\newcommand{\cat}[1]{\mathsf{#1}} \newcommand{\Sets}[0]{{\mathsf{Set}}} \newcommand{\Set}[0]{{\mathsf{Set}}} \newcommand{\sets}[0]{{\mathsf{Set}}} \newcommand{\set}{{\mathsf{Set} }} \newcommand{\Poset}[0]{\mathsf{Poset}} \newcommand{\GSets}[0]{{G\dash\mathsf{Set}}} \newcommand{\Groups}[0]{{\mathsf{Group}}} \newcommand{\Grp}[0]{{\mathsf{Grp}}} % Modifiers \newcommand{\der}[0]{{\mathsf{d}}} \newcommand{\dg}[0]{{\mathsf{dg}}} \newcommand{\comm}[0]{{\mathsf{C}}} \newcommand{\pre}[0]{{\mathsf{pre}}} \newcommand{\fn}[0]{{\mathsf{fn}}} \newcommand{\smooth}[0]{{\mathsf{sm}}} \newcommand{\Aff}[0]{{\mathsf{Aff}}} \newcommand{\Ab}[0]{{\mathsf{Ab}}} \newcommand{\Add}[0]{{\mathsf{Add}}} \newcommand{\Assoc}[0]{\mathsf{Assoc}} \newcommand{\Ch}[0]{\mathsf{Ch}} \newcommand{\Coh}[0]{{\mathsf{Coh}}} \newcommand{\Comm}[0]{\mathsf{Comm}} \newcommand{\Cor}[0]{\mathsf{Cor}} \newcommand{\Corr}[0]{\mathsf{Cor}} \newcommand{\Fin}[0]{{\mathsf{Fin}}} \newcommand{\Free}[0]{\mathsf{Free}} \newcommand{\Tors}[0]{\mathsf{Tors}} \newcommand{\Perf}[0]{\mathsf{Perf}} \newcommand{\Unital}[0]{\mathsf{Unital}} \newcommand{\eff}[0]{\mathsf{eff}} \newcommand{\derivedcat}[1]{\mathbf{D} {#1} } \newcommand{\bderivedcat}[1]{\mathbf{D}^b {#1} } \newcommand{\Cx}[0]{\mathsf{Ch}} \newcommand{\Stable}[0]{\mathsf{Stab}} \newcommand{\ChainCx}[1]{\mathsf{Ch}\qty{ #1 } } \newcommand{\Vect}[0]{{ \mathsf{Vect} }} \newcommand{\kvect}[0]{{ \mathsf{Vect}\slice{k} }} \newcommand{\loc}[0]{{\mathsf{loc}}} \newcommand{\locfree}[0]{{\mathsf{locfree}}} \newcommand{\Bun}{{\mathsf{Bun}}} \newcommand{\bung}{{\mathsf{Bun}_G}} % Rings \newcommand{\Local}[0]{\mathsf{Local}} \newcommand{\Fieldsover}[1]{{ \mathsf{Fields}_{#1} }} \newcommand{\Field}[0]{\mathsf{Field}} \newcommand{\Number}[0]{\mathsf{Number}} \newcommand{\Numberfield}[0]{\Field\slice{\QQ}} \newcommand{\NF}[0]{\Numberfield} \newcommand{\Art}[0]{\mathsf{Art}} \newcommand{\Global}[0]{\mathsf{Global}} \newcommand{\Ring}[0]{\mathsf{Ring}} \newcommand{\Mon}[0]{\mathsf{Mon}} \newcommand{\CMon}[0]{\mathsf{CMon}} \newcommand{\CRing}[0]{\mathsf{CRing}} \newcommand{\DedekindDomain}[0]{\mathsf{DedekindDom}} \newcommand{\IntDomain}[0]{\mathsf{IntDom}} \newcommand{\Domain}[0]{\mathsf{Domain}} \newcommand{\DVR}[0]{\mathsf{DVR}} \newcommand{\Dedekind}[0]{\mathsf{Dedekind}} % Modules \newcommand{\modr}[0]{{\mathsf{Mod}\dash\mathsf{R}}} \newcommand{\modsleft}[1]{\mathsf{#1}\dash\mathsf{Mod}} \newcommand{\modsright}[1]{\mathsf{Mod}\dash\mathsf{#1}} \newcommand{\mods}[1]{{\mathsf{#1}\dash\mathsf{Mod}}} \newcommand{\stmods}[1]{{\mathsf{#1}\dash\mathsf{stMod}}} \newcommand{\grmods}[1]{{\mathsf{#1}\dash\mathsf{grMod}}} \newcommand{\comods}[1]{{\mathsf{#1}\dash\mathsf{coMod}}} \newcommand{\algs}[1]{{{#1}\dash\mathsf{Alg}}} \newcommand{\Quat}[0]{{\mathsf{Quat}}} \newcommand{\torsors}[1]{{\mathsf{#1}\dash\mathsf{Torsors}}} \newcommand{\torsorsright}[1]{\mathsf{Torsors}\dash\mathsf{#1}} \newcommand{\torsorsleft}[1]{\mathsf{#1}\dash\mathsf{Torsors}} \newcommand{\bimod}[2]{({#1}, {#2})\dash\mathsf{biMod}} \newcommand{\bimods}[2]{({#1}, {#2})\dash\mathsf{biMod}} \newcommand{\Mod}[0]{{\mathsf{Mod}}} \newcommand{\Dmod}[0]{{ \mathcal{D}\dash\mathsf{Mod} }} \newcommand{\zmod}[0]{{\mathbb{Z}\dash\mathsf{Mod}}} \newcommand{\rmod}[0]{{\mathsf{R}\dash\mathsf{Mod}}} \newcommand{\amod}[0]{{\mathsf{A}\dash\mathsf{Mod}}} \newcommand{\kmod}[0]{{\mathsf{k}\dash\mathsf{Mod}}} \newcommand{\gmod}[0]{{\mathsf{G}\dash\mathsf{Mod}}} \newcommand{\grMod}[0]{{\mathsf{grMod}}} \newcommand{\gr}[0]{{\mathsf{gr}\,}} \newcommand{\mmod}[0]{{\dash\mathsf{Mod}}} \newcommand{\Rep}[0]{{\mathsf{Rep}}} \newcommand{\Irr}[0]{{\mathsf{Irr}}} \newcommand{\Adm}[0]{{\mathsf{Adm}}} \newcommand{\semisimp}[0]{{\mathsf{ss}}} % Vector Spaces and Bundles \newcommand{\VectBundle}[0]{{ \Bun\qty{\GL_r} }} \newcommand{\VectBundlerk}[1]{{ \Bun\qty{\GL_{#1}} }} \newcommand{\VectSp}[0]{{ \VectSp }} \newcommand{\VectBun}[0]{{ \VectBundle }} \newcommand{\VectBunrk}[1]{{ \VectBundlerk{#1} }} \newcommand{\Bung}[0]{{ \Bun\qty{G} }} % Algebras \newcommand{\Hopf}[0]{\mathsf{Hopf}} \newcommand{\alg}[0]{\mathsf{Alg}} \newcommand{\Alg}[0]{{\mathsf{Alg}}} \newcommand{\scalg}[0]{\mathsf{sCAlg}} \newcommand{\cAlg}[0]{{\mathsf{cAlg}}} \newcommand{\calg}[0]{\mathsf{CAlg}} \newcommand{\liegmod}[0]{{\mathfrak{g}\dash\mathsf{Mod}}} \newcommand{\liealg}[0]{{\mathsf{Lie}\dash\mathsf{Alg}}} \newcommand{\Lie}[0]{\mathsf{Lie}} \newcommand{\kalg}[0]{{\mathsf{Alg}_{/k} }} \newcommand{\kAlg}[0]{{\mathsf{Alg}_{/k} }} \newcommand{\kSch}[0]{{\mathsf{Sch}_{/k}}} \newcommand{\rAlg}[0]{{\mathsf{Alg}_{/R}}} \newcommand{\ralg}[0]{{\mathsf{Alg}_{/R}}} \newcommand{\zalg}[0]{{\mathsf{Alg}_{/\ZZ}}} \newcommand{\CCalg}[0]{{\mathsf{Alg}_{\mathbb{C}} }} \newcommand{\dga}[0]{{\mathsf{dg\Alg} }} \newcommand{\cdga}[0]{{ \mathsf{c}\dga }} \newcommand{\dgla}[0]{{\dg\Lie\Alg }} \newcommand{\Poly}[0]{{\mathsf{Poly} }} \newcommand{\Hk}[0]{{\mathsf{Hk} }} \newcommand{\Grpd}[0]{{\mathsf{Grpd}}} \newcommand{\inftyGrpd}[0]{{ \underset{\infty}{ \Grpd } }} \newcommand{\Algebroid}[0]{{\mathsf{Algd}}} % Schemes and Sheaves \newcommand{\Loc}[0]{\mathsf{Loc}} \newcommand{\Locsys}[0]{\mathsf{LocSys}} \newcommand{\Ringedspace}[0]{\mathsf{RingSp}} \newcommand{\RingedSpace}[0]{\mathsf{RingSp}} \newcommand{\LRS}[0]{\Loc\RingedSpace} \newcommand{\IndCoh}[0]{{\mathsf{IndCoh}}} \newcommand{\Ind}[0]{{\mathsf{Ind}}} \newcommand{\Pro}[0]{{\mathsf{Pro}}} \newcommand{\DCoh}[0]{{\mathsf{DCoh}}} \newcommand{\QCoh}[0]{{\mathsf{QCoh}}} \newcommand{\Cov}[0]{{\mathsf{Cov}}} \newcommand{\sch}[0]{{\mathsf{Sch}}} \newcommand{\presh}[0]{ \underset{ \mathsf{pre} } {\mathsf{Sh} }} \newcommand{\prest}[0]{ {\underset{ \mathsf{pre} } {\mathsf{St} } } } \newcommand{\Descent}[0]{{\mathsf{Descent}}} \newcommand{\Desc}[0]{{\mathsf{Desc}}} \newcommand{\FFlat}[0]{{\mathsf{FFlat}}} \newcommand{\Perv}[0]{\mathsf{Perv}} \newcommand{\smsch}[0]{{ \smooth\Sch }} \newcommand{\Sch}[0]{{\mathsf{Sch}}} \newcommand{\Schf}[0]{{\mathsf{Schf}}} \newcommand{\Sh}[0]{{\mathsf{Sh}}} \newcommand{\St}[0]{{\mathsf{St}}} \newcommand{\Stacks}[0]{{\mathsf{St}}} \newcommand{\Vark}[0]{{\mathsf{Var}_{/k} }} \newcommand{\Var}[0]{{\mathsf{Var}}} \newcommand{\Open}[0]{{\mathsf{Open}}} % Homotopy \newcommand{\CW}[0]{{\mathsf{CW}}} \newcommand{\sset}[0]{{\mathsf{sSet}}} \newcommand{\sSet}[0]{{\mathsf{sSet}}} \newcommand{\ssets}[0]{\mathsf{sSet}} \newcommand{\hoTop}[0]{{\mathsf{hoTop}}} \newcommand{\hoType}[0]{{\mathsf{hoType}}} \newcommand{\ho}[0]{{\mathsf{ho}}} \newcommand{\SHC}[0]{{\mathsf{SHC}}} \newcommand{\SH}[0]{{\mathsf{SH}}} \newcommand{\Spaces}[0]{{\mathsf{Spaces}}} \newcommand{\GSpaces}[1]{{G\dash\mathsf{Spaces}}} \newcommand{\Spectra}[0]{{\mathsf{Sp}}} \newcommand{\Sp}[0]{{\mathsf{Sp}}} \newcommand{\Top}[0]{{\mathsf{Top}}} \newcommand{\Bord}[0]{{\mathsf{Bord}}} \newcommand{\TQFT}[0]{{\mathsf{TQFT}}} \newcommand{\Kc}[0]{{\mathsf{K^c}}} \newcommand{\triang}[0]{{\mathsf{triang}}} \newcommand{\TTC}[0]{{\mathsf{TTC}}} \newcommand{\dchrmod}{{\derivedcat{\Ch(\rmod)} }} % Infty Cats \newcommand{\Finset}[0]{{\mathsf{FinSet}}} \newcommand{\Cat}[0]{\mathsf{Cat}} \newcommand{\Fun}[0]{{\mathsf{Fun}}} \newcommand{\Kan}[0]{{\mathsf{Kan}}} \newcommand{\Monoid}[0]{\mathsf{Mon}} \newcommand{\Arrow}[0]{\mathsf{Arrow}} \newcommand{\quasiCat}[0]{{ \mathsf{quasiCat} } } \newcommand{\inftycat}[0]{{ \underset{\infty}{ \Cat} }} \newcommand{\inftycatn}[1]{{ \underset{(\infty, {#1})}{ \Cat} }} \newcommand{\core}[0]{{ \mathsf{core} }} \newcommand{\Indcat}[0]{ \mathsf{Ind} } % New? \newcommand{\Prism}[0]{\mathsf{Prism}} \newcommand{\Solid}[0]{\mathsf{Solid}} \newcommand{\WCart}[0]{\mathsf{WCart}} % Motivic \newcommand{\Torsor}[1]{{\mathsf{#1}\dash\mathsf{Torsor}}} \newcommand{\Torsorleft}[1]{{\mathsf{#1}\dash\mathsf{Torsor}}} \newcommand{\Torsorright}[1]{{\mathsf{Torsor}\dash\mathsf{#1} }} \newcommand{\Quadform}[0]{{\mathsf{QuadForm}}} \newcommand{\HI}[0]{{\mathsf{HI}}} \newcommand{\DM}[0]{{\mathsf{DM}}} \newcommand{\hoA}[0]{{\mathsf{ho}_*^{\scriptstyle \AA^1}}} \newcommand\Tw[0]{\mathsf{Tw}} \newcommand\SB[0]{\mathsf{SB}} \newcommand\CSA[0]{\mathsf{CSA}} \newcommand{\CSS}[0]{{ \mathsf{CSS} } } % Unsorted \newcommand{\FGL}[0]{\mathsf{FGL}} \newcommand{\FI}[0]{{\mathsf{FI}}} \newcommand{\CE}[0]{{\mathsf{CE}}} \newcommand{\Fuk}[0]{{\mathsf{Fuk}}} \newcommand{\Lag}[0]{{\mathsf{Lag}}} \newcommand{\Mfd}[0]{{\mathsf{Mfd}}} \newcommand{\Riem}[0]{\mathsf{Riem}} \newcommand{\Wein}[0]{{\mathsf{Wein}}} \newcommand{\gspaces}[1]{{#1}\dash{\mathsf{Spaces}}} \newcommand{\deltaring}[0]{{\delta\dash\mathsf{Ring}}} \newcommand{\terminal}[0]{{ \mathscr{1}_{\scriptscriptstyle \uparrow} }} \newcommand{\initial}[0]{{ \mathscr \emptyset^{\scriptscriptstyle \downarrow} }} % Universal guys \newcommand{\coeq}[0]{\operatorname{coeq}} \newcommand{\cocoeq}[0]{\operatorname{eq}} \newcommand{\dgens}[1]{\gens{\gens{ #1 }}} \newcommand{\ctz}[1]{\, {\converges{{#1} \to\infty}\longrightarrow 0} \, } \newcommand{\conj}[1]{{\overline{{#1}}}} \newcommand{\complex}[1]{{ {#1}_{\scriptscriptstyle \bullet}} } \newcommand{\cocomplex}[1]{ { {#1}^{\scriptscriptstyle \bullet}} } \newcommand{\bicomplex}[1]{{ {#1}_{\scriptscriptstyle \bullet, \bullet}} } \newcommand{\cobicomplex}[1]{ { {#1}^{\scriptscriptstyle \bullet, \bullet}} } \newcommand{\floor}[1]{{\left\lfloor #1 \right\rfloor}} \newcommand{\ceiling}[1]{{\left\lceil #1 \right\rceil}} \newcommand{\fourier}[1]{\widehat{#1}} \newcommand{\embedsvia}[1]{\xhookrightarrow{#1}} \newcommand{\openimmerse}[0]{\underset{\scriptscriptstyle O}{\hookrightarrow}} \newcommand{\weakeq}[0]{\underset{\scriptscriptstyle W}{\rightarrow}} \newcommand{\fromvia}[1]{\xleftarrow{#1}} \newcommand{\generators}[1]{\left\langle{#1}\right\rangle} \newcommand{\gens}[1]{\left\langle{#1}\right\rangle} \newcommand{\globsec}[1]{{{\Gamma}\qty{#1} }} \newcommand{\Globsec}[1]{{{\Gamma}\qty{#1} }} \newcommand{\langL}[1]{ {}^{L}{#1} } \newcommand{\equalsbecause}[1]{\overset{#1}{=}} \newcommand{\congbecause}[1]{\overset{#1}{\cong}} \newcommand{\congas}[1]{\underset{#1}{\cong}} \newcommand{\isoas}[1]{\underset{#1}{\cong}} \newcommand{\addbase}[1]{{ {}_{\pt} }} \newcommand{\ideal}[1]{\mathcal{#1}} \newcommand{\adjoin}[1]{ { \left[ \scriptstyle {#1} \right] } } \newcommand{\polynomialring}[1]{ { \left[ {#1} \right] } } \newcommand{\htyclass}[1]{ { \left[ {#1} \right] } } \newcommand{\qtext}[1]{{\quad \operatorname{#1} \quad}} \newcommand{\abs}[1]{{\left\lvert {#1} \right\rvert}} \newcommand{\stack}[1]{\mathclap{\substack{ #1 }}} \newcommand{\powerseries}[1]{ { \left[ {#1} \right] } } \newcommand{\functionfield}[1]{ { \left( {#1} \right) } } \newcommand{\rff}[1]{ \functionfield{#1} } \newcommand{\fps}[1]{{\left[\left[ #1 \right]\right] }} \newcommand{\formalseries}[1]{ \fps{#1} } \newcommand{\formalpowerseries}[1]{ \fps{#1} } \newcommand\fls[1]{{\left(\left( #1 \right)\right) }} \newcommand\lshriek[0]{{}_{!