Correcciones menores
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@ -32,7 +32,7 @@ decorations.markings, calc}
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%opening
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\title[Retraction of the complement of a smooth curve]{Retraction of the complement of smooth projective hypersurfaces to a $n$-dimensional $\Delta$-complex.}
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\author[E. Artal]{Enrique Artal Bartolo}
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\author[E. Artal]{Enrique Artal Bartolo}
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\address{Departamento de Matem\'{a}ticas, IUMA \\
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Universidad de Zaragoza \\
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C.~Pedro Cerbuna 12, 50009, Zaragoza, Spain}
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@ -70,7 +70,7 @@ The retraction is performed in several steps, using the branched cover structure
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\section{Definition of the problem}
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Given $C$ an affine hypersurface in $\CC^2$ (i.e. a curve), Libgober gave in \cite{libgober1986} a construction of a $2$-dimensional CW-complex that is a deformation retract of its complement. However, it is not clear how to extend that construction to the projective case (that is a curve in $\CC\PP^2$).
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In this work, we give a different construction to produce a specific $\Delta$-complex that is a deformation retract of the complement of a Fermat hypersurface in $\CC\PP^n$. Since any smooth hypersurface is isotopic to a Fermat hypersurface, this particular case solves the problemfor a general smooth hypersurface.
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In this work, we give a different construction to produce a specific $\Delta$-complex that is a deformation retract of the complement of a Fermat hypersurface in $\CC\PP^n$. Since any smooth hypersurface is isotopic to a Fermat hypersurface, this particular case solves the problem for a general smooth hypersurface.
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\section{Smooth hypersurfaces}
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@ -97,7 +97,7 @@ We then have a commutative diagram:
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% K & \longhookrightarrow & \PP^n\setminus C_d \\
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% \end{array}
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\begin{tikzcd}
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S_d\rar[hookrightarrow]\dar["\proy{}"]& M_d\dar["\proy{}"]\\
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S_d\rar[hookrightarrow]\dar["\proy{}"]& M_d\dar["\proy{}"]\\
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K\rar[hookrightarrow] & \PP^n\setminus C_d.
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\end{tikzcd}
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\end{equation*}
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@ -119,7 +119,7 @@ i.e. $x^r,x^i\in\RR$ and $x=x^r + i x^i$. This notation also applies to vectors
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% M_1 & \longrightarrow & M_1\cap\RR^3&\longrightarrow&S_1 \\
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% \end{array}
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%\end{equation*}
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We will now define a strong retraction map
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We will now define a strong retraction map
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\begin{equation}
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\begin{tikzcd}[row sep=0pt,/tikz/column 1/.append style={anchor=base east}, /tikz/column 2/.append style={anchor=base west}]
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M_1\times{[0,1]}\rar["r_1"]&M_1\\
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@ -140,7 +140,7 @@ Note that for $(x_0,\dots,x_n)\in M_1$, we already have that
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\sum x^i_j & = & 0 \\
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\end{array}
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\end{equation*}
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and therefore $r_1(x_0,\dots,x_n,t)\in M_1$ for all $t\in[0,1]$, and in particular, we get that $r_1(x_0,\dots,x_n,1)=(x^r_0,\dots,x^r_n)\in M_1\cap\RR^{n+1}$ for all $(x_0,\dots,x_n)\in M_1$.
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and therefore $r_1(x_0,\dots,x_n,t)\in M_1$ for all $t\in[0,1]$, and in particular, we get that $r_1(x_0,\dots,x_n,1)=(x^r_0,\dots,x^r_n)\in M_1\cap\RR^{n+1}$ for all $(x_0,\dots,x_n)\in M_1$.
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Let us summarize it.
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@ -155,21 +155,21 @@ The map $r_1$ is a strong deformation retraction from $M_1$ to $M_1\cap\RR^{n+1}
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Now we will define a strong deformation retraction
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\[
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r_2:(M_1 \cap \RR^{n+1})\times[0, 1]\to M_1 \cap \RR^{n+1}
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r_2:(M_1 \cap \RR^{n+1})\times[0, 1]\to M_1 \cap \RR^{n+1}
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\]
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from $M_1 \cap \RR^{n+1}$ to $S_1$.
