small pice of surgery lecture

images/dehn_twist.pdf_tex

@@ -0,0 +1,61 @@
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lec_5.tex

@@ -114,9 +114,7 @@ So we can calculate:
 \end{proof}
 \begin{corollary}
 If $t$ is not a root of 
-$\det S S^T - $ \\
-????????????????\\
-then 
+$\det (tS - S^T) $, then
 $\vert \sigma_K(t) \vert \leq 2g$.
 \end{corollary}
 \begin{fact}
@@ -157,49 +155,78 @@ was conjecture in the '70 and proved by  P. Kronheimer and T. Mrówka (1994).
 \end{example}
 \begin{proposition}
 $g_4 (T(p, q) \# -T(r, s))$ is in general hopelessly unknown. 
-\\???????????????\\
-essentially $\sup \vert \sigma_K(t) \vert \leq 2 g_n(K)$
 \end{proposition}
+\begin{proposition}
+Supremum of the signature function of the knot is bounded almost everywhere by two times $4$ - genus:
+\[
+\ess \sup \vert \sigma_K(t) \vert \leq 2 g_4(K).
+\]
+\end{proposition}
+\subsection{Topological genus}
 \begin{definition}
-A knot $K$ is called topologically slice if $K$ bounds a topological locally flat disc in $B^4$ (it has tubular neighbourhood). 
+A knot $K$ is called topologically slice if $K$ bounds a topological locally flat disc in $B^4$ (i.e. the disk has tubular neighbourhood). 
 \end{definition}
 \begin{theorem}[Freedman, '82]
-If $\Delta_K(t) \geq 1$, then $K$ is topologically slice, but not necessarily smoothly slice.  
+If $\Delta_K(t) = 1$, then $K$ is topologically slice (but not necessarily smoothly slice).  
 \end{theorem}
 \begin{theorem}[Powell, 2015]
-If $K$ is genus g 
-\\(top. loc.?????????)\\
+If $K$ is genus $g$ 
+(topologically flat) 
 cobordant to $K^\prime$, 
-then $\vert \sigma_K(t)  - \sigma_{K^\prime}(t) \vert \leq 2 g$. \\
-If $g_4^{\mytop}(K) \geq $ ?????ess $\sup \vert \sigma_K(t) \vert$ and ?????????\\
-$\dim \ker (H_1 (Y) \longrightarrow H_1(\Omega)) = \frac{1}{2} \dim H_1(Y)$.
-\end{theorem}
-???????????????
+then 
 \[
-H^1 (B^4 \setminus Y, \mathbb{Z}) = [B^4 \setminus Y, S^1]
+\vert \sigma_K(t)  - \sigma_{K^\prime}(t) \vert \leq 2 g
 \]
+if $g_4^{\mytop}(K) \geq \ess \sup \vert \sigma_K(t) \vert$.
+\end{theorem}
+\noindent
+The proof for smooth category was based on following equality:
+\[
+\dim \ker (H_1 (Y) \longrightarrow H_1(\Omega)) = \frac{1}{2} \dim H_1(Y).
+\]
+For this equality we assumed that there exists a $3$ - dimensional manifold $\Omega$ (as shown in Figure \ref{fig:omega_in_B_4}) which was guaranteed by Pontryagin-Thom Construction.\\
+Pontryagin-Thom Construction relays on taking $\Omega$ as preimage of regular value:
+\[
+H^1 (B^4 \setminus Y, \mathbb{Z}) = [B^4 \setminus Y, S^1], 
+\]
+what relies on Sard's theorem, that the set of regular values has positive measure. But Sard's theorem doesn't work for topologically locally flat category. So there was a gap in the proof for topological locally flat category - the existence of $\Omega$.\\
 \noindent
 Remark: unless $p=2$ or $p = 3 \wedge  q = 4$:
 \[
-g_4^\top (T(p, q)) < q_4(T(p, q))
+g_4^{\mytop} (T(p, q)) < q_4(T(p, q)).
 \]
-%??????????????????????
+% Wilczyński '93 
+%Feller 2014
+%Baoder 2017
+%Lemark
+\\
+\noindent
+From the category of cobordant knots (or topologically cobordant knots) there exists a map to $\mathbb{Z}$ given by signature function. To any element $K$ we can associate a form 
+\[
+(1 - t)S + (1 - \bar{t})S^T) \in W(\mathbb{Z}[t, t^{-1}]).
+\] This association is not well define because id depends on the choice of Seifert form. However, different choices lead ever to congruent forms ($S \mapsto CSC^T$) or induced the change on the form by adding or subtracting a hyperbolic element. 
 \begin{definition}
 The Witt group $W$ of $\mathbb{Z}[t, t^{-1}]$ elements are classes of non-degenerate 
 forms over $\mathbb{Z}[t, t^{-1}]$ under the equivalence relation $V \sim W$ if $V \oplus - W$ is metabolic. 
 \end{definition}
 \noindent
 If $S$ differs from $S^\prime$ by a row extension, then 
-$(1 - t) S + (1 - \bar{t}^{-1}) S^T$ is Witt equivalence to $(1 - t) S^\prime  + (1 - t^{-1})S^T$. 
-%???????????????????????????
+$(1 - t) S + (1 - \bar{t}^{-1}) S^T$ is Witt equivalence to $(1 - t) S^\prime  + (1 - t^{-1})S^T$.
+\\ 
 \noindent
 A form is meant as hermitian with respect to this involution: $A^T = A: (a, b) = \bar{(a, b)}$.
 \\
-????????????????????????????
+$
+W(\mathbb{Z}_p) = \mathbb{Z}_2 \oplus 
+\mathbb{Z}_2$ or 
+$\mathbb{Z}_4$
 \\
+???????????????????????
+\\
+$\sum a_gt^j \longrightarrow \sum a_g t^{-1}$\\
 \begin{theorem}[Levine '68]
 \[
-W(\mathbb{Z}[t^{\pm 1}) 
+W(\mathbb{Z}[t^{\pm 1}]) 
 \longrightarrow \mathbb{Z}_2^\infty \oplus 
 \mathbb{Z}_4^\infty \oplus
 \mathbb{Z} 

