138 lines
3.9 KiB
TeX
138 lines
3.9 KiB
TeX
???????????????????\\
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\begin{theorem}
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Suppose that $K \subset S^3$ is a slice knot (i.e. $K$ bound a disk in $B^4$).
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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:
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\\??????????????? T ????????
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\begin{align}
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PVP^{-1} =
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\begin{pmatrix}
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0 & A\\
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B & C
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\end{pmatrix}, \quad A, C, C \in M_{g \times g} (\mathbb{Z})
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\end{align}
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\end{theorem}
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In other words you can find rank $g$ direct summand $\mathcal{Z}$ of $H_1(F)$ \\
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????????????\\
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such that for any
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$\alpha, \beta \in \mathcal{L}$ the linking number $\Lk (\alpha, \beta^+) = 0$.
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\begin{definition}
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An abstract Seifert matrix (i. e.
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\end{definition}
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Choose a basis $(b_1, ..., b_i)$ \\
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???\\
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of $H_2(Y, \mathbb{Z}$, then $A = (b_i, b_y)$ \\??\\ is a matrix of intersection form:
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\begin{align*}
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\quot{\mathbb{Z}^n}{A\mathbb{Z}^n} \cong H_1(Y, \mathbb{Z}).
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\end{align*}
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In particular $\vert \det A\vert = \# H_1(Y, \mathbb{Z})$.\\
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That means - what is happening on boundary is a measure of degeneracy.
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\begin{center}
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\begin{tikzcd}
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[
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column sep=tiny,
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row sep=small,
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ar symbol/.style =%
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{draw=none,"\textstyle#1" description,sloped},
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isomorphic/.style = {ar symbol={\cong}},
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]
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H_1(Y, \mathbb{Z}) &
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\times \quad H_1(Y, \mathbb{Z})&
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\longrightarrow &
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\quot{\mathbb{Q}}{\mathbb{Z}}
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\text{ - a linking form}
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\\
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\quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] &
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\quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] &\\
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\end{tikzcd}
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$(a, b) \mapsto aA^{-1}b^T$
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\end{center}
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?????????????????????????????????\\
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\noindent
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The intersection form on a four-manifold determines the linking on the boundary. \\
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\noindent
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\begin{fact}
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Let $K \in S^1$ be a knot, $\Sigma(K)$ its double branched cover. If $V$ is a Seifert matrix for $K$, then
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\[H_1(\Sigma(K), \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\ \ ,\] where
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$A = V \times V^T$ and $n = \rank V$.
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\end{fact}
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%\input{ink_diag}
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\begin{figure}[h]
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\fontsize{20}{10}\selectfont
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\centering{
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\def\svgwidth{\linewidth}
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\resizebox{0.5\textwidth}{!}{\input{images/ball_4.pdf_tex}}
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\caption{Pushing the Seifert surface in 4-ball.}
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\label{fig:pushSeifert}
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}
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\end{figure}
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\noindent
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Let $X$ be the four-manifold obtained via the double branched cover of $B^4$ branched along $\widetilde{\Sigma}$.
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\begin{fact}
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\begin{itemize}
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\item $X$ is a smooth four-manifold,
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\item $H_1(X, \mathbb{Z}) =0$,
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\item $H_2(X, \mathbb{Z}) \cong \mathbb{Z}^n$
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\item The intersection form on $X$ is $V + V^T$.
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\end{itemize}
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\end{fact}
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\begin{figure}[h]
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\fontsize{20}{10}\selectfont
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\centering{
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\def\svgwidth{\linewidth}
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\resizebox{0.5\textwidth}{!}{\input{images/ball_4_pushed_cycle.pdf_tex}}
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\caption{Cycle pushed in 4-ball.}
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\label{fig:pushCycle}
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}
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\end{figure}
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\noindent
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Let $Y = \Sigma(K)$. Then:
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\begin{align*}
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H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) &\longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}
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\\
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(a,b) &\mapsto a A^{-1} b^{T},\qquad
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A = V + V^T.
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\end{align*}
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????????????????????????????
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\\
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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 $U = \bigoplus_p U_p$.
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\begin{example}
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\begin{align*}
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\text{If } U &=
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\mathbb{Z}_3 \oplus
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\mathbb{Z}_{45} \oplus
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\mathbb{Z}_{15} \oplus
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\mathbb{Z}_{75}
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\text{ then }\\
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U_3 &=
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\mathbb{Z}_3 \oplus
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\mathbb{Z}_9 \oplus
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\mathbb{Z}_3 \oplus
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\mathbb{Z}_3
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\text{ and }\\
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U_5 &=
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(e) \oplus
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\mathbb{Z}_5 \oplus
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\mathbb{Z}_5 \oplus
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\mathbb{Z}_{25}.
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\end{align*}
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\end{example}
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\begin{lemma}
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Suppose $x \in U_{p_1}$, $y \in U_{p_2}$ and $p_1 \neq p_2$. Then $<x, y > = 0$.
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\end{lemma}
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\begin{proof}
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\begin{align*}
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x \in U_{p_1}
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\end{align*}
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\end{proof}
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\begin{align*}
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H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\\
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A \longrightarrow BAC^T \quad \text{Smith normal form}
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\end{align*}
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???????????????????????\\
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In general
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%no lecture at 29.04
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