CIRCLE-VALUED MORSE THEORY AND NOVIKOV HOMOLOGY ANDREW RANICKI - PDF document

CIRCLE-VALUED MORSE THEORY AND NOVIKOV HOMOLOGY ANDREW RANICKI (Edinburgh) http://www.maths.ed.ac.uk/ � aa r • T raditional Mo rse theo ry deals with di�erentiable real-valued functions : M → R f and o rdina ry homology H ∗ ( M ). • Circle-valued Mo rse theo ry deals with di�erentiable circle-valued functions 1 homology H Nov : M → S and Novik ov ( M ). f ∗ The circle-valued theo ry is new er and ha rder! • The circle-valued theo ry has applications to the structure theo ry of non-simply-connected manifolds, dynamical systems, symplectic top ology , Flo er theo ry , Seib erg-Witten theo ry etc. 1

Novik ov S.P .Novik ov (1938 {), one of the founding • fathers of surgery theo ry . Proved the top ological inva riance of • rational P ontrjagin classes fo r di�erentiable manifolds (1965), fo r which he w as a w a rded the Fields Medal in 1970. Last pap er in surgery theo ry (1969) • fo rmulated the Novik ov conjecture. Intro duced circle-valued Mo rse theo ry in 1981, • motivated b y physical p roblems in electro- magnetism and �uid mechanics. • Autho r of "T op ology" (V olume 12 of Encyclop edia of Mathematical Sciences, Sp ringer, 1996) { the b est intro duction to high-dimensional manifold top ology! 2

The p rogramme • The geometrically de�ned Mo rse-Smale chain complex C MS ( f ) of a real-valued Mo rse : M → R function f is w ell-understo o d. The geometrically de�ned Novik ov chain complex C Nov ( f ) of a circle-valued Mo rse 1 function f : M → S is not so w ell-understo o d. Objective: mak e the Novik ov complex as • w ell-understo o d as the Mo rse-Smale com- plex! F eed algeb ra back into top ology . 1 • The strategy: lift f : M → S to in�- : M → R nite cyclic covers f and compa re C Nov to C MS ( f ) ( f N ), with 1 = f − f N = f | : M N [0 , 1] → [0 , 1] • The general theo ry w o rks fo r a rbitra ry π ( M ). 1 Will concentrate on the 'simply-connected' = Z sp ecial case π ( M ) , π ( M ) = { 1 } . 1 1 3

Real-valued Mo rse functions • A critical p oint of a di�erentiable function : M → R f is a zero p ∈ M of ∇ f : τ M → τ R . • A critical p oint p ∈ M is nondegenerate if f ( p + ( x 1 , x 2 , . . . , x m )) i m � � 2 2 = f ( p ) − ( x j ) + ( x j ) nea r p j =1 j = i +1 with i the index of f . W rite Crit i ( f ) fo r the set of index i critical p oints of f . : M → R A function f is Mo rse if every crit- • ical p oint is nondegenerate. If M is com- pact and non-empt y then a Mo rse f : M → R has a �nite numb er c i ( f ) = | Crit i ( f ) | ≥ 0 of critical p oints with index i . Note that c ( f ) > 0, c m ( f ) > 0 (minimax p rinciple). 0 4

Where do real-valued Mo rse functions come from? • Nature (= geometry) : M → R Mo rse functions f a re dense in the • space of all di�erentiable functions on M . • Mo rse theo ry investigates the relationship b et w een the algeb raic top ology of M and the Mo rse functions on M . T ypical p rob- lem: given M , what a re the minimum num- b er of critical p oints of a Mo rse function : M → R ? As usual, it is easier to �nd f answ er fo r dim ( M ) ≥ 5. 5

Gradient �o w • A vecto r �eld v : M → τ M is gradient-lik e : M → R fo r a Mo rse function f if there exists a Riemannian metric � , � on M with ) ∈ R � v, w � = ∇ f ( w ( w ∈ τ M ) . : R → M • A do wnw a rd v -gradient �o w line γ satis�es γ ′ ( t ∈ R ( t ) = − v ( γ ( t )) ∈ τ M ( γ ( t )) ) . A v -gradient �o w line sta rts at a critical p oint of index i lim = p ∈ Crit i ( f ) t →−∞ and ends at a critical p oint of index i − 1 lim = q ∈ Crit i − ( f ) . 1 t →∞ 6

Mo rse theo ry and surgery : M → R • A critical value of Mo rse f is ) ∈ R f ( p fo r critical p oint p ∈ M . Can assume the critical values a re distinct, and index( p ′ ( p ′ that index ( p ) ≤ ) if f ( p ) < f ). = f − 1 • W rite N a ( a ) ⊂ M fo r any regula r (= value a ∈ R non-critical) . • Theo rem (Thom, 1949) (i) If f : M → [ a, b ] has no critical values then ) ∼ ( M ; N a , N b = N a × ([0 , 1]; { 0 } , { 1 } ) . (ii) If f : M → [ a, b ] has only one critical value c ∈ [ a, b ], of index i , then ( M ; N a , N b ) 1 × D m − i ⊂ N a on S i − is the trace of surgery with 1 × D m − i ) ∪ D i × S m − i − 1 , ( N a \ S i − N b = 1] ∪ D i × D m − i . M = N a × [0 , 7

