sphere as 2 tori$

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lectures_on_knot_theory.tex

@@ -32,6 +32,8 @@
 
 \usepackage{unicode-math}
 
+
+
 \usetikzlibrary{cd}
 
 \hypersetup{
@@ -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] &\\
  \end{tikzcd}
 \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}
 
 \begin{definition}
@@ -312,13 +330,39 @@ ${D^2 \overset{g}\longhookrightarrow M}$ such that:
 g_{\big| \partial D^2} = f_{\big| \partial D^2.}
 \]
 \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{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}