z}[t, t^{-1}]^n}\;, \\
|
||||
&\text{where $V$ is a Seifert matrix.}
|
||||
\end{align*}
|
||||
where $V$ is a Seifert matrix.
|
||||
\end{fact}
|
||||
\begin{fact}
|
||||
\begin{align*}
|
||||
|
|
@ -413,7 +509,7 @@ g\overbar{g}h \equiv 1 \mod{p^{2k}}\\
|
|||
g\overbar{g} \equiv 1 \mod{p^k}
|
||||
\end{align*}
|
||||
???????????????????????????????\\
|
||||
If $p$ has no roots on $S^1$ then $B(z) > 0$ for all $z$, so the assumptions of Lemma \ref{L:coprime polynomials} are satisfied no matter what $A$ is.
|
||||
If $P$ has no roots on $S^1$ then $B(z) > 0$ for all $z$, so the assumptions of Lemma \ref{L:coprime polynomials} are satisfied no matter what $A$ is.
|
||||
\end{proof}
|
||||
?????????????????\\
|
||||
\begin{align*}
|
||||
|
|
@ -448,31 +544,11 @@ $2 \sum\limits_{k_i \text{ odd}} \epsilon_i$. The peak of the signature function
|
|||
%$(\eta_{k, \xi_l^{+}} -\eta_{k, \xi_l^{-}}$
|
||||
\end{theorem}
|
||||
\end{proof}
|
||||
\section{}
|
||||
\begin{flushright}
|
||||
\DTMdate{2019-05-27}
|
||||
\end{flushright}
|
||||
\section{\hfill\DTMdate{2019-05-27}}
|
||||
....
|
||||
\begin{definition}
|
||||
A square hermitian matrix $A$ of size $n$.
|
||||
\end{definition}
|
||||
|
||||
field of fractions
|
||||
|
||||
\section{}
|
||||
In other words:\\
|
||||
Choose a basis $(b_1, ..., b_i)$ \\
|
||||
???\\
|
||||
of $H_2(Y, \mathbb{Z}$, then $A = (b_i, b_y)$ \\??\\ is a matrix of intersection form:
|
||||
\begin{align*}
|
||||
\quot{\mathbb{Z}^n}{A\mathbb{Z}^n} \cong H_1(Y, \mathbb{Z}).
|
||||
\end{align*}
|
||||
In particular $\mid \det A\mid = \# H_1(Y, \mathbb{Z}$.\\
|
||||
That means - what is happening on boundary is a measure of degeneracy.
|
||||
\\
|
||||
\vspace{1cm}
|
||||
\begin{align*}
|
||||
H_1(Y, \mathbb{Z}) \times
|
||||
H_1(Y, \mathbb{Z}) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}} \text{ - a linking form}
|
||||
\end{align*}
|
||||
\end{document}
|
||||
\end{document}
|
||||
|
|
|
|||
Loading…
Reference in New Issue