small progress
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Maria Marchwicka
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@ -277,5 +277,5 @@ H_1(M) \cong \mathbb{Z} \Longrightarrow \lambda \in \ker ( \pi_1(S^1 \times S^1)
|
|||
\label{fig:meridian_and_longitude}
|
||||
\end{figure}
|
||||
Choose a meridian $\mu$ such that $\Lk (\mu, K) = 1$. Recall the definition of linking number via homology group (Definition \ref{def:lk_via_homo}).
|
||||
$[\mu]$ represents the generator of $H_1(S^3\setminus K, \mathbb{X})$. From definition of $\lambda$ we know that $\lambda$ is trivial in $H_1(M)$ ($\Lk(\lambda, K) =0$, therefore $[\lambda]$ was trivial in $pi_1(M)$). If $K$ is non-trivial then $\lambda$ is non-trivial in $\pi_1(M)$, but it is trivial in $H_1(M)$.
|
||||
$[\mu]$ represents the generator of $H_1(S^3\setminus K, \mathbb{Z})$. From definition of $\lambda$ we know that $\lambda$ is trivial in $H_1(M)$ ($\Lk(\lambda, K) =0$, therefore $[\lambda]$ was trivial in $pi_1(M)$). If $K$ is non-trivial then $\lambda$ is non-trivial in $\pi_1(M)$, but it is trivial in $H_1(M)$.
|
||||
\end{proof}
|
||||
|
|
|
|||
10
lec_3.tex
10
lec_3.tex
|
|
@ -1,4 +1,4 @@
|
|||
\subsection{Algebraic knot}
|
||||
\subsection{Algebraic knots}
|
||||
\noindent
|
||||
Suppose $F: \mathbb{C}^2 \rightarrow \mathbb{C}$ is a polynomial and $F(0) = 0$. Let take small small sphere $S^3$ around zero. This sphere intersect set of roots of $F$ (zero set of $F$) transversally and by the implicit function theorem the intersection is a manifold.
|
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The dimension of sphere is $3$ and $F^{-1}(0)$ has codimension $2$.
|
||||
|
|
@ -31,7 +31,7 @@ algebraic link (in older books on knot theory there is another notion of algebra
|
|||
\begin{example}
|
||||
Let $p$ and $q$ be coprime numbers such that $p<q$ and $p,q>1$. \\
|
||||
Zero is an isolated singular point ($\bigtriangledown F(0) = 0$). $F$ is quasi - homogeneous polynomial, so the isotopy class of the link doesn't depend on the choice of a sphere.
|
||||
Consider $S^3 = \{ (z, w) \in \mathbb{C} : \max( \vert z \vert, \vert w \vert ) = \varepsilon$.
|
||||
Consider $S^3 = \{ (z, w) \in \mathbb{C} : \max( \vert z \vert, \vert w \vert )\} = \varepsilon$.
|
||||
The intersection
|
||||
$F^{-1}(0) \cap S^3$ is a torus $T(p, q)$.
|
||||
\\???????????????????
|
||||
|
|
@ -128,9 +128,9 @@ $g_3(K_1 \# K_2) = g_3(K_1) + g_3(K_2)$.
|
|||
\fontsize{12}{10}\selectfont
|
||||
\centering{
|
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\def\svgwidth{\linewidth}
|
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\resizebox{1\textwidth}{!}{\input{images/satellite.pdf_tex}}
|
||||
}
|
||||
\caption{Whitehead double satellite knot. Its pattern knot embedded non-trivially in an unknotted solid torus $T$ (e.i. $K \not\subset S^3\subset T$) and pattern in a companion knot.}
|
||||
\resizebox{1\textwidth}{!}{\input{images/satellite.pdf_tex}}}
|
||||
\caption{Whitehead double satellite knot.\\
|
||||
The pattern knot embedded non-trivially in an unknotted solid torus $T$ (e.i. $K \not\subset S^3\subset T$) on the left and the pattern in a companion knot - trefoil - on the right.}
|
||||
\label{fig:sattelite}
|
||||
\end{figure}
|
||||
\noindent
|
||||
|
|
|
|||
|
|
@ -65,7 +65,7 @@ Remark: $K \sim K^{\prime} \Leftrightarrow K \# -K^{\prime}$ is slice.
|
|||
\def\svgwidth{\linewidth}
|
||||
\resizebox{0.5\textwidth}{!}{\input{images/ball_4_pushed_seifert.pdf_tex}}
|
||||
}
|
||||
\caption{$Y = F \cup \Sigma$ is a smooth close surface.}
|
||||
\caption{$Y = F \cup \Sigma$ is a smooth closed surface.}
|
||||
\label{fig:closed_surface}
|
||||
\end{figure}
|
||||
\noindent
|
||||
|
|
|
|||
134
lec_5.tex
134
lec_5.tex
|
|
@ -1,15 +1,14 @@
|
|||
\subsection{Slice knots and metabolic form}
|
||||
\begin{theorem}
|
||||
\label{the:sign_slice}
|
||||
If $K$ is slice,
|
||||
then $\sigma_K(t)
|
||||
= \sign ( (1 - t)S +(1 - \bar{t})S^T)$
|
||||
is zero except possibly of finitely many points and $\sigma_K(-1) = \sign(S + S^T) \neq 0$.
