Barcodes for closed one form; Alternative to Novikov theory Dan Burghelea 1 Department of mathematics Ohio State University, Columbus, OH Tel Aviv April 2018
Classical MN-theory Input: Output M n smooth manifold, (X, ω) X smooth vector field which is Morse Smale ω closed one form Lyapunov for X κ a field. A field κ(ω), the Novikov field A chain complex (C, ) of κ(ω) vector spaces generated by the rest points and the visible trajectories of X
One of the main results of MN-theory claims that - The homology H r (C, ) calculates the r th Novikov homology, - The complex depends only on ω and is determined up to a non canonical isomorphism by two of the three collections of numbers: β N r := dim H r (C, r ), ρ N r = rank( r ), c N r := dim C r. The chain complex (C, ) 1 provides the exact number of rest points of Morse index r 2 give a good information about the instantons between rest points, 3 establishes the existence of closed trajectories. (when combined with standard Betti numbers) Computer friendly means computable by computer implementable algorithms. κ(ω) and r are not computer friendly
In view of potential interest outside mathematics we like: 1 to derive the collection of numbers β r, ρ r, c r as computer friendly invariants and without any reference to the field κ(ω), 2 to extend the result to a larger class of situations, i.e. a compact ANR X (e.g. finite simplicial complex) instead of manifold M n, TC1- form ω on X instead of a smooth closed one form ω on M n. Done in [1] and implicitly in [3] in case the TC1-form ω is exact and in case ω represents an integral cohomology class. ("TC1-form" abbreviation for "topological closed one form")
TC1-form, first definition Let X be a topological space Note the change in notation Definition A multi-valued map is a systems {U α, f α : U α R, α A} s.t. 1 U α are open sets with X = U α, 2 f α are continuous maps s.t f α f β : U α U β R is locally constant Two multi-valued maps are equivalent if together remain a multi-valued map. A TC1-form is an equivalence class of multi-valued maps A TC1-form ω determines a cohomology class ξ(ω) H 1 (X; R). Denote by: Z 1 (X) the set of all TC1-forms Z 1 (X; ξ) := {ω Z 1 (X) ξ(ω) = ξ}.
TC1-form, second definition Let ξ H 1 (X; R) = Hom(H 1 (X; Z) R) and Γ = Γ(ξ) := (imgξ) R. (Γ Z k ; k degree of irrationality of ξ) The surjective homorphism ξ : H 1 (X; Z) Γ defines the associated Γ principal cover, π : X X i.e. the free action µ : Γ X X with π the quotient map X X/Γ = X. Consider f : X R, Γ equivariant maps, i.e. f (µ(g, x)) = f (x) + g. Definition A TC1- form of cohomology class ξ is an equivalence class of Γ equivariant maps with f g iff f g is locally constant. One denotes: Z 1 (X; ξ) the set of TC-1 forms in the class ξ Z 1 (X) = ξ H 1 (X;R) Z(X; ξ)
The two definitions are equivalent Examples 1 A smooth closed one form on a manifold 2 A simplicial one co-cycle on a simplicial complex
Tameness Let f : X R and t R. Denote by R f (t) := dim H ( X t f, X <t) f + dim H ( X f t >t, X f ). X f t = f 1 ((, t]), X f <t = f 1 ((, t)) = t <t Xt, X f t = f 1 >t ([t, )), X = f 1 t ((t, )) = f t >t X ). Definition 1 t regular value if R f (t) = 0 2 t critical value if R f (t) 0 3 CR(f ) := {t R R f (t) 0} O(f ) := CR(f )/Γ
Definition The Γ equivariant map f : X R is tame if 1 X is a compact ANR. 2 R f (t) < for any t R, 3 the set O(f ) is a finite set. The TC1-form ω Z 1 (X) is tame if one (and then any) f ω is tame. Denote by Ztame 1 ( ) := {ω Z1 ( ) ω tame} Examples of tame TC1-forms: - closed one form which are locally polynomials on a closed manifold - simplicial one-cocycle on a finite simplicial complex
AMN-theory For any tame TC-1 form ω Z(X; ξ), field κ and nonnegative integer r we define two configurations δ ω r : R Z 0 and γ ω r : R + Z 0 and the numbers β ω r, ρ ω r and c ω r, β ω r := t R δ ω r (t), ρ ω r := t R + γ ω r (t) and c ω r = β ω r + ρ ω r + ρ ω r 1, which satisfy the following three theorems.