}} \newcommand\pushf[0]{{}^{*}} \newcommand{\nilrad}[1]{{\sqrt{0_{#1}} }} \newcommand{\jacobsonrad}[1]{{J ({#1}) }} \newcommand{\localize}[1]{ \left[ { \scriptstyle { {#1}\inv} } \right]} \newcommand{\primelocalize}[1]{ \left[ { \scriptstyle { { ({#1}^c) }\inv} } \right]} \newcommand{\plocalize}[1]{\primelocalize{#1}} \newcommand{\sheafify}[1]{ \left( #1 \right)^{\scriptscriptstyle \mathrm{sh}} } \newcommand{\complete}[1]{{ {}_{ \hat{#1} } }} \newcommand{\takecompletion}[1]{{ \overbrace{#1}^{\widehat{\hspace{4em}}} }} \newcommand{\pcomplete}[0]{{ {}^{ \wedge }_{p} }} \newcommand{\kv}[0]{{ k_{\hat{v}} }} \newcommand{\Lv}[0]{{ L_{\hat{v}} }} \newcommand{\twistleft}[2]{{ {}^{#1} #2 }} \newcommand{\twistright}[2]{{ #2 {}^{#1} }} \newcommand{\liesover}[1]{{ {}_{/ {#1}} }} \newcommand{\liesabove}[1]{{ {}_{/ {#1}} }} \newcommand{\slice}[1]{_{/ {#1}} } \newcommand{\coslice}[1]{_{{#1/}} } \newcommand{\quotright}[2]{ {}^{#1}\mkern-2mu/\mkern-2mu_{#2} } \newcommand{\quotleft}[2]{ {}_{#2}\mkern-.5mu\backslash\mkern-2mu^{#1} } \newcommand{\invert}[1]{{ \left[ { \scriptstyle \frac{1}{#1} } \right] }} \newcommand{\symb}[2]{{ \qty{ #1 \over #2 } }} \newcommand{\squares}[1]{{ {#1}_{\scriptscriptstyle \square} }} \newcommand{\shift}[2]{{ \Sigma^{\scriptstyle[#2]} #1 }} \newcommand\cartpower[1]{{ {}^{ \scriptscriptstyle\times^{#1} } }} \newcommand\disjointpower[1]{{ {}^{ \scriptscriptstyle\coprod^{#1} } }} \newcommand\sumpower[1]{{ {}^{ \scriptscriptstyle\oplus^{#1} } }} \newcommand\prodpower[1]{{ {}^{ \scriptscriptstyle\times^{#1} } }} \newcommand\tensorpower[2]{{ {}^{ \scriptstyle\otimes_{#1}^{#2} } }} \newcommand\tensorpowerk[1]{{ {}^{ \scriptscriptstyle\otimes_{k}^{#1} } }} \newcommand\derivedtensorpower[3]{{ {}^{ \scriptstyle {}_{#1} {\otimes_{#2}^{#3}} } }} \newcommand\smashpower[1]{{ {}^{ \scriptscriptstyle\smashprod^{#1} } }} \newcommand\wedgepower[1]{{ {}^{ \scriptscriptstyle\smashprod^{#1} } }} \newcommand\fiberpower[2]{{ {}^{ \scriptscriptstyle\fiberprod{#1}^{#2} } }} \newcommand\powers[1]{{ {}^{\cdot #1} }} \newcommand\skel[1]{{ {}^{ (#1) } }} \newcommand\transp[1]{{ \, {}^{t}{ \left( #1 \right) } }} \newcommand{\inner}[2]{{\left\langle {#1},~{#2} \right\rangle}} \newcommand{\inp}[2]{{\left\langle {#1},~{#2} \right\rangle}} \newcommand{\poisbrack}[2]{{\left\{ {#1},~{#2} \right\} }} \newcommand\tmf{ \mathrm{tmf} } \newcommand\taf{ \mathrm{taf} } \newcommand\TAF{ \mathrm{TAF} } \newcommand\TMF{ \mathrm{TMF} } \newcommand\String{ \mathrm{String} } \newcommand{\BO}[0]{{\B \Orth}} \newcommand{\EO}[0]{{\mathsf{E} \Orth}} \newcommand{\BSO}[0]{{\B\SO}} \newcommand{\ESO}[0]{{\mathsf{E}\SO}} \newcommand{\BG}[0]{{\B G}} \newcommand{\EG}[0]{{\mathsf{E} G}} \newcommand{\BP}[0]{{\operatorname{BP}}} \newcommand{\BU}[0]{\B{\operatorname{U}}} \newcommand{\MO}[0]{{\operatorname{MO}}} \newcommand{\MSO}[0]{{\operatorname{MSO}}} \newcommand{\MSpin}[0]{{\operatorname{MSpin}}} \newcommand{\MSp}[0]{{\operatorname{MSpin}}} \newcommand{\MString}[0]{{\operatorname{MString}}} \newcommand{\MStr}[0]{{\operatorname{MString}}} \newcommand{\MU}[0]{{\operatorname{MU}}} \newcommand{\KO}[0]{{\operatorname{KO}}} \newcommand{\KU}[0]{{\operatorname{KU}}} \newcommand{\smashprod}[0]{\wedge} \newcommand{\ku}[0]{{\operatorname{ku}}} \newcommand{\hofib}[0]{{\operatorname{hofib}}} \newcommand{\hocofib}[0]{{\operatorname{hocofib}}} \DeclareMathOperator{\Suspendpinf}{{\Sigma_+^\infty}} \newcommand{\Loop}[0]{{\Omega}} \newcommand{\Loopinf}[0]{{\Omega}^\infty} \newcommand{\Suspend}[0]{{\Sigma}} \newcommand*\dif{\mathop{}\!\operatorname{d}} \newcommand*{\horzbar}{\rule[.5ex]{2.5ex}{0.5pt}} \newcommand*{\vertbar}{\rule[-1ex]{0.5pt}{2.5ex}} \newcommand\Fix{ \mathrm{Fix} } \newcommand\CS{ \mathrm{CS} } \newcommand\FP{ \mathrm{FP} } \newcommand\places[1]{ \mathrm{Pl}\qty{#1} } \newcommand\Ell{ \mathrm{Ell} } \newcommand\homog{ { \mathrm{homog} } } \newcommand\Kahler[0]{\operatorname{Kähler}} \newcommand\Prinbun{\mathrm{Bun}^{\mathrm{prin}}} \newcommand\aug{\fboxsep=-\fboxrule\!\!\!\fbox{\strut}\!\!\!} \newcommand\compact[0]{\operatorname{cpt}} \newcommand\hyp[0]{{\operatorname{hyp}}} \newcommand\jan{\operatorname{Jan}} \newcommand\curl{\operatorname{curl}} \newcommand\kbar{ { \bar{k} } } \newcommand\ksep{ { k\sep } } \newcommand\mypound{\scalebox{0.8}{\raisebox{0.4ex}{\#}}} \newcommand\rref{\operatorname{RREF}} \newcommand\RREF{\operatorname{RREF}} \newcommand{\Tatesymbol}{\operatorname{TateSymb}} \newcommand\tilt[0]{ {}^{ \flat } } \newcommand\vecc[2]{\textcolor{#1}{\textbf{#2}}} \newcommand{\Af}[0]{{\mathbb{A}}} \newcommand{\Ag}[0]{{\mathcal{A}_g}} \newcommand{\Mg}[0]{{\mathcal{M}_g}} \newcommand{\Ahat}[0]{\hat{ \operatorname{A}}_g } \newcommand{\Ann}[0]{\operatorname{Ann}} \newcommand{\sinc}[0]{\operatorname{sinc}} \newcommand{\Banach}[0]{\mathcal{B}} \newcommand{\Arg}[0]{\operatorname{Arg}} \newcommand{\BB}[0]{{\mathbb{B}}} \newcommand{\Betti}[0]{{\operatorname{Betti}}} \newcommand{\CC}[0]{{\mathbb{C}}} \newcommand{\CF}[0]{\operatorname{CF}} \newcommand{\CH}[0]{{\operatorname{CH}}} \newcommand{\CP}[0]{{\mathbb{CP}}} \newcommand{\CY}{{ \text{CY} }} \newcommand{\Cl}[0]{{ \operatorname{Cl}} } \newcommand{\Crit}[0]{\operatorname{Crit}} \newcommand{\DD}[0]{{\mathbb{D}}} \newcommand{\DSt}[0]{{ \operatorname{DSt}}} \newcommand{\Def}{\operatorname{Def} } \newcommand{\Diffeo}[0]{{\operatorname{Diffeo}}} \newcommand{\Diff}[0]{\operatorname{Diff}} \newcommand{\Disjoint}[0]{\displaystyle\coprod} \newcommand{\resprod}[0]{\prod^{\res}} \newcommand{\restensor}[0]{\bigotimes^{\res}} \newcommand{\Disk}[0]{{\operatorname{Disk}}} \newcommand{\Dist}[0]{\operatorname{Dist}} \newcommand{\EE}[0]{{\mathbb{E}}} \newcommand{\EKL}[0]{{\mathrm{EKL}}} \newcommand{\QH}[0]{{\mathrm{QH}}} \newcommand{\AMGM}[0]{{\mathrm{AMGM}}} \newcommand{\resultant}[0]{{\mathrm{res}}} \newcommand{\tame}[0]{{\mathrm{tame}}} \newcommand{\primetop}[0]{{\scriptscriptstyle \mathrm{prime-to-}p}} \newcommand{\VHS}[0]{{\mathrm{VHS} }} \newcommand{\ZVHS}[0]{{ \ZZ\mathrm{VHS} }} \newcommand{\CR}[0]{{\mathrm{CR}}} \newcommand{\unram}[0]{{\scriptscriptstyle\mathrm{un}}} \newcommand{\Emb}[0]{{\operatorname{Emb}}} \newcommand{\minor}[0]{{\operatorname{minor}}} \newcommand{\Et}{\text{Ét}} \newcommand{\trace}{\operatorname{tr}} \newcommand{\Trace}{\operatorname{Trace}} \newcommand{\Kl}{\operatorname{Kl}} \newcommand{\Rel}{\operatorname{Rel}} \newcommand{\Norm}{\operatorname{Nm}} \newcommand{\Extpower}[0]{\bigwedge\nolimits} \newcommand{\Extalgebra}[0]{\bigwedge\nolimits} \newcommand{\Extalg}[0]{\Extalgebra} \newcommand{\Extcomplex}[0]{\cocomplex{ \Extalgebra} } \newcommand{\Extprod}[0]{\bigwedge\nolimits} \newcommand{\Ext}{\operatorname{Ext} } \newcommand{\FFbar}[0]{{ \bar{ \mathbb{F}} }} \newcommand{\FFpn}[0]{{\mathbb{F}_{p^n}}} \newcommand{\FFp}[0]{{\mathbb{F}_p}} \newcommand{\FF}[0]{{\mathbb{F}}} \newcommand{\FS}{{ \text{FS} }} \newcommand{\Fil}[0]{{\operatorname{Fil}}} \newcommand{\Flat}[0]{{\operatorname{Flat}}} \newcommand{\Fpbar}[0]{\bar{\mathbb{F}_p}} \newcommand{\Fpn}[0]{{\mathbb{F}_{p^n} }} \newcommand{\Fppf}[0]{\mathrm{\operatorname{Fppf}}} \newcommand{\Fp}[0]{{\mathbb{F}_p}} \newcommand{\Frac}[0]{\operatorname{Frac}} \newcommand{\GF}[0]{{\mathbb{GF}}} \newcommand{\GG}[0]{{\mathbb{G}}} \newcommand{\GL}[0]{\operatorname{GL}} \newcommand{\GW}[0]{{\operatorname{GW}}} \newcommand{\Gal}[0]{{ \mathsf{Gal}} } \newcommand{\bigo}[0]{{ \mathsf{O}} } \newcommand{\Gl}[0]{\operatorname{GL}} \newcommand{\Gr}[0]{{\operatorname{Gr}}} \newcommand{\HC}[0]{{\operatorname{HC}}} \newcommand{\HFK}[0]{\operatorname{HFK}} \newcommand{\HF}[0]{\operatorname{HF}} \newcommand{\HHom}{\mathscr{H}\kern-2pt\operatorname{om}} \newcommand{\HH}[0]{{\mathbb{H}}} \newcommand{\HP}[0]{{\operatorname{HP}}} \newcommand{\HT}[0]{{\operatorname{HT}}} \newcommand{\HZ}[0]{{H\mathbb{Z}}} \newcommand{\Hilb}[0]{\operatorname{Hilb}} \newcommand{\Homeo}[0]{{\operatorname{Homeo}}} \newcommand{\Honda}[0]{\mathrm{\operatorname{Honda}}} \newcommand{\Hsh}{{ \mathcal{H} }} \newcommand{\Id}[0]{\operatorname{Id}} \newcommand{\Intersect}[0]{\displaystyle\bigcap} \newcommand{\JCF}[0]{\operatorname{JCF}} \newcommand{\RCF}[0]{\operatorname{RCF}} \newcommand{\Jac}[0]{\operatorname{Jac}} \newcommand{\II}[0]{{\mathbb{I}}} \newcommand{\KK}[0]{{\mathbb{K}}} \newcommand{\KH}[0]{ \K^{\scriptscriptstyle \mathrm{H}} } \newcommand{\KMW}[0]{ \K^{\scriptscriptstyle \mathrm{MW}} } \newcommand{\KMimp}[0]{ \hat{\K}^{\scriptscriptstyle \mathrm{M}} } \newcommand{\KM}[0]{ \K^{\scriptstyle\mathrm{M}} } \newcommand{\Kah}[0]{{ \operatorname{Kähler} } } \newcommand{\LC}[0]{{\mathrm{LC}}} \newcommand{\LL}[0]{{\mathbb{L}}} \newcommand{\Log}[0]{\operatorname{Log}} \newcommand{\MCG}[0]{{\operatorname{MCG}}} \newcommand{\MM}[0]{{\mathcal{M}}} \newcommand{\mbar}[0]{\bar{\mathcal{M}}} \newcommand{\MW}[0]{\operatorname{MW}} \newcommand{\Mat}[0]{\operatorname{Mat}} \newcommand{\NN}[0]{{\mathbb{N}}} \newcommand{\NS}[0]{{\operatorname{NS}}} \newcommand{\OO}[0]{{\mathcal{O}}} \newcommand{\OP}[0]{{\mathbb{OP}}} \newcommand{\OX}[0]{{\mathcal{O}_X}} \newcommand{\Obs}{\operatorname{Obs} } \newcommand{\obs}{\operatorname{obs} } \newcommand{\Ob}[0]{{\operatorname{Ob}}} \newcommand{\Op}[0]{{\operatorname{Op}}} \newcommand{\Orb}[0]{{\mathrm{Orb}}} \newcommand{\Conj}[0]{{\mathrm{Conj}}} \newcommand{\Orth}[0]{{\operatorname{O}}} \newcommand{\PD}[0]{\mathrm{PD}} \newcommand{\PGL}[0]{\operatorname{PGL}} \newcommand{\GU}[0]{\operatorname{GU}} \newcommand{\PP}[0]{{\mathbb{P}}} \newcommand{\PSL}[0]{{\operatorname{PSL}}} \newcommand{\Pic}[0]{{\operatorname{Pic}}} \newcommand{\Pin}[0]{{\operatorname{Pin}}} \newcommand{\Places}[0]{{\operatorname{Places}}} \newcommand{\Presh}[0]{\presh} \newcommand{\QHB}[0]{\operatorname{QHB}} \newcommand{\PHS}[0]{\operatorname{PHS}} \newcommand{\QHS}[0]{\mathbb{Q}\kern-0.5pt\operatorname{HS}} \newcommand{\QQpadic}[0]{{ \QQ_p }} \newcommand{\ZZelladic}[0]{{ \ZZ_\ell }} \newcommand{\QQ}[0]{{\mathbb{Q}}} \newcommand{\QQbar}[0]{{ \bar{ \mathbb{Q} } }} \newcommand{\Quot}[0]{\operatorname{Quot}} \newcommand{\RP}[0]{{\mathbb{RP}}} \newcommand{\RR}[0]{{\mathbb{R}}} \newcommand{\Rat}[0]{\operatorname{Rat}} \newcommand{\Reg}[0]{\operatorname{Reg}} \newcommand{\Ric}[0]{\operatorname{Ric}} \newcommand{\SF}[0]{\operatorname{SF}} \newcommand{\SL}[0]{{\operatorname{SL}}} \newcommand{\SNF}[0]{\mathrm{SNF}} \newcommand{\SO}[0]{{\operatorname{SO}}} \newcommand{\SP}[0]{{\operatorname{SP}}} \newcommand{\SU}[0]{{\operatorname{SU}}} \newcommand{\F}[0]{{\operatorname{F}}} \newcommand{\Sgn}[0]{{ \Sigma_{g, n} }} \newcommand{\Sm}[0]{{\operatorname{Sm}}} \newcommand{\SpSp}[0]{{\mathbb{S}}} \newcommand{\Spec}[0]{\operatorname{Spec}} \newcommand{\Spf}[0]{\operatorname{Spf}} \newcommand{\Spc}[0]{\operatorname{Spc}} \newcommand{\spc}[0]{\operatorname{Spc}} \newcommand{\Spinc}[0]{\mathrm{Spin}^{{ 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\newcommand{\T}[0]{{\mathbf{T}}} \newcommand{\TX}[0]{{\T X} } \newcommand{\TM}[0]{{\T M} } \newcommand{\K}[0]{{\mathsf{K}}} \newcommand{\G}[0]{{\mathsf{G}}} %\newcommand{\H}[0]{{\mathsf{H}}} \newcommand{\D}{{ \mathsf{D} }} \newcommand{\mH}{{ \mathsf{H} }} \newcommand{\BGL}[0]{ \mathbf{B}\mkern-3mu \operatorname{GL} } \newcommand{\proportional}{ \propto } \newcommand{\asymptotic}{ \ll } \newcommand{\RM}[1]{% \textup{\uppercase\expandafter{\romannumeral#1}}% } \DeclareMathOperator{\righttriplearrows} {{\; \tikz{ \foreach \y in {0, 0.1, 0.2} { \draw [-stealth] (0, \y) -- +(0.5, 0);}} \; }} \DeclareMathOperator*{\mapbackforth}{\rightleftharpoons} \newcommand{\fourcase}[4]{ \begin{cases}{#1} & {#2} \\ {#3} & {#4}\end{cases} } \newcommand{\matt}[4]{{ \begin{bmatrix} {#1} & {#2} \\ {#3} & {#4} \end{bmatrix} }} \newcommand{\mattt}[9]{{ \begin{bmatrix} {#1} & {#2} & {#3} \\ {#4} & {#5} & {#6} \\ {#7} & {#8} & {#9} \end{bmatrix} }} \newcommand\stacksymbol[3]{ \mathrel{\stackunder[2pt]{\stackon[4pt]{$#3$}{$\scriptscriptstyle#1$}}{ $\scriptscriptstyle#2$}} } \newcommand{\textoperatorname}[1]{ \operatorname{\textnormal{#1}} } \newcommand\caniso[0]{{ \underset{\can}{\iso} }} \renewcommand{\ae}[0]{{ \text{a.e.