A (computer friendly) alternative to Morse Novikov theory for real and angle valued maps

Transcription

1 A (computer friendly) alternative to Morse Novikov theory for real and angle valued maps Dan Burghelea Department of mathematics Ohio State University, Columbus, OH based on joint work with Stefan Haller

2 CONTENTS 1 Introduction to ANM 2 The configurations C r (f ). 3 The monodromy (V r (f ), T r (f )) 4 The configuration c r (f ) 5 Graph representations and ANM

3 MN theory versus MN-theory starts with (M, g) Riemannian manifold f : M R or S 1 Morse function AMN-theory starts with X a compact ANR f : X R or S 1 weakly tame map and a field κ. Considers Calculates a) Critical points of f a) Critical values of f b) Instantons between b) Bar codes between cr. points of grad g f cr. values of f b) Closed trajectories of grad g f c) Jordan cells of f relates them to relates them to Algebraic Topology 1) Betti numbers, Novikov-Betti numbers 2) Alexander function/ Reidemeister torsion

4 Invariants in AMN-theory 1 critical values/ angles a, b, /θ 1, θ 2, 2 bar codes = intervals of four types: B c r (f ) closed intervals, [a, b], [θ, θ + 2πk] B 0 r 1 (f ) open intervals, (a, b), (θ, θ + 2πk) collected as configuration C r (f ) monic polynomial P f r (z) Br c,o Br c,o (f ) closed-open intervals, [a, b), [θ, θ + 2πk) (f ) open-closed intervals, (a, b], (θ, θ + 2πk] collected as configuration c r (f ) 3 Jordan cells (λ, k), λ κ \ 0, k Z >0, J r (f ) for any r. collected as monodromy (V r (f ), T r (f )), characteristic polynomials D r (z) and the canonical divizors D r (z) D r,1 (z) D r,2 (z)

5 Merits 1 Extends considerably the class of spaces and maps the MN theory considers. (Generality) 2 The invariants produced are finite and are computable via effective algorithms. (Computability) 3 The configurations C r (f ) and c r (f ) are robust to continuous perturbation. (Stability) 4 The configurations and the Jordan cells refine the traditional topological invariants. (Refinment)

6 The configuration C r (f ). for real valued map for angle valued map monic polynomials P f r (z) whose roots are the points of the configuration C r (f )

7 The configuration c r (f ). for real valued map for angle valued map

8 Jordan cells J r (f ) -the monodromy (V r (f ), T r (f )) J = (λ, k) = (κ k, T (λ, k)), λ κ \ 0, k Z 0 λ λ T (λ, k) = λ λ (V r (f ), T r (f )) = J J r J

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10 An Example, f : X = Y φ S 1 φ 1 circle circle 2 circle 3 Y 0 Y Y 1 0 θ 1 θ 2 θ 3 θ 4 θ 5 θ 6 2π map φ circle 1: 1 time around circle 1-3 times around 2, - 2 times around 3 circle 2: 1 time around circle 1, 4 times around 2, 1 time around 3 circle 3: 2 times around 1, 2 times around 2, 2 times around 3 r-invariants dimension bar codes Jordan cells 0 (θ 1, θ 2 ) (1, 1) (θ 6, θ 1 + 2π] (3, 2) 1 [θ 2, θ 3 ], [θ 6, θ 6 ] (θ 4, θ 5 )

11 The configuration C r (f ) : 1 C 0 (f ); no points 2 C 1 (f ); three points ; one outside unit circle, one on unit circle one inside unit circle 3 C 2 (f ); one point inside unit circle The configuration c r (f ) : 1 c 0 (f ); no points 2 c 1 (f ); one point inside unit circle 3 c 2 (f ); no points Jordan cells J r 1 J 0 ; (λ = 1, k = 1) 2 J 1 ; (λ = 3, k = 2)

12 The considers weakly tame and tame maps. Definition A weakly tame map is proper map defined on an ANR whose all levels are ANR s A tame map is a weakly tame map for which the homology of the levels is locally constant except for a discrete collection of values called critical values The invariants are derived from the change in the homology of the levels

13 Configuration C r (f ) for real valued weakly tame map Start with f : X R, Consider X a = f 1 (, a], X a = f 1 [a, ) and F f r (a, b) = (H r (X a ) H r (X)) (H r (X b ) H r (X)), Define F f r : R 2 Z 0 F f r (a, b) = dim F f r (a, b) Z 0

14 For the box B = (a, b] [c, d), witha < b, c < d, define April 1, :35 World Scientific Book - 9in x 6in DANBOOKTEST µ(b) := Fr f (a, c) + Fr f (b, d) Fr f (a, d) Fr f (b, c) 0 32 New Invariants for Real and Angle Valued Maps y-axis d c a b x-axis Fig. 1.2 Dan s New Picture Derive the " jump " function (Dirac measure) B1 B2 δ F,f r B2 : R 2 B1 Z 0 Fig. 1.3 First Pair of Boxes B2 δr F,f B2 B1(x, y) = lim B (x,y) µ(b) B1 Fig. 1.4 Second Pair of Boxes Boxes below diagonal For the system c < d < a < bthe box below diagonal AMN B theory := (a, b] [c, d) 2

15 Define C r (f ) := δ F,f r

16 Configuration C r (f ) for angle valued map Start with f : X S 1 Consider f : X R, the infinite cyclic cover of f JT )r F f r : R 2 Z 0 δr F, f : R 2 Z 0

17 Observe δr F, f (a + 2π, b + 2π) = δr F, f (a, b) Define by and then define δ F,f r : C \ 0 Z 0 δr F,f (z = e (b a)+ia ) := δr F, f (a, b) C r (f ) = δ F,f r.

