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Quindi $f\left(M \setminus \bigcup_{i = 1}^n A_i\right)^c$ è aperto. Si conclude dunque che $V$ è aperto. \smallskip - Su $V$, $\abs{f\inv(\cdot)}$ è necessariamente costante: la prima intersezione assicura che esistano + Su $V$, $\abs{f\inv(-)}$ è necessariamente costante: la prima intersezione assicura che esistano almeno $\abs{f\inv(y)}$ controimmagini, e la sottrazione insiemistica assicura che non possano esisterne di più. \end{proof} @@ -1113,8 +1113,8 @@ Sia $H$ una omotopia $C^\infty$ da $f$ a $g$. Allora, poiché $M$ è compatta, per il Lemma \ref{lem:pila}: \begin{itemize} - \item esiste un intorno $V_1$ di $y \in N$ su cui $\abs{f\inv(\cdot)}$ è costante; - \item esiste un intorno $V_2$ di $y \in N$ su cui $\abs{g\inv(\cdot)}$ è costante. + \item esiste un intorno $V_1$ di $y \in N$ su cui $\abs{f\inv(-)}$ è costante; + \item esiste un intorno $V_2$ di $y \in N$ su cui $\abs{g\inv(-)}$ è costante. \end{itemize} È sufficiente allora mostrare la tesi per un qualsiasi valore $y' \in V_1 \cap V_2$; poiché i valori regolari sono densi per il Teorema di Brown (per le varietà, Corollario \ref{cor:brown}), @@ -1264,6 +1264,22 @@ dove $y$ è un qualsiasi valore regolare di $f$. \end{definition} + \begin{lemma} \label{lem:valori_regolari_aperto} + Siano $M$ e $N$ varietà della stessa dimensione. Sia $M$ chiusa (ovverosia + anche compatta) + e $N$ connessa. Se $f : M \to N$ è liscia, allora i valori regolari di $f$ + formano un aperto di $N$. + \end{lemma} + + \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 $N$. Quindi i valori regolari formano un aperto. + \end{proof} + \begin{theorem} \label{thm:fondamentale_grado_2} Siano $M$ e $N$ varietà della stessa dimensione. Sia $M$ chiusa (ovverosia anche compatta) @@ -1275,17 +1291,12 @@ \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 + \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 + Per il Lemma \ref{lem:valori_regolari_aperto}, i valori regolari di $f$ formano un aperto in $N$. Allora, 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 $2$. + dunque $f$ e $g$ condividono lo stesso grado modulo $2$. \end{proof} \begin{corollary} @@ -1489,7 +1500,7 @@ Dunque $M$ è orientabile secondo l'orientazione canonica di $\RR^m$. \end{proof} - \subsection{Orientazione sul bordo della varietà, semispazio interno o esterno} + \subsection{Semispazio interno o esterno} \begin{remark}[Il semispazio interno è ben definito] L'orientazione locale di una $m$-varietà $M$ con bordo determina sempre la scelta di uno dei semispazi di @@ -1543,6 +1554,33 @@ $T_x M$, e i suoi vettori sono detti \textbf{esterni}. \end{definition} + \begin{lemma} \label{lem:0_interno_1_esterno} + Sia $M \cong [0, 1]$. Si fissi un diffeomorfismo + $\varphi : [0, 1] \to M$. Allora + $\dif \varphi_0(1)$ è un vettore interno, + mentre $\dif \varphi_1(1)$ è esterno. + \end{lemma} + + \begin{proof} + Si fissi $\eps$ tale per cui $0 < \eps < 1$. Poniamo + $I_0 = [0, \eps)$ e $I_1 = (\eps, 1]$. + Osserviamo che $\restr{\varphi}{I_0}$ è una + parametrizzazione locale di $\varphi(0)$, una + volta identificato $I_1$ naturalmente con $H^1$. + + Allora $\dif \varphi_0(1)$ è interno dal momento + che $\dif \varphi_0\inv(\dif \varphi_0(1)) = 1 > 0$. \smallskip + + Per quanto riguarda invece $\varphi(1)$, una parametrizzazione + locale è data su $I_1$ con $H^1$ + secondo $\Psi(t) = \varphi(1-t)$. \smallskip + + Osserviamo che $\dif \Psi_0\inv (\dif \varphi_1(1)) = -1 < 0$, e