% Definitions that use @ : 
\catcode`\@=11

% page heading on even pages gives chapter, but on same side of page
% as for odd pages. 


\newcommand{\ch}{\mbox{$\widehat{\bf C}$}}
\newcommand{\ii}{\mbox{${\mathcal I}$}}
\newcommand{\bp}{\mbox{$B^+$}}
\newcommand{\bn}{\mbox{$B^-$}}
\newcommand{\ag}{\mbox{${\mathcal A}_g$}}
\newcommand{\af}{\mbox{${\mathcal A}_f$}}
\newcommand{\hu}{\mbox{${\mathcal H}^u$}}
\newcommand{\et}{\mbox{$\eta _-$}}
\newcommand{\tsv}{\mbox{$\sigma ^s_e$}}
\newcommand{\hs}{\mbox{${\mathcal H}^s$}}
\newcommand{\hp}{\mbox{${\mathcal H}$}}
\newcommand{\tup}{\mbox{$\Theta ^u_+$}}
\newcommand{\tun}{\mbox{$\Theta ^u_-$}}
\newcommand{\tsp}{\mbox{$\Theta ^s_+$}}
\newcommand{\tsn}{\mbox{$\Theta ^s_-$}}
\newcommand{\vg}{\mbox{$\varphi_G$}}
\newcommand{\vf}{\mbox{$\varphi_F$}}
\newcommand{\vif}{\mbox{$\varphi^{\infty}_F$}}
\newcommand{\vig}{\mbox{$\varphi^{\infty}_G$}}
\newcommand{\vil}{\mbox{$\varphi^{\infty}_L$}}
\newcommand{\vi}{\mbox{$\varphi^{\infty}$}}
\newcommand{\vv}{\mbox{$\varphi _s$}}
\newcommand{\hlf}{\mbox{$X^L_F$}}
\newcommand{\hfl}{\mbox{$X^F_L$}}
\newcommand{\zf}{\mbox{$\zeta_F$}}
\newcommand{\zl}{\mbox{$\zeta_L$}}
\newcommand{\pin}{\mbox{$\partial _{\infty}$}}
\newcommand{\cc}{\mbox{$\mathcal C$}}
\newcommand{\ga}{\mbox{$\gamma $}}
\newcommand{\oo}{\mbox{$\mathcal O$}}
\newcommand{\ooo}{\mbox{$\mathcal J$}}
\newcommand{\ou}{\mbox{${\mathcal J}^u_+$}}
\newcommand{\rrr}{\mbox{$\times {\bf R}$}}
\newcommand{\rrrr}{\mbox{${\bf R}$}}
\newcommand{\rp}{\mbox{$[0,+\infty)$}}
\newcommand{\rr}{\mbox{$\times \{ 0 \}$}}
\newcommand{\pn}{\mbox{$\partial N$}}
\newcommand{\mi}{\mbox{$\widetilde M$}}
\newcommand{\ww}{\mbox{$\widetilde W$}}
\newcommand{\ti}{\mbox{$\partial \widetilde T$}}
\newcommand{\ttt}{\mbox{$\widetilde T$}}
\newcommand{\ppn}{\mbox{$\partial _+ N$}}
\newcommand{\lll}{\mbox{$l _{\epsilon}$}}
\newcommand{\iT}{\mbox{$i _{\widetilde T}$}}
\newcommand{\wwp}{\mbox{$\widetilde \Phi$}}
\newcommand{\w}{\mbox{$\widetilde {\partial _+ C}$}}
\newcommand{\pp}{\mbox{$\varphi$}}
\newcommand{\pw}{\mbox{$\varphi _{\widetilde T}$}}
\newcommand{\lmun}{\mbox{$\Gamma ^s \cup \Gamma ^u$}}
\newcommand{\gs}{\mbox{$\Gamma ^s$}}
\newcommand{\gu}{\mbox{$\Gamma ^u$}}
\newcommand{\ggs}{\mbox{${\mathcal F}^s_T$}}
\newcommand{\ggu}{\mbox{${\mathcal F}^u_T$}}
\newcommand{\gsr}{\mbox{$\Gamma ^s \times {\bf R}$}}
\newcommand{\gur}{\mbox{$\Gamma ^u \times {\bf R}$}}
\newcommand{\se}{\mbox{$S _{\epsilon}$}}
