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sphere as 2 tori$

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Maria Marchwicka 2019-06-14 21:37:33 +02:00
parent 6575000384
commit bcf43402a6
9 changed files with 1522 additions and 225 deletions
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images/ball_4_pushed_seifert.pdf
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images/ball_4_pushed_seifert.pdf_tex
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@ -36,7 +36,7 @@
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69
lectures_on_knot_theory.tex
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@ -32,6 +32,8 @@
\usepackage{unicode-math} \usepackage{unicode-math}
\usetikzlibrary{cd} \usetikzlibrary{cd}
\hypersetup{ \hypersetup{
@ -135,12 +137,12 @@ Let us consider a long exact sequence of cohomology of a pair $(S^3, S^3 \setmin
&\mathbb{Z} &\mathbb{Z}
\\ \\
& H^0(S^3) \ar[u,isomorphic] \to & H^0(S^3) \ar[u,isomorphic] \to
&H^0(S^3 \setminus N) \to &H^0(S^3 \setminus N) \to
\\ \\
\to H^1(S^3, S^3 \setminus N) \to \to H^1(S^3, S^3 \setminus N) \to
& H^1(S^3) \to & H^1(S^3) \to
& H^1(S^3\setminus N) \to & H^1(S^3\setminus N) \to
\\ \\
& 0 \ar[u,isomorphic]& & 0 \ar[u,isomorphic]&
\\ \\
@ -155,13 +157,29 @@ Let us consider a long exact sequence of cohomology of a pair $(S^3, S^3 \setmin
& \mathbb{Z} \ar[u,isomorphic] &\\ & \mathbb{Z} \ar[u,isomorphic] &\\
\end{tikzcd} \end{tikzcd}
\end{center} \end{center}
\[ \begin{align*}
H^* (S^3, S^3 \setminus N) \cong H^* (N, \partial N) N \cong & D^2 \times S^1\\
\] \partial N \cong & S^1 \times S^1\\
H^1(N, \partial N) \cong & \mathbb{Z} \oplus \mathbb{Z}
\end{align*}
\begin{align*}
H^* (S^3, S^3 \setminus N) &\cong H^* (N, \partial N)\\
\\ \\
?????????????? H^ 1 (S^3\setminus N) &\cong H^1(S^3\setminus K) \cong \mathbb{Z}
\\ \end{align*}
\begin{equation*}
\begin{tikzcd}[row sep=huge]
H^1(S^3 \setminus K) \arrow[r,] \arrow[d,"\widetilde{\Theta}"] &
H^1(N \setminus K) \arrow[d,"\Theta"] \\
{[S^3 \setminus K, S^1]} \arrow[r,]&
{[N \setminus K, S^1]}
\end{tikzcd}
\end{equation*}
\noindent
$\Sigma = \widetilde{\Theta}^{-1}(X)$ is a surface, such that $\partial \Sigma = K$, so it is a Seifert surface.
%
%
% Thom isomorphism,
\end{proof} \end{proof}
\begin{definition} \begin{definition}
@ -312,13 +330,39 @@ ${D^2 \overset{g}\longhookrightarrow M}$ such that:
g_{\big| \partial D^2} = f_{\big| \partial D^2.} g_{\big| \partial D^2} = f_{\big| \partial D^2.}
\] \]
\end{lemma} \end{lemma}
\section{} \noindent
Remark: Dehn lemma doesn't hold for dimension four.\\
Let $M$ be connected, compact three manifold with boundary.
Suppose $\pi_1(\partial M) \longrightarrow \pi_1(M)$ has non-trivial kernel. Then there exists a map $f: (D^2, \partial D^2) \longrightarrow (M, \partial M)$ such that $f\big| \partial D^2$ is non-trivial loop in $\partial M$
\begin{theorem}[Sphere theorem]
Suppose $\pi_1(M) \ne 0$. Then there exists an embedding $f: S^2 \hookrightarrow M$ that is homotopy non-trivial.
\end{theorem}
\begin{problem}
Prove that $S^3 \ K$ is Eilenberg–MacLane space of type $K(\pi, 1)$.
\end{problem}
\begin{corollary}
Suppose $K \subset S^3$ and $\pi_1(S^3 \setminus K)$ is infinite cyclic ($\mathbb{Z})$. Then $K$ is trivial.
\end{corollary}
\subsection*{Construction}
We know that $3$ - sphere can be obtained by gluing two solid tori:
$S^3 = (D^2 \times S^1) \cup (S^1 \times D^2)$. So the complement of solid torus in $S^3$ is another solid torus.\\
Take $(z_1, z_2) \in \mathbb{C}$ such that $max(\mid z_1 \mid, \mid z_2, \mid) = 1
$
\begin{figure}[h]
\centering{
\def\svgwidth{\linewidth}
\resizebox{0.3\textwidth}{!}{\includegraphics[width=0.3\textwidth]{sphere_as_torus.png}}
\caption{The complement of solid torus in $S^3$ is another solid torus.}
\label{fig:sphere_as_tori}
}
\end{figure}
\begin{example} \begin{example}
\begin{align*} \begin{align*}
&F: \mathbb{C}^2 \rightarrow \mathbb{C} \text{ a polynomial} \\ &F: \mathbb{C}^2 \rightarrow \mathbb{C} \text{ a polynomial} \\
&F(0) = 0 &F(0) = 0
\end{align*} \end{align*}
\end{example} \end{example}
???????????? ????????????
\\ \\
@ -348,6 +392,7 @@ Figure 8 knot is negative amphichiral.
% %
% %
% %
\section{Concordance group \hfill\DTMdate{2019-03-18}} \section{Concordance group \hfill\DTMdate{2019-03-18}}
\begin{definition} \begin{definition}
Two knots $K$ and $K^{\prime}$ are called (smoothly) concordant if there exists an annulus $A$ that is smoothly embedded in ${S^3 \times [0, 1]}$ such that Two knots $K$ and $K^{\prime}$ are called (smoothly) concordant if there exists an annulus $A$ that is smoothly embedded in ${S^3 \times [0, 1]}$ such that
@ -1064,6 +1109,10 @@ H_{p, i} = \quot{\mathbb{Z}[t, t^{-1}]}{p^{k_i} \mathbb{Z} [t, t^{-1}]}
\noindent \noindent
The proof is the same as over $\mathbb{Z}$. The proof is the same as over $\mathbb{Z}$.
\noindent \noindent
%Add NotePrintSaveCiteYour opinionEmailShare
%Saveliev, Nikolai
%Lectures on the Topology of 3-Manifolds
%An Introduction to the Casson Invariant
\end{document} \end{document}