dunque + $\dif \varphi_1(1)$ è invece esterno. + \end{proof} + + \subsection{Orientazione sul bordo della varietà} + \begin{remark}[L'orientazione indotta sul bordo è ben definita -- \textit{esistenza}] Sia $x \in \partial M$ un punto della $m$-varietà orientata $(M, \Theta)$, con $m > 1$. Sia $g : U \subseteq H^m \to g(U)$ una parametrizzazione locale di $x$ con $g(u) = x$ che @@ -1648,5 +1686,123 @@ in $\RR^{n+1}$. \end{proof} - %\section{Teoria del grado su \texorpdfstring{$\ZZ$}{ℤ}} + \section{Teoria del grado su \texorpdfstring{$\ZZ$}{ℤ}} + + \subsection{Grado intero rispetto a un valore regolare} + + \begin{definition}[Segno di un differenziale in un punto] + Sia $M$ una varietà orientata con $\Theta^M$ e $N$ orientata + con $\Theta^N$ e $\dim M = \dim N$. Se $f : M \to N$ è una mappa liscia, si definisce + il \textbf{segno di $\dif f_x$} per un punto regolare $x \in M$ come: + \[ + \boxed{\sgn(\dif f_x) = \begin{cases} + +1 & \text{se } \dif f_x(\Theta^M) = \Theta^N, \\ + -1 & \text{se } \dif f_x(\Theta^M) = -\Theta^N. + \end{cases}} + \] + \end{definition} + + \begin{definition}[Grado intero di un valore regolare] + Sia $M$ una varietà chiusa e sia $N$ una varietà connessa con $\partial N = \emptyset$ + e $\dim M = \dim N$. Sia $M$ orientata con $\Theta^M$ e sia $N$ orientata con $\Theta^N$. \smallskip + + Se $y \in N$ è regolare per $f : M \to N$ liscia, si definisce il \textbf{grado di $f$ rispetto a $y$} come: + \[ + \boxed{\deg(f; y) \defeq \sum_{x \in f\inv(y)} \sgn(\dif f_x).} + \] + \end{definition} + + \begin{remark}[Il grado intero è localmente costante] \label{rmk:grado_loc_costante} + Se $y \in N$ è regolare, possiamo scegliere per il Lemma \ref{lem:pila} un intorno $I_1$ + sul quale $\abs{f\inv(-)}$ è costante. \smallskip + + Inoltre, per il Teorema della permanenza del + segno applicato sul determinante dello jacobiano di $h\inv \circ f \circ g$, dove + $g : U \to g(U)$ parametrizza localmente $x \in f\inv(y)$ e $h : V \to h(V)$ + un intorno di $y$, esiste un intorno $I_2$ di $y$ sul quale le orientazioni sono + preservate allo stesso modo in cui lo sono preservate dalle controimmagini di $y$. \smallskip + + Poiché i valori regolari sono aperti (vd. Lemma \ref{lem:valori_regolari_aperto}), + possiamo allora prendere un intorno aperto di valori regolari + in $I_1 \cap I_2$ entro cui $\deg(f; -)$ è costante; quindi il grado intero è \textit{localmente} + costante. + \end{remark} + + \begin{lemma}[Il grado di una mappa estendibile dal bordo è nullo] + Sia $X$ una varietà compatta e orientata con bordo non nullo. Sia $N$ + una varietà connessa, orientata, senza bordo e con $\dim X = \dim N + 1$. + Sia $F : X \to N$ una mappa liscia. \smallskip + + Se $y \in N$ è un valore regolare per $f \defeq \restr{F}{\partial X}$, allora + $\deg(f; y) = 0$. + \end{lemma} + + \begin{proof} + Denotiamo con $n$ la dimensione di $X$. \smallskip + + Grazie all'Osservazione \ref{rmk:grado_loc_costante}, sappiamo che esiste un intorno + di $y$ fatto di valori regolari di $f$ su cui $\deg(f; -)$ è costante. Possiamo prendere allora in questo intorno, + per il Teorema di Brown (per varietà, Corollario \ref{cor:brown}), un valore regolare + $z$ di $F$. Per definizione, $z$ è valore regolare sia di $F$ che di $f$; mostrando che + $\deg(f; z) = 0$, mostriamo dunque anche che $\deg(f; y) = 0$, per costruzione. \smallskip + + Per il Teorema \ref{thm:varietà_bordata_da_valore_regolare}, allora $F\inv(z)$ è una $1$-varietà + compatta e chiusa