} }} \newcommand\eqae[0]{\underset{\ae}{=}} \newcommand{\sech}[0]{{ \mathrm{sech} }} %\newcommand{\strike}[1]{{\enclose{\horizontalstrike}{#1}}} \DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} # Tuesday, January 11 :::{.remark} References: - - - [PLC's Notes](https://outlookuga-my.sharepoint.com/:f:/g/personal/pbl20394_uga_edu/EjT_H9wRyAhAsZodxgXOgU0BSIYbqWav8X1jZY5v3RxqJA?e=n6dfVJ) Emphasis for the course: applications to low-dimensional topology, lots of examples, and ways to construct contact structures. The first application is critical to 4-manifold theory: ::: ## Application 1 :::{.theorem title="Cerf's Theorem"} Every diffeomorphism $f: S^3\to S^3$ extends to a diffeomorphism $\BB^4\to \BB^4$. ::: :::{.remark} This isn't true in all dimensions! This is essentially what makes Kirby calculus on 4-manifolds possible without needing to track certain attaching data. ::: :::{.remark} There is a standard contact structure on $S^3$: regard $\CC^2 \cong \RR^4$ and suppose $f:S^3\to S^3$. There is an intrinsic property of contact structures called *tightness* which doesn't change under diffeomorphisms and is fundamental to 3-manifold topology. :::{.theorem title="Eliashberg"} There is a unique tight contact structure $\xi_\std$ on $S^3$. ::: So up to isotopy, $f$ fixes $\xi_\std$. ::: :::{.remark} A useful idea: tiling by holomorphic discs. This involves taking $S^1$ and foliating the bounded disc by geodesics -- by the magic of elliptic PDEs, this is unobstructed and can be continued throughout the disc just using convexity near the boundary. In higher dimensions: $\BB^4$ is foliated by a 2-dimensional family of holomorphic discs. ::: ## Application 2 :::{.remark} Another application: monotonic simplification (?) of the unknot. Given a knot $K \injects S^3$, a theorem of Alexander says $K$ can be braided about the $z\dash$axis, which can be described by a word $w\in B_n$, the braid group \[ B_n = \ts{ \sigma_1,\cdots, \sigma_{n-1} \st [\sigma_i, \sigma_j] = 1 \,\abs{i-j}\geq 2,\, \sigma_i \sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i \sigma_{i+1}\, i=1,\cdots, n-2 } .\] This captures positive vs negative braiding on nearby strands, commuting of strands that are far apart, and the Reidemeister 3 move. Write $K = K(\beta)$ for $\beta$ a braid for the braid closure. ::: :::{.remark} Markov's theorem: if $K = K(\beta_1), K(\beta_2)$ where $\beta_1 \in B_n$ and $\beta_2\in B_m$ with $m,n$ not necessarily equal, then there is a sequence of Markov moves $\beta_1$ to $\beta_2$. The moves are: - Stabilization and destabilization: ![](figures/2022-01-11_11-41-29.png) - Conjugation in $B_n$: ![](figures/2022-01-11_11-39-52.png) - Braid isotopy, which preserves braid words in $B_n$. ::: :::{.remark} A theorem of Birman-Menasco: if $K(\beta) = U$ is the unknot for $\beta \in B_n$, then there is a sequence of braids $\ts{\beta_i}_{i\leq k}$ with $\beta_k = 1 \in B_1$ such that - $\beta_i\in B_{n_i}$ - $K(\beta_i) = U$ - $n_1\geq n_2\geq \cdots \geq n_k = 1$ - $\beta_i \to \beta_{i+1}$ is either a Markov move or an exchange move. Here an exchange move is ![](figures/2022-01-11_11-46-31.png) ::: ## Application 3 :::{.remark} Genus bounds. A theorem due to Thurston-Eliashberg: if $\xi$ is either a taut foliation or a tight contact structure on a 3-manifold $Y$ and $\Sigma \neq S^2$ is an embedded orientable surface in $Y$, then there is an Euler class $e(\xi) \in H^2(Y)$. Then \[ \abs{ \inner{ e(\xi)}{ \Sigma } } \leq -g(\Sigma) ,\] which after juggling signs is a lower bound on the genus of any embedded surface. ::: :::{.remark} Taut foliations: the basic example is $F\cross S^1$ for $F$ a surface. The foliation carries a co-orientation, and the tangencies at critical points of an embedded surface will have tangent planes tangent to the foliation, so one can compare the co-orientation to the outward normal of the surface to see if they agree or disagree and obtain a sign at each critical point. Write $c_\pm$ for the number of positive/negative elliptics and $h_\pm$ for the hyperbolics. Then \[ \chi = (e_+ + e_-) - (h_+ + h_-) ,\] by Poincaré-Hopf. On the other hand, $\inner{e(\xi)}{\Sigma} = (e_+ - h_+) - (e_- - h_-)$, so adding this yields \[ \inner{e(\xi)}{\Sigma} +\chi = 2(e_+ - h_+) \leq 0 .\] Isotope the surface to cancel critical points in pairs to get rid of caps/cups so that only saddles remain. ::: ## Contact Geometry :::{.definition title="?"} A contract structure on $Y^{2n+1}$ is a hyperplane field (a codimension 1 subbundle of the tangent bundle) $\xi = \ker \alpha$ such that $\alpha \wedge (d\alpha)\wedgepower{n} > 0$ is a positive volume form. ::: :::{.example title="?"} On $\RR^3$, \[ \alpha = dz - ydx \implies d\alpha = -dy \wedge dx = dx\wedge dy ,\] so \[ \alpha \wedge d\alpha = (dz-ydx)\wedge(dx\wedge dy) = dz\wedge dx\wedge dy = dx\wedge dy\wedge dz .\] ::: :::{.exercise title="?"} On $\RR^5$, set $\alpha = dz-y_1 dx_1 - y_2 dx_2$. Check that \[ \alpha\wedge (d\alpha)^2 = 2(dz \wedge dx_1 \wedge dy_1 \wedge dx_2 \wedge dy_2) .\] ::: # Contact Forms and Structures (Thursday, January 13) :::{.definition title="Contact form"} A **contact form** on $Y^3$ is a 1-form $\alpha$ with $\alpha \wedge d\alpha > 0$. A **contact structure** is a 2-plane field $\xi = \ker \alpha$ for some contact form. ::: :::{.remark} Forms are more rigid than structures: if $f>0$ and $\alpha$ is contact, then $f\cdot \alpha$ is also contact with $\ker( \alpha ) = \ker(f \alpha)$. ::: ## Examples of Contact Structures :::{.example title="Standard contact structure"} On $\RR^3$, a local model is $\alpha\da \dz - y\dx$. :::{.exercise title="?"} Show $\alpha \wedge d\alpha = dz\wedge dx \wedge dy$. ::: Write $\xi = \spanof_\RR(\del y, y \del z + \del x)$, which yields planes with a corkscrew twisting. Verify this by writing \( \alpha = 0 \implies \dd{z}{x} = y \), so the slope depends on the $y\dash$coordinate. ::: :::{.example title="Rotation of the standard structure"} On $\RR^3$, take \( \alpha_2 \da dz + x\dy \) and check \( \alpha_2 \wedge d \alpha_2 = dz \wedge dx \wedge dy \). This is a rigid rotation by $\pi/2$ of the previous \( \alpha \), so doesn't change the essential geometry. ::: :::{.example title="Radially symmetric contact structure"} Again on $\RR^3$, take \( \alpha_3 = dz + {1\over 2}r^2 d\theta \). Check that \( d \alpha_3 = r\dr \wedge \dtheta \) and \( \alpha_3 \wedge d \alpha_3 = r\dz \wedge \dr \wedge \dtheta \). Then $\xi = \spanof_\RR(\del r, {1\over 2} r^2 \del z + \del \theta )$. Note that as $r\to \infty$, the slope of these planes goes to infinity, but doesn't depend on $z$ or $\theta$. ::: :::{.example title="Rectangular version of radially symmetric structure"} Set \( \alpha_4 = \dz + {1\over 2}(x\dy - y\dx) \), then this is equal to \( \alpha_3 \) in rectangular coordinates. ::: :::{.example title="Overtwisted"} Set \[ \alpha_4 = \cos(r^2)\dz + \sin(r^2)\dtheta .\] :::{.exercise title="?"} Compute the exterior derivative and check that this yields a contact structure. ::: Now note that \[ \alpha = 0 \implies \dd{z}{\theta} = -{\sin(r^2)\over \cos(r^2)} = -\tan(r^2) ,\] which is periodic in $r$. So a fixed plane does infinitely many barrel rolls along a ray at a constant angle $\theta_0$. This is far too twisty -- to see the twisting, consider the graph of $(r, \tan(r^2))$ and note that it flips over completely at odd multiples of $\pi/2$. In the previous examples, the total twist for $r\in (-\infty, \infty)$ was less than $\pi$. ::: :::{.definition title="Contactomorphisms"} A **contactomorphism** is a diffeomorphism \[ \psi: (Y_1^3, \xi_1) \to (Y_2^3,\xi_2) \] such that $\phi_*(\xi_1) = \xi_2$ (tangent vectors push forward). A strict contactomorphism is a diffeomorphism \[ \phi: (Y_1^3, \ker \alpha_1) \to (Y_2^3, \ker \alpha_2) .\] such that $\phi^*( \alpha_2) = \alpha_1$ (forms pull back). ::: :::{.remark} Strict contactomorphisms are more important for dynamics or geometric applications. ::: :::{.exercise title="?"} Prove that \( \alpha_1,\cdots, \alpha_4 \) are all contactomorphic. ::: :::{.remark} Recall that $X$ has a cotangent bundle $\T\dual X \mapsvia{\pi} X$ of dimension $2 \dim X$. There is a canonical 1-form $\lambda \in \Omega^1(\T\dual X)$, i.e. a section of $T\dual(T\dual X)$. Given any smooth section $\beta\in \Gamma(\T\dual X\slice X)$ there is a unique 1-form $\lambda$ on $\T\dual X$ such that $\beta^*( \lambda) = \beta$, regarding $\beta$ as a smooth map on the left and a 1-form on the right. In local coordinates $(x_1,\cdots, x_n)$ on $X$, write $y_i = dx_i$ on the fiber of $\T\dual X$. Why this works: the fibers are collections of covectors, so if $x_i$ are horizontal coordinates there is a dual vertical coordinate in the fiber: ![](figures/2022-01-13_11-50-52.png) So we can write \[ \lambda = \sum y_i \dx_i \in \Omega^1(\T\dual X) ,\] regarding the $y_i$ as functions on $\T\dual X$ and $\dx_i$ as 1-forms on $\T\dual X$. :::{.exercise title="?"} Find out what $\beta = \sum a_i \dx_i$ is equal to as a section of $\T\dual X$. ::: ::: :::{.remark} To get a contact manifold of dimension $2n+1$, consider the 1-jet space $J^1(X) \da T\dual X \times \RR$. Write the coordinates as $(x,y)\in \T\dual X$ and $z\in \RR$ and define $\alpha = \dz - \lambda$, the claim is that this is contact. For dimension $2n-1$, choose a cometric on $X$ and take $\SS\T\dual X$ the unit cotangent bundle of unit-length covectors. Then $\alpha \da -\ro{\lambda}{\SS \T\dual X}$ is contact. ::: :::{.exercise title="?"