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\label{prop:orientazione_immersa_Rm} Sia $M$ una $m$-varietà immersa in $\RR^m$. Allora $M$ è orientabile secondo l'orientazione canonica di $\RR^m$. @@ -1500,6 +1544,23 @@ Dunque $M$ è orientabile secondo l'orientazione canonica di $\RR^m$. \end{proof} + \subsection{Orientazione nel prodotto di due varietà orientate} + + \begin{definition}[Orientazione prodotto per varietà] + Siano $(M, \Theta^M)$ e $(N, \Theta^N)$ due varietà orientate. + Si definisce l'\textbf{orientazione prodotto} su $M \times N$ + come l'orientazione $\Theta^{M \times N}$ tale per cui: + \[ + \boxed{\Theta_{(x, y)}^{M \times N} = \Theta_x^M \times \Theta_y^N, \quad \forall x \in M, y \in N} + \] + dove $\Theta_x^M \times \Theta_y^N$ è l'orientazione prodotto su + $T_{(x, y)} M \times N \cong T_x M \times T_y N$ indotta da + $\Theta_x^M$ e $\Theta_y^N$. \smallskip + + Il prodotto di parametrizzazioni locali compatibili induce parametrizzazioni + locali compatibili con l'orientazione prodotto appena definita. + \end{definition} + \subsection{Semispazio interno o esterno} \begin{remark}[Il semispazio interno è ben definito] @@ -1728,7 +1789,9 @@ costante. \end{remark} - \begin{lemma}[Il grado di una mappa estendibile dal bordo è nullo] + \subsection{Grado di una mappa estendibile dal bordo} + + \begin{lemma}[Il grado di una mappa estendibile dal bordo è nullo] \label{lem:grado_mappa_estendibile} 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 @@ -1805,4 +1868,97 @@ 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} + + \subsection{Passaggio per omotopia e buona definizione del grado intero di una mappa} + + \begin{lemma}[di omotopia, per il grado intero] \label{lem:omotopia_intero} + Siano $M$ e $N$ due varietà orientate con $M$ chiusa e $\dim M = \dim N$. + Sia $F : M \times [0, 1] \to N$ un'omotopia $C^\infty$ con $f \defeq F_0$ + e $g \defeq F_1$. Se $y \in N$ è un valore regolare comune a $f$ e $g$, + allora: + \[ \boxed{\deg(f; y) = \deg(g; y).} \] + \end{lemma} + + \begin{proof} + Dal momento che $M \times [0, 1]$ è un prodotto, allora acquisisce l'orientazione + prodotto di quella di $M$ con quella canonica di $[0, 1]$. Grazie + all'Osservazione \ref{rmk:grado_loc_costante} e al Teorema di Brown (per varietà, Corollario \ref{cor:brown}), + possiamo assumere senza perdita di generalità che $y$ sia un valore regolare anche di $F$. \smallskip + + Sia $\basis_x$ una base positiva di $T_x M$ per $x \in M$. Allora, + per l'Osservazione \ref{rmk:regola_segni} la base $\basis_x \cup \{1\}$ + è positiva per $T_{(x, t)} M \times [0, 1] \cong T_x M \times T_t [0, 1]$. \smallskip + + Ricordiamo che il bordo di $M \times [0, 1]$ è $M \times \{0\} \sqcup M \times \{1\}$ + per \ref{rmk:m_01_varietà}. Studiamo l'orientazione indotta su tale bordo. + Poiché $(0, 1) \in T_{(x, 0)} M \times [0, 1]$ è interno per il Lemma \ref{lem:0_interno_1_esterno}, + $\{-\basis_x\} \cup \{-1\}$ è positiva per $(x, 0)$, e quindi $\basis_x$ è una base negativa di $(x, 0)$ sul + bordo. Dunque $M \times \{0\} \cong M$ è orientata come $-M$. Analogamente, $(0, 1) \in T_{(x, 1)} M \times [0, 1]$ è + esterno, quindi $M \times \{1\} \cong M$ è orientata come $M$. \smallskip + + Allora, per il Lemma \ref{lem:grado_mappa_estendibile}, si ha: + \[ + \deg(g; y) - \deg(f; y) = \deg(\restr{F}{\partial(M \times [0, 1])}; y) = 0, + \] + da cui la tesi. + \end{proof} + + \begin{theorem}[Il grado intero è 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: + \[ + \boxed{\deg(f; y) = \deg(f; z).} + \] + \end{theorem} + + \begin{proof} + Per il Lemma \ref{lem:omogeneità}, esiste un diffeomorfismo $h : N \to N$ + con $h(y) = z$ isotopo all'identità $\id_N$. Sia $H : N \times [0, 1] \to N$ + una tale isotopia. \smallskip + + Consideriamo la mappa $f(t) = \sgn(\dif (H_t)_y (\Theta^N_y))$. Poiché + $H_1 = \id_N$, si ha chiaramente $f(1) = 1$. Inoltre $f$ è continua e localmente + costante, utilizzando l'usuale argomento sulla permanenza del segno + sul determinante dello jacobiano, una volta scelta una parametrizzazione locale. + Dunque, poiché $[0, 1]$ è connesso, $f$ deve essere costantemente uguale a $1$. + Quindi $\dif (H_t)_y (\Theta^N_y) = \Theta^N_{H_t(y)}$ per ogni $t \in [0, 1]$. \smallskip + + Segue quindi che $\dif h_y (\Theta^N_y) = \Theta^N_z$, da cui per la regola della catena: + \[ + \deg(h \circ f; z) = \deg(f; y). + \] + D'altra parte $h \circ f$ e $f$ sono $C^\infty$-omotope tramite $H \circ (f \times \id_{0, 1})$, + e quindi, per il Lemma \ref{lem:omotopia_intero}: + \[ + \deg(f; y) = \deg(h \circ f; z) = \deg(f; z). + \] + \end{proof} + + \begin{definition}[Grado intero di una mappa liscia] + Sia $M$ una varietà chiusa e sia $N$ una varietà connessa. + Siano $M$ e $N$ della stessa dimensione e orientate. Se + $f : M \to N$ è una mappa liscia, si definisce il suo + \textbf{grado intero} come: + \[ + \boxed{\deg(f) \defeq \deg(f; y),} + \] + dove $y$ è un valore regolare qualsiasi di $f$. + \end{definition} + + \begin{theorem} \label{thm:fondamentale_grado_intero} + Sia $M$ una varietà chiusa e sia $N$ una varietà connessa. + Siano $M$ e $N$ della stessa dimensione e orientate. + Se $f$ e $g$ sono due mappe lisce $C^\infty$-omotope da $M$ in $N$, + allora: + \[ + \boxed{\deg(f) = \deg(g).} + \] + \end{theorem} + + \begin{proof} + 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_intero}, + dunque $f$ e $g$ condividono lo stesso grado intero. + \end{proof} \end{multicols*}