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
\noindent
|
||||
We will use the following lemma.
|
||||
\begin{lemma}
|
||||
\label{lem:metabolic}
|
||||
If $V$ is a Hermitian matrix ($\bar{V} = V^T$), $V$ is of size $2n \times 2n$ and
|
||||
If $V$ is a Hermitian matrix ($\bar{V} = V^T$), $V$ is of size $2n \times 2n$,
|
||||
$
|
||||
V = \begin{pmatrix}
|
||||
0 & A \\
|
||||
|
|
@ -25,11 +24,16 @@ $\begin{pmatrix}
|
|||
\end{pmatrix}$ with half-dimensional null-space.
|
||||
\end{definition}
|
||||
\noindent
|
||||
In other words: non-degenerate metabolic hermitian form has vanishing signature.\\
|
||||
We note that $\det(S + S^T) \neq 0$. Hence $\det ( (1 - t) S + (1 - \bar{t})S^T)$ is not identically zero on $S^1$, so it is non-zero except possibly at finitely many points. We apply the Lemma \ref{lem:metabolic}. \\
|
||||
Let $t \in S^1 \setminus \{1\}$. Then:
|
||||
Theorem \ref{the:sign_slice} can be also express as follow:
|
||||
non-degenerate metabolic hermitian form has vanishing signature.
|
||||
\begin{proof}
|
||||
\noindent
|
||||
We note that $\det(S + S^T) \neq 0$. Hence $\det ( (1 - t) S + (1 - \bar{t})S^T)$ is not identically zero on $S^1$, so it is non-zero except possibly at finitely many points. We apply the Lemma \ref{lem:metabolic}.
|
||||
\\
|
||||
Let $t \in S^1 \setminus \{1\}$.
|
||||
Then:
|
||||
\begin{align*}
|
||||
&\det((1 - t) S + (1 - \bar{t}) S^T) =
|
||||
\det((1 - t) S + (1 - \bar{t}) S^T) =&
|
||||
\det((1 - t) S + (t\bar{t} - \bar{t}) S^T) =\\
|
||||
&\det((1 - t) (S - \bar{t} - S^T)) =
|
||||
\det((1 -t)(S - \bar{t} S^T)).
|
||||
|
|
@ -37,71 +41,118 @@ Let $t \in S^1 \setminus \{1\}$. Then:
|
|||
As $\det (S + S^T) \neq 0$, so $S - \bar{t}S^T \neq 0$.
|
||||
\end{proof}
|
||||
\begin{corollary}
|
||||
If $K \sim K^\prime$ then for all but finitely many $t \in S^1 \setminus \{1\}: \sigma_K(t) = \sigma_{K^\prime}(t)$.
|
||||
If $K \sim K^\prime$ then for all but finitely many $t \in S^1 \setminus \{1\}: \sigma_K(t) = -\sigma_{K^\prime}(t)$.
|
||||
\end{corollary}
|
||||
\begin{proof}
|
||||
If $ K \sim K^\prime$ then $K \# K^\prime$ is slice.
|
||||
\[
|
||||
\sigma_{-K^\prime}(t) = -\sigma_{K^\prime}(t)
|
||||
\]
|
||||
\\??????????????\\
|
||||
The signature give a homomorphism from the concordance group to $\mathbb{Z}$.\\
|
||||
??????????????????\\
|
||||
The signature gives a homomorphism from the concordance group to $\mathbb{Z}$.
|
||||
Remark: if $t \in S^1$ is not algebraic over $\mathbb{Z}$, then $\sigma_K(t) \neq 0$
|
||||
(we can is the argument that $\mathscr{C} \longrightarrow \mathbb{Z}$ as well).
|
||||
(we can use the argument that $\mathscr{C} \longrightarrow \mathbb{Z}$ as well).