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Note that $M_1\cap\RR^{n+1}$
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is a hypersurface in $\RR^{n+1}$, in Figure~\ref{fig:triangulo} we can see the example $n=2$.
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is a hypersurface in $\RR^{n+1}$, in Figure~\ref{fig:triangulo} we can see the example $n=2$.
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\begin{lemma}
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The space $S_1$ is a $n$-simplex.
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\end{lemma}
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\begin{proof}
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Let $e_i=(0,\dots,1,\dots,0)$ be the vectors of the canonical base. Then
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Let $e_i=(0,\dots,1,\dots,0)$ be the vectors of the canonical base. Then
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\begin{equation*}
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\begin{array}{lllll}
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S_1&=&\{(x_0,\dots,x_n)\in\RR^{n+1}|\sum x_i=1, x_i\geq1\}&& \\
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&=&\{x_0e_0+\dots+x_n|\sum x_i=1, x_i\geq1\}&=&\Delta_n\\
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&=&\{x_0e_0+\dots+x_ne_n|\sum x_i=1, x_i\geq1\}&=&\Delta_n\\
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\end{array}
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\qedhere
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\end{equation*}
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@ -227,13 +227,13 @@ S_1&=&\{(x_0,\dots,x_n)\in\RR^{n+1}|\sum x_i=1, x_i\geq1\}&& \\
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\end{center}
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\end{figure}
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As we can see in Figure \ref{fig:triangulo}, the hypersurfaces defined by $\{x_i=0\}$ divide the space in $2^{n+1}-1$ sections. We denote each of these sections as $(\pm,\dots,\pm)$ indicating which coordinates are greater or equal to $0$ and which are lower or equal to $0$. For example, the triangle corresponding to $S_1$ would be $(+,\dots,+)$.
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Note that there is no section $(-,\dots,-)$.
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As we can see in Figure \ref{fig:triangulo}, the hyperplanes defined by $\{x_i=0\}$ divide the space $S_1$ in $2^{n+1}-1$ regions. We denote each of these regions as $(\pm,\dots,\pm)$ indicating which coordinates are greater or equal to $0$ and which are lower or equal to $0$. For example, the triangle corresponding to $S_1$ would be $(+,\dots,+)$.
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Note that there is no region $(-,\dots,-)$.
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The retraction in $(+,\dots,+)$ is the identity for any value of $t$. For the rest of the parts, we define a retraction:
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\begin{equation}
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\begin{tikzcd}[row sep=0pt,/tikz/column 1/.append style={anchor=base east}, /tikz/column 2/.append style={anchor=base west}]
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(M_1\cap\RR^{n+1})\times{[0,1]}\rar["r_2"]&M_1\cap\RR^{n+1}\times [0,1]\\
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(M_1\cap\RR^{n+1})\times{[0,1]}\rar["r_2"]&M_1\cap\RR^{n+1}\\
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(Q, t)\rar[mapsto]& t P_Q + (1 - t) Q,
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\end{tikzcd}
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\end{equation}
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@ -243,7 +243,7 @@ The retraction in $(+,\dots,+)$ is the identity for any value of $t$. For the re
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% (x,y,z, t) & \longrightarrow & (1-t)(x,y,z)+tP_{x,y,z} \\
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% \end{array}
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%\end{equation}
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where $P_Q$ is a point depending on $Q\in M_1\cap\RR^{n+1}$.
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where $P_Q$ is a point depending on $Q\in M_1\cap\RR^{n+1}$.
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In general, for a point $Q$ in a region $(\pm,\dots,\pm)$, we want the retraction to take the negative coordinates to $0$ and to change the positive ones such that at the end the sum is $1$. Therefore, we can write the point $P_Q$ as:
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\begin{equation}
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@ -380,8 +380,8 @@ Then, we can see that $G(z,t)^d=\bar{r}(z^d,t)$, since:
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We can the write the lifting of $r_1$ as $\tilde{r}_1((x_0,\dots,x_n),t)=(G(x,t), \dots, G(x_n,t))$.