\newcommand{\gus}{\mbox{$\Gamma ^u \downarrow S_{\epsilon}$}}
\newcommand{\vt}{\mbox{$\overline \varphi _{\widetilde T}$}}
%\newcommand{\vv}{\mbox{$\varphi _{\widetilde T}$}}
\newcommand{\hh}{\mbox{${\bf H}^2$}}
\newcommand{\hhh}{\mbox{${\bf H}^3$}}
\newcommand{\lmS}{\mbox{$\Gamma ^s \downarrow S$}}
\newcommand{\lmU}{\mbox{$\Gamma ^u \downarrow S$}}
\newcommand{\lms}{\mbox{$\Gamma ^s \downarrow S_{\epsilon}$}}
\newcommand{\lmu}{\mbox{$\Gamma ^u \downarrow S_{\epsilon}$}}
\newcommand{\wall}{\mbox{$L \times \R$}}
\newcommand{\lap}{\mbox{$\Gamma ^s \cup \Gamma ^u \cup Q$}}
\newcommand{\tz}{\mbox{$\widetilde T \times \{ 0 \}$}}
\newcommand{\wt}{\mbox{$\widetilde T$}}
\newcommand{\wss}{\mbox{$\widetilde S$}}
\newcommand{\wn}{\mbox{$\widetilde N$}}
\newcommand{\si}{\mbox{$S^2_{\infty}$}}
\newcommand{\su}{\mbox{$S^1_{\infty}$}}
\newcommand{\sss}{\mbox{$S^s_{\infty}$}}
\newcommand{\sus}{\mbox{$S^u_{\infty}$}}
\newcommand{\s}{\mbox{$S ^1$}}
\newcommand{\fol}{\mbox{$\mathcal F$}}
\newcommand{\fn}{\mbox{$\mathcal \widetilde F$}}
\newcommand{\ls}{\mbox{${\mathcal L} ^s$}}
\newcommand{\lu}{\mbox{${\mathcal L} ^u$}}
\newcommand{\lns}{\mbox{$\widetilde {\mathcal L} ^s$}}
\newcommand{\lnu}{\mbox{$\widetilde {\mathcal L} ^u$}}
\newcommand{\fs}{\mbox{${\mathcal F} ^s$}}
\newcommand{\fu}{\mbox{${\mathcal F} ^u$}}
\newcommand{\fss}{\mbox{${\mathcal F} ^{ss}$}}
\newcommand{\fuu}{\mbox{${\mathcal F} ^{uu}$}}
\newcommand{\fnss}{\mbox{$\widetilde {\mathcal F} ^{ss}$}}
\newcommand{\fnuu}{\mbox{$\widetilde {\mathcal F} ^{uu}$}}
\newcommand{\fns}{\mbox{${\widetilde {\mathcal F}}^s $}}
\newcommand{\fnu}{\mbox{${\widetilde {\mathcal F}}^u $}}
\newcommand{\fsS}{\mbox{${\mathcal F} ^s_S$}}
\newcommand{\fuS}{\mbox{${\mathcal F} ^u_S$}}
\newcommand{\fsT}{\mbox{${\mathcal F} ^s_T$}}
\newcommand{\fuT}{\mbox{${\mathcal F} ^u_T$}}
\newcommand{\fSS}{\mbox{$\widetilde{\mathcal F} ^s_{\mathcal S}$}}
\newcommand{\fUS}{\mbox{$\widetilde{\mathcal F} ^u_{\mathcal S}$}}
\newcommand{\fnuS}{\mbox{$\widetilde {{\mathcal F}}^u_S $}}
\newcommand{\fnsS}{\mbox{$\widetilde {{\mathcal F}}^s_S $}}
\newcommand{\ws}{\mbox{$\widetilde  W^s$}}
\newcommand{\wu}{\mbox{$\widetilde  W^u$}}
\newcommand{\hnt}{homotopically non trivial}
\newcommand{\up}{\mbox{${\mathcal U} _+$}}
\newcommand{\un}{\mbox{${\mathcal U} _-$}}
\newcommand{\Dp}{\mbox{$\Delta ^+$}}
\newcommand{\Dn}{\mbox{$\Delta ^-$}}
\newcommand{\fg}{\mbox{$\phi _g$}}
\newcommand{\h}{\mbox{$\bf H$}}

 


\catcode`\@=12