|
||||
\end{proof}
|
||||
\subsection{Four genus}
|
||||
\begin{figure}[h]
|
||||
\fontsize{20}{10}\selectfont
|
||||
\centering{
|
||||
\def\svgwidth{\linewidth}
|
||||
\resizebox{0.5\textwidth}{!}{\input{images/genus_2_bordism.pdf_tex}}
|
||||
\resizebox{0.7\textwidth}{!}{\input{images/genus_2_bordism.pdf_tex}}
|
||||
}
|
||||
\caption{$K$ and $K^\prime$ are connected by a genus $g$ surface of genus.}\label{fig:genus_2_bordism}
|
||||
\caption{$K$ and $K^\prime$ are connected by a genus $g$ surface.}\label{fig:genus_2_bordism}
|
||||
\end{figure}
|
||||
???????????????????????\\
|
||||
|
||||
\begin{proposition}[Kawauchi inequality]
|
||||
If there exists a genus $g$ surface as in Figure \ref{fig:genus_2_bordism}
|
||||
then for almost all $t \in S^1 \setminus \{1\}$ we have $\vert \sigma_K(t) - \sigma_{K^\prime}(t) \vert \leq 2 g$.
|
||||
then for almost all
|
||||
$t \in S^1 \setminus \{1\}$ we have
|
||||
$\vert
|
||||
\sigma_K(t) - \sigma_{K^\prime}(t)
|
||||
\vert \leq 2 g$.
|
||||
\end{proposition}
|
||||
% Kawauchi Chapter 12 ???
|
||||
% Borodzik 2010 Morse theory for plane algebraic curves
|
||||
\begin{lemma}
|
||||
If $K$ bounds a genus $g$ surface $X \in B^4$ and $S$ is a Seifert form then ${S \in M_{2n \times 2n}}$ has a block structure $\begin{pmatrix}
|
||||
0 & A\\
|
||||
B & C
|
||||
\end{pmatrix}$, where $0$ is $(n - g) \times (n - g)$ submatrix.
|
||||
\end{lemma}
|
||||
??????????????????????\\
|
||||
\begin{align*}
|
||||
\dim H_1(Z) = 2 n\\
|
||||
\dim H_1 (Y) = 2 n + 2 g\\
|
||||
\dim (\ker (H_1, Y) \longrightarrow H_1(\Omega)) = n + g\\
|
||||
Y = X \sum \Sigma
|
||||
\end{align*}
|
||||
|
||||
\begin{proof}
|
||||
\begin{figure}[h]
|
||||
\fontsize{20}{10}\selectfont
|
||||
\centering{
|
||||
\def\svgwidth{\linewidth}
|
||||
\resizebox{0.5\textwidth}{!}{\input{images/genus_bordism_zeros.pdf_tex}}
|
||||
}
|
||||
\caption{There exists a $3$ - manifold $\Omega$ such that $\partial \Omega = X \cup \Sigma$.}\label{fig:omega_in_B_4}
|
||||
\end{figure}
|
||||
\noindent
|
||||
If $\alpha, \beta \in \ker(H_1(\Sigma \longrightarrow H_1(\Omega))$, then ${\Lk(\alpha, \beta^+) = 0}$.
|
||||
Let $K$ be a knot and $\Sigma$ its Seifert surface as in Figure \ref{fig:omega_in_B_4}.
|
||||
There exists a $3$ - submanifold
|
||||
$\Omega$ such that
|
||||
$\partial \Omega = Y = X \cup \Sigma$
|
||||
(by Thom-Pontryagin construction).
|
||||
If $\alpha, \beta \in \ker (H_1(\Sigma) \longrightarrow H_1(\Omega))$,
|
||||
then ${\Lk(\alpha, \beta^+) = 0}$. Now we have to determine the size of the kernel. We know that
|
||||
${\dim H_1(\Sigma) = 2 n}$. When we glue $\Sigma$ (genus $n$) and $X$ (genus $g$) along a circle we get a surface of genus $n + g$. Therefore $\dim H_1 (Y) = 2 n + 2 g$. Then:
|
||||
\[
|
||||
\dim (\ker (H_1(Y) \longrightarrow H_1(\Omega)) = n + g.
|
||||
\]
|
||||
So we have $H_1(W)$ of dimension
|
||||
$2 n + 2 g$
|
||||
- the image of $H_1(Y)$
|
||||
with a subspace
|
||||
corresponding to the image of $H_1(\Sigma)$ with dimension $2 n$ and a subspace corresponding to the kernel
|
||||
of $H_1(Y) \longrightarrow H_1(\Omega)$ of size $n + g$.
|
||||
We consider minimal possible intersection of this subspaces that corresponds to the kernel of the composition $H_1(\Sigma) \longrightarrow H_1(Y) \longrightarrow H_1(\Omega)$. As the first map is injective, elements of the kernel of the composition have to be in the kernel of the second map.
|
||||
So we can calculate:
|
||||
\[
|
||||
\dim \ker (H_1(\Sigma) \longrightarrow H_1(\Omega)) = 2 n + n + g -2 n - 2 g = n - g.
|
||||
\]
|
||||
\end{proof}
|
||||
\begin{corollary}
|
||||
If $t$ is nota ???? of $\det $ ????