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Lastly, we are going to see how to lift $r_2$ in order to get a retraction from $M_d\cap\{x^d_0, y^d, x^d_n\in\RR\}$ into $S_d$.
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For a point $P=(x_0\xi_0,\dots,x_n\xi_n)\in M_d\cap\{x^d_0, y^d, x^d_n\in\RR\}$ (where $(x_0,\dots,x_n)$ are real and $\xi_i$ are $d$-th roots of unity), the corresponding point in $M_1\cap\RR^{n+1}$ is $(x^d_0,\dots,x^d_n)$ and therefore the goal point of the retraction $r_2$ will be
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Lastly, we are going to see how to lift $r_2$ in order to get a retraction from $M_d\cap\{x^d_0, \ldots , x^d_n\in\RR\}$ into $S_d$.
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For a point $P=(x_0\xi_0,\dots,x_n\xi_n)\in M_d\cap\{x^d_0, \ldots , x^d_n\in\RR\}$ (where $(x_0,\dots,x_n)$ are real and $\xi_i$ are $d$-th roots of unity), the corresponding point in $M_1\cap\RR^{n+1}$ is $(x^d_0,\dots,x^d_n)$ and therefore the goal point of the retraction $r_2$ will be
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\begin{equation}
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P^d_Q=\left(\frac{x^d_0\chi_0}{1-\sum x^d_j(1-\chi_j)}, \dots,\frac{x^d_n\chi_n}{1-\sum x^d_j(1-\chi_j)} \right)
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\end{equation}
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@ -427,7 +427,7 @@ We have then proved that $S_d$ is a strong deformation retract of $M_d$.
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Since $\lambda$ is a $d-$th root of unity, we get that $G(x,t)=G(\lambda x,t)$. Therefore, we get that $\tilde{r}_1(p,t)=\tilde{r}_1(\lambda p,t)$. It's just left to check that $r_2$ also sends them to the same class.
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If $[x^d_0: \dots: x^d_n]$ is in $(+,\dots,+)$, the retraction is the identity, and therefore both representatives will stay the same and therefore go to the same class.
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For the rest of the regions, we have
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\begin{equation}
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\begin{array}{lll}
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@ -436,7 +436,7 @@ We have then proved that $S_d$ is a strong deformation retract of $M_d$.
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\end{equation}
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Then, for the point $\lambda p$ we get
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Then, for the point $\lambda p$ we get
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\begin{equation}
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\begin{array}{lll}
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\tilde{r}_2([\lambda x_0\xi_0:\dots:\lambda x_n\xi_n], t)&=&\left[\sqrt[d]{(1-t)\lambda^d x^d_0+t\frac{\lambda^d x^d_0\chi_0}{1-\sum \lambda^d x^d_j(1-\chi_j)}}\xi_0:\dots:\sqrt[d]{(1-t)\lambda^d x^d_n+t\frac{\lambda^d x^d_n\chi_n}{1-\sum \lambda^d x^d_j(1-\chi_j)}}\xi_n\right]\\
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@ -461,8 +461,8 @@ We denote as $\Delta$ the $\Delta-$complex $[v_0, \dots, v_n]$ that has $n+1$ ve
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% & & K & \hookrightarrow & \PP^2\setminus C_d \\
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% \end{array}
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\begin{tikzcd}
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\Delta \rar[leftrightarrow]& S_1&\\
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\Delta^d\uar\rar[leftrightarrow]&S_d\uar\rar[hookrightarrow]\dar& M_d\dar\\
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\Delta \rar[leftrightarrow]& S_1&\\
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\Delta^d\uar\rar[leftrightarrow]&S_d\uar\rar[hookrightarrow]\dar& M_d\dar\\
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&K \rar[hookrightarrow]& \PP^n\setminus C_d,
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\end{tikzcd}
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\end{equation*}
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@ -485,7 +485,7 @@ First, we give a detailed description of $\Delta$. We will denote the $0$-simpli
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The face maps of $X$ and $X^{(j_0,\dots,j_i)}$ are:
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\begin{equation}
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\delta^i_k(X^{(j_0,\dots,j_i)}) =[v_{j_0},\dots,\hat{v},\dots,v_{j_i}]
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\delta^i_k(X^{(j_0,\dots,j_i)}) =[v_{j_0},\dots,\hat{v},\dots,v_{j_i}]
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\end{equation}
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where $\hat{v}$ indicates that we take out the vertix in position $k$, for $k\leq i$.
