From f64008811ffa75d43dfe3f67ab2535d9e50f6236 Mon Sep 17 00:00:00 2001 From: Joshua Moerman Date: Mon, 17 Jun 2013 14:53:24 +0200 Subject: [PATCH] Htp: Added some figures --- thesis/5_Homotopy.tex | 30 ++- thesis/images/simplicial_eqrel.pdf | Bin 0 -> 5710 bytes thesis/images/simplicial_eqrel.svg | 197 ++++++++++++++++++ thesis/images/simplicial_htp.svg | 315 +++++++++++++++++++++++++++++ thesis/images/simplicial_htp1.pdf | Bin 0 -> 4437 bytes thesis/images/simplicial_htp2.pdf | Bin 0 -> 3203 bytes thesis/symbols.tex | 8 +- 7 files changed, 544 insertions(+), 6 deletions(-) create mode 100644 thesis/images/simplicial_eqrel.pdf create mode 100644 thesis/images/simplicial_eqrel.svg create mode 100644 thesis/images/simplicial_htp.svg create mode 100644 thesis/images/simplicial_htp1.pdf create mode 100644 thesis/images/simplicial_htp2.pdf diff --git a/thesis/5_Homotopy.tex b/thesis/5_Homotopy.tex index 004de5e..0fe70d5 100644 --- a/thesis/5_Homotopy.tex +++ b/thesis/5_Homotopy.tex @@ -18,13 +18,35 @@ When dealing with homotopy in a topological space $X$ we always need a base-poin We will call $y$ the \emph{homotopy} and notate $y: x \sim x'$. \end{definition} -Of course we would like $\sim$ to be an equivalence relation, however this is not true for all simplicial sets. For example there is in general no reason for symmetry, existence of a $1$-simplex $y$ from $x$ to $x'$ does not give us a $1$-simplex $y'$ from $x'$ to $x$. One can give an precise condition on when it is a equivalence relation, the so called Kan-condition. In our case of abelien groups, however, we can prove this directly. - -\todo{Htp: Discuss/picturize Kan-condition?} +Of course we would like $\sim$ to be an equivalence relation, however this is not true for all simplicial sets. For example there is in general no reason for symmetry, existence of a $1$-simplex $y$ from $x$ to $x'$ does not give us a $1$-simplex $y'$ from $x'$ to $x$. One can give an precise condition on when it is a equivalence relation, the so called \emph{Kan-condition}. In our case of simplicial abelien groups, however, we can