|
||||
then $\vert \sigma_K(t) \vert \leq 2g$.\\
|
||||
If $t$ is not a root of
|
||||
$\det S S^T - $ \\
|
||||
????????????????\\
|
||||
then
|
||||
$\vert \sigma_K(t) \vert \leq 2g$.
|
||||
\end{corollary}
|
||||
\noindent
|
||||
If there exists cobordism of genus $g$ between $K$ and $K^\prime$ like shown in Figure \ref{fig:genus_2_bordism}, then $K \# -K^\prime$ bounds a surface of genus $g$ in $B^4$.
|
||||
\begin{fact}
|
||||
If there exists cobordism of genus $g$ between $K$ and $K^\prime$ like shown in Figure \ref{fig:proof_for_bound_disk}, then $K \# -K^\prime$ bounds a surface of genus $g$ in $B^4$.
|
||||
\end{fact}
|
||||
\begin{figure}[H]
|
||||
\fontsize{20}{10}\selectfont
|
||||
\centering{
|
||||
\def\svgwidth{\linewidth}
|
||||
\resizebox{0.7\textwidth}{!}{\input{images/genus_bordism_proof.pdf_tex}}
|
||||
}
|
||||
\caption{If $K$ and $K^\prime$ are connected by a genus $g$ surface, then $K \# -K^\prime$ bounds a genus $g$ surface.}\label{fig:proof_for_bound_disk}
|
||||
\end{figure}
|
||||
|
||||
\begin{definition}
|
||||
The (smooth) four genus $g_4(K)$ is the minimal genus of the surface $\Sigma \in B^4$ such that $\Sigma$ is compact, orientable and $\partial \Sigma = K$.
|
||||
\end{definition}
|
||||
\noindent
|
||||
Remark: $3$ - genus is additive under taking connected sum, but $4$ - genus is not.
|
||||
Remarks:
|
||||
\begin{enumerate}[label={(\arabic*)}]
|
||||
\item
|
||||
$3$ - genus is additive under taking connected sum, but $4$ - genus is not,
|
||||
\item
|
||||
for any knot $K$ we have $g_4(K) \leq g_3(K)$.
|
||||
\end{enumerate}
|
||||
\begin{example}
|
||||
\begin{itemize}
|
||||
\item Let $K = T(2, 3)$. $\sigma(K) = -2$, therefore $T(2, 3)$ isn't a slice knot.
|
||||
\item Let $K$ be a trefoil and $K^\prime$ a mirror of a trefoil. $g_4(K^\prime) = 1$, but $g_(K \# K^\prime) = 0$.
|
||||
\\?????????????????????\\
|
||||
\item Let $K$ be a trefoil and $K^\prime$ a mirror of a trefoil. $g_4(K^\prime) = 1$, but $g_4(K \# K^\prime) = 0$, so we see that $4$-genus isn't additive,
|
||||
\item
|
||||
?????????????\\
|
||||
The equality:
|
||||
the equality:
|
||||
\[
|
||||
g_4(T(p, q) ) = \frac{1}{2} (p - 1) (g -1)
|
||||
\]
|
||||
was conjecture in the '70 and proved by P. Kronheimer and T. Mrówka.
|
||||
was conjecture in the '70 and proved by P. Kronheimer and T. Mrówka (1994).
|
||||
% OZSVATH-SZABO AND RASMUSSEN
|
||||
\end{itemize}
|
||||
\end{example}
|
||||
\begin{proposition}
|
||||
|
|
@ -142,4 +193,15 @@ If $S$ differs from $S^\prime$ by a row extension, then
|
|||
$(1 - t) S + (1 - \bar{t}^{-1}) S^T$ is Witt equivalence to $(1 - t) S^\prime + (1 - t^{-1})S^T$.
|
||||
%???????????????????????????
|
||||
\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: $A^T = A: (a, b) = \bar{(a, b)}$.
|
||||
\\
|
||||
????????????????????????????
|
||||
\\
|
||||
\begin{theorem}[Levine '68]
|
||||
\[
|
||||
W(\mathbb{Z}[t^{\pm 1})
|
||||
\longrightarrow \mathbb{Z}_2^\infty \oplus
|
||||
\mathbb{Z}_4^\infty \oplus
|
||||
\mathbb{Z}
|
||||
\]
|
||||
\end{theorem}
|
||||
|
|
|
|||
147
lec_6.tex
147
lec_6.tex
|
|
@ -0,0 +1,147 @@
|
|||
$X$ is a closed orientable four-manifold. Assume $\pi_1(X) = 0$ (it is not needed to define the intersection form). In particular $H_1(X) = 0$.
|
||||
$H_2$ is free (exercise).
