LECTURE 1: ADVERTISEMENT LECTURE. (0) x i + 1 2! x i x j

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1 LECTUE 1: ADVETISEMENT LECTUE. PAT III, MOSE HOMOLOGY, What are nice functions? We will consider the following setu: M = closed 1 (smooth) m-dimensional manifold, f : M a smooth function. Locally, f : m near = 0: f(x) = f(0)+ i f x i (0) x i + 1 2! i,j 2 f x i x j + = f(0)+df 0 x+ 1 2 xt Hess 0 (f) x+ (matrix notation) Hoe: Want few M with df = 0 critical oint (e.g. max,min) Note: these are m conditions, so we hoe (i) finite Crit(f) = {critical oints of f} At critical, we want a good next order term: Fact. (ii) (i) (ii) dethess (f) 0 nondegenerate Def. f : M is Morse if all critical oints are nondegenerate Examle. M = torus: standing vertically lying flat Morse Modern ersective df T M grah of df (section of T M) M = zero section f is Morse circle of maxima hence not Morse the section df is transverse to the zero section of T M Date: May 1, 2011, c Alexander F. itter, Trinity College, Cambridge University. 1 closed = comact and no boundary. 1

2 2 PAT III, MOSE HOMOLOGY, L1 Idea of what transverse means: transverse non-transverse Transverse objects are the generic ones in geometry: small erturbation non-transverse transverse almost all functions are Morse Fact. All manifolds arise as submanifolds of some k. We ll rove that: almost any height function 2 is Morse on M k. it is easy to find Morse functions How do you relate Morse f to the toology? Two natural geometrical objects to look at, given a function f: level sets f = r sublevel sets f r Consider the torus standing vertically: r Level set Sublevel set Observe that the toology changes when you cross the critical values f(), where df = 0. Otherwise the toology does not change: 2 height function = linear functional k.

3 PAT III, MOSE HOMOLOGY, L1 3 f 1 (r +ε) f 1 (r) flow in the direction of f to homeomorh f r+ε onto f r We will rove: assing through a critical oint changes the toology by attaching a k-cell, where k = #negative eigenvalues of Hess f index Examle: M = torus, saddle f locally f(x,y) = x 2 +y 2 ( ) 2 0 Hess 0 f = 0 2 so = 1 Sublevel set ass through critical value f() New sublevel set hy euiv 1-cell 1-handle diffeo = homeo Can reconstruct the mfd u to hy euivalence Hwk 8: f Morse with 2 critical oints M = S m homeomorhic. Warning. Not diffeomorhic, exotic S 7 homeo but not diffeo to the usual S 7 8 (roved by Milnor, using the result of Hwk 8). Is it easy to recover the homology from f? Classical aroach (Morse 1930, Thom, Smale, Milnor 1960,...) Pick a self-indexing Morse function (meaning index() = f()). the above cell-attachments define a CW structure on M. recover cellular homology of M Examle: M = S 1

4 4 PAT III, MOSE HOMOLOGY, L1 f Consider the unstable cells x U(x) = {oints flowing down from the critical oint x} U() = 1-cell U() = 0-cell The cellualar boundary is: 3 U() = U() U() = 0 so H cell (S 1 ) is generated by cells U(),U() in degrees =0,1, as exected. Why is the classical aroach bad? If M is -dimensional, then U() is usually -dimensional, hence not a cell. Also, the flow is often not defined, so U() is not even well-defined. You may ask: who cares about -dimensional manifolds? Actually, these nowadays arise uite naturally in geometry. For examle: Morse theory article theory modern research Floer theory string theory Modern aroach (Witten, Floer, ) Consider the moduli sace 4 M = sace of all loos is an dimensional mfd M(, ) = { f flowlines from to }/rearametrization Examle: M = S 1 γ 1 γ 2 M(,) = {γ 1,γ 2 } 3 Note that orientation signs are a subtle issue: if we got the sign wrong, then suddenly U() would no longer be zero. To avoid such technical subtleties, we will work over Z/2 in this course. 4 these flowlines are called instantons or tunneling aths in hysics.

