descripcion del complejo simplicial asociado a Sd

generaldegree.tex

@@ -272,4 +272,23 @@ All that is left to see is that the formulas for each case coincide on the borde
 The first root corresponds to $\tilde{r}_2$ in $(+,+,-)$ and the last one corresponds to the lifting in $(-,+,-)$,so we see that it coincides. It would be similar in the rest of borders.
 \end{proof}
 
+We have then proved that $S_d$ is a strong deformation retract of $M_d$. We denote as $\Delta$ the 2-simplex $[v_0, v_1, v_2]$, then, we can identify $S_1$ with $\Delta$, and we get a diagram:
+
+\begin{equation*}
+	\begin{array}{ccccc}
+		\Delta & \longleftrightarrow & S_1 & & \\
+		\uparrow& & \uparrow & & \\
+		\Delta^d & \longleftrightarrow & S_d & \longrightarrow & M_d \\
+		& & \downarrow & & \downarrow \\
+		& & K & \longrightarrow & \PP^2\setminus C_d \\
+	\end{array}
+\end{equation*}
+where $\Delta^d$ is a simplicial complex that we are trying to find and identify with $S_d$, equivalently as what we have for $d=1$.
+
+Let $\phi: S_d\longleftarrow S_1$ be the map given by $\phi(x,y,z)=(x^d, y^d, z^d)$, then necessarily $\phi^{-1}(\Delta)=\Delta^d$.
+In particular, the $d^3$ vertices of the simplicial complex $\Delta^d$ will be identified to $(\xi_d,0,0)$, $(0,\xi_d,0)$, $(0,0,\xi_d)$ for all $\xi_d$ the $d$-$th$ roots of unity.
+
+The edges of $\Delta$ can be identified with the lines of $S_1$ parametrized as $(1-t,t,0)$, $(1-t,0,t)$ and $(0,t,1-t)$ with $t\in[0,1]$. Then, the preimages of these lines by $\phi$ are of the form $(\sqrt[d]{1-t}\xi_i, \sqrt[d]{t}\xi_j, 0)$ in the first case and analogous in the rest, where we denote $\sqrt[s]{x}$ the positive real $d$-root for $x$ real, and where $\xi_i$ and $\xi_j$ are two any $d$-$th$ roots of unity. For each of these preimages we can define an edge between the corresponding vertices in $\Delta^d$.
+
+The $2$-faces of the simplicial complex $\Delta^d$ will then be identified to the areas between the lines $(\sqrt[d]{1-t}\xi_i, \sqrt[d]{t}\xi_j, 0)$, $(\sqrt[d]{1-t}\xi_i,0, \sqrt[d]{t}\xi_k)$ and $(0,\sqrt[d]{t}\xi_j, \sqrt[d]{1-t}\xi_k)$.
 \end{document}