prove directly that $\sim$ is an equivalence relation. + +In figure~\ref{fig:simplicial_htp} it is shown why the definition of homotopy makes sense for $n=1$. Two homotopic $1$-simplices from $Z_n(X)$ are depicted in two ways. The first way only shows the structure we have, indicating what the boundaries are (as described by the face maps). In the second figure we collapsed all occurences of $0$ into a single point. This way of drawing a homotopy should remind the reader of homotopy (between paths) in a topological space. + +\begin{figure}[h!] +\begin{subfigure}{.5\textwidth} + \centering + \includegraphics{simplicial_htp1} +\end{subfigure}% +\begin{subfigure}{.5\textwidth} + \centering + \includegraphics{simplicial_htp2} +\end{subfigure} +\caption{In the figure on the left two homotopic $1$ simplices $x, x' \in Z_n(X)$ are shown. The fact that $d_2(y) = \ast$ is depicted by crossing out the bottom line. The right image shows exactly the same structure if we would draw the $0$-simplex $0$ only once (and hence also collapse the degenerate $1$-simplex $d_2y$).} +\label{fig:simplicial_htp} +\end{figure} \begin{lemma} The relation $\sim$ as defined above is an equivalence relation on $Z_n(X)$. Furthermore it is compatible with addition. \end{lemma} + +Before proving this, one should have a look at figure~\ref{fig:simplicial_eqrel}. In this figure we show what we want to proof in degree $n=0$ (i.e. the simplices of interest are points, and the homotopies are paths). + +\begin{figure}[h!] +\includegraphics{simplicial_eqrel} +\caption{The three properties of an equivalence relation: reflexivity, symmetry and transitivity. The dashed lines show which homotopy we should construct.} +\label{fig:simplicial_eqrel} +\end{figure} + \begin{proof} \emph{Reflexivity}. Let $x \in Z_n(X)$, define $y = s_0 x$. By considering the simplicial identities $d_0 s_0 = \id$ and $d_1 s_0 = \id$, it follows that $d_0 y = d_1 y = x$. Furthermore $d_i y = d_i s_0 x = s_0 d_{i-1} x = 0$ for all $i > 1$, because $x \in Z_n(X)$. @@ -78,7 +100,7 @@ Recall construction of the singular chain complex in