small pice of surgery lecture

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@ -114,9 +114,7 @@ So we can calculate:
\end{proof} \end{proof}
\begin{corollary} \begin{corollary}
If $t$ is not a root of If $t$ is not a root of
$\det S S^T - $ \\ $\det (tS - S^T) $, then
????????????????\\
then
$\vert \sigma_K(t) \vert \leq 2g$. $\vert \sigma_K(t) \vert \leq 2g$.
\end{corollary} \end{corollary}
\begin{fact} \begin{fact}
@ -157,49 +155,78 @@ was conjecture in the '70 and proved by P. Kronheimer and T. Mrówka (1994).
\end{example} \end{example}
\begin{proposition} \begin{proposition}
$g_4 (T(p, q) \# -T(r, s))$ is in general hopelessly unknown. $g_4 (T(p, q) \# -T(r, s))$ is in general hopelessly unknown.
\\???????????????\\
essentially $\sup \vert \sigma_K(t) \vert \leq 2 g_n(K)$
\end{proposition} \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} \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} \end{definition}
\begin{theorem}[Freedman, '82] \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} \end{theorem}
\begin{theorem}[Powell, 2015] \begin{theorem}[Powell, 2015]
If $K$ is genus g If $K$ is genus $g$
\\(top. loc.?????????)\\ (topologically flat)
cobordant to $K^\prime$, cobordant to $K^\prime$,
then $\vert \sigma_K(t) - \sigma_{K^\prime}(t) \vert \leq 2 g$. \\ then
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}
???????????????
\[ \[
H^1 (B^4 \setminus Y, \mathbb{Z}) = [B^4 \setminus Y, S^1] \vert \sigma_K(t) - \sigma_{K^\prime}(t) \vert \leq 2 g
\] \]
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 \noindent
Remark: unless $p=2$ or $p = 3 \wedge q = 4$: 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} \begin{definition}
The Witt group $W$ of $\mathbb{Z}[t, t^{-1}]$ elements are classes of non-degenerate 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. forms over $\mathbb{Z}[t, t^{-1}]$ under the equivalence relation $V \sim W$ if $V \oplus - W$ is metabolic.
\end{definition} \end{definition}
\noindent \noindent
If $S$ differs from $S^\prime$ by a row extension, then 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 \noindent
A form is meant as hermitian with respect to this involution: $A^T = A: (a, b) = \bar{(a, b)}$. 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] \begin{theorem}[Levine '68]
\[ \[
W(\mathbb{Z}[t^{\pm 1}) W(\mathbb{Z}[t^{\pm 1}])
\longrightarrow \mathbb{Z}_2^\infty \oplus \longrightarrow \mathbb{Z}_2^\infty \oplus
\mathbb{Z}_4^\infty \oplus \mathbb{Z}_4^\infty \oplus
\mathbb{Z} \mathbb{Z}

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@ -88,6 +88,7 @@
\DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\ord}{ord}
\DeclareMathOperator{\ess}{ess}
\DeclareMathOperator{\mytop}{top} \DeclareMathOperator{\mytop}{top}
\DeclareMathOperator{\Gl}{GL} \DeclareMathOperator{\Gl}{GL}
\DeclareMathOperator{\Sl}{SL} \DeclareMathOperator{\Sl}{SL}
@ -121,27 +122,28 @@
%\input{myNotes} %\input{myNotes}
\section{Basic definitions \hfill\DTMdate{2019-02-25}} \section{Basic definitions \hfill\DTMdate{2019-02-25}}
\input{lec_1.tex} %\input{lec_1.tex}
\section{Alexander polynomial \hfill\DTMdate{2019-03-04}} \section{Alexander polynomial \hfill\DTMdate{2019-03-04}}
\input{lec_2.tex} %\input{lec_2.tex}
%add Hurewicz theorem? %add Hurewicz theorem?
\section{Examples of knot classes \section{Examples of knot classes
\hfill\DTMdate{2019-03-11}} \hfill\DTMdate{2019-03-11}}
\input{lec_3.tex} %\input{lec_3.tex}
\section{Concordance group \hfill\DTMdate{2019-03-18}} \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}} \section{Genus $g$ cobordism \hfill\DTMdate{2019-03-25}}
\input{lec_5.tex} %\input{lec_5.tex}
\section{\hfill\DTMdate{2019-04-08}} \section{\hfill\DTMdate{2019-04-08}}
\input{lec_6.tex} \input{lec_6.tex}
\section{\hfill\DTMdate{2019-04-15}} \section{\hfill\DTMdate{2019-04-15}}
???????????????????\\
\begin{theorem} \begin{theorem}
Suppose that $K \subset S^3$ is a slice knot (i.e. $K$ bound a disk in $B^4$). 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: 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. A = V + V^T.
\end{align*} \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*} \begin{align*}
H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\\ H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\\
A \longrightarrow BAC^T \quad \text{Smith normal form} A \longrightarrow BAC^T \quad \text{Smith normal form}
@ -511,7 +545,7 @@ $H_1(\bar{X}$
field of fractions field of fractions
\section{\hfill\DTMdate{2019-06-03}} \section{Surgery \hfill\DTMdate{2019-06-03}}
\begin{theorem} \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: 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. Recall that if $u(K)=u$ then $K$ bounds a disk $\Delta$ with $u$ ordinary double points.
\\ \\
\noindent \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} \end{proof}
???????????????????\\ %Tim D. Cochran and Peter Teichner
\begin{example} \begin{example}
The knot $8_{20}$ is slice: $\sigma \equiv 0$ almost everywhere but $\sigma(e^{\frac{ 2\pi i}{6}}) = + 1$. The knot $8_{20}$ is slice: $\sigma \equiv 0$ almost everywhere but $\sigma(e^{\frac{ 2\pi i}{6}}) = + 1$.
\end{example} \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 %Journal-ref: Comment. Math. Helv. 79 (2004) 105-123
\subsection*{Surgery} \subsection*{Surgery}
%Rolfsen, geometric group theory, Diffeomorpphism of a torus, Mapping class group %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. 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 homology group: Consider an induced map on the homology group:
\begin{align*} \begin{align*}
H_1(S^1 \times S^1, \mathbb{Z}) \ni \phi_* (\alpha) &= p\alpha + q \beta, \quad p, q \in \mathbb{Z},\\ 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}, \\ \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} \begin{pmatrix}
p & q\\ p & q\\
r & s r & s
\end{pmatrix} \end{pmatrix}.
\end{align*} \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 $\phi_* \in \Sl(2, \mathbb{Z})$. 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} \end{theorem}
\vspace{10cm} \vspace{10cm}
\begin{theorem} \begin{theorem}
Every such a matrix can be realized as a torus. Every such a matrix can be realized as a torus.
@ -564,6 +597,15 @@ S^1 \times \pt &\longrightarrow \pt \times S^1 \\
\item \item
\end{enumerate} \end{enumerate}
\end{proof} \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}