} Check that $\RR^3 = J^1(\RR)$ and $\SS \T\dual(\RR^2) = \RR^2 \cross S^1$. ::: :::{.remark} A neat theorem: the contact geometry of $\SS\T\dual \RR^3$ is a perfect knot invariant. This involves assigning to knots unique Legendrian submanifolds. ::: ## Perturbing Foliation :::{.example title="?"} Define \[ \alpha_t = \dz - ty\dx \qquad t\in \RR \] to get a 1-parameter family of 1-forms. Check that \( \alpha_t \wedge d \alpha_t = t(\dz \wedge \dx \wedge \dy) \). Consider $t\in (-\eps, \eps)$: - $t>0 \implies \alpha = \dz - y\dx$ yields a positive contact structure, - $t>0 \implies \alpha = \dz$ is a foliation, - $t<0 \implies \alpha = \dz + y\dx$ is a negative contact structure. ::: :::{.remark} What is a (codimension $r$) foliation on an $n\dash$manifold? A local diffeomorphism $U\cong \RR^n \times \RR^{n-r}$ with *leaves* $\pt\cross \RR^{n-r}$. For example, $\RR^3\cong \RR \cross \RR^2$ with coordinates $t$ and $(x, y)$. We're leaving out a lot about how many derivatives one needs here! > For a fiber bundle or vector bundle to admit an interesting foliation, one needs a flat connection. ::: :::{.definition title="Integrability"} Any $\xi \da \ker \alpha$ is **integrable** iff for all vector fields $X, Y \subseteq \xi$, their Lie bracket $[X, Y] \subseteq \xi$. ::: :::{.theorem title="Frobenius Integrability"} For $\alpha$ nonvanishing on $Y^3$, $\ker \alpha$ is tangent to a foliation by surfaces iff $\alpha \wedge d\alpha = 0$. ::: :::{.example title="?"} Consider $\alpha = \dz - y\dx$, so $\ker \alpha = \spanof_\RR\ts{\del y, y\del z + \del x}$ which bracket to $\del z \not \in \ker \alpha$. This yields a non-integrable contact structure. On the other hand, for $\alpha = \dz$, $\ker \alpha = \spanof_\RR\ts{\del x, \del y}$ which bracket to zero. So this yields a foliation. ::: :::{.remark} A theorem of Eliashberg and Thurston: taut foliations can be perturbed to a (tight) positive contact structure. ::: # Tuesday, January 18 :::{.remark} Refs: - Geiges, Intro to Contact - Ozbogi-Stipsicz - Etnyre lecture notes - Massot - Sivck ::: :::{.definition title="Standard contact structure"} For $S^3 \subseteq \CC^2$, define a form on $\RR^4$ as \[ \alpha \da -y_1 dx_1 + x_1 dy_1 - y_2 dx_2 + x_2 dy_2 .\] Then then **standard contact form** on $S^3$ is \[ \xi_{\std} \da \ker \ro{\alpha}{S^3} .\] ::: :::{.exercise title="?"} Show that $\alpha$ defines a contact form. ::: :::{.solution} Write $f = x_1^2 + y_1^2 + x_2^2 + y_2^2$, then \[ \ro{ \alpha}{S^3} \wedge \ro{ d\alpha}{S^3} > 0 \iff df\wedge d \alpha \wedge d \alpha > 0 .\] Check that - $d\alpha = 2(dx_1 \wedge dy_1) + 2(dx_2 + dy_2)$ - $df = 2(x_1 dx_1 + y_1 dy_1) + 2(x_2 dx_2 + y_2 dy_2)$. ::: :::{.remark} Note that at $p= \tv{1,0,0,0} \subseteq S^3$, $\T_pS^3 = \spanof\ts{\del y_1, \del x_2, \del y_2}$. and $\alpha_p = -0 dx_1 + 1dy_1 -0 dx_2 + 0dy_2 = dy_1$ and $\xi_p = \ker dy_1 = \spanof\ts{\del x_1, \del y_2}$. ![](figures/2022-01-18_11-29-05.png) Then $\xi_p \leq \T_p \CC^2 = \spanof\ts{\del x_1, \del y_1, \del x_2, \del y_2} \cong \CC^4$ is a distinguished complex line. ::: :::{.definition title="Almost complex structures"} An **almost complex structure** on $X$ is a bundle automorphism $J: \T X\selfmap$ with $J^2 = -\id$. ::: :::{.example title="?"} For $X = \CC^2$, take \[ \del x_1 &\mapsto \del y_1 \\ \del y_1 &\mapsto - \del x_1 \\ \del x_2 &\mapsto \del y_2 \\ \del y_2 &\mapsto -\del x_2 .\] ::: :::{.exercise title="?"} Show that $f: \CC\to \CC$ is holomorphic if $df \circ J = J \circ df$, which corresponds to the Cauchy-Riemann equations. ::: :::{.lemma title="?"} Given $J:W\to W$, an $\RR\dash$subspace $V \leq W$ is a $\CC\dash$subspace iff $J(V) = V$. ::: :::{.definition title="?"} The field of $J\dash$complex tangents is the hyperplane field \[ \xi_p \da \T S^3 \intersect J(\T S^3) .\] ::: :::{.example title="?"} Consider $\T_p S^3$ for $p=\tv{1,0,0,0}$, then \[ J(\spanof\ts{\del y_1, \del x_1, \del y_2}) = \spanof\ts{-\del x_1, \del y_2, -\del x_2} ,\] so $\xi_p = \spanof{ \del x_1, \del y_2}$ is the intersection and coincides $\xi_{\std}$. ::: :::{.question} Where does $\alpha$ come from? Let $\rho = \sum x_i \del x_i + \sum y_i \del y_i$ be the radial vector field, so $\rho = {1\over 2}\grad\tv{\sum x_i^2 + \sum y_i^2}$. Setting $\omega \da \Wedge dx_i \wedge \Wedge dy_i$, then $\alpha = \iota_p\omega \da \omega(p, \wait)$ is the interior product of $\omega$. Then \[ \alpha = dx_1 \wedge dy_1 (x_1 \del x_1 + y_1 \del y_1 + \cdots ) + \cdots = x_1 dy_1 -y_1 dx_1 + \cdots .\] So the contact form comes from pairing the symplectic form against a radial vector field. ::: :::{.remark} Recall $f \da \sum x_i^2 + \sum y_i^2$ satisfies $df = 2\sum x_i dx_i + 2\sum y_i dy_i$. Note that $J$ acts on 1-forms by $J^*(dx)(\wait) = dx(J(\wait))$. For $J = i$, - $\delta x: dx(J \del x) = dx(\del y) = 0$, - $\del y: dx(J \del y) = dx(-\del x) = -1$. So $J^*(dx) = -dy$, and \[ J^*(df) = 2x_1 (-dy_1) + 2y_1 (dx_1) + 2x_2 (-dy_2) + 2y_2 (dx_2) = -2 \alpha .\] Thus $J^*(df)$ is a rotation of $df$ by $\pi/2$. ::: :::{.example title="?"} The field of complex tangencies along $Y = f\inv (0)$ is the kernel of $\ro{ df(J(\wait)) } {Y}$. ::: :::{.remark} Methods of getting contact structures: for a vector field $X$, being contact comes from $\mcl_X \omega = \omega$. For functions $f:\CC^2 \to \RR$, being contact comes from $\alpha = d^\CC f$ being contact. See strictly plurisubharmonic functions and Levi pseudoconvex subspaces. ::: :::{.example title="?"} The standard contact structure is orthogonal to the Hopf fibration: define a map \[ \CC^2\smz &\to \CP^1 \cong S^2 \\ \tv{z, w} &\mapsto \tv{z: w} ,\] which restricts to a map $S^3\to S^2$ defining the Hopf fibration. If $L$ is a complex line through 0, then $L \intersect S^3$ is a Hopf fiber that is homeomorphic to $S^1$. ![](figures/2022-01-18_12-11-38.png) ![](figures/2022-01-18_12-15-08.png) ![](figures/2022-01-18_12-17-02.png) Take \[ \CC^2 &\to \RR^2 \\ (z_1, z_2)&\mapsto (\abs{z_1}, \abs{z_2}) .\] Consider the image of $S^2 = \ts{\abs{z_1}^2 + \abs{z_2}^2 = 1}$: ![](figures/2022-01-18_12-20-49.png) The preimage is $S^1\times S^1$. This can be realized as a tetrahedron with sides identified: ![](figures/2022-01-18_12-24-27.png) There are Hopf fibers on the ends, and undergo a $\pi/2$ twist as you move through the tetrahedron. ![](figures/2022-01-18_12-28-04.png) ::: # Darboux and Gromov Stability (Thursday, January 20) :::{.remark} Almost-complex structures: weaker than an actual complex structure, but not necessarily integrable. Useful for studying pseudoholomorphic curves. A necessary and sufficient condition for integrability: the Nijenhuis tensor $N_J = 0$ iff $J$ is integrable. In real dimension 2, all $J$ are integrable. ::: :::{.theorem title="Darboux"} If $(Y^3, \xi)$ is contact then for every point $p$ there is a chart $U$ with coordinates $x,y,z$ where $\xi = \ker (\dz -y\dx) = \ker (\alpha_\std)$. ::: :::{.slogan} Locally, all contact *structures* (not necessarily forms) look the same. The mantra: local flexibility vs global rigidity. ::: ## Proof of Darboux :::{.remark} Two proofs: - Geometric, due to Giroux - PDEs, which generalizes. This uses Moser's trick. ::: :::{.proof title="1"} Locally write $\xi = \ker \alpha$ with $\alpha \wedge d \alpha > 0$. Pick a contact plane $\xi_p$ and let $S$ be a transverse surface, so $\T_p S \transverse \xi_p$. This produces a set of curves in $S$ which are tangent to $\xi_p$ everywhere, called the *characteristic foliation*. ![](figures/2022-01-20_11-24-33.png) Then $\ro{\alpha}{S} = \dz$, which is a 1-form that is nonvanishing near $p$ and is locally integrable. Sending $\alpha \to X$ a vector field along $S$ yields a set of integral curves tracing out the characteristic foliation. This yields an $x$ direction and a $z$ direction on $S$ by flowing $t\in (-\eps, \eps)$ around $p$ along $X$. Choose a vector field $\del t$ which is transverse to $S$ and contained in $\xi$. Then $\alpha(\del t) = 0$, so we can write \[ \alpha = f\dx + g\dz + h\dt = f\dx + g\dz .\] Since $g(p) = 1$, replace $\alpha$ with ${1\over g}\alpha$ which is positive near $p$ and doesn't change the contact structure $\xi$. So write \[ \alpha = f\dx + \dz \implies \alpha \wedge d \alpha = \alpha \wedge \qty{ f_t \dy\wedge \dx + f_z \dz \wedge \dx } = -f_t \dx \wedge \dt \wedge \dz > 0 ,\] meaning $f_t <0$ and we can set $y = f(x,z,t)$. This yields \[ \alpha = \dz + f\dx = \dz - y\dx .\] ::: :::{.proof title="2, Moser's Trick"} By a linear change of coordinates, choose $x,y$ along $\xi$ to write $\alpha_p = \dz$ and $\xi_p = \spanof{\del x, \del y}$: ![](figures/2022-01-20_11-43-26.png) Write $(\alpha_0)_p$ for the original form and $\alpha_1 = \dz - y\dx$ the standard form, then the claim is that $\alpha_0 \homotopic \alpha_1$ through a path of contact forms. :::{.lemma title="?"} In a neighborhood of $p$, there is a family $\alpha_t$ for $t\in [0, 1]$. To obtain this, interpolate: \[ d\alpha_t = t d\alpha_1 + (1-t) d\alpha_0 \implies \alpha_t \wedge d\alpha_t = t^2 \alpha_1 \wedge d\alpha_1 + t(1-t) (\alpha_0 \wedge d\alpha_1 + \alpha_1 \wedge d\alpha_0) + (1-t)^2 \alpha_0 \wedge d\alpha_0 .\] The first and last terms are positive since the $\alpha_i$ are contact. For the middle term, $\alpha_0 = \alpha_1$ near $p$, so by continuity this is positive in some neighborhood of $p$. ::: :::{.remark} Note that $\dot\alpha_t \da \dd{}{t} \alpha_t$, so \[ \dd{}{t} \qty{ t\alpha_1 + (1-t) \alpha_0} = \alpha_1 - \alpha_0 .\] ::: We'll assume that there is a time-dependent vector field $V_t \in \xi_t$ with flow $\Phi_t$ such that $(\Phi_t)_*(\xi_t) = \xi_0$. We'll also require $\xi_t = \ker \alpha_t$, so this is a contactomorphism for each $t$. The goal is to show $(\Phi_1)_*(\xi_0) = \xi_1$, or equivalently $\Phi_t^* \alpha_t = f_t \alpha_0$ with $f_t > 0$. Take $\dd{}{t}$ of both sides here to get \[ \Phi_t^*( \dot \alpha_t + \mcl_{V_t} \alpha_t ) = \dot f_t \alpha_0 .\] > See Prop 6.4 in Cannas da Silva. :::{.remark} ✨Cartan's magic formula✨: \[ \mcl_V(\alpha) = d(\iota_V \alpha) + \iota_V(d\alpha) ,\] so \[ \mcl_{V_t}(\alpha_t) = d(\alpha_t(V_t)) + d\alpha_t(V_t, \wait) = 0 + d\alpha_t(V_t, \wait) .\] ::: We can thus write this equation as \[ \Phi_t^*(\dot \alpha_t + d\alpha_t(V_t, \wait)) = \dot f_t \alpha_0 = \dot f_t \qty{\Phi_t^*(\alpha_t) \over f_t } .\] Applying $(\Phi_t^*)\inv$ yields \[ \dot \alpha_t + d\alpha_t(V_t, \wait)= {\dot f_t \over f_t }\alpha_t .