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Introduction: Overview of Morse Homology

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Introduction: Overview of Morse Homology Introduction: Overview of Morse homology 1 2 INTRODUCTION: OVERVIEW OF MORSE HOMOLOGY What are nice functions? We will consider the following setup: M = closed1 (smooth)m-dimensional manifold, f:M→R a smooth function. Locally,f:R m →R nearp=0: ∂f 1 ∂2f f(x) =f(0)+ (0)·x + (0)·x x +··· ∂x i ! ∂x ∂x i j �i i �i,j i j 1 T =f(0)+ df 0 ·x+ x · "ess0(f)·x+··· (matrix notation) "ope: • Want fewp∈M with dfp = 0 ←−p critical point (e.g. ma#,min) $ote: these arem conditions, so we hope (i) %nite &rit(f)={critical points off} • 't criticalp, we want a good ne#t order term: (ii) det"essp(f)�= 0 ←−p nondegenerate Fact. (ii)⇒ (i) Def. f:M→R is Morse if all critical points are nondegenerate Example.M= torus: standing )ertically R lying *at R f = height f = height � circle of ma#ima (orse hence not Morse Modern perspective ∗ T M graph of df ∗ dfp (section ofT M) the section df is trans)erse f is Morse⇔ ∗ to the +ero section ofT M p M = +ero section Idea of what -trans)erse. means: trans)erse non-trans)erse /rans)erse o01ects are the -generic. ones in geometry: 1closed = compact and no boundary. HOW DO YOU RELATE MORSEf TO THE TOPOLO GY? 3 small perturbation non-trans)erse trans)erse ⇒ almost all functions are (orse Fact. All manifolds arise as submanifolds of someR k. We’ll prove that: almost any “height function”2 is Morse onM⊂R k. ⇒ it is easy to %nd Morse functions How do you relate Morsef to the topology? /wo natural geometrical o01ects to loo2 at, given a functionf: level setsf=r sublevel setsf≤r &onsider the torus standing )ertically: Level set Sublevel set R r∈ ∅ f = height ∅ ∅ 30ser)e that the topology changes when you cross the critical valuesf(p), where dfp = 0. 3therwise the topology does not change: f −1(r+ε) −1 flow in the direction of −∇f f (r) to homeomorphf≤r+ε ontof≤r We will pro)e: passing through a critical point changes the topology 0y attaching ak-cell, where k = 4negative eigen)alues of "esspf ←− index|p| Example:M = torus,f = height 2height function = linear functionalR k →R. 4 INTRODUCTION: OVERVIEW OF MORSE HOMOLOGY saddle −∇f locallyf(x, y)=−x 2 +y 2 − 0 "ess0f= � 0 � p so|p|=1 1-cell 5u0le)el set $ew su0le)el set pass through � hpy e6uiv 1-handle critical )aluef(p) diffeo =∼ homeo ⇒ &an reconstruct the mfd up to hpy e6uivalence Hw !:f (orse with critical p oints⇒M =∼ Sm homeomorphic. Warning. $ot diffeomorphic,∃ -e#otic.S 7 homeo 0ut not diffeo to the usual S7 ⊂R 8 (pro)ed 0y Milnor, using the result of "w2 8). "s it easy to recover the homology fromf? #lassical approach $Morse∼ 19:0% &hom% Smale% Miln or∼ 19;0% ...( <ic2 a self-inde#ing Morse function (meaning inde#(p)=f(p)). ⇒ the a0o)e cell-attachments de%ne a &W structure onM. ⇒ reco)er cellular homology ofM =#ample:M=S 1 p R −∇f f = height q &onsider the unsta0le cells x �→ U(x) ={points *owing down fro m the critical pointx} p U(p)= 1-cell U(q)= 0-cell q IS IT EASY TO RECOVER THE HOMOLOGY FROMf? 5 /he cellular 0oundary is:3 ∂U(p)=U(q)−U(q)=0 c !! 1 soH ∗ (S ) is generated 0y cellsU(q),U(p) in degrees∗ =0,1, as e#pected. Why is the classical approach bad? IfM is∞-dimensional, thenU(p) is usually∞-dimensional, hence no t a cell. 'lso, the -*ow. is often not de%ned, soU(p) is not e)en well-de%ne d. >ou may as2: who cares a0out∞-dimensional manifolds ? 'ctually, these nowadays arise 6uite naturally in geometry. @or e#ample: p p M = space of all loops q modern q is an∞ dimensional mfd research Morse theory Floer theory “particle theory” “string theory” Modern approach $Witten, Floer%...∼ 1980( &onsider the moduli space4 M(p, q)= {−∇f *owlines fromp toq}/reparametrization Aemar2: we identify the reparametrized *owlinesu(s) andu(s + constant). =#ample:M=S 1 p γ1 γ2 M(p, q)={γ 1, γ2} q Be%ne a chain comple#, called Morse complex, MC∗(f)=Z 2 ·p⊕Z 2 ·q⊕··· where p,q,... are the critical points, and we wor2 o)erZ 2 =Z/ to a)oid signs. /he grading∗ refers to the (orse ind e#, soMC k(f) has aZ 2 summand for each critical point of inde#k. d:MC k →MC k−1 dp= q(4elements inM(p, q))·q � =#ample:M = hot-dog ( =∼ S2) 3Note that orientation signs are a subtle issue if we got the sign wrong! then suddenly∂U(p) would no longer be "ero. #o a$oid such technical subtleties! we will wor% o$erZ/& in this course. 