Topological properties Theorem Let ω Ztame 1 (X). One has 1 δ ω r (t) 0 or γ ω r (t) 0 implies that for any f ω, t = c c with c, c CR(f ), 2 β ω r = t R δ ω r (t) = β N r (X; ξ) 3 for any f ω and a o o O(f ), c ω r = δ ω r (t) + γ ω r (t) + γ ω r 1 (t) = t R t R + t R + = dim H r ( X a f o, X <a f o ) o O(f ) Recall X f ao = f 1 ((, a 0 ]), X f <ao = f 1 ((, a 0 )).
Note that if X is a smooth closed manifold and ω is a smooth Morse form then the number of critical points (= the rest points of the vector field) of Morse index r is equal to o O(f ) dim H r ( X f a o, X f <a o ). Corollary The Novikov complex of the Morse form ω in Hodge form is given by C r = Cr C r + H r with H r = κ βω r, Cr = C + r 1 = κρω r 0 0 0 r = Id 0 0. 0 0 0
Relation to other work [4]and [3] The points in suppδ ω r suppγr ω with their multiplicity correspond bijectively to UZ verbose r bar codes. In case ξ(ω) integral to the BD- closed r barcodes plus BD-open (r 1) barcodes plus closed-open r barcodes cf [2] or [3]. In the case ξ(ω) is trivial the points in suppγ ω with their r multiplicity correspond bijectively to the finite barcodes in classical ELZ-persistence and in the case ξ(ω) is integral to closed-open r barcodes in angle valued persistence.
Poincaré Duality properties Theorem Suppose X is a closed topological n dimensional manifold and ω Z 1 tame (M). 1 δ ω r (t) = δ ω n r ( t), 2 γ ω r (t) = γ ω n r 1 (t).
The set of TC-1 forms Z 1 (X; ξ) is an R vector space equipped with the distance ω ω := inf { f ω g ω sup f (x) g(x). x X The space of configurations Conf N (Y ) := {δ : Y Z 0 δ(y) = N} is equipped with the obvious collision topology. For K closed subset of Y the space of configurations Conf (Y \ K ) is equipped with the bottleneck topology provided by the pair K Y.
Stability properties Theorem 1 the assignment Ztame 1 (X; ξ) ω δω r Conf β N r (X;ξ)(R) is continuous. 2 the assignment Ztame 1 (X; ξ) ω γω Conf ([0, ) \ 0) is r continuous.
Definition of δ ω r and γ ω r. Let ω Z 1 tame (X) and f ω, f : X R with f (µ(g, x)) = f (x) + g. First one defines the maps δr f : R 2 Z 0 and γr f : R 2 + Z 0. R 2 + := {(x, y) x < y} Denote: X a = f 1 ((, a]), X<a = f 1 ((, a)) = a <a Xa, X a = f 1 ([a, )), X >a = f 1 ((a, )) = a >a X a.
Define 1 I f a(r) := img(h r ( X a ) H r ( X)), I f a(r) H r ( X) I f <a(r) := a <ai f a (r), I f (r) = a I f a(r), 2 I a f (r) := img(h r ( X a ) H r ( X)), I a f (r) H r ( X) I >a f (r) := a >ai a f (r), I + f (r) = a I a f (r), 3 (For a < b T f r (a, b) := ker(h r ( X a ) H r (X b )), T r (a, < b) := b <bt r (a, b ) T r (a, b), 4 For a < a let i : T r (a, b) T r (a, b) the induced map and T r (< a, b) := a <ai(t r (a, b)) T r (a, b).
Next define for a, b R ˆδ f r (a, b) := I a (r) I b (r) I <a (r) I b (r) + I a (r) I >b (r), δf r (a, b) := dim ˆδ f r (a, b) for a, b R, a < b ˆγ f r (a, b) := T r (a, b) T r (< a.b) + T r (a, < b), γf r (a, b) := dim ˆγ f r (a, b). and for t R F f r (t) := { a b t, a, b CR(f ) I a (r) I b (r)
One can show that: Proposition 1 δ f r (a, b) = δ f r (a + g, b + g) and γ f r (a, b) = γ f r (a + g, b + g) for any g Γ, 2 for a regular value suppδ f r (a R) and suppγ f r (a (a, )) is empty, 3 for a critical value suppδ f r (a R) and suppγ f r (a (a, )) is finite, 4 for b regular value suppδ f r (R b) and suppγ f r ((, b) b) is empty, 5 for b critical value suppδ f r (R b) and suppγ f r ((, b) b) is finite.