\] Now try to solve this for $V_t$. Let $R_t$ be the **Reeb vector field** of $\alpha_t$, which satisfies - $\alpha_t(R_t) = 1$ - $d\alpha_t(R_t, \wait) = 0$. ![](figures/2022-01-20_12-11-26.png) Then \[ \dot \alpha_t (R_t) = {\dot f_t \over f_t} = \dd{}{t} \log(f_t) \da \mu_t ,\] so $\dot\alpha_t(R_t)$ determines $f_t$ by first integrating and exponentiating. We now need to solve \[ \ro {d\alpha_t(V_t, \wait)}{\xi_t} = \ro{\mu_t \alpha_t - \dot \alpha_t}{\xi_t} .\] Since \( d \alpha_t \) is a volume form on $\xi_t$, it identifies vector fields in $\xi_t$ with 1-forms on $\xi_t$ using the happy coincidence that $n=2$ so $1\mapsto n-1 = 1$. So $V_t$ is uniquely determined by the solution to the above equation. ::: # Gray Stability (Tuesday, January 25) :::{.remark} A homotopy of contact structures o $Y^3$ is a smooth family $\ts{\phi_t}$ of contact structures. Similarly, an **isotopy** of structures such that $\ts{D\phi_t(\xi_0)}$ for an isotopy $\phi_t: Y\to Y$ with $\phi_0 = \id$. If $Y^3$ is closed then every homotopy of contact structures is an isotopy. Theorem: contact structures mod isotopy is discrete, which critically uses closedness. ::: :::{.lemma title="?"} For $\phi_t$ an isotopy generated by the flow of $X_t$ and $\alpha_t$ a family of 1-forms, \[ \dd{}{t} \phi^*_t(\alpha_t) \mid_{t=t_0} = \phi_{t_0}^*(\dot \alpha_{t_0} + \mcl_{X_{t_0}} \alpha_{t_0} ) .\] ::: :::{.proof title="?"} Write \[ \phi_x^*(\alpha_y) = \dd{}{x} ? + \dd{}{y}? = \phi_{x_0}^* \mcl_{X_0} \mcl_X \alpha_{y_0} + \phi_{x_0}^* \alpha_y ,\] and proceed similarly to the proof of Darboux's theorem. Pick $\ts{\phi_t}$ a homotopy, one can choose $\alpha_t$ with $\xi_t = \ker \alpha_t$ for all $t$. Apply Moser's trick: assume there exists a $\phi_t$ with $\phi_t^*(\alpha_t) = \lambda_t \alpha_0$ and try to find $v_t$ generating it, where $\lambda_t: Y\to \RR_+$. What does $\phi_t$ need to look like? Differentiate in $t$: \[ \phi^*_{t_0}(\cdot \alpha_{t_0} + \mcl_{V_{t_0}} \alpha_{t_0} ) = \dot \lambda_t \alpha_0 = \dot \lambda_t \qty{ \phi^*_{t_0} (\alpha_t) \over \lambda_t } .\] Apply $(\phi^*_{t_0})\inv$: \[ \cdot\alpha_t + \mcl_{V_t}\alpha_t = \mu_t \alpha_t\qquad \mu_t = (\phi^*_{t_0})\inv(\dot\lambda_t \over \lambda_t) .\] Use that $V_t$ is always tangent to the contact structure, so $V_t \in \xi_t$, to assume $\alpha_t(V_t) =0$. Apply Cartan: \[ \dot \alpha_t + d\alpha_t(V_t) + \iota_{V_t} d\alpha_t = \mu_t \alpha_t ,\] and $d\alpha_t(V_t) = 0$, so \[ \iota_{V_t}d\alpha_t = \mu_t \alpha_t - \dot \alpha_t .\] Plug in the Reeb vector field $R_t$, then $\alpha_t(R_t) = 0$ so $\mu_t = \dot \alpha_t (R_t)$. ::: :::{.corollary title="?"} Let $Y$ be n $S^3 \subseteq \CC^2$ that is transverse to the radial vector field. Then \[ \alpha = x_1 dy_1 - y_1 dx_1 + x_2 dy_2 - y_2 dx_2\mid_y \] defines the standard tight contact structure. ::: :::{.proof title="?"} Write $Y \subseteq \RR\times S^3$ in coordinates $(f(x), x)$ as the graph of a function $f: S^3\to \RR$. Take an isotopy $Y_t = (tf(x), x) \subseteq \RR\times S^3$ to get a family of contact forms where $\alpha_0 = \alpha_\std$ and $\alpha_1$ is some unknown form. By Gray stability, the contact structures are isotopic. ::: ## Legendrian Links :::{.definition title="Legendrian and transverse knots"} Let $Y$ be a contact 3-manifold and $L \injects Y$ a link. Then $L$ is a **Legendrian knot** iff it is everywhere tangent to $\xi$, so $\alpha(L) = 0$: ![](figures/2022-01-25_12-10-49.png) This is a closed condition. $L$ is **transverse** if it is everywhere transverse to $\xi$, so $\alpha(L) > 0$: ![](figures/2022-01-25_12-14-25.png) This is an open condition. ::: :::{.remark} Every Legendrian knot has a transverse pushoff (up to transverse isotopy). Every transverse knot has a Legendrian approximation. ::: :::{.example title="?"} Take $\RR^3$ and $\alpha_\std = dz-ydx$, then the $y\dash$axis $L_1 \da\ts{\tv{0,t,0}}$ is Legendrian. Similarly the $x\dash$axis $L_2$ is Legendrian, checking that $\T L_2 = \spanof\ts{\tv{1,0,0}}$. However the slight pushoff $L_3 \da \ts{\tv{t, -\eps, 0}}$ is transverse since $\ro{\alpha}{L_3} = \eps dx >0$. ::: :::{.theorem title="Neighborhood theorem, Darboux for Legendrian/transverse knots"} Every Legendrian has a neighborhood contactomorphic to the zero section in $J_1 S^1 = \T S^1 \cross \RR$. Every transverse has a neighborhood contactomorphic to the $z\dash$axis in $\RR\times S^1$ with $\alpha \da \dz + r^2 \dtheta$. ::: # Thursday, January 27 :::{.remark} Goal: classify Legendrian knots up to (Legendrian) isotopy. Recall a knot $\gamma: S^1 \injects Y$ satisfies $\gamma^*(\alpha) = 0$, and a Legendrian isotopy is a 1-parameter family $\gamma_t$ which are Legendrian for all $t$. ::: :::{.example title="?"} $\gamma(s) = \tv{x(s), y(s), z(s)}$ and $\xi = \ker \alpha, \alpha = \dz - y\dx$. Then $\gamma^*(\alpha) = z' \ds - yx^1\ds = (z' - yx')\ds$, which is Legendrian iff $y=z'/x'$. ::: :::{.example title="?"} Let $f:\RR \to \RR$ and take the 1-jet $\gamma(s) = \tv{s, f'(s), f(s)}$ of the graph of $f$ -- this is like the graph of the 1st order Taylor expansion. This is Legendrian since $s'=1$ implies $z'/x' = f'/s' = f'$. ::: :::{.remark} There are two projections: - $\tv{x,y,z} \to \tv{x,z}$, a wave front projection, plotted with $y$ into the board, - $\tv{x,y,z} \to \tv{x,y}$, Lagrangian projection. ::: :::{.example title="?"} Let $\gamma(s) = \tv{s^2, {3\over 2}s, s^3}$, then the two projections are as follows: ![](figures/2022-01-27_11-25-37.png) ::: :::{.remark} The front projection uniquely determines $L$, since the $y$ coordinate can be recovered as $y=z'/x'$. So for example, there is no ambiguity about crossing order: the more negatively sloped line in a diagram is the over-crossing: ![](figures/2022-01-27_11-30-01.png) ::: :::{.example title="?"} A front diagram of the unknot: ![](figures/2022-01-27_11-34-42.png) ::: :::{.theorem title="?"} Every knot $K \injects \RR^3$ can be $C^0$ approximation by a Legendrian knot $L$. Idea: zigzags in an $\eps$ tube in the knot diagram, which will be Legendrian. How to measure: $\sup_{s\in I} \abs{\gamma_1(s) - \gamma_2(s)} \leq \eps$? ::: :::{.remark} Note that $\Lie(\SO_3) \da \T_e(\SO_3) = \liesu_2$, spanned by roll, pitch, and yaw generators: ![](figures/2022-01-27_11-48-58.png) So measuring the number of rotations along each generator after traversing $L$ in a full loop yields integer invariants. ::: :::{.definition title="The Thurston–Bennequin number"} A **framing** of a knot $K$ is a trivialization of its normal bundle, so an identification of $\nu(K) \cong S^1\times \DD^2$. The potential framings are in $\pi_1(\SO_2) \cong \pi_1(S^1) \cong \ZZ$, since a single vector field normal (?) to the knot determines the framing by completing to an orthonormal basis. The Reeb vector field is never tangent to a Legendrian knot, so this determines a framing called the **contact framing**. The **Thurston–Bennequin number** is the different between the 0-framing and the contact framing. The 0-framing comes from a Seifert surface. This is an invariant of Legendrian knots, since Legendrian isotopy transports frames. Note that adding zigzags adds cusps, and thus decreases this number. ::: :::{.remark} How to compute: take a pushoff and compute the linking number: ![](figures/2022-01-27_12-07-19.png) ::: :::{.proposition title="?"} \[ \mathrm{tb}(L) = w(L) - {1\over 2}C(L) ,\] where $w(L)$ is the writhe and $C(L)$ is the number of cusps. ::: :::{.proof title="?"} The linking number is ${1\over 2}(c_+(L) - c_-(L))$, half of the signed number of crossings. Here all 4 crossing have the same sign: ![](figures/2022-01-27_12-12-35.png) ::: :::{.example title="?"} TB for the knots from before: - The 3 unknots: - 2 cusps, so $-1$ - 4 cusps, so $-2$ - 4 cusps, so $-2$ - The 2 trefoils: - $3-{1\over 2}4 = 1$ - $3 - {1\over 2}6 = -6$. ::: :::{.remark} Since adding zigzags decreases $\mathrm{tb}$, define $\mathrm{TB}$ to be the max over all Legendrian representatives of $K$. This distinguishes mirror knots. In fact $\mathrm{tb}(L) \leq 2g_3(L) - 1$ (the Bennequin bound), involving the 3-genus. ::: :::{.definition title="Rotation number"} The **rotation number** of $L$ is the *turning number* $\rot(L)$ in the Lagrangian projection, i.e. how many times a tangent vector spins after traversing the knot. ::: :::{.example title="?"} ![](figures/2022-01-27_12-26-39.png) It turns out that \[ \rot(L) = {1\over 2}\qty{ \size\text{down cusps} - \size\text{up cusps}} .\] ::: # Tuesday, February 01 :::{.remark} Last time: front diagrams $\tv{x,y,z}\mapsto \tv{x,z}$, where $\alpha = \dz -y\dx$ forces $y=\ds/\dx$ can be recovered as the slope in the projection. Note that we can also recover crossing information from the Legendrian condition, since $y$ always points into the board, so more negative slopes go on top. Some invariants: - Thurston-Bennequin invariant: a contact framing with respect to the Reeb vector field ![](figures/2022-02-01_11-15-00.png) - Equal to writhe minus half the number of cusps. - Rotation numbers: Turning number of $L$ with respect to $\xi$, after fixing a trivialization of $\xi$. Equal to ${1\over 2}(D-U)$, the number of down/up cusps respectively. ::: :::{.remark} Disallowed moves: ![](figures/2022-02-01_11-20-07.png) Allowed moves: ![](figures/2022-02-01_11-24-46.png) ::: :::{.remark} Geography problem: given a smooth knot $K$, which pairs $(t, r) \in \ZZ^2$ are realized as $(\mathrm{tb}(L), \rot(L))$ for $L$ a Legendrian representative of $K$? Botany problem: given $(t, r) \in \ZZ^2$, how many inequivalent $L$ representing $K$ realize $(t,r) = (\mathrm{tb}(L), \rot(L))$? ::: :::{.example title="?"} For $K$ the unknot: ![](figures/2022-02-01_11-37-46.png) So these numerical pairs fall into a cone: ![](figures/2022-02-01_11-39-30.png) ::: :::{.proposition title="?"} For $L \subseteq \RR^3$ a Legendrian knot, \[ \mathrm{tb}(L) + \rot(L) \equiv 1 \mod 2 .\] ::: :::{.remark} Note that $\chi(S) \equiv 1\mod 2$ for $S$ a Seifert surface. ::: :::{.theorem title="Bennequin-Thurston inequality"} For any Seifert surface $S$, \[ \mathrm{tb}(L) + \abs{ \rot(L) } \leq -\chi(S) .