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differenziale/Scheda/sections/4-varieta-e-teoria-del-grado.tex b/Corsi/Geometria e topologia differenziale/Scheda/sections/4-varieta-e-teoria-del-grado.tex index dc299eb..5b4fb21 100644 --- a/Corsi/Geometria e topologia differenziale/Scheda/sections/4-varieta-e-teoria-del-grado.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda/sections/4-varieta-e-teoria-del-grado.tex @@ -1903,7 +1903,7 @@ da cui la tesi. \end{proof} - \begin{theorem}[Il grado intero è ben definito] + \begin{theorem}[Il grado intero è ben definito] \label{thm:grado_intero_ben_definito} Siano $M$ e $N$ varietà con $M$ chiusa, $N$ connessa e $\dim M = \dim N$. Se $f : M \to N$ è liscia e $y$, $z \in N$ sono suoi valori regolari, allora: \[ @@ -1974,7 +1974,7 @@ dal momento che $\deg(\id_M) = 1$ e $\deg(c_x) = 0$ (infatti $c_x$ non è surgettiva). \end{proof} - \begin{proposition}[Il grado è moltiplicativo] + \begin{proposition}[Il grado è moltiplicativo] \label{prop:grado_moltiplicativo} Sia $M$ una varietà orientata, chiusa e connessa. Se $f$ e $g$ sono due mappe lisce da $M$ in sé stessa, allora: \[ @@ -1983,10 +1983,34 @@ \end{proposition} \begin{proof} - ... + Sia $z$ un valore regolare di $f \circ g$. Sia $x \in (f \circ g)\inv(z)$. + Allora, per la regola della catena: + \[ + \dif (f \circ g)_x = \dif f_{g(x)} \circ \dif g_x. + \] + Dal momento che $z$ è regolare, $x$ è un punto regolare e dunque + $\dif (f \circ g)_x$ è un isomorfismo. Dunque necessariamente + $\dif f_{g(x)}$ e $\dif g_x$ sono isomorfismi, ovverosia + $x$ è regolare anche per $g$, mentre $g(x)$ lo è per $f$. \smallskip + + Sia $f\inv(z) = \{y_1, \ldots, y_n\}$. + Da quanto appena visto, ogni $y_i$ è valore regolare. + Allora: + \[ + g\inv(f\inv(z)) = \bigsqcup_{i = 1}^n g\inv(y_i). + \] + + Allora, sfruttando anche l'Osservazione \ref{rmk:regola_segni}: + \[ + \deg(f \circ g) = \sum_{x \in g\inv(f\inv(z))} \sgn(\dif f_{g(x)}) \sgn(\dif g_x). + \] + Raccogliendo i termini utilizzando $f\inv(z)$ si ottiene dunque, anche usando il Teorema \ref{thm:grado_intero_ben_definito}: + \[ + \deg(f \circ g) = \sum_{y_i \in f\inv(y_i)} \sgn(\dif f_{y_i}) \deg(g) = \deg(f) \deg(g). + \] \end{proof} - \subsection{Applicazioni immediate della teoria del grado intero} + \subsection{Grado di \texorpdfstring{$z^k$}{zᵏ}, delle riflessioni e della mappa antipodale su \texorpdfstring{$S^1$}{S¹}} \begin{lemma} \label{lem:grado_zk} Sia $f_k : S^1 \to S^1$ tale per cui: @@ -2036,4 +2060,87 @@ Si conclude facilmente allora che $\deg(f) = \deg(f; 1) = k$. L'ultima affermazione è conseguenza del Lemma \ref{lem:grado_mappa_estendibile}. \end{proof} + + \begin{remark}[Un diffeomorfismo ha grado $1$ o $-1$] \label{rmk:grado_diffeomorfismo} + Per un diffeomorfismo $f : M \to M$ su $M$ chiusa e connessa, + possono esistere solo due gradi, $+1$ o $-1$, dacché l'insieme + controimmagine di un valore regolare contiene un singolo elemento. \smallskip + + In particolare, $\deg(f) = 1$ se e solo se per un elemento $x \in M$, $\dif f_x$ + preserva