For ω Ztame 1 (X) define O(ω) := f ω O(f )/, O(f 1 ) o 1 o 2 O(f 2 ), f 2 = f 1 + s, iff θ(s) : O(f 1 ) O(f 2 ), the bijective correspondence induced the translation θ(s)(t) = t + s satisfies θ(s)(o 1 ) = o 2. Observation For a o o O(f ), f ω, δ f r (a o, a o + t) and γ f r (a o, a o + t) are independent on a o and f.
Define δ ω o,r (t) := δr f (a o, a o + t) and then δ ω r (t) = δ ω o,r (t). o O(ω) Define γ ω o,r (t) := γf r (a o, a o + t) and then γ f r (t) = γ f o,r (t). o O(ω)
About the proof H r ( X), I f (r), I f (r) H r ( X) are f.g. κ[γ] module. Denote by TH r ( X) the submodule of torsion elements. For f ω Z 1 tame (X) I (r) = F f r ( ) F f r (t ) F f r (t) F f r ( ) = H r ( X), t < t, is a filtration of f.g. κ[γ] modules. There are finitely many t s where the rank of F r (t) jumps.
First theorem (Topological properties) follows essentially from Proposition 1 I f (r) = I f (r) = TH r ( X) 2 F f r (t)/th r ( X) { a b t, a, b CR(f ) ˆδ f r (a, b) 3 if a is a regular value then H r ( X f a, X f <a) = 0 and if a o O(f ) then dim H r ( X a, X <a ) = δ ω o,r (t) + γ ω o,r (t) + γ ω o,r 1 (t) t R t R + t R +
To prove the second theorem (PD property) use Borel-Moore homology to define the analogues BM ˆδ and BM ˆγ of ˆδ and ˆγ and then BM ˆδ and BM ˆγ. Prove Proposition BM δ f r (a, b) = δ f r (a, b) and BM γ f r (a, b) = γ f r (a, b). Derive the result using similar arguments as in the proof of Theorems 5.6 and 6.2 in [1] (which contain Theorem 7 as a particular case ω of degree of irrationality zero or one.
To prove third theorem (Stability properties) proceed as follows: Suppose f ω 1, g ω 2, ω 1, ω 2 Z tame (X; ξ) and verify Proposition If f g < ɛ then F f r (t) F g r (t + 2ɛ) F f r (t + 4ɛ) The finiteness of the support of δ ω r implies analogues of Proposition 5.7 in [1], Adapt the arguments of Theorem 5.2 in [1]. These steps establish item 1.
Consider the configuration λ ω r : R + Z 0 defined by λ r (t) = δ ω r (t) + γ ω r (r) + γω r 1 (t), and establish the continuity of the assignment Z tame (X; ξ) ω λ ω r Conf ([0, ) \ 0). Use Item 1 to conclude the continuity for the assignment given by the restriction of the configuration δ ω r to R + denoted by δ ω r +. Observe that the configuration λ r can be derived from the restriction of the map f ω to a compact fundamental domain of µ : Γ M M. These steps establish Item 2. This last fact is also established in [4]
Applications 1. MN theory on M S1 2. Data analysis 3. Dynamics
Dan Burghelea, New topological invariants for real- and angle-valurd maps; an altrnative to Morse-Novikov theory Word scientific Publishing Co. Pte. Ltd, 2017 D. Burghelea and T. K. Dey, Persistence for circle valued maps. Discrete Comput. Geom.. 50 2013 pp69-98 ; arxiv:1104.5646 Dan Burghelea, Stefan Haller, Topology of angle valued maps, bar codes and Jordan blocks, J. Appl. and Comput. Topology, vol 1, pp 121-197, 2017; arxiv:1303.4328; Max Plank preprints Michael Usher and Juo Zhang, Persistent homology and Floer-Novikov theory Geom. Topol. Volume 20, Number 6 (2016), 3333-3430.