\] ::: :::{.remark} This solves the geography problem: this cone contains all of the possible pairs. ::: :::{.theorem title="Eliashberg-Fraser"} The unknot is **Legendrian simple**: if $\mathrm{tb}(L_1) = \mathrm{tb}(L_2)$ and $\rot(L_1) = \rot(L_2)$, then $L_1$ is isotopic to $L_2$. ::: :::{.remark} This solves the botany problem: every red dot has exactly one representative. ::: :::{.remark} Other knots are Legendrian simple, e.g. the trefoil. A theorem of Checkanov says the following $5_2$ knots are not Legendrian isotopic: ![](figures/2022-02-01_11-53-16.png) ::: :::{.remark} This all depended on the standard contact form. Consider instead the overtwisted disc: take $\RR^3$ with $\alpha = \cos(r)\dz + \sin(r) \dtheta$. Take the curve $\tv{r,\theta, z} = \gamma(t) \da \tv{1, t, 0}$, a copy of $S^1$ in the $x,y\dash$plane. Then $\gamma' = \tv{0,1,0}$, and at $\theta=\pi, \alpha = \cos(\pi)\dz + \sin(\pi) \dtheta = -\dz$, but at $r=0$ $\alpha = \dz$, so traversing a ray from $0$ to $-1$ in the $x,y\dash$plane forces the contact plane to flip: ![](figures/2022-02-01_12-02-08.png) One can check that $\mathrm{tb}$ is given my $\lk(L, L') = 0$ where $L'$ is a pushoff of $L$, and can be made totally disjoint from $L$ in this case by moving in the $z\dash$plane. ::: :::{.definition title="Overtwisted discs"} An **overtwisted disc** in $(Y^3, \xi)$ that is locally contactomorphic to this local model. $Y$ is **overtwisted** if it contains an overtwisted disc, and is tight otherwise. ::: :::{.theorem title="Bennequin"} $(\RR^3, \xi_\std)$ is a tight contact structure. ::: :::{.theorem title="Eliashberg"} For every closed oriented $Y^3$, every homotopy class of 2-plane fields on $Y$ contains a unique (up to isotopy) overtwisted contact structure. ::: ## Transverse Knots :::{.definition title="Self-linking"} The **self-linking number** $\mathrm{sl}(T, S)$ of a transverse knot rel a Seifert surface $S$ is $\lk(T, T')$ for $T'$ a pushoff of $T$ determined by a trivialization of $\ro{\xi}{S}$. ::: :::{.remark} In this case, $\xi$ restricts to an $\RR^2$ bundle over $\Sigma$, which is trivial since $\Sigma$ is closed with boundary and $e(\xi) \in H^2(S) = 0$. To see this, use $H^2(S) \cong H_0(S, \bd S) = 0$ by Lefschetz duality. This yields a section of the frame bundle over $S$, which gives a pushoff direction along the first basis vector: ![](figures/2022-02-01_12-27-33.png) This turns out to be well-defined: it's independent of the surface $S$ chosen and the trivialization of $\xi$. The difference of two trivializations gives a map $\pi_1(S) \to \ZZ$, which factors through $\pi_1(S)^\ab = H_1(S)$. The difference in surfaces is measured by $\inp{e(S)}{ \Sigma_1 \disjoint_T \Sigma_2 }$, which is a glued surface. ::: # Thursday, February 03 :::{.remark} Last time: self-linking of transverse knots. Today: surfaces with transverse boundary. Let $\Sigma$ be a surface embedded in $(Y, \xi)$ with $\bd \Sigma$ transverse to $\xi$. Let $F$ be the characteristic foliation, the singular foliation on $\Sigma$ induced by $\ro{\xi}{\Sigma}$. Equivalently, if $\xi = \ker \alpha$, consider the 1-form $\ro{\alpha}{\Sigma}$. Generically, $\ro{\ker \alpha}{\T\Sigma}$ is 1-dimensional except at finitely many points where $\alpha_p = 0$, i.e. $\xi$ is tangent to $\Sigma$. This line field integrates to a singular foliation. Recall that $\mathrm{sl}(L)$ is the self-linking number. ::: :::{.example title="?"} Take $\alpha = \dz +x\dy - y\dx$ and $\Sigma = S^2$, then the singular foliation is given by ![](figures/2022-02-03_11-21-06.png) ::: :::{.remark} Two possible types of singularities, the local models: ![](figures/2022-02-03_11-24-34.png) There are also two numerical invariants: - $e_{\pm}$: the number of positive (resp. negative) elliptics - $h_{\pm}$: the number of positive (resp. negative) hyperbolics A theorem \[ \inp{c(\Sigma)}{ \Sigma} = (e_+ - h_+) - (e_- - h_-) .\] If $\Sigma$ is transverse, $\mathrm{sl}(\bd\Sigma, \Sigma) = -(e_+ - h_+) + (e_- - h_-)$. ::: ## Local Model 1: Elliptic :::{.remark} $\sigma$ is the $x,y\dash$plane and $\xi = \ker (\dz + x\dy - y\dx)$ with $\ro \alpha \Sigma = x\dy - y\dx$. Set $V: x\del_x + y\del_y$ and $L' = \gens{x\del_y - y\del_x}$, and $\alpha(i) = x^2+y^2 = 1 > 0$. ![](figures/2022-02-03_11-32-36.png) Here $\mathrm{sl} = 1$. To compute $\mathrm{sl}$: - Trivialize $\ro{\xi}{\Sigma}$ to get $\tau = \gens{e_1, e_2}$ a fiberwise basis for $\xi$. - Let $\tilde L$ be a pushoff in the $e_1$ direction. - Compute $\mathrm{sl} = \lk(L, \tilde L)$. Set - $e_1 = \del_x + y\del_z$ - $e_2 = \del_y - x\del_z$ - $\rho = x \del_x + y\del_y$ - $\theta = x\del_y - y\del_x$. Then \[ x\rho - y\theta = x(x\del_x + y\del_y) - y(-y \del_x + x\del_y) = (x^2+y^2)\dx .\] Then - $c_1 = x\rho - y\theta + y\del_z$ - $\bar{c_1} = x\rho + y\del_z = \cos(\rho) + \sin(\theta) \del_z$. Example: - $\theta = 0\implies e_1 = \rho$ - $\theta = \pi/4 \implies e_1 = {\sqrt 2\over 2}(\rho + \del_z)$ - $\theta = \pi/2 \implies e_1 = \del_z$ So here $\lk(U, \tilde U) = -1$: ![](figures/2022-02-03_11-43-46.png) ::: ## Local Model 2: Hyperbolic :::{.remark} Here $\xi$ is the $x,y\dash$plane, so $\xi = \ker (\dz + 2x\dy + y\dx)$ with $\ro{\alpha}{\Sigma} = 2x\dy + y\dx$ and $V = y\del_y + dx\del_x\in \ker(\ro \alpha \Sigma)$. ::: :::{.remark} The Euler class of a real vector bundle $E \mapsvia{\pi} X$ is the obstruction to finding a nonvanishing section $s$ of $E$, given by $e(E) \in H^k(X)$. It is Poincare dual to $[s\inv(0)] \in H_{n-k}(X, \bd X)$. For the tangent bundle, $e(\T X)\in H^{n}(X)$, and \[ \inp{e(\TX)}{[X]} = \chi(X) .\] Since a section of $\T X$ is a vector field, $e(\T X)$ is an obstruction to finding a nonvanishing vector field. If $\bd X \neq \emptyset$ and $t$ is a section of $\ro{E}{\bd X}$, there is a relative Euler class $e(E, t)\in H^k(X, \bd X) \cong H_{n-k}(X)$. Similarly, \[ \inp{e(\T X, t)}{[X]} = \chi(X) .\] ::: :::{.example title="?"} Note $\chi(\DD) = 1$, so any vector field has a singularity? ![](figures/2022-02-03_11-59-11.png) ::: :::{.proposition title="?"} The total class is the sum of the relative obstructions. If $\sigma = \Sigma_1 \glue{\bd} \Sigma_2$ and $\tau$ is a nonvanishing section of $\ro{\Sigma}{\bd \Sigma_1} = \ro{\Sigma}{\bd\Sigma_2}$, then \[ c(E) = e(\ro{E}{\Sigma_1}, \tau) + c(\ro E {\Sigma_2}, \tau) .\] ![](figures/2022-02-03_12-01-38.png) ::: ## More Contact Geometry :::{.remark} Let $\Sigma$ have transverse boundary with characteristic foliation $F$, and let $V$ be the vector field directing $F$, so $V \in \xi \intersect \T \Sigma$. We can assume $V$ is outward-pointing along $\bd \Sigma$. Check that - $\chi(\Sigma) = e(\T\Sigma, V) \in H^2(\Sigma, \bd \Sigma) \cong H_0(\Sigma)$ - $\mathrm{sl}(\bd\Sigma, \Sigma) = e(\xi, V) \in H^2(\Sigma, \bd \Sigma)$ ::: :::{.fact} \envlist - $e_+ + e_-$ correspond to $+1$ in $e(\T\Sigma, V)$, - $h_+, h_-$ correspond to $-1$ in $e(\T\Sigma, V)$. Proof: near a zero, $V$ determines a map $S^1\to S^1$ and the contribution to $e$ is the degree of this map. - $e_+$ contributes $-1$ to $e(\xi, V)$, by the same computation of $\mathrm{sl}(U)$ for $U$ the unknot. - $e_-$ contributes $-(-1) = +1$ to $e(\xi, V)$. - $h_+$ contributes $+1$ to $e(\xi, V)$ - $h_-$ contributes $-(+1) = -1$ to $e(\xi, V)$. Proof: exercise. ::: :::{.remark} Bennequin inequality: \[ \mathrm{sl}(T, \Sigma) \leq -\chi(\Sigma) \implies e_+ + h_+ + e_- + h_- \leq -(e_+ + e_- - h_+ - h_-) \iff e_- \leq h_- .\] Try to cancel in pairs: ![](figures/2022-02-03_12-24-10.png) The inequality follows if we can cancel every $e_-$ with some $h_-$. ::: # Tuesday, February 08 :::{.remark} Topics for talks: - Thom-Pontryagin - Brieskorn spheres - Milnor fibrations - Lens spaces ::: :::{.theorem title="?"} Every closed oriented 3-manifold $Y$ admits a (positive) contact form. ::: :::{.remark} Three proofs: - Lickorish-Wallace, using that $Y$ is Dehn surgery on a link in $S^3$, - Birman-Hildon, using that $Y$ is a branched cover of $S^3$, - Alexander, using that $Y$ admits an open book decomposition. ::: :::{.remark} Dehn surgery for slope $p/q$: for $K \injects S^3$, cut out $\nu(K) \cong S^1\times \DD$ and re-glue by a map $\bd(S^1\times \DD) \to \bd \nu(K)$ such that $[\ts{0} \times \bd \DD] = p[m] + q[\ell] \in H^1(\bd \nu (K) )$. Use that $\nu(K) \cong S^1\times \DD$ and $\bd \nu(K) \cong S^1\times S^1 = T^2$. Idea: wrapped $p$ times longitudinally, $q$ times around the meridian. ::: :::{.remark} Recall: - Every knot $K$ can be $C^0$ approximated by a transverse knot - Every link $L$ can be $C^0$ approximated by a transverse link - Neighborhood theorem: for every transverse knot $K$, there is a $w(K)$ and a contactomorphism to a standard model: $S^1\times \DD$ in coordinates $(\phi, r, \theta)$ with $0\leq r\leq \delta$ and $\alpha = d\phi + r^2\dtheta$. Re-gluing corresponds to the map $\tv{0, \delta, \theta}\mapsto \tv{q\theta, \delta, p\theta}$. \[ \tv{0, \delta, \bar\theta} &\mapsto \tv{q\bar\theta, \delta, p\bar \theta} \\ \tv{\bar\pi, \bar r, \bar\theta} &\mapsto \tv{\phi,r,\theta} .\] If $p, q$ are coprime there exist $m,n$ with $pm-qn = 1$. So define \[ \psi: \tv{\bar\pi, \bar r, \bar\theta} &\mapsto \tv{\phi,r,\theta} ,\] so \[ \psi^*(\alpha) = d(\alpha\bar\theta + m\bar\phi) + r^2d(p\bar\theta + n\bar\phi) = (q+r^2p)d\bar\theta + (m+r^2n)d\bar\phi .\] We want $\alpha = h_1(r) d\bar\theta + h_2(r) d\bar\theta$ to be contact and satisfy $(h_1, h_2) = (r^2, 1)$ near $r=0$ and $(q+r^2 p, m+r^2 n)$ near $r=\delta$. This requires \[ d\alpha = h_1' \dr \wedge d\bar\phi + h_2' \dr \wedge d\bar\theta = (h_2 h_1' - h_1 h_2') dr \wedge d\bar \theta \wedge d\bar\phi ,\] which happens iff \[ \det \begin{bmatrix} h_2 & h_2' \\ h_1 & h_1' \end{bmatrix} > 0 .\] Think of $\tv{h_2, h_1}$ as a path with tangent vector $\tv{h_2', h_1'}$. This requires moving counterclockwise. ![](figures/2022-02-08_11-57-58.png) ::: :::{.definition title="?"} An **open book decomposition** of $Y$ is a pair $(B, \pi)$ where - $B$ is a link in $Y$, called the **binding** - $\pi: Y\sm B\to S^1$ is a locally trivial fibration of relatively compact fibers **pages** ![](figures/2022-02-08_12-09-13.png) ::: :::{.remark} An open book decomposition is determined by its monodromy map $\phi: \Sigma_0\to \Sigma_0$, which determines a class $[\phi] \in \MCG(\Sigma_0)$. Form \[ Y\sm \nu(B) \cong {\Sigma \times I \over \phi(x) \cross \ts{0} \sim x\cross \ts{1}} ,\] which is a glued cylinder: ![](figures/2022-02-08_12-14-04.png) ::: :::{.definition title="Open book decompositions supporting a contact structure"} An open book decomposition **supports** a contact structure $\xi$ iff there exists a contact form $\alpha$ such that $d\alpha$ is an area form on each page and $B$ is a transverse link in $(B, \xi)$. ::: :::{.theorem title="Thurston-Winkelnkemper"} Every open book decomposition admits a contact structure. ::: :::{.theorem title="Giroux"} Every $(Y^3, \xi)$ with $Y$ closed has a supporting open book decomposition. ::: :::{.proposition title="?"} If an open book decomposition supports $\xi_1$ and $\xi_2$, then $\xi_1$ is isotopic to $\xi_2$. ::: :::{.proof title="?"