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an the boundary maps are given by:
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@ -506,11 +506,11 @@ Let $\mu_d=\{\xi| \xi^d=1\}$. We can express $S_d$ as a union
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\end{equation}
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where $X_{(\xi_0, \dots, \xi_n)}=\{(x_0,\dots,x_n)| \sum x^d_i=1 \text{ and } \xi_0x, \dots, \xi_nx_n\geq0\}$.
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Then, we can see that the faces of the simplex are given by $\mathring{X}_{(\xi_0, \dots, \xi_n)}$, while the $j-$simplices are given by
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Then, we can see that the faces of the simplex are given by $\mathring{X}_{(\xi_0, \dots, \xi_n)}$, while the $j-$simplices are given by
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\[
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X_{(\xi_0, \dots,\xi_n)}\cap\{x_{i_1},\dots,x_{i_j}=0\},
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\]
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where $\xi_{i_1},\dots,\xi_{i_j}=1$, and the vertices by
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where $\xi_{i_1},\dots,\xi_{i_j}=1$, and the vertices by
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\[
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X_{(1,\dots,\xi_i, 1, 1)}\cap\{x_j=1\forall j\neq i\}.
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\]
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@ -520,7 +520,7 @@ Let
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\begin{array}{rll}
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X^{(j_0,\dots,j_i)}_{(1,\dots,1)} & = & X_{(1, \dots, 1)}\cap\{x_{j_k}\neq0 \forall k=0,\dots i\text{ and }x_{i}=0\text{ for the rest}\}\\
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X^j_{(1,\dots,1)} & = & X_{(1, \dots, 1)}\cap\{x_i=0\forall i\neq j\}\\
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\end{array}
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\end{equation}
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We can describe the chain complexes. They are abelian groups but they are also $R_d$-complexes
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@ -530,7 +530,7 @@ C_{n}(\Delta^d) = R_d\langle X_{(1,\dots,1)}\rangle,
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\]
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which is a free principal $R_d$-module, i.e. a free abelian group of rank~$d^{n+1}$. For the $i$-chains, with $i<n$ we deal with non-free modules.
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Let us denote as $t_0, \dots, t_n$ the variables of $R_d$; note that $t_0^d=\dots=t_n^d=1$.
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Then
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Then
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\[
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C_i(\Delta^d) = \bigoplus_{(t_{j_1},\dots,t_{j_{n-i}})}\frac{R_d}{(t_{j_1} - 1,\dots,t_{j_{n-i}} - 1)} \langle X^{(k_0,\dots,k_i)}_{(1,\dots,1)}\rangle,
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\]
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@ -540,7 +540,7 @@ C_0(\Delta^d) = \bigoplus_{t_j}\frac{R_d}{(t_0 - 1,\dots,t_{j-1}-1,t_{j+1}-1,\do
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\]
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a free abelian group of rank~$(n+1)d$.
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%Then, the $2-$simplices, $1-$simplices and $0-$simplices of $\Delta^d$ are given by
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%\begin{equation}
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%\begin{equation}
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%\begin{array}{rll}
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%C_2(\Delta^d) & = & \ZZ[\mu^3_d]<X_{(1,1,1)}>\\
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%C_1(\Delta^d) & = & \ZZ[\mu^3_d]/\ZZ[\mu_d]<L^z_{(1,1,1)}> \bigoplus \ZZ[\mu^3_d]/\ZZ[\mu_d]<L^y_{(1,1,1)}> \bigoplus \ZZ[\mu^3_d]/\ZZ[\mu_d]<L^x_{(1,1,1)}>\\
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@ -548,7 +548,7 @@ a free abelian group of rank~$(n+1)d$.