: M → R A Mo rse function f determines a • handleb o dy decomp osition of M m � � D i × D m − i . M = i =0 c i ( f ) The Mo rse-Smale transversalit y condition • Theo rem (Smale, 1962) F o r every Mo rse : M → R f there is a class GT ( f ) of gradient- lik e vecto r �elds v fo r f such that there is only a �nite numb er n ( p, q ) of v -gradient �o w lines from p to q whenever index( q ) = index( p ) − 1 . GT ( f ) is dense in the space of all gradient- lik e vecto r �elds on M . 8

The Mo rse-Smale complex = C MS • The Mo rse-Smale complex C ( M, f, v ) : M → R fo r Mo rse f and v ∈ GT ( f ) is a free Z based f.g. -mo dule chain complex = Z with C i [Crit i ( f )]. • The di�erentials a re given b y the signed numb ers of v -gradient �o w lines � d : C i → C i − ; p �→ n ( p, q ) q . 1 q ∈ Crit i − 1 ( f ) • The Mo rse-Smale complex is the cellula r chain complex of the CW structure on M with one i -cell fo r each critical p oint of f index i , C MS of ( M, f, v ) = C ( M ), so ( C MS H ∗ ( M, f, v )) = H ∗ ( M ) . de�ne C MS ; N, N ′ • Can also fo r Mo rse f : ( M ) → with C MS ([0 , 1]; { 0 } , { 1 } ), ( M, f, v ) = C ( M, N ). 9

The Mo rse inequalities • The Betti numb ers of a �nite CW complex M a re de�ned b y ( M ) = dim Z ( H i ( M ) /T i ( M )) , b i q i ( M ) = minimum no. generato rs of T i ( M ) with 0 ∈ Z } T i ( M ) = { x ∈ H i ( M ) | nx = 0 fo r some n � = the to rsion subgroup of H i ( M ). Theo rem (Mo rse, 1927) The numb er c i ( f ) • of index i critical p oints of a Mo rse function : M → R f is b ounded b elo w b y ( f ) ≥ b i ( M ) + q i ( M ) + q i − ( M ) . c i 1 free Z Pro of A f.g. -mo dule chain complex with H ∗ ( C ) = H ∗ ( M ) must have C dim Z ( C i ) ≥ b i ( M ) + q i ( M ) + q i − ( M ) . 1 = C MS In pa rticula r, this applies to C ( M, f, v ). 10

The Mo rse inequalities a re sha rp fo r π ( M ) = { 1 } 1 • Theo rem (Smale, 1962) An m -dimensional manifold M with m ≥ 5 and π ( M ) = { 1 } 1 : M → R admits a Mo rse function f with c i ( f ) = b i ( M ) + q i ( M ) + q i − ( M ) . 1 • Proved b y handle cancellation. The situation is much mo re complicated • fo r π ( M ) � = { 1 } . Need algeb raic K -theo ry 1 of C MS the Z of [ π ( M )]-mo dule version ( M, f, v ) 1 to give sha rp b ounds on minimum num- : M → R b er of critical p oints of Mo rse f (Sha rk o). 11

Circle-valued Mo rse functions A critical p oint of a di�erentiable function • 1 f : M → S is zero p ∈ M of ∇ f : τ M → τ S . 1 • A critical p oint p ∈ M is nondegenerate if f ( p + ( x 1 , x 2 , . . . , x m )) i m � � 2 2 = f ( p ) − ( x j ) + ( x j ) nea r p j =1 j = i +1 with i the index of f . A function f is Mo rse if every critical p oint is nondegenerate. • If M is compact and non-empt y then a 1 Mo rse f : M → S has a �nite numb er c i ( f ) ≥ 0 of critical p oints with index i . Can de�ne gradient-lik e v : M → τ M , GT ( f ) • etc., as fo r the real-valued case. 12

Where do circle-valued Mo rse functions come from? • Nature, cohomology , and knot theo ry . 1 • Mo rse functions f : M → S a re dense in the space of all di�erentiable functions on 1 1 M rep resenting �xed c ∈ H ( M ) = [ M, S ]. 1 • T ypical p roblem: given c ∈ H ( M ) what a re the minimum numb ers c i ( f ) of critical 1 p oints of a Mo rse function f : M → S with f ∗ 1 (1) = c ∈ H ( M )? F o r m ≥ 6 can apply the cancellation metho d • of real-valued Mo rse theo ry , but the alge- b raic b o ok-k eeping is much ha rder. • Circle-valued Mo rse theo ry extends to the Mo rse theo ry of closed 1-fo rms, rep resent- 1 ; R ing classes c ∈ H ( M ). 13