section~\ref{sec:Chain Comp $$ C_n(X) = \Z[\Hom{\cat{Top}}{\Delta^n}{X}]. $$ Where the boundary map was given as an alternating sum. Looking more closely we see that this construction decomposes as: $$ C: \Top \tot{\text{Sing}} \sSet \tot{\Z^\ast} \sAb \tot{C} \Ch{\Ab}, $$ -where the last functor is the \emph{unnormalized chain complex}. All the categories involved have a notion of homotopy. In topological spaces this is the known notion where $f, g:X \to Y$ are homotopic if there exists a homotopy $H:I \times X \to Y$ with the appropriate properties. In simplicial sets (or simplicial abelian groups) we only saw the notion of homotopy groups, but there exists a more general notion of homotopy, as discussed in the overview of Friedman \cite{friedman}. And finally in chain complexes we saw homology groups, but this category also has a more general notion of chain homotopy, which can be found in any book on homological algebra such as in the book of Weibel \cite{weibel}. +where the last functor is the \emph{unnormalized chain complex}. All the categories involved have a notion of homotopy. In topological spaces this is the known notion where $f, g:X \to Y$ are homotopic if there exists a homotopy $H:I \times X \to Y$ with the appropriate properties. In simplicial sets (or simplicial abelian groups) we only saw the notion of homotopy groups, but there exists a more general notion of homotopy, as discussed in the overview of Friedman \cite{friedman}. And finally in chain complexes we saw homology groups, but this category also has a more general notion of chain homotopy, which can be found in any book on homological algebra such as in the book of Rotman \cite{rotman}. It is known that for any simplicial abelian group both the normalized and unnormalized chain complex have the same homology groups. More precisely for any simplicial abelian group $X$ we have: $$ H_n(N(X)) \iso H_n(C(X)) \quad\text{for all } n \in \N. $$ diff --git a/thesis/images/simplicial_eqrel.pdf b/thesis/images/simplicial_eqrel.pdf new file mode 100644 index 0000000000000000000000000000000000000000..8fdf62142b293bf774869bb4d05accb2383bfba2 