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%\end{array}
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%\end{equation}
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The face maps are $R_d$-morphisms defined as
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The face maps are $R_d$-morphisms defined as
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\begin{equation}
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\begin{aligned}
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\delta^n_i(X_{(1,1,1)}) & =X^{(0,\dots,i-1,i+1,\dots,n)}_{(1,\dots,1)} &&& \delta^j_i(X^{(k_0,\dots,k_j)}_{(1,\dots,1)})&= X^{(k_0,\dots,k_{i-1},k_{i+1},\dots,k_n)}_{(1,\dots,1)} .
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@ -588,8 +588,8 @@ We have then defined $\Delta^d$, the simplicial complex with which we can identi
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% & & K & \hookrightarrow & \PP^2\setminus C_d \\
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% \end{array}
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\begin{tikzcd}
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\Delta \rar[leftrightarrow]& S_1&\\
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\Delta^d\uar\rar[leftrightarrow]\dar&S_d\uar\rar[hookrightarrow]\dar& M_d\dar\\
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\Delta \rar[leftrightarrow]& S_1&\\
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\Delta^d\uar\rar[leftrightarrow]\dar&S_d\uar\rar[hookrightarrow]\dar& M_d\dar\\
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\Delta^d_\textrm{proj}\rar[leftrightarrow]&K \rar[hookrightarrow]& \PP^2\setminus C_d,
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\end{tikzcd}
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\end{equation*}
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@ -604,10 +604,10 @@ We have then defined $\Delta^d$, the simplicial complex with which we can identi
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%\end{equation*}
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The map $\Delta^d\longrightarrow\Delta^d_\text{proj}$ is given by $(x_0,\dots,x_n)\longrightarrow[x_0:\dots:x_n]$. Since we are passing to the projective plane, two points $(x_0\xi_0, \dots, x_n\xi_n)$ and $(x_0\xi_0\xi, \dots, x_n\xi_n\xi)$ will have the same image.
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The $n$-cells $X_{(\xi_0, \dots, \xi_n)}$ and $X_{(\xi_0\xi, \dots, \xi_n\xi)}$ of $\Delta^d$, project the same $n$-cell of $\Delta^d_\textrm{proj}$, which will be denoted as $X_{[\xi_0: \dots: \xi_n]}$.
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The $n$-cells $X_{(\xi_0, \dots, \xi_n)}$ and $X_{(\xi_0\xi, \dots, \xi_n\xi)}$ of $\Delta^d$, project the same $n$-cell of $\Delta^d_\textrm{proj}$, which will be denoted as $X_{[\xi_0: \dots: \xi_n]}$.
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The same convention is followed for the rest of $i-$simplices. Note that for the vertices, since all coordinates but one are $0$, we have only $n+1$ vertices $X^0, \dots, X^n$.
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The group $\mu_d^{n+1}$ also acts, but the action of the diagonal vectors is trivial. Let us denote
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The group $\mu_d^{n+1}$ also acts, but the action of the diagonal vectors is trivial. Let us denote
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\[
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T_d:=\ZZ\left[\frac{\mu_d^{n+1}}{\mu_d\langle(\xi,\dots,\xi)\rangle}\right];
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\]
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@ -615,7 +615,7 @@ we use variables $t_0,\dots,t_n$ as for $R_d$ and we have to add the relation $t
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We have the following chain complexes:
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\begin{align*}
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C_n(\Delta^d_\text{proj}) &= T_d\langle X_{[1:\dots:1]}\rangle,\\
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C_i(\Delta^d_\text{proj}) &= \bigoplus_{(t_{j_1},\dots,t_{j_{n-i}})}\frac{T_d}{(t_{j_1} - 1,\dots,t_{j_{n-i}} - 1)} \langle X^{(k_0,\dots,k_i)}_{[1:\dots:1]}\rangle,\\
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C_i(\Delta^d_\text{proj}) &= \bigoplus_{(t_{j_1},\dots,t_{j_{n-i}})}\frac{T_d}{(t_{j_1} - 1,\dots,t_{j_{n-i}} - 1)} \langle X^{(k_0,\dots,k_i)}_{[1:\dots:1]}\rangle,\\
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C_0(\Delta^d) &= \bigoplus_{t_j}\frac{T_d}{(t_0 - 1,\dots,t_{j-1}-1,t_{j+1}-1,\dots, t_n - 1)} \langle X^j_{[1:\dots:1]}\rangle.