lectures_on_knot_theory.tex

@@ -88,6 +88,7 @@
 \DeclareMathOperator{\rank}{rank}
 \DeclareMathOperator{\coker}{coker}
 \DeclareMathOperator{\ord}{ord}
+\DeclareMathOperator{\ess}{ess}
 \DeclareMathOperator{\mytop}{top}
 \DeclareMathOperator{\Gl}{GL}
 \DeclareMathOperator{\Sl}{SL}
@@ -121,27 +122,28 @@
 %\input{myNotes}
 
 \section{Basic definitions \hfill\DTMdate{2019-02-25}}
-\input{lec_1.tex}
+%\input{lec_1.tex}
 
 \section{Alexander polynomial \hfill\DTMdate{2019-03-04}}
-\input{lec_2.tex}
+%\input{lec_2.tex}
 %add Hurewicz theorem?
 
 
 \section{Examples of knot classes
 \hfill\DTMdate{2019-03-11}}
-\input{lec_3.tex}
+%\input{lec_3.tex}
 
 \section{Concordance group \hfill\DTMdate{2019-03-18}}
-\input{lec_4.tex}
+%\input{lec_4.tex}
 
 \section{Genus $g$ cobordism \hfill\DTMdate{2019-03-25}}
-\input{lec_5.tex}
+%\input{lec_5.tex}
 
 \section{\hfill\DTMdate{2019-04-08}}
 \input{lec_6.tex}
 
 \section{\hfill\DTMdate{2019-04-15}}
+???????????????????\\
 \begin{theorem}
 Suppose that $K \subset S^3$ is a slice knot (i.e. $K$ bound a disk in $B^4$). 
 Then if $F$ is a Seifert surface of $K$ and $V$ denotes the associated Seifet matrix, then there exists $P \in \Gl_g(\mathbb{Z})$ such that:
@@ -236,6 +238,38 @@ H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) &\longrightarrow \quot{\mathbb{Q}}{
 A = V + V^T.
 \end{align*}
 ????????????????????????????
+\\
+We have a primary decomposition of $H_1(Y, \mathbb{Z}) = U$ (as a group). For any $p \in \mathbb{P}$ we define $U_p$ to be the subgroup of elements annihilated by the same power of $p$. We have $U = \bigoplus_p U_p$. 
+\begin{example}
+\begin{align*}
+\text{If } U &= 
+\mathbb{Z}_3 \oplus
+\mathbb{Z}_{45} \oplus
+\mathbb{Z}_{15} \oplus
+\mathbb{Z}_{75} 
+\text{ then }\\
+U_3 &= 
+\mathbb{Z}_3 \oplus
+\mathbb{Z}_9 \oplus
+\mathbb{Z}_3 \oplus
+\mathbb{Z}_3
+\text{ and }\\
+U_5 &= 
+(e) \oplus
+\mathbb{Z}_5 \oplus
+\mathbb{Z}_5 \oplus
+\mathbb{Z}_{25}.
+\end{align*}
+\end{example}
+
+\begin{lemma}
+Suppose $x \in U_{p_1}$, $y \in U_{p_2}$ and $p_1 \neq p_2$. Then $<x, y > = 0$.
+\end{lemma}
+\begin{proof}
+\begin{align*}
+x \in U_{p_1} 
+\end{align*}
+\end{proof}
 \begin{align*}
 H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\\
 A \longrightarrow BAC^T \quad \text{Smith normal form}
@@ -511,7 +545,7 @@ $H_1(\bar{X}$
 