} Two steps: - Form a mapping cylinder of the monodromy map $\phi$, - Extend over the binding, using the same idea as in Dehn surgery. Choose an area form $\omega$ on $\Sigma$ and a primitive $\beta$ with $d\beta = \omega$. Let $\beta_1 \da \phi^*\beta$ and $\beta_0 = \beta$, then set \[ \beta_t = t\beta_1 + (1-t)\beta_0 .\] This yields a 1-form on $\Sigma\times I$ that extends to the mapping cylinder. Moreover $d\beta_t = td\beta_1 + (1-t)d\beta_0$ is an area form on $\sigma\times \ts{t}$ and $\alpha = \dt + \eps \beta_t$ is a contact form for small $\eps > 0$. Then $d\alpha = \eps d\beta_t + \eps \dt \wedge \dot{\beta}_t$ and $\alpha \wedge d\alpha = \eps dt \wedge d\beta_t + \bigo(\eps^2)$. ::: # Tuesday, February 15 Missed due to orthodontic appointment! Please send me notes. :) # Thursday, February 17 :::{.remark} Let $\Sigma \subseteq (Y^3, \xi)$. - Characteristic foliation: $F = \xi \intersect \T\Sigma$, complicated but necessary - Dividing set: a multicurve, simpler ::: :::{.theorem title="?"} If $\Sigma$ is convex with a dividing set $\Gamma$ and $F$ is any foliation divided by $\Gamma$, there is a $C^0\dash$small isotopy $\phi_t$ wt - $\phi_0(\Sigma) = \Sigma, \phi_t(\Gamma) = \Gamma$ - $\phi_t(\Sigma)$ is convex for all $t \in [0, 1]$ - The characteristic foliation of $\phi_1(\Sigma)$ is $F$. ::: :::{.remark} Idea: dividing sets give ways to detect overtwisted contact structures. ::: :::{.remark} If $\Sigma = S^2$ and $\size \Gamma \geq 2$, then $(Y, \xi)$ is overtwisted. Recall that an overtwisted disc is an embedded $D^2$ with Legendrian boundary such that $\tb(\bd D) = 0$ and $\mathrm{tw}(\xi, \bd D)$. ![](figures/2022-02-17_11-24-46.png) Spheres can have exactly one dividing component. ::: :::{.exercise title="?"} Generalize to an arbitrary number of components $\size \Gamma = n$. ::: :::{.remark} Same if $\Sigma \neq S^2$ and $\Gamma$ contains a contractible curve. Contrapositively, if $(Y, \xi)$ is tight, then either - $\Sigma = S^2$ and $\Gamma$ is connected, or - $\Sigma \neq S^2$ and $\Gamma$ has no contractible components. ::: :::{.exercise title="?"} Consider tight contact structures on $S^3$. Choose Darboux $B^3$ neighborhoods at the ends, and note the interior is $S^2\times [0, 1]$: ![](figures/2022-02-17_11-37-35.png) The $S^3\times \ts{t_0}$ slices can be perturbed to be complex. So there is only one tight contact structure on $S^3$. ::: :::{.remark} What can $F$ look like on an $S^2$ in a tight $(Y,\xi)$? $F$ can be perturbed to be Morse-Smale. - There are a finite number of elliptic/hyperbolic singularities - There are nondegenerate periodic orbits, either attracting or repelling - There are no saddle-saddle arcs - The limit sets are singularities or periodic orbits Dimension 3: strange attractors! Two types of limit sets: - $\omega$ limit sets: $x\in Y$ where there exists a sequence $\ts{t_1 < \cdots }$ with $\phi(t_k)\to x$. - $\alpha$ limit sets: $x\in Y$ where there exists a sequence $\ts{t_1 > \cdots }$ with $\phi(t_k)\to x$. ::: :::{.remark} For $S^2$, take $S^+$ with an outward pointing vector field. ![](figures/2022-02-17_11-50-05.png) There are no periodic orbits since $(Y, \xi)$ is tight. The only limit sets are singular points. $\chi(D) = 1 = \size e - \size h$. Stable manifold of $h$: $\Stab_h$ are $x\in D^2$ such that there exists a flow like with $\phi(0) = x$ and $\phi(t) \to h$ Form a 1-complex $\Union_h \cl_X(\Stab_h)$ -- this contains no cycles, thus this is a tree, and the dividing set is a neighborhood of the tree. ![](figures/2022-02-17_11-58-12.png) ::: :::{.proposition title="?"} If $F$ on $\Sigma$ is Morse-Smale, then it admits dividing curves. ::: :::{.proof title="?"} Let $G = \Union_h \cl(\Stab_h) \union \Union_e e_t$ along with all of the repelling periodic orbits. Then $\Gamma = \bd \nu(G)$ divides $F$. ::: :::{.theorem title="?"} If $\Sigma$ is orientable, then there is a $C^\infty$ small perturbation of $F$ such that it is Morse-Smale. ::: :::{.proposition title="?"} Every oriented $\Sigma \subseteq (Y, \xi)$ can be perturbed to be convex. ::: :::{.proof title="?"} Near $\Sigma$, $\alpha = \beta_t + \alpha_t \dt$ and $\beta_0$ define $F$. By Peixoto there exists $\tilde \beta_t$ such that$\tilde \beta_t$ defines a Morse-Smale $F$. For $\norm{\beta - \tilde\beta}_{C^\infty} \ll \eps$, $\tilde \alpha = \tilde \beta_t + \alpha_t \dt$ is contact. Then $\alpha_s = s\tilde\alpha + (1-s)\alpha$ is a path of contact forms, so by Gray stability there is an isotopy $\phi_s$ such that $\phi_s^*(\alpha_s) = \lambda_s \alpha$ and we can take $\phi_1(\Sigma)$ to be our surface. ::: :::{.proposition title="?"} If $(\Sigma, \tilde F)$ admits dividing curves, then it is convex. ::: # Thursday, February 24 :::{.remark} Last time: there is a unique tight contact structure on $S^3$, using the existence of a contact structure on $S^3\times I$. Next: tight contact structures on - $T^2\times I$ - $S^1\times \DD^2$ - $L(p, q)$ - $T^3$ Given dividing sets of $\Gamma_0, \Gamma_1 \in T^2\times I$, how can contact structures vary in a family. Tightness implies no contractible components in $\Gamma$, so $\Gamma$ consists of $2n$ embedded curves of slow $p/q$. So the dividing set is governed by two parameters. ::: :::{.remark} The only change to the dividing set in a generic family can be: - Retrograde saddle-saddle, yielding by pass moves. ![](figures/2022-02-24_11-26-51.png) ::: :::{.proposition} Given any contact structure on $\Sigma \times I$ with dividing sets $\Gamma_0, \Gamma_1$, $\xi$ is determined by a finite number of bypass moves. ::: :::{.proof title="?"} Diagrams? $\vdots$ ::: :::{.remark} Given $\Gamma_0$ with slope $p/q$ and $\Gamma_1$ with slope $r/s$, form a Farey graph: ![](figures/2022-02-24_12-00-24.png) $\vdots$ ::: :::{.proposition title="Legendrian Darboux"} If $L$ is a Legendrian knot in $M$, then a neighborhood of $L$ is contactomorphic to a neighborhood of a zero section in $J(S^1) \cong \RR\times \T\dual S^1 \cong S^1\times \RR^2$.. ::: :::{.remark} Write this in coordinates as $(z, (x, y))$, so $\alpha = \dz -y\dx$ with $x\in \RR/\ZZ$. Then $v(L) = \ts{y^2+z^2\leq \eps}$, $y=r\cos\theta, z=r\sin\theta$. $T^2 = \ts{x, \theta}, \ro{\alpha}{T^2} = y\dtheta -y\dx = \eps\cos\theta (\dtheta - \dx)$. Unwrap: ![](figures/2022-02-24_12-15-41.png) Note that $\dalpha > 0$ at $\pi/2$ and $\dalpha < 0$ at $3\pi/2$. Idea: given two unrelated surfaces with their own foliations, how do they interact at the boundary? Dividing sets on each can be extended into the annulus, and this reduces to a combinatorial problem of how to connected arcs: ![](figures/2022-02-24_12-26-17.png) ::: # Tuesday, March 15 > See :::{.remark} Last time: classifying tight contact structures on $T^3$. Some contact structure: \[ \xi_n = \ker( \cos(2\pi n z) \dx - \sin(2\pi n z)\dy ) .\] Realize $T^3$ as a cube with faces glued, then moving in the $z$ direction twists $n$ times as you traverse the cube. We can reduce this to $\xi_1$ using $\tv{x,y,z}\mapsto \tv{x,y,nz}$. ::: :::{.remark} Goal: classify tight contact structures on lens spaces $L_{p, q} = T^2\times I/\sim$. We can discretize the contact structure on $\Sigma\times I$ into a finite number of *bypass moves* on the dividing sets. The basic move: ![](figures/2022-03-15_11-30-20.png) ::: :::{.definition title="Basic slice"} A **basic slice** is a contact structure on $T^2$ such that - $T^2\times\ts{0}$ is convex with 2 dividing curves of slope 0 - $T^2\times\ts{1}$ is convex with 2 dividing curves of slope -1 - $\xi$ is tight - $\xi$ is minimally twisting, so if $T^2 \subseteq T^2\times I$ is convex then $\slope(r) \in [-1, 0]$. ::: :::{.proposition title="?"} There are exactly 2 basic slices. Both embed in $(T^3, \xi_1) = \ker(\cos(2\pi z)\dx - \sin(2\pi z)\dy) = T^2\times I/\sim$, and are given by - $(T_2\times [0, 1/8], \xi_1)$ - $(T_2\times [1/2, 5/8], \xi_1)$ ::: :::{.proof title="?"} Step 1: There are at most 2 basic slices. Reduce to $S^1\times D^2$ by removing a convex annulus. Note that $T^2\times I\sm (S^1\times I) \cong S^1\times I^2 \cong S^1\times D^2$. ![](figures/2022-03-15_11-40-54.png) Since the boundary is convex, we can make the foliations on both of the ruling curves of slope $\infty$. > ? Take an annulus $A$ with some condition on $\bd A$, perturb to be convex? Something contradicts the "minimally twisting" assumption, involving these pics: ![](figures/2022-03-15_11-51-21.png) Smooth corners? > ? ::: :::{.definition title="Relative Euler class"} Let $(M,\xi)$ be a contact 3-manifold with $\ro{\xi}{\bd M}$ trivial. Let $s$ be a nonvanishing section of $\ro{\xi}{\bd M}$, then the **relative Euler class** $e(\xi, s) \in H^2(M, \bd M;\ZZ) \cong H_1(M)$ (by Lefschetz duality) is the dual of the vanishing set of an extension of $s$ to a section of $\xi$ on $M$. ::: :::{.remark} In this case $\dim s\inv = \dim M - \dim \xi$. ::: :::{.lemma title="?"} If $\Sigma \embeds (M,\xi)$ is a properly embedded convex surface and $s$ is a section of $\ro{\xi}{\bd M}$ that is tangent to $\bd \Sigma$ with the correct orientation, then \[ \inp{e(\xi, s)}{\Sigma} = \chi(\Sigma_+) - \chi(\Sigma_-) .\] where $\inp\wait\wait: H^2(M, \bd M;\ZZ) \times H_2(M, \bd M; \ZZ) \to \ZZ$. ::: :::{.remark} Note $H_2(T^2\times I, \bd; \ZZ) = \gens{[\alpha\times I], [\beta\times I]}$ where $H_2(T^2) = \gens{\alpha, \beta}$. ::: # Tuesday, March 22 ## Farey Graphs :::{.remark} Build a graph on the hyperbolic plane in the Poincare disc model: ![](figures/2022-03-22_11-20-24.png) Here every midpoint corresponds to adding numerators and denominators respectively. Associate slopes: - $0/1 \leadsto 1\alpha + 0 \beta$ - $1/0 \leadsto 0\alpha + 1 \beta$ - $1/1 \leadsto 1\alpha + 1 \beta$ Any pair of these is a $\ZZ\dash$basis for $H^1(T^2; \ZZ)\cong \ZZ\cartpower{2}$. Use $\SL_2(\ZZ) \embeds \PSL_2(\CC)$ to realize any change of basis as an isometry of $\mfh$. This makes the interior/exterior of any tile isometric to the full upper/lower half-disc. ::: :::{.remark} Basic moves: bypasses ![](figures/2022-03-22_11-29-28.png) The first case corresponds to slopes $r\in (-\infty , -1)$ and the second to $r\in (-1, -1/2)$. Idea: the resulting dividing set is locally constant in perturbations of $r$, provided one doesn't cross the endpoints of the curve for the bypass move. This produces a continued fraction defined inductively by $r_0 = \floor{-{p\over q}}$, writing $-{p\over q} = r_0 -{1\over {p'/q'}} = - {q'\over p'}$ with $-p/q < -p'/q' < -1$ and thus $0 < -{p\over q} - r_0 < 1$, so set $r_1 = \floor{-{p'\over q'}}$. This yields \[ -{p\over q} = r_0 - {1\over r_1 - {1\over r_2 - \cdots}} = [r_0, r_1,\cdots, r_m] ,\] which terminates in finitely many steps since $p/q$ is rational. Note that $r_i \leq -1 \implies \floor{r_i}\leq -2$. ::: :::{.proposition title="?"