5 PAT III, MOSE HOMOLOGY, L1 5 Define a chain comlex, called Morse comlex, MC (f) = Z 2 Z 2 where,,... are the critical oints, and we work over Z 2 = Z/2 to avoid signs. Examle: M = hot-dog ( = S 2 ) 2 a α d : MC k MC k 1 d = (#elements in M(,)) c 1 β 2 b γ ε Then: e 0 MC 2 = Z 2 a Z 2 b da = #{α} c = c, db = #{β} c = c MC 1 = Z 2 c dc = #{γ,ε} = 0 (mod 2) MC 0 = Z 2 e de = 0 Morse homology = ker = MH (f) = Z 2 e Z 2 (a b) im = 0 1 Observe this is the same as H (S 2 ) (over Z/2). Theorem. MH (f) = H (M) Cor. #(critical oints of a Morse function) = #(generators of MC (f)) #(generators of MH (f)) = dimh i (M) (Betti numbers) Examle. A generic f : has 2+2 genus = 6 critical oints. 5 Geometry is functional analysis We made two tacit assumtions when defining MC,MH : (1) need #M(,) finite for = 1. ehrasing: M(, ) is a comact 0-dimensional manifold (2) need d 2 = d d = 0 to define homology. 5 Non-examinable: Algebraic toology tells you χ(m) = ( 1) i dimmc i (f), via the intersection number: grah(df) 0 T M = χ(m). So for a torus it just redicts 0.

6 6 PAT III, MOSE HOMOLOGY, L1 Idea of roof of (2): d 2 = d( #M(,) ) =,r #M(,) #M(,r) r Now #M(,) #M(,r) counts broken flowlines from to to r. Hence: d 2 = 0 once-broken flowlines arise in airs Hoe: a 1-family of flowlines joining two broken flowlines: 1 2 View the flowlines as oints in the moduli sace, then: Hoe: broken flowline M(, r) is a non-comact 1-mfd natural way of making it comact: r 1-family M(,r) broken flowline M(,r) = M(,r) M(,r) ( M(,r) = {broken flowlines}) M(,r) comact 1-mfd M(,r) = disjoint union of circles and comact intervals M(,r) = even number of oints, so = 0 mod 2 d 2 = 0 d 2 = 0 Idea of roof of (1): Functional Analysis (3) transversality roblem: M(, ) are smooth manifolds for a generic metric g (which defines f by g( f, ) = df), and dimm(,) = 1. (4) comactness roblem: M(, ) can be comactified by broken flowlines. Most modern homology theories involve these two roblems Morse homology is a erfect layground!

7 PAT III, MOSE HOMOLOGY, L1 7 Idea to solve (3): consider the Banach vector bundle {smooth vector fields along u} section F = s u f(u) {smooth aths u : M,u(s), as s,+ } Observe: F = 0 s u = f u M(,) M(,) = intersection of a section of a Banach vector bundle with 0 section F erturb the metric g non-transverse transverse M(,) mfd! Geometry is algebra Define the Morse cohomology MH (f) = H (M) using MC = Z 2 Z 2 δ : MC k MC k+1 δ = #M(,) (where = +1) Then Poincaré duality H (M) = H m (M) (over Z 2 ) 6 is just the symmetry: MH (f) = MH m ( f) M(,;f) = M(,; f) f flowline u(s) f flowline u( s) The switch in grading is because fliing the sign of f flis the sign of the Hessian. Poincaré duality is just reversal of flowlines in Morse theory! If you use a height function, Poincaré duality is the intuitive idea look at the manifold uside down!. The Künneth isomorhism H (M 1 M 2 ) = H (M 1 ) H (M 2 ) can be roved uite simly now by the observation: Morse functions f 1 : M 1, f 2 : M 2 give naturally rise to the Morse function f 1 +f 2 : M 1 M 2, and the flowlines are just the combined flowline for f 1,f 2 on the resective factors of M 1 M 2. The cu roduct H a (M) H b (M) H a+b (M) can also be described Morse theoretically: you count flows along a Y-shaed Feynman grah, flowing by a Morse function along each of the three edges of the grah and you reuire that the flow converges to the inuts, at the to, and to r at the bottom. This solution then contributes = r+ to the roduct. 6 this also works over Z, but then one needs to assume M is orientable, which is secretly hidden in the orientation signs that define MC, MC.