From bcf43402a6d2a990936364e99dd83dbfd84b21d0 Mon Sep 17 00:00:00 2001 From: Maria Marchwicka Date: Fri, 14 Jun 2019 21:37:33 +0200 Subject: [PATCH] sphere as 2 tori$ --- images/ball_4_pushed_seifert.pdf | Bin 5150 -> 4461 bytes images/ball_4_pushed_seifert.pdf_tex | 19 +- images/ball_4_pushed_seifert.svg | 146 +-- images/seifert_matrix.pdf | Bin 18198 -> 18107 bytes images/seifert_matrix.pdf_tex | 16 +- images/seifert_matrix.svg | 280 +++--- images/sphere_as_torus.png | Bin 0 -> 30255 bytes images/sphere_as_torus.svg | 1217 ++++++++++++++++++++++++++ lectures_on_knot_theory.tex | 69 +- 9 files changed, 1522 insertions(+), 225 deletions(-) create mode 100644 images/sphere_as_torus.png create mode 100644 images/sphere_as_torus.svg diff --git a/images/ball_4_pushed_seifert.pdf b/images/ball_4_pushed_seifert.pdf index f8a5e1219ae20b20a07a8637a4b601b1d9c00fb0..dd1e3e8fa7f4a46f69fe9486eb04b04d82d549a1 100644 GIT binary patch delta 2615 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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{align*} &F: \mathbb{C}^2 \rightarrow \mathbb{C} \text{ a polynomial} \\ &F(0) = 0 \end{align*} - \end{example} ???????????? \\ @@ -348,6 +392,7 @@ Figure 8 knot is negative amphichiral. % % % + \section{Concordance group \hfill\DTMdate{2019-03-18}} \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 @@ -1064,6 +1109,10 @@ H_{p, i} = \quot{\mathbb{Z}[t, t^{-1}]}{p^{k_i} \mathbb{Z} [t, t^{-1}]} \noindent The proof is the same as over $\mathbb{Z}$. \noindent +%Add NotePrintSaveCiteYour opinionEmailShare +%Saveliev, Nikolai +%Lectures on the Topology of 3-Manifolds +%An Introduction to the Casson Invariant \end{document}