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$H(-, 1) = g$. \end{itemize} + Definiamo inoltre: + \[ + \boxed{H_t \defeq H(-, t).} + \] Due funzioni $f$ e $g$ per le quali esiste un'omotopia da $f$ a $g$ si dicono \textbf{$C^\infty$-omotope}. @@ -1083,7 +1087,7 @@ relazione di equivalenza per le funzioni lisce da $M$ a $N$. \end{remark} - \begin{lemma}[di omotopia] + \begin{lemma}[di omotopia] \label{lem:omotopia} Siano $f$ e $g$ due funzioni $C^\infty$-omotope da una varietà $M$ in una $N$, con $M$ compatta e $\dim M = \dim N$. Se $y \in N$ è un valore regolare sia per $f$ che per $g$, allora: @@ -1143,14 +1147,62 @@ \begin{lemma} \label{lem:classi_equivalenza_aperte} Le classi di equivalenza della relazione ``essere isotopi'' sui - punti di una varietà sono aperti della varietà. + punti di una varietà (senza bordo) $M$ sono aperti della varietà. \end{lemma} \begin{proof} - ... + Sia $x$ un punto di $M$. Supponiamo $M$ sia una $(n+1)$-varietà. + Usando le carte locali, eventualmente riparametrizzate + per avere come immagine una palla di centro $0$, è sufficiente mostrare + che per ogni $r$, esiste $0 < r_0 \leq r$ tale per cui + tutti i punti di $B_{r_0}(0) \subseteq B_r(0) \subseteq \RR^{n+1}$ sono isotopi a $0$. \smallskip + + Possiamo ridurre il problema ulteriormente concentrandoci sui punti + della forma $z = (a, 0, \ldots, 0)$ con $a < r_0$, dal momento che una rotazione + permette poi di rendere isotopi tutti gli altri punti di $\partial B_a(0)$. Se + infatti $H$ è un isotopia a supporto compatto che porta $z$ in $0$, allora: + \[ H'(-, t) \defeq R_\theta\inv \circ H(-, t) \circ R_\theta \] + è l'isotopia cercata, dove $R_\theta$ è l'opportuna rotazione scelta. \smallskip + + Siano $\rho : \RR \to [0, 1]$ e $\sigma : \RR^n \to [0, 1]$ due funzioni di test + (\textit{bump function}) lisce con $\rho(0) = \sigma(0) = 1$ e: + \[ \abs{x} > 1 \implies \rho(x) = 0, \quad \norm{y} > 1 \implies \sigma(y) = 0. \] + Osserviamo che: + \[ + \abs{x} > \eps \implies \rho\left(\frac{x}{\eps}\right) = 0, \quad \norm{y} > \delta \implies \sigma\left(\frac{y}{\delta}\right) = 0. + \] + + Scelti allora $\eps$ e $\delta$ di modo che $\eps^2 + \delta^2 < r_0^2$, definiamo allora l'omotopia $C^\infty$ $H$ tale per cui: + \begin{equation*} + \begin{aligned} + H : \overbrace{\mathbb{R} \times \mathbb{R}^n}^{\mathclap{\mathbb{R}^{n+1}}} \times [0, 1] & \to \overbrace{\mathbb{R} \times \mathbb{R}^n}^{\mathclap{\mathbb{R}^{n+1}}}, \\ + H(x, y, t) & = \left(x + a t \rho\left(\frac{x}{\varepsilon}\right) \sigma\left(\frac{y}{\delta}\right), y\right). + \end{aligned} + \end{equation*} + + Osserviamo subito che $H_0 = \id_N$, $H_1(0) = z$ e che $H_t$ fissa tutto ciò che è fuori + da $B_{r_0}(0)$. Mostriamo che per $r_0$ sufficientemente piccolo $H_t$ è un diffeomorfismo. \smallskip + + Lo jacobiano di $H_t$ è il seguente: + \[ + J H_t = \begin{pmatrix} + 1 + \frac{a t}{\varepsilon} \sigma\left(\frac{y}{\delta}\right) \rho'\left(\frac{x}{\varepsilon}\right) & \vline & * \\[0.06in] + \hline + 0 & \vline & I_n + \end{pmatrix}. + \] + Si verifica facilmente che si può scegliere $r_0$, e successivamente $\eps$ e $\delta$ in modo tale che: + \[ + \abs{\frac{a t}{\varepsilon} \sigma\left(\frac{y}{\delta}\right) \rho'\left(\frac{x}{\varepsilon}\right)} < 1, \quad \forall a, x, y, + \] + e conseguentemente $\det(J H_t) > 0$, da cui si deduce che $H_t$ è un diffeomorfismo locale. Analogamente, $H_t$ è + bigettiva sulle rette $\{y = \text{cost.