topologicamente con bordo $F\inv(z) \cap \partial M = f\inv(z)$. Per il + Corollario \ref{cor:classificazione_dim_1}, $F\inv(z)$ è unione di archi (con $2$ punti sul bordo ciascuno) + e cerchi (che non hanno bordo). Per mostrare la tesi è sufficiente mostrare che per ogni arco + i punti di bordo abbiano segno opposto sul differenziale, in modo tale che la somma complessiva dei + segni sia ancora nulla. \smallskip + + Sia $A \subseteq F\inv(z) \subseteq X$ un tale arco. Fissiamo un diffeomorfismo $\varphi : [0, 1] \to A$. + Poniamo $v_1(t) \defeq \dif \varphi_t (1)$. Allora vale: + \[ + T_{\varphi(t)} X = \underbrace{T_{\varphi(t)} A}_{\mathclap{=\, \Span(v_1(t))}} \oplus (T_{\varphi(t)} A)^\perp, + \] + dove ricordiamo che, per una generalizzazione della Proposizione \ref{prop:tangente_valore_regolare}, + $\restr{\dif F_{\varphi(t)}}{(T_{\varphi(t)} A)^\perp} : (T_{\varphi(t)} A)^\perp \to T_z N$ è un isomorfismo. \smallskip + + Fissiamo una base positiva $\{w_2, \ldots, w_n\}$ di $T_y N$. Possiamo allora porre: + \[ + v_i(t) = \restr{\dif F_{\varphi(t)}}{(T_{\varphi(t)} A)^\perp}\inv(w_i), \quad i = 2, \ldots, n. + \] + Poiché $v_1(t)$ è ortogonale agli altri $v_i(t)$, $\{v_i(t)\}_i$ è una base + di $T_{\varphi(t)} X$. \smallskip + + Se poniamo $A \defeq \{t \mid \{v_i(t)\}_i \text{ positiva}\} \subseteq [0, 1]$, + allora $A$ è un aperto di $[0, 1]$. Infatti, per il Teorema della permanenza del segno applicato + allo jacobiano di una parametrizzazione locale, in un + intorno di $t_0$ si mantiene la stessa orientazione. + Analogamente anche il complementare + di $A$ è aperto. Dacché $[0, 1]$ è connesso, allora uno tra $A$ e il suo complementare è vuoto: in altre + parole $\{v_i(t)\}_i$ è sempre positiva per ogni $t \in [0, 1]$, o altrimenti è sempre negativa. \smallskip + + A meno di invertire l'orientazione di $\varphi$ usando al suo posto $\varphi(1 - t)$, possiamo + assumere che $\basis(t) \defeq \{v_i(t)\}_i$ sia sempre positiva. \smallskip + + Mostriamo che $\sgn(\dif f_{\varphi(0)}) = -1$. Sia $\basis' \defeq \{v_1(0), v_2'(0), \ldots, v_n'(0)\}$ + una base di $T_{\varphi(0)} \partial X$ tale per cui $\{ \dif F_{\varphi(0)} (v_i'(0)) \}_{i \geq 2} $ è + una base positiva di $T_z N$. Se mostriamo che $\basis$ e $\basis'$ hanno la stessa orientazione, + dal momento che $\basis$ è positiva e $v_1(0)$ è \underline{interno} per il Lemma \ref{lem:0_interno_1_esterno}, + si deduce dalla definizione di orientazione di bordo che + $\{ v_2'(0), \ldots, v_n'(0)\}$ è negativa; in particolare $\dif f_{\varphi(0)}$ invertirebbe l'orientazione, + e quindi avrebbe segno $-1$. \smallskip + + Consideriamo la matrice di cambio di base da $\basis$ a $\basis'$, della seguente forma: + \[ + M \defeq \begin{pmatrix} + 1 & \vline & * \\ + \hline + 0 & \vline & B + \end{pmatrix}. + \] + Osserviamo che $B$ è la matrice di cambio di base ottenuta togliendo ad entrambi le + basi il vettore $v_1(0)$. Per costruzione, queste tali basi hanno la stessa orientazione + dal momento che $\dif F_{\varphi(0)}$ mantiene invariate le orientazioni. Dunque + $\det(B) > 0$, da cui si deduce che $\det(M) > 0$, e quindi che $\basis$ e + $\basis'$ hanno effettivamente la stessa orientazione. \smallskip + + Per $\varphi(1)$ il ragionamento è del tutto analogo, ma essendo $v_1(1)$ \underline{esterno} per + il Lemma \ref{lem:0_interno_1_esterno}, il segno sarà invece $+1$. Questo conclude la dimostrazione + per le osservazioni fatte in precedenza. + \end{proof} \end{multicols*}