|
||||
|
||||
\begin{align*}
|
||||
H_2(X, \mathbb{Z}) \xrightarrow{\text{Poincar\'e duality}} H^2(X, \mathbb{Z} ) \xrightarrow{\text{evaluation}}\Hom(H_2(X, \mathbb{Z}), \mathbb{Z})
|
||||
\end{align*}
|
||||
|
||||
Intersection form:
|
||||
$H_2(X, \mathbb{Z}) \times
|
||||
H_2(X, \mathbb{Z}) \longrightarrow \mathbb{Z}$ is symmetric and non singular.
|
||||
\\
|
||||
Let $A$ and $B$ be closed, oriented surfaces in $X$.
|
||||
\\
|
||||
\begin{figure}[h]
|
||||
\fontsize{20}{10}\selectfont
|
||||
\centering{
|
||||
\def\svgwidth{\linewidth}
|
||||
\resizebox{0.5\textwidth}{!}{\input{images/intersection_form_A_B.pdf_tex}}
|
||||
}
|
||||
\caption{$T_X A + T_X B = T_X X$
|
||||
}\label{fig:torus_alpha_beta}
|
||||
\end{figure}
|
||||
???????????????????????
|
||||
\begin{align*}
|
||||
x \in A \cap B\\
|
||||
T_XA \oplus T_X B = T_X X\\
|
||||
\{\epsilon_1, \dots , \epsilon_n \} = A \cap C\\
|
||||
A \cdot B = \sum^n_{i=1} \epsilon_i
|
||||
\end{align*}
|
||||
\begin{proposition}
|
||||
Intersection form $A \cdot B$ doesn't depend of choice of $A$ and $B$ in their homology classes:
|
||||
\[
|
||||
[A], [B] \in H_2(X, \mathbb{Z}).
|
||||
\]
|
||||
\end{proposition}
|
||||
\noindent
|
||||
\\
|
||||
|
||||
If $M$ is an $m$ - dimensional close, connected and orientable manifold, then $H_m(M, \mathbb{Z})$ and the orientation if $M$ determined a cycle $[M] \in H_m(M, \mathbb{Z})$, called the fundamental cycle.
|
||||
\begin{example}
|
||||
If $\omega$ is an $m$ - form then:
|
||||
\[
|
||||
\int_M \omega = [\omega]([M]), \quad [\omega] \in H^m_\Omega(M), \ [M] \in H_m(M).
|
||||
\]
|
||||
|
||||
\end{example}
|
||||
????????????????????????????????????????????????
|
||||
\begin{figure}[h]
|
||||
\fontsize{20}{10}\selectfont
|
||||
\centering{
|
||||
\def\svgwidth{\linewidth}
|
||||
\resizebox{0.8\textwidth}{!}{\input{images/torus_alpha_beta.pdf_tex}}
|
||||
}
|
||||
\caption{$\beta$ cross $3$ times the disk bounded by $\alpha$.
|
||||
$T_X \alpha + T_X \beta = T_X \Sigma$
|
||||
}\label{fig:torus_alpha_beta}
|
||||
\end{figure}
|
||||
\begin{example}
|
||||
?????????????????????????\\
|
||||
Let $X = S^2 \times S^2$.
|
||||
We know that:
|
||||
\begin{align*}
|
||||
&H_2(S^2, \mathbb{Z}) =\mathbb{Z}\\
|
||||
&H_1(S^2, \mathbb{Z}) = 0\\
|
||||
&H_0(S^2, \mathbb{Z}) =\mathbb{Z}
|
||||
\end{align*}
|
||||
We can construct a long exact sequence for a pair:
|
||||
\begin{align*}
|
||||
&H_2(\partial X) \to H_2(X)
|
||||
\to H_2(X, \partial X) \to \\
|
||||
\to &H_1(\partial X) \to H_1(X) \to H_1(X, \partial X) \to
|
||||
\end{align*}
|
||||
????????????????????\\
|
||||
Simple case $H_1(\partial X)$ \\????????????\\
|
||||
is torsion.
|
||||
$H_2(\partial X)$ is torsion free (by universal coefficient theorem),\\
|
||||
???????????????????????\\
|
||||
therefore it is $0$.
|
||||
\\?????????????????????\\
|
||||
We know that $b_1(X) = b_2(X)$. Therefore by Poincar\'e duality:
|
||||
\begin{align*}
|
||||
b_1(X) =
|
||||
\dim_{\mathbb{Q}} H_1(X, \mathbb{Q})
|
||||
\overset{\mathrm{PD}}{=}
|
||||
\dim_{\mathbb{Q}} H^2(X, \mathbb{Q}) =
|
||||
\dim_{\mathbb{Q}} H_2(X, \mathbb{Q}) = b_2(X)
|
||||
\end{align*}
|
||||
???????????????????????????????\\
|
||||
$H_2(X, \mathbb{Z})$ is torsion free and
|
||||
$H_2(X_1, \mathbb{Q}) = 0$, therefore $H_2(X, \mathbb{Z}) = 0$.