18 Results about configuration C r (f ), f real valued. Theorem 1 If f : X R then δ F,f r (a, b) = β r = dim H r (M) hence C r (f ) S βr (C) = C βr. 2 The assignment C wt (X, R) f C r (f ) C βr extends to a continuous map from C(X, R) to C βr 3 If X is a (κ-) orientable closed topological manifold then C r (f )(z) = C n r ( f )( z)

19 Results about configuration C r (f ), f angle valued. Theorem 1 If f : X S 1 with f representing ξ H 1 (X; Z) then δ F,f r (a, b) = β N r = dim H N r (M, ξ), hence C r (f ) S βn r (C \ 0) = C βn r 1 C \ 0 C βn r 2 The assignment C ξ wt (X, S1 ) f C r (f ) C βn r extends to a continuous map from C ξ (X, S 1 ) to C βn r 3 If X is a (κ-) orientable closed topological manifold then C r (f )(z) = C n r (f )(z 1 )

20 Theorem For an open and dense set of real or angle valued maps (w.r. to compact open topology) δ F,f r (a, b) = 0 or 1.

21 Linear relations. κ a fixed field. A linear relation R : V 1 V 2 is provided by the collection {V 1, V 2, V V 1 V 2 }; one writes v 1 Rv 2 for (v 1, v 2 ) V Composition of relations : R : V 1 V 2, R : V 2 V 3 R R : V 1 V 3, v 1 (R R )v 3 { v 2 V 2 v 1 R v 2, v 2 R v 3 }. Transpose of relation: R : V 1 V 2 R : V 2 V 1 v 2 R v 1 v 1 Rv 2.

22 α : V 1 V 2 linear map defines R(α) : V 1 V 2 v 1 R(α)v 2 {v 2 = α(v 1 )} V α β 1 W V 2 α, β linear maps define R(α, β) : V 1 V 2 v 1 R(α, β)v 2 {α(v 1 ) = β(v 2 )}. To α : V V linear one assign α reg : V reg = a linear isomorphism. V n ker α n V n ker α n = V reg

23 Similarly to R : V V one can assign T reg (R) : V reg V reg linear isomorphism. R reg := R(T reg (R)) Theorem If R : V 1 V 2 and R : V 2 V 1 then T reg (R R ) and T reg (R R ) are similar (conjugate) linear isomorphisms.

24 the r monodromy of an angle valued map Start with f : X S 1 Consider the infinite cyclic cover f and choose t R and θ S 1 with p(t) = θ X f R t (1) X π f p S 1 θ Consider i t+2π 1 X θ = X t i t Xt,t+2π Xt+2π = X θ 1 Xt = f 1 (t), X t,t = f 1 [t, t ], t t

25 Lt** Consider H r (X θ ) = H r ( X t ) i t (r) which defines the linear relation H r ( X i t+2π (r) [t,t+2π] ) H r ( X t+2π ) = H r (X θ ) R f θ (r) := R(i t(r), i t+2π (r))

26 Theorem 1 (R f θ 1 (r)) reg = (R f θ 2 (r)) reg = (V r (f ), T r (f ) : V r (f ) V r (f )). 2 If f, g : X S 1 are homotopic then (V r (f ), T r (f )) similar to (V r (g), T r (g)). 3 (V r (f ), T r (f )) = (ker(h r ( X) H r ( X) κ[t 1,t] κ[t 1, t]]), τ r ) τ r induced by the deck transformation τ. The pair (V r (f ), T r (f )) is the r monodromy The Jordan cells J r (f ) are the "Jordan components" of T r (f ).

27 D f r (z) the characteristic polynomial of T r (f ) The Alexander function of (X, ξ f ) is A(X, ξ f ) = r=0, dim X D f r (z) ( 1)r If ϕ : Y Y and f : V I ϕ S 1 then A(X, ξ f ) is the Lefschetz zeta function. If K S 3 a knot, N = open disc neighborhood of K, X = S 3 \ K and ξ H 1 (X : Z) the Alexander dual of K in S 3 then z/a(x; ξ)(z) = Alexander polynomial of the knot.

28 Configuration c r (f ) for f real valued. Start with f : X R Consider X a = f 1 (, a], X a = f 1 [a, ) Define and T f r (a, b) = { ker H r (X a ) H r (X b ) if a < b ker H r (X a ) H r (X b ) if a > b T f r : R 2 \ Z 0 T f r (a, b) := dim T f r (a, b).

