A COMPUTATIONAL ALTERNATIVE TO MORSE-NOVIKOV THEORY

Transcription

1 A COMPUTATIONAL ALTERNATIVE TO MORSE-NOVIKOV THEORY Dan Burghelea Department of Mathematics Ohio State University, Columbus, OH Shanghai, July 2017

2 summary A brief summary of what AMN-theory does The AMN theory invariants From AMN invariants to the algebraic topology Complements of complex hyper surfaces via AMN theory Computational results Description of AMN invariants

3 The AMN-theory considers: f : X R and f : X S 1 X a compact ANR, f a tame map Examples: finite simplicial complex, compact topological manifold, simplicial map local polynomial maps

4 x 2 x 1 x 3 x 4 f f c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8

5 f f

6 AMN invariants for real-valued maps f : X R Fix a field κ For f : X R a) critical values Cr(f ) := c i < c i+1 < c i+2 b) barcodes 1 closed barcodes B c r (f ), [a, b] R, a, b Cr(f ) regarded as a complex number z = a + ib C 2 open barcodes B o r (f ), (a, b) R, a, b Cr(f ) regarded as a complex number z = b + ia C 3 closed-open barcodes B c,o r (f ), [a, b) R, a, b Cr(f ) regarded as a complex number z = a + ib C 4 open-closed barcodes B o,c r (f ), (a, b] R, a, b Cr(f ) regarded as a complex number z = b + ia C

7 AMN invariants for angle-valued maps f : X S 1 For f : X S 1 a) critical values Cr(f ) := 0 θ 1 < θ 2,, < θ N < 2π b) barcodes 1 closed barcodes B c r (f ), [a, b] R 0 a, b Cr(f ) mod 2π regarded as a complex number z = e i(a+b)+(b a) C 2 open barcodes Br o (f ), (a, b), R 0 a, b Cr(f ) regarded as a complex number z = e i(a+b)+(a b) C 3 closed-open barcodes Br c,o (f ), [a, b) R 0 a, b Cr(f ) regarded as a complex number z = e i(a+b)+(b a) C 4 open-closed barcodes Br o,c (f ), (a, b] R 0 a, b Cr(f ) regarded as a complex number z = e i(a+b)+(a b) C c) Jordan cells J r (f ) := {(λ, k) λ κ \ 0, k Z 1 }

8 (λ, k) corresponds to a Jordan matrix T (λ, k) = λ λ λ λ (1) if k 2, and T (λ, 1) = (λ) if k = 1.

9 algebraic topology of f f : X S 1 ξ f H 1 (X; Z) Betti numbers β r (X) = dim H r (X) Twisted Betti numbers β r (X : (ξ f, u)) for u κ \ 0 Novikov-Betti numbers βr N (X, ξ) = rank FH r ( X) Monodromy [T r ] := (V r, T r ) = Tor H r ( X) ξ X X H r ( X) a f.g. κ[t 1, t]-module, H r ( X) = FH r ( X) Tor H r ( X) u κ \ 0, ξ H 1 (X; Z) (ξ, u) : H 1 (X; Z) ξ Z û κ \ 0 with û(n) = u n Tor H r ( X) = (V (r), T (r)) f.g. torsion module hence, V (r) a f.d κ-vector space, T (r) : V (r) V (r) linear isomprphism.

10 AMN-invariants of f : X R/S 1 Algebraic topology of X or(x, ξ f ) Dynamics of a flow on X Ψ with f Lyapunov Dynamics= rest points, instantons, closed trajectories

11 Results AMN-invariants Algebraic topology Theorem For f : X S 1 the following holds: 1 βr N (X, ξ f ) = Br c (f ) + Br 1 o (f ) 2 β r (X, (ξ f, u)) = βr N (X, ξ f ) + J r,u (f ) + J (r 1),1/u (f ) 3 β r (X) = βr N (X, ξ f ) + J r,u (f ) + J (r 1),1/u (f ) 4 [T r (ξ f )] = [ (λ,k) Jr (f )T (λ, k)] 5 For any θ S 1 we have β r (f 1 (θ)) (λ,n) J r (f ) n The Morse or the Novikov complex associated with f (up to isomorphism) w.r. to a field is completely determined by the closed, open and closed-open bar codes.

12 Complement of hyper surfaces P(z 1, z 2, z n ) polynomial, z = (z 1, z 2, z n ) C n. V = P 1 (0) C n, B P := {z 1, z 2, z N }, z i C n the bifurcation set Y := C n \ V, P : Y C \ 0, f 1 : Y R >0, f 1 (z) = P(z) f 2 : Y S 1, f 2 (z) := P(z)/ P(z) X R := {z C n z R} \ {z P(z) < 1/R}, P : C n \ P 1 (B) C \ B P is a smooth bundle.

