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zX6D!%8Y&h^m9DfT9Y8QU)$fJgAj!0%8H>{P&vv@g6~&9-!WMyyg`r{^dE{-$SA)Se z8VEL06ifj$B`9E16bPbf4r00#1_a42hvMH91IoO_79LfnuJ9M(p!LKN-NN47#nssY WDr169h{3}z$cau*FQ+b#{(k_@0$} (o $m$-varietà) se per ogni @@ -76,6 +76,10 @@ $f_x$ si dice invece \textbf{parametrizzazione locale} di $x$ in $M$. \end{definition} + \begin{definition}[Varietà chiusa] + Si dice \textbf{varietà chiusa} una varietà (senza bordo) che è compatta. + \end{definition} + \begin{remark} Le varietà di dimensione zero sono esattamente le unioni di punti isolati. \end{remark} @@ -779,7 +783,7 @@ \begin{theorem} \label{thm:varietà_bordata_da_valore_regolare_su_R} Sia $f : M \to \RR$ una mappa liscia tra varietà, dove $M$ è una $m$-varietà senza bordo. Se $0$ è un valore regolare per $f$, allora $\{ f \geq 0 \}$ è una - $m$-varietà con bordo $f\inv(0)$. + $m$-varietà con bordo $f\inv(0) = \{f = 0\}$. \end{theorem} \begin{proof} @@ -932,7 +936,7 @@ Si veda l'appendice di [Milnor, J. (1965). \textit{Topology from the Differentiable Viewpoint}. University Press of Virginia]. \end{proof} - \begin{corollary}[Classificazzione delle $1$-varietà compatte con bordo] \label{cor:classificazione_dim_1} + \begin{corollary}[Classificazione delle $1$-varietà compatte con bordo] \label{cor:classificazione_dim_1} Una $1$-varietà compatta con bordo è necessariamente un'unione di copie di $S^1$ e di intervalli chiusi di $\RR$. \end{corollary} @@ -942,6 +946,15 @@ utilizzando l'ipotesi di compattezza. \end{proof} + \begin{corollary} \label{cor:punti_pari_su_1varietà} + Una $1$-varietà compatta con bordo ha un numero pari di punti sul bordo + \end{corollary} + + \begin{proof} + Deriva immediatamente dal Corollario \ref{cor:classificazione_dim_1}, dal momento che + il segmento chiuso ha due punti sul bordo e $S^1$ non ne ha nessuno. + \end{proof} + \begin{lemma}[di non retrazione sul bordo] \label{lem:non_retrazione} Sia $M$ una varietà compatta con bordo $\partial M \neq \emptyset$. Allora \underline{non} esistono mappe lisce $f$ da $M$ in $\partial M$ che fissano il bordo, ovverosia @@ -963,7 +976,7 @@ Osserviamo che il bordo di $f\inv(y)$ si semplifica a $\{y\}$ dacché $\restr{f}{\partial M} = \id_{\partial M}$. Tuttavia questo è un assurdo, dal momento che le $1$-varietà compatte con bordo finito hanno un numero pari di elementi sul - bordo per il Corollario \ref{cor:classificazione_dim_1}. Quindi $f$ \underline{non} può esistere. + bordo per il Corollario \ref{cor:punti_pari_su_1varietà}. Quindi $f$ \underline{non} può esistere. \end{proof} \begin{lemma} \label{lem:punto_fisso_cinf} @@ -1042,6 +1055,16 @@ \subsection{Omotopie \texorpdfstring{$C^\infty$}{C∞}} + \begin{remark}[{$M \times [0,1]$} è una varietà con bordo] + Sia $M$ una $m$-varietà senza bordo. Allora, per la Proposizione \ref{prop:prodotto_varietà}, + $M \times \RR$ è una $(m+1)$-varietà senza bordo. \smallskip + + Consideriamo la mappa liscia $f : M \times \RR \to \RR$ tale per cui + $f(x, t) = t(t-1)$. Allora per il Teorema \ref{thm:varietà_bordata_da_valore_regolare_su_R} + $\{f \leq 0\} = M \times [0, 1]$ è una $(m+1)$-varietà con bordo + $\{f = 0\} = M \times \{0\} \sqcup M \times \{1\}$. + \end{remark} + \begin{definition}[Omotopia $C^\infty$ e funzioni $C^\infty$-omotope] Siano $f$ e $g$ due funzioni da una varietà $M$ in una $N$. Un'\textbf{omotopia $C^\infty$} da $f$ a $g$ è una funzione liscia @@ -1055,6 +1078,11 @@ da $f$ a $g$ si dicono \textbf{$C^\infty$-omotope}. \end{definition} + \begin{remark} + È immediato verificare che ``essere $C^\infty$-omotope'' è una + relazione di equivalenza per le funzioni lisce da $M$ a $N$. + \end{remark} + \begin{lemma}[di 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 @@ -1072,10 +1100,11 @@ \item esiste un intorno $V_2$ di $y \in N$ su cui $\abs{g\inv(\cdot)}$ 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, possiamo prendere $y'$ valore regolare di $H$. \smallskip + poiché i valori regolari sono densi per il Teorema di Brown (per le varietà, Corollario \ref{cor:brown}), + possiamo prendere $y' \in V_1 \cap V_2$ valore regolare di $H$. \smallskip Allora $H\inv(y')$ è una varietà di dimensione $1$, il cui bordo ha un numero pari - di punti per il Corollario \ref{cor:classificazione_dim_1}. \smallskip + di punti per il Corollario \ref{cor:punti_pari_su_1varietà}. \smallskip Osserviamo che $\partial (M \times [0, 1]) = M \times \{0\} \sqcup M \times \{1\}$. Allora: \[ @@ -1112,13 +1141,24 @@ È immediato osservare che la relazione ``essere isotopi'' sui punti di una varietà è una relazione di equivalenza. \end{remark} + \begin{lemma} \label{lem:classi_equivalenza_aperte} + Le classi di equivalenza della relazione ``essere isotopi'' sui + punti di una varietà sono aperti della varietà. + \end{lemma} + + \begin{proof} + ... + \end{proof} + \begin{lemma}[di omogeneità] Sia $N$ una varietà connessa e siano $y$, $z$ due suoi punti. Allora $y$ e $z$ sono isotopi. \end{lemma} \begin{proof} - ... + Poiché $N$ è connessa, per il Lemma \ref{lem:classi_equivalenza_aperte} esiste + allora un'unica classe di equivalenza per la relazione ``essere isotopi'', da + cui segue immediatamente la tesi. \end{proof} \subsection{Grado modulo \texorpdfstring{$2$}{2} e buona definizione}