l'orientazione. + \end{remark} + + \begin{lemma} \label{lem:grado_riflessione} + Sia $r_i : S^n \to S_n$ la riflessione tale per cui: + \[ + r_i(x_1, \ldots, x_i, \ldots, x_{n+1}) \defeq (\underbrace{x_1, \ldots}_{\textnormal{invariati}}, -x_i, \underbrace{\ldots, x_{n+1}}_{\textnormal{invariati}}). + \] + Allora $\deg(r_i) = -1$. + \end{lemma} + + \begin{proof} + Per l'Osservazione \ref{rmk:grado_diffeomorfismo} è sufficiente studiare $\dif (r_i)_{e_i}$. + Per il Corollario \ref{cor:sn_è_varietà}, + si ha: + \[ + T_{e_i} S^n = e_i^\perp = T_{-e_i} S^n. + \] + $r_i$ si estende con la stessa formula a un diffeomorfismo $\tilde{r_i}$ su $D^{n+1}$. Poiché + $\tilde{r_i}$ è lineare e $T_{e_i} D^{n+1} = \RR^{n+1}$, si ha + $\dif \tilde{r_i}_{e_i} = \tilde{r_i}$. Tale differenziale manda la base $\{e_i, e_1, \ldots, e_{n+1}\}$ + in $\{-e_i, e_1, \ldots, e_{n+1}\}$. Queste due basi hanno orientazione diversa, e quindi solo + una di queste induce l'orientazione canonica su $\RR^{n+1}$. \smallskip + + Osserviamo che $e_i$ è esterno per sé stesso; allo stesso modo $-e_i$ è esterno per sé stesso. + Segue allora facilmente che $\{e_1, \ldots, e_{n+1}\}$ cambia orientazione tramite $r_i$, + e quindi $\deg(r_i) = -1$. + \end{proof} + + \begin{lemma} \label{lem:grado_antipodale} + Sia $A : S^n \to S^n$ la mappa antipodale, ossia tale per cui $A(x) = -x$. + Allora $\deg(A) = (-1)^{n+1}$. + \end{lemma} + + \begin{proof} + Segue immediatamente dalla Proposizione \ref{prop:grado_moltiplicativo} e il Lemma \ref{lem:grado_riflessione}. + \end{proof} + + \subsection{Campi vettoriali tangenti su \texorpdfstring{$S^n$}{Sⁿ} e pettinabilità} + + \begin{definition}[Campo vettoriale (tangente)] + Sia $M \subseteq \RR^k$ una varietà liscia, con o senza bordo. Un + \textbf{campo vettoriale (tangente)} è una mappa liscia + $v : M \to \RR^k$ tale per cui: + \[ + \boxed{v(x) \in T_x M, \quad \forall x \in M.} + \] + \end{definition} + + \begin{definition}[Pettinabilità] + Una varietà liscia, con o senza bordo, si dice \textbf{pettinabile} + se ammette un campo vettoriale mai nullo. + \end{definition} + + \begin{theorem}[di pettinabilità della sfera] \label{thm:pettinabilità_sfera} + $S^n$ è pettinabile se e solo se $n$ è dispari. + \end{theorem} + + \begin{proof} + Se $n$ è dispari, un campo vettoriale mai nullo è il seguente: + \[ + f(x_1, x_2, \ldots, x_n, x_{n+1}) = (-x_2, x_1, \ldots, -x_{n+1}, x_n) \in x^\perp. + \] + Sia ora $n$ pari. + Supponiamo $v$ sia un campo vettoriale mai nullo. Senza + perdita di generalità, possiamo considerarlo unitario. Allora + possiamo costruire un omotopia liscia dalla mappa + antipodale $A$ a $\id_{S^n}$ nel seguente modo: + \[ + H : S^n \times [0, 1] \to S^n, \quad H(x, t) = \cos(\pi t) x + \sin(\pi t) v(x). + \] + Tuttavia una tale omotopia \underline{non} può esistere per + il Teorema \ref{thm:fondamentale_grado_intero}: l'identità + ha grado $1$, mentre la mappa antipodale ha grado $(-1)^{n+1} = -1$ per + il Lemma \ref{lem:grado_antipodale}. Quindi $v$ non può esistere per $n$ pari. + \end{proof} \end{multicols*}