GIT binary patch literal 5710 zcma)gcQ~8v`#$v&jo7nhP>QMFoXaZ0l{FfsFtfc7J(t`?rIMy-+Exe5tk-Y8Sv&9B!dv=BG8_c( zu|ej<5o|N_thVlhZbg9&7u7$$P{Wkd{?v1N5j*rQDB0bX0KtrORuxH_?Z z6+Sh6{dxB&?j)K^cQdu=b7B7_jC2Y~CY)%p>-=-JK7}b(1^>=WD#m*?rrUJwOI4ln z;2@KH%am1y-JnOXRAZf!I6QOUyMu(>dAHMfnTJv|tJzj;=YS~HfzRcTDMtzh>yeMSOQyALM#RManSACl zzW;$-yZa;WGcg`z2ung_f(dYi-VCt?DGh&n1g!WJ?OZ~)ttYYxqtNPn*LvH`cd9KP z)r{RznpfoIAx<@%l{lMuCUiuc+KHZ`0@cJqK?C+ezrG}x!Wo%xO9KtNJkQZt2*y_< zQV9CF!|1qz-eOEc*{E3f>>O(>bgs;-HcTzL$Ap8YZ{N2w!MYdYTR7&vXy+RIfZM$D zWhqZ1X<<5%Gn}aXJe+zDdp0t-=@j6cTTGpibSgLKtqs{R{dJmZ7F60SCVXYM^^)a+ zxOqM~1@Gl$qG^T9McES~P6qvn({F5X$*cLn_0^8j^hjXEV1eJOJZf1-*`{=P;SM(` zz5iEAoaCHidQK0httn;mk0hJHKFN@%T`-={vp&5 zoBB3OZSX|k;I-i*t|(BiH;%(`h7=LEMcy^Y28s>`+x^t)QTo1f(^H~Y8n&Gdj4%mW zvln*8_#8Q~#~11gM;eNluRzn!1!&@(4VMcNPd_9Pxq|5&HtjkZVTZhs^zCcOVyoqfJ$ps^38-fd-PzFT^=A)YzrHwO6@bZR5647Nxw9kMt zcY4|vyf;*1?ywyj_`RT45p!*!-3G@DD;z2o>M)-J9p%7Y6%__NpC5qHDeqXGsJ~m!T&KBiA6sd=BC01w% zqDClpxP!{qs;^wxH2FR*mmVs=?naK~C+$@4GikWN0)p2nC{_c&0mGXQ6`f(zc@Ad{ zM=GoK;aXNefI5AeJ#mp{5?-y6NXH@Mpy4YEBbCvBm#nnTALs2X5$~gZo!ggBh5o_K z@S=X41kCNWiPEo?h<>Ras)-<>0}3)>>UGEzVhm$&ap{5{r3)O8$9e#&*mFtAV;Uk# zoCQvCtI&~=C^`Ky%2C)+*$HqjA+(Xjv6s?Xp&PMl_IP@?IWAx)FLCquW1M(M_x4NuH4ZT;8?R~`Wzz*p$;>P>)16~Xflt&SR?EP&CF-0_)2-%xI zILII7!1uUP^}W`^UL0UpocG~-B_uKW+G68LZOlnoGOI)G2l#s1qJvOMEi4A=R|e&} zPoAY;5y)Yh&Z>3d8dVhbhJnp#L5o*OC1;s61kX>&FRDUjrugIv=e=gxICOrD2cUKE z^Ye8MrD{@2G#0BW(_5*0MG15JS^1bv3_F^{tSc%xG5>pA3pU|_Afmg_b97z%=ID|} zorgB@&Lp$4z1yhu1A&DPo3KdoXs+O1Mu{d4^m&jUaQcU|Q$c;y65rBRJ2Y{}e`jD&5NwM)^tgZP(Z6G2~8Q z-V6Rui;KbqVs%6oOb?x@p2n*~NKF|`j9uP_v)vD(sS=LOx~BoW5pUWOSKCw46CEF( zq{=c+5}KCh&qlvEY8yqioCm}~I&k9}??FlJ1<9~?6Ls~hy#a2wBsVO--wALLq}vO7 zZW8q1DNS~$79nbiQO#E0i@D8$J4y@Rw@-(8v2n4Q zc-Ykq>1M-)m}5H!zn>EEZ7z3A6?2P6rWtm?kR+H7u8Gxo+EFF~_F0)7W7k-aN*0Q|`l=2s#0a4rJ zP~knEK6=N?qfxoy5^opGCKU2=KOW=3@uf?TC_E2I<73*^RcKhp@s*gK0&_-EgFWhQ zXA*q#3Ji`9rg~e(HCd}T%^yZWj9iE(0tmozPrZ6zi&RHP#$3Scrbf{+Yn=VZ@%Xs8 ze5tOX;*F2b=5M#TP>v-g#XJYus&B6t$$X=r(VlyI>>_E%d<)0G@nQYRVQb8^Q*Nq9 zg#aQAzi`jRtMKmZ)&qZSTLTjw-|XEp@K*;sTBcJ&)N-JgksHKwg6%>-+P>*qo$JBC zI&7bNkpI>1a>R~?4+Z!