|
||||
The map
|
||||
$H_2(X, \mathbb{Z}) \longrightarrow H_2(X, \partial X, \mathbb{Z})$ is a monomorphism. \\??????????\\ (because it is an isomorphism after tensoring by $\mathbb{Q}$.
|
||||
\\
|
||||
Suppose $\alpha_1, \dots, \alpha_n$ is a basis of $H_2(X, \mathbb{Z})$.
|
||||
Let $A$ be the intersection matrix in this basis. Then:
|
||||
\begin{enumerate}
|
||||
\item
|
||||
A has integer coefficients,
|
||||
\item
|
||||
$\det A \neq 0$,
|
||||
\item
|
||||
$\vert \det A \vert =
|
||||
\vert H_1 (\partial X, \mathbb{Z}) \vert =
|
||||
\vert \coker H_2(X) \longrightarrow H_2(X, \partial X) \vert$.
|
||||
\end{enumerate}
|
||||
\end{example}
|
||||
???????????????????\\
|
||||
If $CVC^T = W$, then for
|
||||
$\binom{a}{b} = C^{-1} \binom{1}{0}$ we have $\binom{a}{b} $ \\
|
||||
????????????????\\
|
||||
$\omega \binom{a}{b} = \binom{1}{0} u \binom{1}{0} = 1$.
|
||||
|
||||
\begin{theorem}[Whitehead]
|
||||
Any non-degenerate form
|
||||
\[
|
||||
A : \mathbb{Z}^4 \times \mathbb{Z}^4 \longrightarrow \mathbb{Z}
|
||||
\]
|
||||
can be realized as an intersection form of a simple connected $4$-dimensional manifold.
|
||||
\end{theorem}
|
||||
??????????????????????????
|
||||
\begin{theorem}[Donaldson, 1982]
|
||||
If $A$ is an even definite intersection form of a smooth $4$-manifold then it is diagonalizable over $\mathbb{Z}$.
|
||||
\end{theorem}
|
||||
??????????????????????????
|
||||
??????????????????????????
|
||||
??????????????????????????
|
||||
??????????????????????????
|
||||
\begin{definition}
|
||||
even define
|
||||
\end{definition}
|
||||
Suppose $X$ us $4$ -manifold with a boundary such that $H_1(X) = 0$.
|
||||
|
||||
%$A \cdot B$ gives the pairing as ??
|
||||
\begin{proof}
|
||||
Obviously:
|
||||
\[H_1(\partial X, \mathbb{Z}) = \coker H_2(X) \longrightarrow H_2(X, \partial X) = \quot{H_2(X, \partial X)}{H_2(X)}.
|
||||
\]
|
||||
Let $A$ be an $n \times n$ matrix. $A$ determines a \\
|
||||
??????????????/\\
|
||||
\begin{align*}
|
||||
\mathbb{Z}^n \longrightarrow \Hom (\mathbb{Z}^n, \mathbb{Z})\\
|
||||
a \mapsto (b \mapsto b^T A a)\\
|
||||
\vert \coker A \vert = \vert \det A \vert
|
||||
\end{align*}
|
||||
all homomorphisms $b = (b_1, \dots, b_n) $???????\\?????????\\
|
||||
|
||||
\end{proof}
|
||||
|
|
@ -8,6 +8,7 @@
|
|||
|
||||
\usepackage[english]{babel}
|
||||
|
||||
\usepackage{caption}
|
||||
\usepackage{comment}
|
||||
\usepackage{csquotes}
|
||||
|
||||
|
|
@ -23,6 +24,7 @@
|
|||
\usepackage{mathtools}
|
||||
|
||||
\usepackage{pict2e}
|
||||
\usepackage[section]{placeins}
|
||||
\usepackage[pdf]{pstricks}
|
||||
|
||||
\usepackage{tikz}
|
||||
|
|
@ -84,9 +86,9 @@
|
|||
|
||||
\DeclareMathOperator{\Hom}{Hom}
|
||||
\DeclareMathOperator{\rank}{rank}
|
||||
\DeclareMathOperator{\coker}{coker}
|
||||
\DeclareMathOperator{\ord}{ord}
|
||||
\DeclareMathOperator{\mytop}{top}
|
||||
|
||||
\DeclareMathOperator{\Gl}{GL}
|
||||
\DeclareMathOperator{\Sl}{SL}
|
||||
\DeclareMathOperator{\Lk}{lk}
|
||||
|
|
@ -126,80 +128,18 @@
|
|||
%add Hurewicz theorem?
