From 60da9d6543f33d9148aaef0d7bf66c50967ce5b7 Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Fri, 3 Apr 2026 11:21:20 -0500 Subject: [PATCH] update graph --- content/Math4202/Math4202_L30.md | 28 ++++++++++++++++++ .../Math4202/Seifert-Van-Kampen-Theorem.png | Bin 0 -> 22464 bytes 2 files changed, 28 insertions(+) create mode 100644 content/Math4202/Math4202_L30.md create mode 100644 public/Math4202/Seifert-Van-Kampen-Theorem.png diff --git a/content/Math4202/Math4202_L30.md b/content/Math4202/Math4202_L30.md new file mode 100644 index 0000000..ebccca4 --- /dev/null +++ b/content/Math4202/Math4202_L30.md @@ -0,0 +1,28 @@ +# Math4202 Topology II (Lecture 30) + +## Algebraic Topology + +We skipped a few chapters about Jordan curve theorem, which will be your final project soon. LOL, I will embedded the link once I'm done. + +### Seifert-Van Kampen Theorem + +#### The Seifert-Van Kampen Theorem + +Let $X=U\cup V$ be a union of two open subspaces. Suppose that $U\cap V$, $U,V$ are path connected. Fix $x_0\in U\cap V$. + +Let $H$ be a group (arbitrary). And now we assume $\phi_1,\phi_2$ be a group homomorphism, and $\phi_1:\pi_1(U,x_0)\to H$, and $\phi_2:\pi_1(V,x_0)\to H$. + +![Seifert-Van Kampen Theorem](https://notenextra.trance-0.com/Math4202/Math4202_L30/Seifert-Van-Kampen-Theorem.png) + +Let $i_1,i_2,j_1,j_2,i_{12}$ be group homomorphism induced by the inclusion maps. + +Assume this diagram commutes. + +$$ +\phi_1\circ i_1=\phi_2\circ i_2 +$$ + +There is a group homomorphism $\Phi:\pi_1(X,x_0)\to H$ making the diagram commute. $\Phi\circ j_1=\phi_1$ and $\Phi\circ j_2=\phi_2$. + +We may change the base point using conjugations. + diff --git a/public/Math4202/Seifert-Van-Kampen-Theorem.png b/public/Math4202/Seifert-Van-Kampen-Theorem.png new file mode 100644 index 0000000000000000000000000000000000000000..9d5ccc59307abf7c607b7cfa78a4587c176bd535 GIT binary patch literal 22464 zcmce8cQ}^s|M#s$l-00FL`6hp?~KZdl$DV!*|Jw9k=; zulw^moK z*$(`R>#+QN{I%ENg03@xKv_rpAIVc5N+tr~2;ri|7oNh)2UxqGYj=wGdR=8XlT-`66MWP(mz-3KT{+c>B1Dxue?XS5$MhWp>&pP7X1o;-SFQv9j4b-pb{T~kw2+->GZYb!g43c-^cm&_r)hamIs zeP-s{R;$;qU)M$9zw;WDK3PdgPrqei^18fSVl!{1ufoRE^d6g+r>9+4?)@sq=_YfDSE zU9T7k$(_lLtjF|w$CY2jlgcFg9sTw)G<2VUKu&eFjHIOG+LfiHrMlq?uYcb=R_QZi z1WZ5G1s~|5wbzg4)qS0uJTf-+EH+lfmG@PIe=L4~Xl5py+H+%NK9u2%URhv7g#F6g zpo;o2PR=6%KYsiuF8RFeL`_4}RcNcPsv1^SCjRu&F>&S=I^vyhl97?Qugu-Jc=4hQ zU20m|x0k}rb#-sff85&q*B=}l{PykJD=BKW!?|5~mJw{yl_T@%>FM1?_94N+Q6et) z@4R|-L~gvV!i$fO54XyNk6mvi5NWd#OEiJ7(qgyj+=N1;uxM&fmv1SDQI8kS=Q>QX|8=pLR;_c%ju9X`RLB;BOcT#cR_wV0_hKIZP z*m-y;*v%{~+7s8m^PD37`4}O5tJ4AdZ*I=P-rk=1Dz}BAwY7B^v#8*)Vn?WWh*CIy@KWPec_RSYZ-b!^)75|Y`BkG&>8)zxla*ng-_5obXfgsE}^KXug|2ON-yE=aOo0x z+xH`5ckJZ1T6hU-DPO)^3W$9C_`ue3{kA-ruAbh|l!84Dt_XX<=vZ??@krziHP$-~ z9vSn?%XYVJO$~$?hx@)O%gZ}c+f`Rrr}|PzBkf0yY5k{9pDMjqH7{Sz#(|j|thJ?6 zE>|G9F#eVxG&ug7F^~|B-AXul@}zP!ueHZK&fceRmc~mt@^=<<^YSMD{FyU#c6NTN zj$_Rw*YM-V!3c&4zEi{VOG`MGw1*B!It~S-rl$J)SI3Gv&uu&`92*=gBJpQ6uO?2kSI4la7^RA&)VV%4K~}0y1FHkId^<1 zomdG|*K~C!Mn?mwkAAvIYyAqJRuKFrIyyS8(zM(8>-%@_93(;xpVGW?WpdFVn~vWw zARvHxdvS5`jkLawPUokGCv6js{ki|YOWFCSx6JME;lsryN};nyJm2BB>cd&Qu;ZiE zy>ZH!KAt>rqJVi-PN(n|LuMbYP0HhMz?PPW$LhxF!u$8{W5R{m@us-xB%c$cgK_8N)1$js2CjJAVEwkWil5-`LzNZP^ry z?TTAkTy)vs_4W1T=g(@l*xFoZmX?r^=;`ShYl<`Vlr=X`e*C!Y?rw@*yLRo{zu#|j z?dkL97o?TdFJC^*$4ANk<8DOi`}YUbxBre%I3DX;j&PXmll1of`0$WzWdR|37jdl* z(9qne`hdz<>9?)M+}zyEeb@&TzUkXH+tnUd=f1b9FXwj8&(EJ~{`B!p-P4e{98+Qt|fo15R~e=g;-Hrrn{FS}M@7#14(<;xebsD+@9RwxL+f~X0H zKm0N!WJd%Cf1V$%pARP%lDhfC^A8ji6naaXN!jz==S_C+zOQz$(sy$W&E?$LvszkO z`4Tq9#?(wqOsuRkqodcZUsqx7M;CZ_hzAe-*KvEp+bfq{Xtva&11K|y46blNr{kzq{2;WvU%+3l(2Y}(TqM4dP( zW=ROs?BWwU2s+>GBqJ1)x;i^iC9({Ar`_uBU8iIgX-Tx)+*mzx^yuBx)f+Efyx6x? 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