{$2(kC0KXPTvfz!JK6tX29Vr8kXdbd3)Y-x~X-f%n=67Ur zvxwxh(;GPk>2v`Wknnd~DHb!{Q|i|&wgZfNUoTBnWLuF_PcS2|QJsYY0V?{LR4VZ= z_43P%UTE{;X?6{T%F<*Q$!bwR`Ed3T0V_9qsfOncwktP!h-n&wvriYbv^@a45=2r@ z+Y^)BJ9MMN(iT2G8dhi`aWdt&Z)U<_6S^~Ds6fHIf6t*O1hO2q|3ajOQT&8Roc$Z% zmUo~N{N|j*bp;i5dM`ywP2echy3Ap)R#c@-YjU@Z)-JH4YUDZj5#s+3KnYZgpTC_(jEO8i33nWW+WZ-4bY>eiDP{3z2KIcXVDW1cE3g@J2{z3 zTO=<0?CQp5_p~%(Dw+H6V=oc;`4AI+P)Hw{iFPWuJ3Hfb%rigE4sd5orXk)7T|*ZQ z`6#in6&qPHswvKqA1?*L8^RU5Hy;vXu8WMmt9n{nb=Qdbu1bnwm}hW`O1 z8>S5`D&-ZCLLR`hDcmFuZt3^DW*s~f!S$M-EI!?}ACvVW}M zb!MfbuEXt2OFF(a74Y`vZ=zmhu7Y&~MKXdEXG=MAM{1XHT&+%~6{04dXxby(1@v+Iym=Dw1@ z@nDrXYilxe^~KY5jTzW@p|@fYRADF@g>*m5a=op^3wy1!8HmI*(U^+Um82yUwiQI# zUP|e!*_oD30?@Z&X@Engp>3W`+5M@`RzjLeih@}0jJksrn=Yd!iDtUN0GUHoyeIm0+soT{PXSyjf8{)|)(e+3TKu z?|?|9{4SpQN;~Pe0o7ym6|bLLF#K&U>QePz@;}=0PhKh4sdIXogC@qZvs)IU$5b>` z(+ZwBMLjL<6N^uZp%6$vD00*54i@WwnyvNx$uDruN%LaemN@a9Bv|ltlRHT>mwZ+m zScHv}CdR95w<(1DF;&?&5+0zwq8ictkjxYlCusu&CRXqAq}xr~^$-@d#m&aKeyg{< z01BI8y4oicnK_ZG8!>wINAgAE_jI59e12ETIII&7(si8OEX^sB4JJ3@B63Y)Ug8HK zN^UnR*j2v-7mgwuKUs@V$sDYcbcQ{y#I&PV1h((?u`PGyJw{Dghzy_-M&tO$G~QP* znUzY)NGBoD$-Wsrk?pLLPo6UA*>VE&i`I&-mFJfzH0?iJ2tWC@gI67jEFSR5=L=h2 z42WtJ^>Ph6p)z zJ@7s{Pkt$t+BJvTip$d0Pes_y@@Gj07q>RFtE;<$LvndDvWGeLJyG*rrV)C#{KtD4 z@g(-;2cuWWCKSnt&vvA9ppHnFY?|S8ssgScPA~?11 ztp?J|1>r{YpE|Cg;;Ih{x(9q^(Q<9|>xwgtMf-30b3ccNB1J3GKs53F1`E?Lw2S*%cOeVFE)x*yFY z0)MrRdYZ{llFxi2Csexfkm9w(R8@}wIDp&7Sq+_yThE!2ifgqrA3snlf`e9$^2)Wh1G{-w@;o0e?exA)BYA>e4y;dP6y^a@vOO90? z#v0A2zCa90SmshodL7ka)Jj#U{!w74)ybnOKNoJviH@>$7TYN~2G3q}}g;^sLZ*>{p z7*bpLN>icy*cx33cx8>_spaiYoxtoYaKcX3NAHwmMI6oTp2EwZHxS{H{bex!o9X;c zx`@v~GdfGpU#k!=`~uNl;x?rllQ;<}#f5Y@w_0pG-YeZxp3uZ`0}&{vza%cg*M&HC zp@?1V1WgPgD*?H&%a7DQe!!w8_t6NjC=Bj``0W!0#~@Hxg4y4aoB_hg6|Uy#2eu+WVqghz zh=@1@A|@*bhRBIX%1DY!NP=yCo2q-FumoFg@P)+pr=F*c@$^CeE4x2~{YL*z*qD$4 z29EMZUj%l%|2M2oz%>x}TpbZc+G-czpEN{`5Z<0X7)OLR_@Wp_{~~pf^54SZ|44cN zPEF6%$(umng16B>+4@}Y{Fejie>qSm2##1!3|PPs?uzjQi;IYhiAaD2kXS5QUX)N* z0_cK)qmizT-XfkD7eT?