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\end{align*}
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%As for the points where one coordinate is $0$, say, for example the first one, since the group action is then by $\mu^3_2/\mu_2$, we write the elements of $\mu^3_2/\mu_2$ as $(1,\xi_1,\xi_2)$, where the $1$ is on the coordinate that is $0$. Then, passing to the projective space, we will say that $L^x_{[1:\xi_1,\xi_2]}$ and $L^x_{[1:\xi'_1,\xi'_2]}$ are the same if $[1:\xi'_1,\xi'_2]=[1:\xi_1\xi,\xi_2\xi]$ for some $\xi\in\mu_2$. The rest of $1-$simplices would be identified similarly. Lastly, for the $0-$simplices, since we have $2$ zero coordinates, we have that all $0-$simplices with the same nonzero coordinate will be identified to each other. As a result, $\Delta^d_\text{proj}$ will only have $3$ $0-$simplices, $3d$ $1-$simplices and $d^2$ $2-$simplices.
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@ -782,7 +782,7 @@ The code for the function is:
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def DeltaComplex_hypersurface(n,d):
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resdic = [{tuple(i*[0]+[1] + (n-i)*[0]): None for i in range(n+1) }]+ [{} for i in range(n)]
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for nzerosi in range(n+1):
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for nzerosi in range(n+1):
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for t in Tuples(range(d+1),n-nzerosi ):
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T = nzerosi*[0] + [1] + list(t)
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dimT = len(T) - T.count(0) - 2
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@ -795,9 +795,9 @@ def DeltaComplex_hypersurface(n,d):
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canonicalize(CT)
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caras.append(tuple(CT))
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resdic[dimT+1][tuple(T)] = caras
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DC = [[tuple() for _ in range(n+1)]]
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for i in range(1,n+1):
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for i in range(1,n+1):
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lis = []
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for p,v in resdic[i].items():
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lisp = []
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||||
@ -805,15 +805,15 @@ def DeltaComplex_hypersurface(n,d):
|
||||
lisp.append(list(resdic[i-1].keys()).index(s))
|
||||
lis.append(lisp)
|
||||
DC.append(lis)
|
||||
|
||||
dc=DeltaComplex(DC)
|
||||
|
||||
dc=DeltaComplex(DC)
|
||||
return dc
|
||||
\end{python}
|
||||
|
||||
This function uses an auxiliary function, named $canonicalize$ in the code.
|
||||
This function uses an auxiliary function, named $canonicalize$ in the code.
|
||||
$canonicalize$ is used to identify the simplices in the projective space. The code to this function is:
|
||||
\begin{python}
|
||||
def canonicalize(t):
|
||||
def canonicalize(t):
|
||||
v = next(filter(lambda a : a != 0, t))
|
||||
correct = v - 1
|
||||
for i in range(len(t)):
|
||||
@ -848,7 +848,7 @@ Delta_Complex = DeltaComplex_hypersurface(4,6)
|
||||
Once we have gotten the $\Delta$-complex, we can compute the fundamental group or the cohomology of the space, for example.
|
||||
\begin{python}
|
||||
Dc = SimplicialSet(Delta_Complex)
|
||||
dc_punto = it commit Dc.set_base_point(Dc.all_n_simplices(0)[0])
|
||||
dc_punto = Dc.set_base_point(Dc.all_n_simplices(0)[0])
|
||||
dc_punto.fundamental_group()
|
||||
-> Finitely presented group < e0 | e0^6 >
|
||||
dc_punto.cohomology()
|
||||
|
||||
Loading…
Reference in New Issue
Block a user