|
||||
|
||||
|
||||
\section{\hfill\DTMdate{2019-03-11}}
|
||||
\section{Examples of knot classes
|
||||
\hfill\DTMdate{2019-03-11}}
|
||||
\input{lec_3.tex}
|
||||
|
||||
\section{Concordance group \hfill\DTMdate{2019-03-18}}
|
||||
\input{lec_4.tex}
|
||||
|
||||
|
||||
\section{\hfill\DTMdate{2019-03-25}}
|
||||
\section{Genus $g$ cobordism \hfill\DTMdate{2019-03-25}}
|
||||
\input{lec_5.tex}
|
||||
|
||||
\section{\hfill\DTMdate{2019-04-08}}
|
||||
%
|
||||
%
|
||||
$X$ is a closed orientable four-manifold. Assume $\pi_1(X) = 0$ (it is not needed to define the intersection form). In particular $H_1(X) = 0$.
|
||||
$H_2$ is free (exercise).
|
||||
\begin{align*}
|
||||
H_2(X, \mathbb{Z}) \xrightarrow{\text{Poincar\'e duality}} H^2(X, \mathbb{Z} ) \xrightarrow{\text{evaluation}}\Hom(H_2(X, \mathbb{Z}), \mathbb{Z})
|
||||
\end{align*}
|
||||
Intersection form:
|
||||
$H_2(X, \mathbb{Z}) \times
|
||||
H_2(X, \mathbb{Z}) \longrightarrow \mathbb{Z}$ - symmetric, non singular.
|
||||
\\
|
||||
Let $A$ and $B$ be closed, oriented surfaces in $X$.
|
||||
\begin{proposition}
|
||||
$A \cdot B$ doesn't depend of choice of $A$ and $B$ in their homology classes:
|
||||
\[
|
||||
[A], [B] \in H_2(X, \mathbb{Z}).
|
||||
\]
|
||||
\end{proposition}
|
||||
\noindent
|
||||
\\
|
||||
|
||||
If $M$ is an $m$ - dimensional close, connected and orientable manifold, then $H_m(M, \mathbb{Z})$ and the orientation if $M$ determined a cycle $[M] \in H_m(M, \mathbb{Z})$, called the fundamental cycle.
|
||||
\begin{example}
|
||||
If $\omega$ is an $m$ - form then:
|
||||
\[
|
||||
\int_M \omega = [\omega]([M]), \quad [\omega] \in H^m_\Omega(M), \ [M] \in H_m(M).
|
||||
\]
|
||||
|
||||
\end{example}
|
||||
????????????????????????????????????????????????
|
||||
\begin{figure}[h]
|
||||
\fontsize{20}{10}\selectfont
|
||||
\centering{
|
||||
\def\svgwidth{\linewidth}
|
||||
\resizebox{0.8\textwidth}{!}{\input{images/torus_alpha_beta.pdf_tex}}
|
||||
}
|
||||
\caption{$\beta$ cross $3$ times the disk bounded by $\alpha$.
|
||||
$T_X \alpha + T_X \beta = T_Z \Sigma$
|
||||
}\label{fig:torus_alpha_beta}
|
||||
\end{figure}
|
||||
|
||||
\begin{theorem}
|
||||
Any non-degenerate form
|
||||
\[
|
||||
A : \mathbb{Z}^n \times \mathbb{Z}^n \longrightarrow \mathbb{Z}
|
||||
\]
|
||||
can be realized as an intersection form of a simple connected $4$-dimensional manifold.
|
||||
\end{theorem}
|
||||
??????????????????????????
|
||||
\begin{theorem}[Donaldson, 1982]
|
||||
If $A$ is an even defined intersection form of a smooth $4$-manifold then it is diagonalizable over $\mathbb{Z}$.
|
||||
\end{theorem}
|
||||
??????????????????????????
|
||||
??????????????????????????
|
||||
??????????????????????????
|
||||
??????????????????????????
|
||||
\begin{definition}
|
||||
even define
|
||||
\end{definition}
|
||||
Suppose $X$ us $4$ -manifold with a boundary such that $H_1(X) = 0$.
|
||||
|
||||
%$A \cdot B$ gives the pairing as ??
|
||||
|
||||
\input{lec_6.tex}
|
||||
|
||||
\section{\hfill\DTMdate{2019-04-15}}
|
||||
\begin{theorem}
|
||||
|
|
@ -341,14 +281,17 @@ a &\mapsto (a, \_) H_2(M, \mathbb{Z})
|
|||
\end{align*}
|
||||
has coker precisely $H_1(Y, \mathbb{Z})$.
|
||||
\\???????????????\\
|
||||
Let $K \subset S^3$ be a knot, \\
|
||||
$X = S^3 \setminus K$ - a knot complement, \\
|
||||
$\widetilde{X} \xrightarrow{\enspace \rho \enspace} X$ - an infinite cyclic cover (universal abelian cover).