_VPF_fCm%C;N=i}+>tV1BE<#;*PdLdqA@YUJsO1z!;Vb6j*#&Ys|l aHvI0ZHx`b;{$?lzk(LGW@oC=D0{$Pan>C^U literal 0 HcmV?d00001 diff --git a/thesis/images/simplicial_eqrel.svg b/thesis/images/simplicial_eqrel.svg new file mode 100644 index 0000000..069ea13 --- /dev/null +++ b/thesis/images/simplicial_eqrel.svg @@ -0,0 +1,197 @@ + + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/thesis/images/simplicial_htp.svg b/thesis/images/simplicial_htp.svg new file mode 100644 index 0000000..b0922e2 --- /dev/null +++ b/thesis/images/simplicial_htp.svg @@ -0,0 +1,315 @@ + + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/thesis/images/simplicial_htp1.pdf b/thesis/images/simplicial_htp1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..af838334a30ed49422cd700d18915b95fae71c7d GIT binary patch literal 4437 zcma)Ac{r5&`?e%YwnQ?-AZubQGr}aYl`LbdlSYj(m@LE06jDh;WU^)7v#0F)+Cs8s z%@#uTLCL=6H|m_uIp4p2pXkG#>p9Hp3)CVR}y3HdekfVt0j@;bs z^~(|mn=wKi7k`Q-UEH6&JODi)XZzIq9H{=>*i=N$@6DoSQ#x;S=EfYny;z=)&D^~E zT3x6=--Ln~h!?Idmh;YCa9cK$sq=-c8*U}N*(MmSo>V?@w<0)`?ah-?LyRU=+N>N zK1<4UyXR-Ms#g#x&gN+RbRUzh2qEL=?P~C(soH9_h8}UjF{4|8$v#ZZ!5w+Cgia6$ zo%e}Yu2m>YD8Nir_8+^J&!A+)6ai4OgQKyqABf?;UXL*30Nsf#Vl)U#OYXx4whP4h z03a_xpIXPxups+bR0hlyy!;^w>LdnDl0^~;7;1D(#)`+b`@`6o>#|po`ZT>F(8p%s zGjZ>vJ>~~_I?M_baBGJ3g_bR)?T!(*H10LSo&kmp(=AV#o98n((q(QS{UpqUyIbTL z49O??Kfc(22ip|yJbN}TxEwc(7tFqqokG5s<2)9?^%>N2ql*>Rxx%sFB3kXZG&bp0 zD*W?JXyCf}bWKF6eIWqqB4Q+SShu}He@ zLvg8I?DFJW`dcg@@X+A%#!$a?{Hr$R7#HopcU!`jwqTbj^J1AAMoj&&G;{3aM?B5# zF3unhL}vxkIpe6!>pkByyFQE48v!&JSl+SuucdTkhphJNRcuN8k6oP9(;89~`2v)} zLhc9!o&zvXRNY9nbeak~&+O6s^n%*vb^S{k;iY^(n#dge@OZr(X8Q*7W$mzb~ zGR#$Y-|uzmwk3lUE9_c_yyy|D96`Ru2qk|fS=RH9-pd8{MoRq9@Tb{+vt4cxXd#V@ zkHyoCPsp&pSEo3wE3qPYg#)(%i0>ik62({MVF-x8F}#9-ON!k5rW5Iijww!Ajhk+09@)C zUgNJb+Lv0595gHS_lZRrdpZogF3u#h(;}ihkL=u=pDV7jzFy8-7_?8Y4>axtB1hbzk7yZBX2zr$`G4tnm}Pp@XZX}|y+8=qt}~7sRM(Oc8ycea1#8p0 zy2uWN-WO{foLchLj%Ai!en2!X*#p}RL%1Y7f=ym7C8lm^?k)>roQLNcoVlWHic}l zdC`)(f5Jn-0#Q4aCIoU~i|X23A{7$>EIkaq>)WM3L$iHP&f-_10fRyE=HQ5}pSzA< z-)i>i?v~|;?rj2bQSqlX)mRzq#JMjm*I%#P>Vc5#iVEp8dYQwr>xG5ICG*Ng(sbX$ zDR1sO9MpMzmPCY(UFM9#%^OJ8{5bcSx7>N~1GiLugon%hoH}F1_}I_HNa5~I=Uc44 zEF;m^lD|Y%`&&sWcZCl`^Tt?VoG&_hoi4WcdT()Gc17mRqN*I`Q?G`qekq?H0y5Nj zTLfRreS)TB?DCRvhqArL+D}YPMffv80b=%G(o7m+cn(HrGVQX-7xqF$nDJy9XL z`xd^xm;bu|29Gdsg_rZYIPN^j86E=M3@m2EfCYW~j=bH9W2=l-zJ?L4Yft`)Sjvyq zeq8rCg0s%zw7Y(L8)&AehOeY^rAqrYrg>Wj&*+~J&Ow?