4these flowlines are called instantons or tunneling paths in physics. " INTRODUCTION: OVERVIEW OF MORSE HOMOLOGY a b R α β 1 c f = height γ ε e 0 /hen: MC2 =Z 2 ·a⊕Z 2 · b da=4{α}·c=c, db=4{β}·c=c MC1 =Z 2 · c dc=4{γ,ε}=0(mod ) MC0 =Z 2 · e de=0 2er∂ Morse homology= =MH ∗(f) =Z 2 ·e⊕Z 2(a−b) im∂ ∗=0 2 30ser)e this is the same asH ∗(S ) (o)erZ/ ). &heorem.MH ∗(f) =∼ H∗(M) #or. 4(critical points of a Morse function) = 4(generators ofMC ∗(f)) ≥ 4(generators ofMH ∗(f)) = dimH i(M)(Betti numbers) � =#ample. ' genericf: →R has≥ + · genus = ; critical poin ts.5 )eometry is functional analysis We made two tacit assumptions when de%ningMC ∗,MH∗: (1) need 4M(p, q) %nite for|q|=|p|− 1. Aephrasing: M(p, q) is a compact 0-dimensional manifold ( ) needd 2 =d◦d = 0 to de%ne homology . "dea of proof of $*(: 2 d p=d( q 4M(p, q)·q) = �q,r 4M(p, q)·4M(q, r)·r $ow 4M(p, q)·4M(q, r) counts� -0ro2en. *owl ines fromp toq tor. "ence: d2 = 0⇔ once-bro2en *owlines a rise in pairs "ope:∃ a 1-family of *owlines 1oining two 0ro2en *owlines: 5 i Non-examinable: 'lgebraic topology tells you≥|χ(M)|=| �(−() dimMC i(f)|! $ia the intersection number graph(df)·) T ∗M =−χ(M). *o for a torus it +us t predicts≥ ). GEOMETRY IS FUNCTIONAL ANALYSIS 7 p q( q& r Ciew the *owlines as points in the moduli space, then: 1-family ⊂ M(p, r) 0ro2en flowline 0ro2en flowline "ope: •M(p, r) is a non-compact 1-mfd • ∃ natural way of ma2ing it compact: M(p, r)=M(p, r)∪-∂M(p, r). (∂M(p, r)={bro2en *owlines}) ⇒ M(p, r) compact 1-mfd ⇒ M(p, r) = dis1oint union of circles and compact inter)als ⇒∂ M(p, r) = e)en num0er of points, so =0 mod ⇒d 2p=0 ⇒d 2 = 0� "dea of proof of $+(: Functional ,nalysis (:) transversality problem:M(p, q) are smooth manifolds for a -generic. metricg (which de%nes∇f 0yg(∇f,·) = df), and dimM(p, q)=|p|−|q|−1. (D) compactness problem:M(p, q) can 0e compactified 0y 0ro2en flow- lines. (ost modern homology theories in)olve these two pro0lems ⇒ (orse homology is a perfect playground! "dea to solve $-(: consider the ./anach0 vector bundle {smooth )ector %elds alongu} � sectionF=∂ su−∇f(u) � {smooth pathsu:R→M,u(s)→p,q ass→−∞,+∞} 30ser)e: F=0⇔∂ su=−∇f⇔u∈M(p, q) 8 INTRODUCTION: OVERVIEW OF MORSE HOMOLOGY ⇒ M(p, q) = intersection of a section of a Eanach )ector 0undle with 0 section F perturb the metricg non-trans)erse trans)erse ⇒ M(p, q) mfd! )eometry is algebra ∗ ∗ Be%ne the Morse cohomologyMH (f) =∼ H (M) using ∗ MC =Z 2 ·p⊕Z 2 ·q⊕··· :MC k →MC k#1 p= q 4M(q, p)·q (where|q|=|p|+1) � m−∗ " /hen oincar!edualityH ∗(M) =∼ H (M) (o)erZ 2) is 1ust the symmetry: m−∗ MH∗(f) =∼ MH (−f) p �→p M(p, qFf) =∼ M(q, pF−f) −∇f *owlineu(s) �→ ∇f *owlineu(−s) /he switch in grading is 0ecause *ipping the sign off *ips the sign of the "e ssian. <oincarGeduality is 1ust re)ersal of *owlines in (orse theory! If you use a height function, <oincarGeduality is the intuitive idea -loo2 at the manifold upside down!.. The "#unneth isomorphismH ∗(M1 ×M 2) =∼ H∗(M1)⊗H ∗(M2) can 0e pro)ed 6uite simply now 0y the o0ser)ation: (orse functionsf 1 :M 1 →R,f 2 :M 2 →R give naturally rise to the (orse functionf 1 +f 2 :M 1 ×M 2 →R, and the *owlines are 1ust the com0ined *owline forf 1, f2 on the respective factors ofM 1 ×M 2. The cup productH a(M)⊗H b(M)→H a#b(M) can also 0e described (orse theoretically: you count *ows along a >-shaped @eynman graph, *owing 0y a (orse function along each of the three edges of the graph and you re6uire that the *ow con)erges to the inputs p,q at the top, and tor at the 0ottom. /his so lution then contributesp·q=r+··· to the product. "this also wor%s o$erZ! but then one needs to assumeM is orientable! which is secretly ∗ hidden in the orientation signs that de,neMC ∗!MC ..
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