29 For a < b < c < d and B = (a, b] (c, d] or for c < d < a < b and B = [a, b) [c, d) define µ T,f (B) := T f r (a, c) + T f r (b, d) T f r (a, d) T f r (b, c) 0 d y c d y c a x b a x b

30 Derive the "jump" function δ T,f r : R 2 \ Z 0 δ T,f r (x, y) = lim B (x,y) µ(b) Define the configuration of points in R 2 \ c r (f ) := δ T,f r.

31 The configuration c r (f ). for real valued map for angle valued map

32 Bottle-neck (b-n) topology on C(Y K ) Y locally compact K Y closed C(Y K ), finite configuration of points in Y \ K Bottle neck topology on C(Y K ) defined by U(U 1, U 2, U k, V ; n 1, n 2, n k ) = {δ : Y \ K Z 0, { suppδ Ui V p U k δ(p) = n k C t (X, R) and C t (X, S 1 ) the tame maps with the c-o topology. C(R 2 ) and C(C \ 0 S 1 ) configurations with the b-n topology.

33 Results about configurations c r (f ) Theorem 1 (C-E-H) The assignment C t (X, R) f c r (f ) C(R 2 ) is continuous. 2 If X is an orientable closed topological n manifold then c r (f )(z) = c n 1 r ( f )( z) (z = a + ib). Theorem 1 The assignment C ξ t (X, S1 ) f c r (f ) C(C \ 0 S 1 ) is continuous. 2 If X is an orientable closed topological n manifold then c r (f )(z) = c n 1 r (f )( z).

34 Figure 2: Example of r-invariants for a circle va Alternative description of bar codes and Jordan cells 4 Representation theory and r-invariants The invariants for the circle valued map are derived from the representat are directed graphs. The representation theory of simple quivers such described by Gabriel [8] and is at the heart of the derivation of the in persistence in [4]. For circle valued maps, one needs representation the edges. This theory appears in the work of Nazarova [14], and Donovan a One consider the oriented infinite graph Z and the finite oriented graph G 2m. b a i 1 i v 2i 1 can find b i a refined treatment in Kac [15]. v 2i v 2i+1 v 2i+2 v 2i+3 Let G 2m be a directed graph with 2m vertices, x 1, x 1, x 2m. Its simple cycle. The The graph directed Z edges in G 2m are of two types: forward a i backward b i : x 2i+1 x 2i, 1 i m 1, b m : x 1 x 2m. x 4 b 2 x 2m 2 a 2 x 3 x 2m 1 b 1 a m x 2 a1 x 1 b m x 2m able representations The graph unique G 2m up to isomorphisms. We think of this graph as be tered at the origin o in the plane. A representation ρ on G 2m V x to each vertex x and a linear m edge e = {x, y}. Two representa each vertex x there exists an isom of ρ to the vector space V x of ρ, the linear maps V x V y and V tion assigns at least one vector sp A representation is indecompos sum of two nontrivial representa each representation has a decom

35 An oriented graph representation ρ, associates to a vertex v i a vector space V i and to an oriented arrow a linear map between the corresponding vector spaces. To f tame one associates ρ r (f ): a Z representation for f real valued map a G 2m representation for f angle valued map defined as follows:

36 ρ r (f ) for f real valued. Let s 0 < s 1 < s 2 < s 3 s N 1 < s N be all critical values of f, s 0 = inf f, s N = sup f. Choose regular values t 0, t 1, t N, s i 1 < t i < s i. Define 1 V 2i = H r (X si ), V 2i 1 = H r (X ti ) 2 α i : V 2i 1 V 2i, β i : V 2i+1 V 2i induced in homology by the maps derived from the tameness of f. ρ r (f ) for f angle valued. Let 0 θ 1 < θ 2 θ m 1 < θ m < 2π be all critical angles. Choose regular angles t 0, t 1, t N, θ i < t i+1 < θ i+1, t 2m < 2π. Define 1 V 2i = H r (X si ), V 2i 1 = H r (X ti ) 2 α i : V 2i 1 V 2i, β i : V 2i+1 V 2i induced in homology by the maps derived from the tameness of f.

37 The r bar codes and the r Jordan cells are provided by the indecomposable components of ρ r (f ).

38 Applications 1 (Differential Geometry) Refinement of Hodge theory for Riemannian manifolds 2 (Computational topology) Effective algorithms of the the Betti numbers of homology with local coefficients and of Novikov-Betti numbers- alternative definitions (without coverings). 3 (Dynamics) Establish existence of instantons and closed trajectories for some flows on metric spaces which admit Lyapunov real or angle valued function 4 (Data Analysis) Shape recognition

39 References and S.Haller, Topology of angle valued maps bar codes and Jordan blocks. arxiv: D. Burghelea and T. K. Dey, Persistence for circle valued maps. (arxiv: , ), 2011, Discrete and Computational Geometry. 50 Number , pp D. Cohen-Steiner, H. Edelsbrunner, and J. L. Harer. Stability of persistence diagrams. Discrete Comput. Geom. 37 (2007),