 field of fractions
 
-\section{\hfill\DTMdate{2019-06-03}}
+\section{Surgery \hfill\DTMdate{2019-06-03}}
 \begin{theorem}
 Let $K$ be a knot and $u(K)$ its unknotting number. Let $g_4$ be a minimal four genus of a smooth surface $S$ in $B^4$ such that $\partial S = K$. Then:
 \[
@@ -521,9 +555,9 @@ u(K) \geq g_4(K)
 Recall that if $u(K)=u$ then $K$ bounds a disk $\Delta$ with $u$ ordinary double points. 
 \\
 \noindent
-Remove from $\Delta$ the two self intersecting and glue the Seifert surface for the Hopf link. The reality surface $S$ has Euler characteristic $\chi(S) = 1 - 2u$. Therefore $g_4(S) = u$ .
+Remove from $\Delta$ the two self intersecting disks and glue the Seifert surface for the Hopf link. The reality surface $S$ has Euler characteristic $\chi(S) = 1 - 2u$. Therefore $g_4(S) = u$.
 \end{proof}
-???????????????????\\
+%Tim D. Cochran and Peter Teichner
 \begin{example}
 The knot $8_{20}$ is slice: $\sigma \equiv 0$ almost everywhere but $\sigma(e^{\frac{ 2\pi i}{6}}) = + 1$.
 \end{example}
@@ -532,8 +566,8 @@ The knot $8_{20}$ is slice: $\sigma \equiv 0$ almost everywhere but $\sigma(e^{\
 %Journal-ref: Comment. Math. Helv. 79 (2004) 105-123
 \subsection*{Surgery}
 %Rolfsen, geometric group theory, Diffeomorpphism of a torus, Mapping class group
-Recall that $H_1(S^1 \times S^1, \mathbb{Z}) = \mathbb{Z}^3$. As generators for $H_1$ we can set ${\alpha = [S^1 \times \pt]}$ and ${\beta=[\pt \times S^1]}$. Suppose ${\phi: S^1 \times S^1 \longrightarrow S^1 \times S^1}$ is a diffeomorphism. 
-Consider an induced map on homology group: 
+Recall that $H_1(S^1 \times S^1, \mathbb{Z}) = \mathbb{Z}^2$. As generators for $H_1$ we can set ${\alpha = [S^1 \times \pt]}$ and ${\beta=[\pt \times S^1]}$. Suppose ${\phi: S^1 \times S^1 \longrightarrow S^1 \times S^1}$ is a diffeomorphism. 
+Consider an induced map on the homology group: 
 \begin{align*}
 H_1(S^1 \times S^1, \mathbb{Z}) \ni \phi_* (\alpha) &= p\alpha + q \beta, \quad p, q \in \mathbb{Z},\\
 \phi_*(\beta) &= r \alpha + s \beta, \quad r, s \in \mathbb{Z}, \\
@@ -541,11 +575,10 @@ H_1(S^1 \times S^1, \mathbb{Z}) \ni \phi_* (\alpha) &= p\alpha + q \beta, \quad
   \begin{pmatrix}
      p & q\\
      r & s
-  \end{pmatrix}
+  \end{pmatrix}.
 \end{align*} 
 As $\phi_*$ is diffeomorphis, it must be invertible over $\mathbb{Z}$. Then for a direction preserving diffeomorphism we have  $\det \phi_* = 1$. Therefore $\phi_* \in \Sl(2, \mathbb{Z})$.
 \end{theorem}
-
 \vspace{10cm}
 \begin{theorem}
 Every such a matrix can be realized as a torus. 
@@ -564,6 +597,15 @@ S^1 \times \pt  &\longrightarrow \pt \times S^1 \\
 \item
 \end{enumerate}
 \end{proof}
+\begin{figure}[h]
+\fontsize{20}{10}\selectfont
+\centering{
+\def\svgwidth{\linewidth}
+\resizebox{0.5\textwidth}{!}{\input{images/dehn_twist.pdf_tex}}
+\caption{Dehn twist.}
+\label{fig:dehn_twist}
+}
+\end{figure}