#-4&38GI^n0-3-QLu< zO#Te`s~H>RfQ*yZRl7{?`h6`iC=ew#s3|YC^qXtQ6fQRbdh=BHR!(T~op~b=WX>Uw z#)FV~y=K3W1u)?w9~aUxpVIsos-_&`c6<^Y(8P5*K%`xL8DRm$KNIV+vo@}HM_ zY2O!-fqDfRUt-qyg`5qNz@5sV0Fh8kbQ zs~r5nGu9oc!i$&fG6TH9lGD-3wo-g6rJiq1=6Zy4YtFYAj}09Y>vjsYqg!|zQRlg} z{4jvWqA#)mKyaRa@lEo9*8T-X+k~#uDc&j~{tSe=)YPf9n;eN@u6GSGCBHO~2t|1Y z$938y?v%_i71ip)hrhEbJ&7J;)4StxLD&9NlzV}AqmcK>i*g2*>TI*K*`BZw2NHQk zB6}*z8(aT#l~;uwQSjYYAVRv>W|LGtKqHa@ewQvdt|L*%9?9xEqt}xY)N=JnEu`nl z<@5#bvYojnNEuFtzB5X9R~xM7^)DHGdBZetig&={J!ypmH>s}{0d zKIu|d9iE42!U!~QEQD@5p$>|>b&o(w<-sLC_<1Qag*aT_(*nQld9GfH74>SBP03rM zpfTVqy8{NBpzb?yMPHG)P6=7dcXjz@o_t|Ts>#{3iKhptdS{{a%z|%@tgM82MCc^h zCJUMym-S?*o$|1mbEL}i$874;UL~@J7uOWAP-SvT9gWNum z&GOz&UITaD*2rw{a4qE1=$rNKV_LR9Aw^yGL(Laxiv_)_R5yE1r@JDe5tXs%YnfZ& zDZJrlaTf$=Gr=Lvn=6C7+)i_EdjlSod(P~-B<~VSbA9DjWxn&!*QbX0Bp8nt(o$Tm zR+m3=m%!4lRL5vp&o!OTX4EK8s-_?Bz%e}uA*IkS!$PlhE`q_lXyYzW7{SOG2y2@GZg)ZNV%Ht#?pYUyY!TUXk=rB2GkW zeUV44O$=`i?lTk#ouE~lw z9_xKtdaHp!z$S{hsBd`}oSGHEGlU&EWPYKffXx*%iH)hvOjq7Y_CM6qVbB64BwbakcwP>QV0wMdG0{JiJo1p&}TpaJb>{QQ?n< z{G}RoJc)XyM5|5v$^egYCAb~NMSJ}XYf*7+%pEKm1J}EB z2>#AOjykWn+jumF06eS({9id8GX7g%|C1Bp7qzQcI|7yeVcPINY;PZ?`7Z~Me>u>i z>TV(q50pfsuy`C$URE9?s{oXABof_J<)~$)f)03;nx6xD` z|Nl4przgt)3jdqG7K(^+#yR|UP3;fiude)KgFvX!{)>S?^3?MD7gK~NQ3vsVFl9xR|HYI*^3>w}cb}3xbwvJPL_7-X zjKTl<`C*E^i=k3J99B3EM+6=w{(D@mxZ2}@hi&-PR{{}*C;rM%NnVBe7e!3RSeNDh E0IdkvkpKVy literal 0 HcmV?d00001 diff --git a/thesis/images/simplicial_htp2.pdf b/thesis/images/simplicial_htp2.pdf new file mode 100644 index 0000000000000000000000000000000000000000..913fa4e3fc05778ce2f4aaa0bce9b3756b9976c3 GIT binary patch literal 3203 zcma)9c|4R`A8*LTxQwDgTu&m}m>Gt#W?#lOk!(?8j6uvWGh~^RE&IM?$=+!6x*|*V ztBgXWgvef#x+F#=YxT}(z4!g+J)h_EoO8a<`TfrKcfRM3?<0pcG(HMbM1bY$-&Spa zl>sO~bn^ylX#q+WI0ByH38=6k8!!L>O2%G36dZ}&`(P+ILmZaqjsxrHfXNgR4&w_B zx{~B@ze-Mh$N0wKIo{^#g0L{q&YcP2a&F<^<+{h?T^$iq@-k1VG|uxbizk&Y#@#IF zOQa&-o;J{0ojUzFX!E)B((2l=eTB?brS{MKU%njPe7k6(#aw!XPCh36*B@pXf3d7{ zUn?