} If $r=-p/q = [r_0, \cdots, r_k]$ in a continued fraction expansion and $s=a/b$ is the first point connected to $p/q$ while moving counterclockwise from $0/1$ on the Farey graph, then $-a/b = [r_0, \cdots, r_{k}+1]$. ::: :::{.remark} This gives the minimal graph path from $p/q$ back to $0/1$ by jumping the maximal distance along the circle to $a/b$. Noting that $[r_1, \cdots, r_{k-1}, -1] = [r_1, \cdots, r_{k-1} + 1]$ which is a shorter continued fraction. ::: :::{.example title="?"} Let $p/q = 53/17$, then - $r_0 = -4 = -68/17$ - $r_1 = -2 = -30/15$ - $r_2 = -2 = -26/13$ - $r_3 = -2 = \cdots$ So this yields $[-4, -2, \cdots_7, -2, -3]$. ::: :::{.remark} Idea: decompose $p/q = [r_0, \cdots, r_k]$ surgery into integer surgeries on a link with $k$ components. ::: # Tuesday, March 29 :::{.remark} Goal: classification of tight contact structures on lens spaces. Lens spaces: $L_{p, q} = S^3/C_p$ where the action is $\tv{z_1, z_2} \mapsto \tv{e^{2\pi i\over p}, e^{2\pi iq \over p}}$ which has order $p$. Note $L_{p, q} \cong L_{p, q'}$ when $q\equiv q'\mod p$, so we can assume $-p ![](figures/2022-03-29_11-31-57.png) All lens spaces can be generated by genus 1 Heegaard splittings? ::: :::{.remark} $-p/q$ Dehn surgery is equivalent to a sequence of linked unknots with numbers $r_1,\cdots, r_k$. When can this be done in a way that preserves the contact structure? Idea: Legendrian surgery, which removes a Legendrian knot and reglues. ::: :::{.remark} Let $L$ be Legendrian and $\nu(L)$ is a standard neighborhood (so standard contact structure). Then $\bd \nu(L) \cong T^2$ is convex with 2 dividing curves, where "slope" is the contact framing. For $\tv{\theta, x, y} \in S^1\times \RR^2$, set $\alpha = \dx + y\dtheta$. Then $\tv{\theta, 0, 0}$ is Legendrian. When can we extend $\xi$ uniquely across surgery $S^1\times \DD^2$? Need to attach handles along integer framing (choice of integer in $\pi_1 \SO_2(\RR) \cong \ZZ$ corresponding to trivializing the normal bundle $\nu (K)$ in an embedding). Need good surgery slopes: $\ts{n}_{n\in \ZZ} \intersect \ts{1\over k}_{k\in \ZZ} = \ts{\pm 1}$, relative to the tb-framing. So $\mathrm{tb}-1$ is the best framing.: ![](figures/2022-03-29_12-11-25.png) Stabilize up to $r_K+1$ on each Legendrian knot. Fact: yields a Stein fillable thing, implies tight contact structure. ::: :::{.remark} There are $-r_0-1$ ways to perform $-r_0 - 2$ stabilizations. E.g. for $-r_0-2 = 3$, break into positive and negative stabilizations: - $(3, 0)$ - $(2, 1)$ - $(1, 2)$ - $(0, 3)$ So there are $\prod_{1\leq i\leq k} (-r_k - 1)$ tight contact structures on $-p/q = \tv{r_0, \cdots, r_k}$. ::: # Tuesday, April 05 ## Symplectic Fillings :::{.example title="Properties of the standard contact structure on $S^3$"} Consider $(S^3, \xi_\std) \subseteq \CC^2$; some things that are true: - There is a symplectic form $\omega = dx_1 \wedge dy_1 + dx_2 \wedge dy_2$ where $d\omega = 0$ and $\omega \wedge \omega =2 d\mathrm{Vol} > 0$. Write $\phi = \sum x_i^2 + \sum y_i^2$, then $S^3 = \phi\inv(1)$. - Letting $\rho = \sum x_i \del x_i + \sum y_i \del y_i = {1\over 2}\grad \phi$ in the standard metric yields a contact form $\alpha = \omega(\rho, \wait)$. - Since $\ro{\omega}{\xi_\std} > 0$, this yields an area form on contact planes. - There is also a complex structure $J: \T_p \CC^2 \to \T_p \CC^2$ where $J(\del x_i) = \del y_i$ and $J(\del y_i) = - \del x_i$ with a compatibility $g(x, y) = \omega(x, Jy)$. ::: :::{.definition title="Fillings"} A complex symplectic manifold $(X^4, \omega, J)$ is a filling of $(Y^3, \xi)$ if $Y = \bd X$, - **Stein filling**: $(X^4, J)$ is a Stein manifold, and $\xi = \T Y \intersect J(\T Y)$. - **Strong filling**: if there is an outward pointing (Liouville) vector field $\rho$ with $\mcl_\rho \omega = \omega$ with $\xi \ker (\omega(\rho, \wait))$ (which is always contact). Note $\mcl_\rho \omega = d(\iota_\rho \omega) + \iota_\rho(d\omega)$ where the 2nd term vanishes for a symplectic form. - **Weak filling**: $\ro{\omega}{\xi} > 0$. > Note that we aren't defining what "Stein" means here. ::: :::{.theorem title="?"} There are strict implications - Stein $\implies$ - Strong $\implies$ - Weak $\implies$ - Tight. Note that the last implication is the harder part of the theorem. ::: :::{.problem title="?"} Given $(Y, \xi)$, classify all fillings. ::: :::{.example title="?"} Consider $(T^3, \xi_n)$ -- if $n=1$, this is Stein fillable, and for $n\geq 2$ these are weakly fillable but not strongly fillable. In this case, all of the filling manifolds are $T^2 \times \BB^2$. ::: :::{.example title="?"} For lens spaces $L_{p, q}$ all tight contact structures are Stein fillable with the same smooth filling. Take the linear plumbing $X$ of copies of $S^2$ corresponding to $-p/q = \tv{r_1, r_2, \cdots, r_k}$ as a continued fraction expansion. They're distinguished by Chern classes $c_1(T_X, J)$. ::: :::{.example title="?"} Brieskorn spheres are examples of fillings, related to Milnor fibers. For $p,q,r \geq 2$, define \[ \Sigma(p,q,r) \da \ts{F_{p,q,r}(x,y,z) = x^p + y^q + z^r = \eps} \intersect S^5 \subseteq \CC^3 = \spanof_\CC\ts{x,y,z} .\] In this case, we have: ![](figures/2022-04-05_11-58-26.png) Note that $\eps = 0$ yields a singular variety, while $\eps>0$ small yields a smooth manifold. ::: :::{.exercise title="?"} Show $\Sigma_{p,q,r}$ is the $r\dash$fold cyclic branched cover of $S^3$ over the torus knot $T_{p, q}$. ::: :::{.remark} Let $J: \T X\to \T X$ with $J^2 = -\id$, so the eigenvalues are $\pm i$. So consider complexifying to $\T_\CC X \da \T X \tensor_\RR \CC$, so e.g. $\del x_k \mapsto (a_k + i b_k) \del x_k$. This splits into positive (holomorphic) and negative (antiholomorphic) eigenspaces $\T^{1, 0}_\CC X \oplus \T^{0, 1}_\CC X$. Take a change of basis $\tv{x_1, y_1, x_2, y_2} \mapsto \tv{z_1, \bar{z}_1, z_2, \bar{z}_2}$ which yields $\del z = {1\over 2}\qty{\del x - i \del y}$ and $\delbar z = {1\over 2}\qty{\del x + i \del y}$. ::: :::{.exercise title="?"} Let $f(z) = \abs{z}^2$ and check - $\del \delbar f = \del(z d\bar{z}) = dz \wedge d\zbar = -2i (dx \wedge dy)$. - $d = \del + \delbar$ Practicing this type of change of variables is important! ::: :::{.definition title="Levi forms and plurisubharmonicity"} Let $\phi: X\to \RR$ for $X$ a complex manifold, then the **Levi form** of $\phi$ is \[ \mcl \phi = \del\delbar \phi = \sum_{i, j} {\del^2 \over \del z_j \delbar \bar{z}_k} dz_j \wedge d\bar{z}_k ,\] generalizing the Hessian. The function $\phi$ is **plurisubharmonic** if $\mcl \phi$ is positive semidefinite at every point. ::: :::{.example title="?"} Consider $\phi: \CC\to \RR$, then \[ \mcl \phi &= \del\delbar \phi \\ &= 2\qty{{1\over 2}\qty{\phi_x + i\phi_y}}\\ &= {1\over 2}\qty{ {1\over 2}\qty{\phi_{xx} - \phi_{xy}} + {1\over 2}\qty{ \phi_{yx} - i \phi_{yy}}} \\ &= {1\over 4}\qty{\phi_{xx} + \phi_{yy}}\\ &= {1\over 4}\laplacian \phi ,\] so plurisubharmonic implies positive Laplacian. Note that in 1 dimension, $\laplacian f = 0 \implies f'' = 0$, so $(x, f(x))$ is a straight line. In higher dimensions, $f''>0$ forces convexity, so secant lines are under the straight lines, hence the "sub" in subharmonic. ::: :::{.proposition title="?"} If $\phi: X\to \RR$ is plurisubharmonic and $0$ is a regular value, then $(\phi\inv(0), \xi)$ (where $\xi$ is its complex tangencies) forms a contact structure and the sub-level set $\phi\inv(-\infty, 0]$ is a Stein filling. ::: :::{.example title="A basic example of a plurisubharmonic function"} The radical function $\phi: \CC^3\to \RR$ where $\phi(z_1,z_2,z_3) = \sum \abs{z_i}^2$ is plurisubharmonic, as is its restriction to any submanifold of $\CC^3$, including any filling of $\Sigma_{p,q,r}$. Hard theorem: any Stein manifold and any Stein filling essentially comes from this construction. ::: # Thursday, April 21 > Note: student talks in previous weeks! :::{.remark} Possible topics for the remainder of the class: - Open book decompositions - Every $(Y^3, \xi)$ is homotopic to a contact structure. - Seifert fibered spaces ::: ## Seifert Fibered Spaces :::{.remark} Brieskorn spheres $\Sigma(p,r,q) \da \ts{x^p + y^q + z^r = 0} \intersect S_\eps^5 \subseteq \CC^3$ are 3-manifolds *foliated* by $S^1$. Note that $S_1\to X \to S^2$ for $X = S^3$ or $L(p, q)$ are actual fibrations. Idea: a foliation by $F$'s is a decomposition $X = \disjoint F \to B$ which is a fibration with ramification in some fibers. ::: :::{.definition title="Seifert fibered spaces"} A **Seifert fibered space** associated to $(\Sigma, (p_1/q_1, \cdots, p_n/q_n))$ with $p_i/q_i\in \QQ$ and $\Sigma$ an orbifold surface is a 3-manifold $Y$ and knots $L_1,\cdots, L_n$ with neighborhoods $\nu L_i$ such that - $Y\sm\union_i\nu(L_i) = (\Sigma\sm\ts{\pt_1,\cdots, \pt_n}) \times S^1$ - $\nu L_i = S^1\times \DD^2$ is glued in by $p_i/q_i$ Dehn surgery. ::: :::{.example title="?"} $L(p, q)$ is $-p/q$ surgery on $S^1$, or by a slam-dunk move: ![](figures/2022-04-21_11-46-26.png) $\Sigma(p,q,r)$: ![](figures/2022-04-21_11-48-50.png) ::: :::{.exercise title="?"} Show that for $\Sigma(p,q,r)$, removing the axes in $\CC^3$ yields a trivial fibration by copies of $S^1$ over $S^2\sm{\pt_1,\pt_2,\pt_3}$ and check the surgery slopes. ::: :::{.exercise title="?"} Prove that $\Sigma(p,q,r)$ comes from the plumbing diagram for the Milnor fibration using Kirby calculus. ::: # Tuesday, April 26 :::{.remark} Recall that $\PHS^3 = \Sigma(2,3,5)$ has a Stein-fillable (and hence tight) contact structure. ::: :::{.theorem title="?"} The negative $-\Sigma(2,3,5)$ admits no tight contact structures. ::: :::{.remark} Let $S=S^3\smts{\pt_1,\pt_2,\pt_3}$ be a pair of pants and consider $X = S\cross S^1$. Note $\bd X = T^2\union T^2\union T^2$: ![](figures/2022-04-26_11-20-51.png) Note $-\Sigma(2,3,5) = \Sigma(2,-3,-5)$, since $\PHS^3$ is $-1$ surgery on the trefoil. Glue in 3 solid torii by \[ A_1 = \matt 2 {-1} 1 0, \quad A_2 = \matt 3 1 {-1} 0, \quad A_3 = \matt 5 1 {-1} 0 .\] acting on $\tv{m, \lambda}$ in $S^1\times \DD^2$: ![](figures/2022-04-26_11-24-08.png) ::: :::{.exercise title="?"} Show via Kirby calculus: ![](figures/2022-04-26_11-25-37.png) ::: :::{.lemma title="?"} There exist Legendrian representatives $F_2, F_3$ with twisting numbers $m_2, m_3 = -1$. ::: :::{.proof title="?"} Idea: by stabilization, we can assume $m_2,m_3 < 0$, and the claim is that we can destabilize them back up to $-1$ simultaneously using bypass moves. Reduce to studying dividing sets on $T^2\times I$ or $S^1\times \DD^2$. Check that the dividing set has slope $-1/2$, which implies that there is an overtwisted disc. Reduce to $2\cdot 3\cdot 5 = 30$ cases, check that an overtwisted disc can be found in each case. :::