|
||||
Let $K \subset S^3$ be a knot, $X = S^3 \setminus K$ a knot complement and
|
||||
$\widetilde{X} \xrightarrow{\enspace \rho \enspace} X$ an infinite cyclic cover (universal abelian cover). By Hurewicz theorem we know that:
|
||||
\begin{align*}
|
||||
\pi_1(X) \longrightarrow \quot{\pi_1(X)}{[\pi_1(X), \pi_1(X)]} = H_1(X, \mathbb{Z} ) \cong \mathbb{Z}
|
||||
\end{align*}
|
||||
????????????????????????????????????????????????????????????????????????\\
|
||||
????????????????????????????????????????????????????????????????????????\\
|
||||
????????????????????????????????????????????????????????????????????????\\
|
||||
????????????????????????????????????????????????????????????????????????\\
|
||||
$C_{*}(\widetilde{X})$ has a structure of a $\mathbb{Z}[t, t^{-1}] \cong \mathbb{Z}[\mathbb{Z}]$ module. \\
|
||||
$H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}])$ - Alexander module, \\
|
||||
Let $H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}])$ be the Alexander module of the knot $K$ with an intersection form:
|
||||
\begin{align*}
|
||||
H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \times
|
||||
H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}[t, t^{-1}]}
|
||||
|
|
@ -365,27 +308,31 @@ H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \longrightarrow \quot{\mathbb{Q}}{\mat
|
|||
\begin{align*}
|
||||
H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \times
|
||||
H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) &\longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}[t, t^{-1}]}\\
|
||||
(\alpha, \beta) &\mapsto \alpha^{-1}(t -1)(tV - V^T)^{-1}\beta
|
||||
(\alpha, \beta) \quad &\mapsto \alpha^{-1}(t -1)(tV - V^T)^{-1}\beta
|
||||
\end{align*}
|
||||
\end{fact}
|
||||
\noindent
|
||||
Note that $\mathbb{Z}$ is not PID. Therefore we don't have primer decomposition of this moduli. We can simplify this problem by replacing $\mathbb{Z}$ by $\mathbb{R}$. We lose some date by doing this transition.
|
||||
Note that $\mathbb{Z}$ is not PID.
|
||||
Therefore we don't have primary decomposition of this module.
|
||||
We can simplify this problem by replacing $\mathbb{Z}$ by $\mathbb{R}$. We lose some date by doing this transition, but we can
|
||||
\begin{align*}
|
||||
&\xi \in S^1 \setminus \{ \pm 1\}
|
||||
\quad
|
||||
\xi \in S^1 \setminus \{ \pm 1\}
|
||||
&\quad
|
||||
p_{\xi} =
|
||||
(t - \xi)(t - \xi^{-1}) t^{-1}
|
||||
\\
|
||||
&\xi \in \mathbb{R} \setminus \{ \pm 1\}
|
||||
\quad
|
||||
\xi \in \mathbb{R} \setminus \{ \pm 1\}
|
||||
&\quad
|
||||
q_{\xi} = (t - \xi)(t - \xi^{-1}) t^{-1}
|
||||
\\
|
||||
&
|
||||
\xi \notin \mathbb{R} \cup S^1 \quad
|
||||
q_{\xi} = (t - \xi)(t - \overbar{\xi})(t - \xi^{-1})(t - \overbar{\xi}^{-1}) t^{-2}\\
|
||||
&
|
||||
\Lambda = \mathbb{R}[t, t^{-1}]\\
|
||||
&\text{Then: } H_1(\widetilde{X}, \Lambda) \cong \bigoplus_{\substack{\xi \in S^1 \setminus \{\pm 1 \}\\ k\geq 0}}
|
||||
\xi \notin \mathbb{R} \cup S^1
|
||||
&\quad
|
||||
q_{\xi} = (t - \xi)(t - \overbar{\xi})(t - \xi^{-1})
|
||||
(t - \overbar{\xi}^{-1}) t^{-2}
|
||||
\end{align*}
|
||||
Let $\Lambda = \mathbb{R}[t, t^{-1}]$. Then:
|
||||
\begin{align*}
|
||||
H_1(\widetilde{X}, \Lambda) \cong \bigoplus_{\substack{\xi \in S^1 \setminus \{\pm 1 \}\\ k\geq 0}}
|
||||
( \quot{\Lambda}{p_{\xi}^k })^{n_k, \xi}
|
||||
\oplus
|
||||
\bigoplus_{\substack{\xi \notin S^1 \\ l\geq 0}}
|
||||
|
|
@ -557,7 +504,9 @@ $2 \sum\limits_{k_i \text{ odd}} \epsilon_i$. The peak of the signature function
|
|||
|
||||
....
|
||||
\begin{definition}
|
||||
A square hermitian matrix $A$ of size $n$.
|
||||
A square hermitian matrix $A$ of size $n$ with coefficients in \\
|
||||
the Blanchfield pairing if:
|
||||
$H_1(\bar{X}$
|
||||
\end{definition}
|
||||
|
||||
field of fractions
|
||||
|
|
|
|||
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Reference in New Issue