13 Figure : X R

14 Proposition Suppose R > sup i=1,2, N { z i }. Then 1 X R is a compact manifold with boundary, with Y \ IntX R is a collar of the boundary (X R ), i.e.y X R X R [0, ɛ) 2 the critical values of f 1 are { z i, z i B P \ 0} 3 the critical angles go f 2 are {argz i.z i B P \ 0} 4 If P is good at then f 2 : X R S 1 is a bundle. Good at means V is in general position at cf. L.Maxim (i.e., its projective completion of V intersects the hyperplane at infinity transversally) or is tame at cf. Nemethi-Zaharia.

15 Theorem Let P a polynomial as above and let ξ = ξ f2 H 1 (Y ; Z) the induced cohomology class. Suppose that P is good at 1 β N r (Y ; ξ) = { 0 if r n χ(y ) if r = n (2) 2 J r (f ) = for r n 3 For any u κ \ 0 0, r > n β r (X : (ξ, u)) = J r,u (f ) + J (r 1),1/u (f ), r < n χ(y ) + J (n 1),1/u (f ), r = n 4 For θ S 1, β r (f 1 2 (θ)) (λ,k) J r (f ) k

16 Graphs and graph-representations An oriented graph has: Vertices : {x α, α A} a α,β Oriented edges; { x α x β }, A graph representations ρ assigns: x α V α finite dimensional κ-vector space a α,β V α V β, κ-linear map

17 The graph Z b i 1 x 2i 1 a i b x 2i i x 2i+1 a i+1 b x 2i+2 i+1 Figure : The graph Z. Compactly supported indecomposable representations are Barcodes: indexed by four type of Intervals in R { [a, b], (a, b), with ends, a b, a, b Z [a, b), (b, a]

18 the graph G 2m x 2 x 3 b 1 a 1 x 1 x 4 a 2 x 2m 3 b m 1 b m x 2m a m x 2m 1 a m 1 x 2m 2 Figure : The graph G 2m.

19 The spiral z(t) = { x(t) = (t + 1) cos t y(t) = (t + 1) sin(t), t [0, )

20 Indecomposable representations are Barcodes four type of Intervals [0, ) { [a, b], (a, b), [a, b), (b, a] with a {2πk/m 1 k m}, b {2πk/m + 2πr 1 k m, r Z 0 } Jordan cells pairs (λ, k), λ κ \ 0, k Z 1

21 The AMN invariants for f : X R Construct the representation ρ r (f ), r 0, 1 Find c i < c i+1 <,, < c i+m < critical values 2 Choose, < t i < t i+1,, t i+m, c i < t i < c i+1, 3 Denote X 2i 1 = f 1 (t i 1 ), X 2i = f 1 ([t i 1, t i ]) 4 Define ρ r (f ): V 2i 1 = H r (X 2i 1 ) a i (r) b i (r) H r (X 2i ) = V 2i H r (X 2i+1 ) = V 2i+1 a i (r), b i (r) induced by the inclusions X 2i 1 X 2i X 2i+1 The r-barcodes of f are the barcodes of ρ r (f ) with the ends i and j replaced by c i and c j

22 The AMN invariants for f : X S 1 Construct the representation ρ r (f ), r 0, 1 Find 0 θ 1 < θ 2 <,, < θ m < 2π critical angles 2 Choose t 1 < t 2,, t m, θ i < t i < θ i+1, θ m < t m < 2π 3 Denote by X 2i 1 = f 1 (t i 1 ), X 2i = f 1 ([t i 1, t i ]) 4 Define ρ r (f ): V 2i 1 = H r (X 2i 1 ) a i (r) b i (r) H r (X 2i ) = V 2i H r (X 2i+1 ) = V 2i+1 a i (r), b i (r) induced by the inclusions X 2i 1 X 2i X 2i+1 The r-barcodes of f are the barcodes of ρ r (f ) with the ends i and j + mk replaced by θ i and θ j + 2πk The Jordan cells J r (f ) are the Jordan cells of ρ r (f ).

23 Linear relations monodromy Jordan cells A better way to calculate J r (f ). Linear relation - Regular part V V reg α reg β α W V W reg β reg V reg (V reg, T reg ), T reg := β 1 reg α reg.

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25 Jordan cells (of f ) Use any θ S 1 Consider i +(r) V r = H r (X θ+2π ) i (r) W r = H r (X θ ) V r = H r (X θ ) (V r ) reg i (r) reg (W r ) reg i +(r) reg (V r ) reg Define (V (r), T (r)) := (V (r) reg, i + (r) 1 reg i (r) reg ). J r (f ) are the Jordan cells of T (r)

26 References 1.Dan Burghelea, Stefan Haller, Topology of angle valued maps, bar codes and Jordan blocks arxiv: to appear Dan Burghelea, New topological invariants for real- and angle-valued maps. An alternative to Morse-Novikov theory chapter 8 (book) World Scientific 2017 see (