}\}$ dal momento che la derivata sulle rette è strettamente + positiva. Dunque $H_t$ è bigettiva e diffeomorfismo locale, e quindi è diffeomorfismo. \smallskip + + Si conclude che $H$ è l'isotopia cercata, e quindi ogni classe di isotopia è aperta. \end{proof} - \begin{lemma}[di omogeneità] + \begin{lemma}[di omogeneità] \label{lem:omogeneità} Sia $N$ una varietà connessa e siano $y$, $z$ due suoi punti. Allora $y$ e $z$ sono isotopi. \end{lemma} @@ -1164,14 +1216,29 @@ \subsection{Grado modulo \texorpdfstring{$2$}{2} e buona definizione} \begin{theorem} - Siano $M$ e $N$ varietà della stessa dimensione. Sia $M$ compatta - e $N$ connessa. Siano $y$ e $z$ due valori regolari di una + Siano $M$ e $N$ varietà della stessa dimensione. Sia $M$ chiusa (ovverosia + anche compatta) e $N$ connessa. Siano $y$ e $z$ due valori regolari di una funzione $f : M \to N$ liscia. Allora: \[ \abs{f\inv(y)} \equiv \abs{f\inv(z)} \pmod{2}. \] \end{theorem} \begin{proof} - ... + Poiché $N$ è connessa, per il Lemma \ref{lem:omogeneità} esiste un diffeomorfismo + $h : N \to N$ con $h(y) = z$ e un'isotopia a supporto compatto + $H : N \times [0, 1] \to N$ da $\id_N$ a $h$. + \[\begin{tikzcd} + {M \times [0, 1]} && {N \times [0, 1]} && N + \arrow["{f \times \id_{[0, 1]}}", from=1-1, to=1-3] + \arrow["H", from=1-3, to=1-5] + \end{tikzcd}\] + Poiché $y$ è regolare per $f$ e $h$ è diffeomorfismo, si deduce che + $z$ è regolare per $h \circ f$. Consideriamo l'omotopia + $H' = H \circ (f \times \id_{[0,1]})$. Osserviamo che + $H'_0 = f$ e che $H'_1 = h \circ f$. + Quindi, per il Lemma \ref{lem:omotopia}, si conclude che: + \[ + \abs{f\inv(z)} \equiv \abs{(h \circ f)\inv(z)} \equiv \abs{f\inv(y)} \pmod{2}. + \] \end{proof} \begin{definition}[Grado modulo $2$ di una funzione liscia] @@ -1185,7 +1252,8 @@ \end{definition} \begin{theorem} \label{thm:fondamentale_grado_2} - Siano $M$ e $N$ varietà della stessa dimensione. Sia $M$ compatta + Siano $M$ e $N$ varietà della stessa dimensione. Sia $M$ chiusa (ovverosia + anche compatta) e $N$ connessa. Se $f$ e $g$ sono due mappe lisce $C^\infty$-omotope da $M$ in $N$, allora: \[ @@ -1193,13 +1261,28 @@ \] \end{theorem} + \begin{proof} + Dall'Osservazione \ref{rmk:punti_regolari_formano_aperto} sappiamo che + i punti regolari di $f$ formano un aperto di $M$; dunque i punti critici + formano un chiuso di $M$. Dacché $M$ è compatta, in particolare i punti + critici formano un compatto di $M$. Tramite $f$, si deduce che i valori + critici formano un compatto di $N$, che, essendo $N$ T2, è dunque + chiuso in $N$. \smallskip + + Allora i valori regolari di $f$ formano un aperto in $N$: per il Teorema + di Brown (per varietà, Corollario \ref{cor:brown}) esiste in questo + aperto anche un valore regolare di $g$. Per il Lemma \ref{lem:omotopia}, + allora $f$ e $g$ condividono lo stesso grado modulo $2$. + \end{proof} + \begin{corollary} La mappa costante $c_{x_0} : S^n \to S^n$ \underline{non} è $C^\infty$-omotopa a $\id_{S^n}$. \end{corollary} \begin{proof} - Poiché $c_{x_0}$ non è surgettiva, $\deg_2 c_{x_0} = 0$; mentre + Poiché $c_{x_0}$ non è surgettiva, $\deg_2 c_{x_0} = 0$ (ogni punto diverso da $x_0$ è + valore regolare con controimmagine vuota); mentre $\deg_2 \id_{S^n} = 1$. Quindi per il Teorema \ref{thm:fondamentale_grado_2} le due mappe non possono essere $C^\infty$-omotope. \end{proof}