{rG^NoRyPC6d|&i=2{ugnddTYFvp15UzS%7Zu-?u zO5mAODcC|9jhUdgYGD!|cyhkK&-LEY#9;lv8z+M3Bd*8`)ddA*PxoilI^B_o6$^M$ zr^ONV{>l5XUkI53=}zSx-N$%NW$s@lkE(JT*RQ*>v<5DY zgZ=AZT_STH4rvLJT!P`HJ>_uap~C6;KkqF0NY(Pj(yRRDN9aWeJq#=CWnlTJ!mWE& zyKKxBXJk%>XdJ;dyc~TdQVMbs>=&A4Rhh;dXYmO41j{2YEAK9Ab7`^OuP?6qV*5dY~$ODvLIF6b2LOj4+E6h~^a-{w*NT@>DeoiH_3RCbN6m zFp{|lBZIveYEi-$Ze^aw?}mC#ovG@}s+Q!I+Mr5$67mdev$}4sy&F?WxENv~sy1e% z%`7GbThw;bANE=p7k+fJDUlfl%}j@`;!3*uv@dlxUD*5g%(Kji%sIP)a@ph33a+Xm zpqy7nby34IqLocK{nmY@P02Yf?N^Jz69C+Mw|8R6OU#IsvLhxqQlQe!cvNiY$x3_% z<3ZJW#++z0f4$c;45`N{+WqW{E~7)V-A>1aNPiNcse{a(i#1$hanE80*NIj>ZQ3GP z#=>xgcG#PkkyDzsY5V^CTpEmtaL!JNWkLFycFhsp% z(cGh9w0b#N-GP81hO`zpIHvq}x ziTill@%+u^YoT=>Mb6uOP=qWTvOXxMlP@zpUZ-z9P*ORpHK?hDw&Ix}({twB$Ybqu zQV@?)?h5=x>fqwh#`uneF`;TkgK7F+@Akfs11`(tfRdQn&=Vzr`}wX)KDv->WtJXb zSNObtebaSRqcSXU{m@7G-bo|Ji3*p3O9-cIKLIs6q@)MtA}maIa+iT`Bj60KwUsYL{0N1I)>VA-AMJ>(&?i7xu3>1?gC%bki6w zj3TZ}>rd&}-?~xqyzb0m458+|d1q6iZTNYas|aFW`0tnFd9xU)F34$vPPf=O+fL-w zy&HWs9Y#6Mp?7y$lU*z75?6##2T8h5v;k&$bO=kp|4MIP9O_dRlFc%3-DVK z!3K}D9BczTzZ6&H)%tbB@JEB5O|S@$>pp<;_MUC#Ez6UYd1=%CDay^XPws)Hp2$9< zw|~_vh=L~9`WbV{4yB`l*gHELb@&cbqN5~k>iG&DP#w~IGJHj4 za9bh=%gX@_JKP)!A?%`|WX+p_;CJ;A`u%K&R zi{Y_+sd)L6xn9l-44STzqCT8$=b>lmQPdc-=ULJ5Q2*`Cw`G$Zs69DJ&L|fQLldEW zT(no$r!3s`S8{1V?hhHgXpX?IWS4oglU>+o2MtGSzdVW<0IR8Y1B@qjNmS} zUf6i~HDN-r=ad59b*FTz1-DJU)+MW4Lb?6#?}x7~0lD{p^TxaI{n`V&qC_d=G%6gH zWO>6>!|1n?0pn{eHnPT6R+^7@zq8pm<7D1s`%%HLrOHTvCxt)sQvRDFEq#KhU5;8Z zNqC3|mrskx=a=cfdzl`c_OXB2+6-~K8E&sd-LSm9vEGVn4;?U=9d^M-#Lp6CYG)UU z@!pE}X7xR%pkfq1k$Jb(yln*wbjX_Cj`rDQDz6|XOw&#p^1sWR zuqRL16Hpq_P#)!^Z{FEK-lB-T9$ zl{)`ALuyKD8TF75yC;xZ7^02S6ht5y&X)b=^lQD=&oB5{wXwiBg8MgP#2#@h6~{+$ zdbX2-E5neS1)M4ThvVAvSVln<6I%+~UjbTL9Dv=FFi`dcP%_X1V1N=9!+x_}7yASI z&3?Y+LHx)AD4huQ!vRWY3?8?Y6OAF^2o!eMcOz$sbN9mN6N3O}HUtG!l@*aH2&jrO zprXzm!Bth20hg^v10sRK4kZH|U-;DonvjSAe!r0Wy4M!^ov$sM0SQAO`*HTg27iYQ z*tj7s&W1ggIzFkrh>3XudTU@=}KA^=l_K^5VEf+vOIr=i3?RyK$yVf;M3uw+Fd34ie5 z);4Go(LDgmcD?_9*iTK=e%byje*+8!<3q%MP0iMa{7sdgF+n7p2N((h= z5r7Bq4TGz}*qJ#EApF2ol-U~n6N5rw?Bn?-23LWxCI1(Ofd20~H6&Y&f2~u8s{h0& zB#f62j