Graphs Representations and the Topology of Real and Angle valued maps (an alternative approach to Morse-Novikiov theory) Dan Burghelea Department of mathematics Ohio State University, Columbus, OH Tianjin, May 8
Based on joint work with TAMAL K. DEY Department of Computer Sciences Ohio State University, Columbus, OH STEFAN HALLER Univ. of Vienna, Austria Preceeded by work on Persistent Homology of H.Edelsbruner, D.Letscher, A. Zamorodian, G.Carlsson, V. de Silva
Consider TAME maps f : X R and f : X S 1 Our theory recovers the topology of the UNDERLYING SPACE X from computable invariants associated with a TAME map f In our theory the key elements are: critical values, bar codes between critical values Jordan cells and the canonical long exact sequence. Morse theory recovers the topology of the UNDERLYING manifold M from elemets associated with a Morse map f : M R, f : M S 1 In Morse theory the key elements are: critical points, instantons between critical points, closed trajectories and the Morse complex).
CONTENTS Background Topology Tame (real and angle valued) maps Bar codes and Jordan cells Graph representations Definitions and Results Invariants for a tame map The results The mathematics behind About the proof Others (computation, what the bar codes tell)
Topology κ a field, κ its algebraic closure. κ[t, t 1 ] resp. κ[[t, t 1 ] the ring resp. field of Laurent polynomials resp. formal power series. κ[t, t 1 ] κ[[t, t 1 ]. X be a compact ANR. H r (X) the singular homology with coefficients in κ Betti numbers β r (X) = dim H r (X).
(X, ξ), ξ H 1 (X, Z). X X the infinite cyclic cover associated with ξ. T : X X the deck transformation. H r ( X) is a κ[t, t 1 ] module with the multiplication by t induced by T. NH r (X, ξ) = H r ( X) κ[t,t 1 ] κ[[t, t 1 ] the Novikov homology is a vector space over the field κ[[t, t 1 ]. Novikov- Betti numbers βn r (X; ξ) = dim κ[[t,t 1 ] NH r (X; ξ).
Consider V (ξ) := ker{h r ( X) NH r (X; ξ)} induced by tensoring H r ( X) by κ[t, t 1 ] κ[[t, t 1 ]. V (ξ) is a finite dimensional vector space over κ T (ξ) : V (ξ) V (ξ) is induced by the multiplication by t, κ linear isomorphism. The monodromy associated with ξ is the pair (V (ξ), T (ξ)).
Tame maps Definition A continuous map f : X R resp. f : X S 1, X a compact ANR, is tame if: 1 Any fiber X θ = f 1 (θ) is the deformation retract of an open neighborhood. 2 Away from a finite set of numbers/angles Σ = {θ 1,, θ m } R, resp. S 1 the restriction of f to X \ f 1 (Σ) is a fibration. To f : X R resp. f : X S 1 a tame map one associates via graph representations bar codes resp. bar codes and Jordan cells.
Bar codes and Jordan cells Bar codes are finite intervals I of real numbers of four types: 1 Type 1, closed, [a, b] with a b, 2 Type 2, open, (a, b) with a < b, 3 Type 3, left open right closed(a, b] with a < b, 4 Type 4, left closed right open [a, b) with a < b. Jordan cells are pairs J = (λ κ, n Z 0 ) = (κ n, T (λ, n) It should be interpreted as a a matrix λ 1 0 0. 0 λ 1... T (λ, n) =. 0 0..... 0....... λ 1 0 0 0 λ
Graphs and graph representations The oriented graph Z : b i 1 x 2i 1 The oriented graph G 2m : a i b x 2i i x 2i+1 a i+1 b x 2i+2 i+1 x 3 x 4 b 2 a 2 b 1 x 2 a1 x 1 b m x 2m 2 a m x 2m x 2m 1
A Z or a G 2m representation ρ is given by x i V i ρ = a i α i : V 2i 1 V 2i b i β i : V 2i+1 V 2i A finitely supported Z-representation 1, or a G 2m -representation ρ can be uniquely decomposed as a sum of indecomposable representations. For the graph Z the indecomposable representations with finite support are indexed by the four types of bar codes intervals with ends i, j Z : closed, open, left-closed right-open and left-open right-closed. 1 i.e. all but finitely many vector spaces V x have dimension zero
The indecomposable Z representations with finite support : 1 ρ([i, j]), i j has V r = κ for r = {2i, 2i + 1, 2j} and V r = 0 if r [2i, 2j], 2 ρ((i, j)), i < j has V r = κ for r = {2i + 1, 2i + 2, 2j 1} and V r = 0 if r [2i + 1, 2j 1], 3 ρ((i, j]), i < j has V r = κ for r = {2i + 1, 2i + 2, 2j} and V r = 0 if r [2i + 1, 2j], 4 ρ([i, j)), i < j has V r = κ for r = {2i, 2i + 1, 2j 1} and V r = 0 if r [2i, 2j 1], with all α i and β i the identity provided that the source and the target are both non zero.
For G 2m the indecomposable representations are indexed by similar intervals (bar codes) with ends i, j + mk, 1 i, j m, k Z 0, i j with 1 i m and by Jordan cells. bar codes: For any triple of integers {i, j, k}, 1 i, j m, k 0, we have the representations denoted by 1 ρ I ([i, j]; k) ρ I ([i, j + mk]), 2 ρ I ((i, j]; k) ρ I ((i, j + mk]), 3 ρ I ([i, j); k) ρ I ([i, j + mk)), 4 ρ I ((i, j); k) ρ I ((i, j + mk)), Jordan cell: ρ II (λ, k) = {V r = λ k, α 1 = T (λ, k), α i = Id i 1, β i = Id}.
i i and β i : V x2i+1 V x2i be the linear maps defined on bases and extended by linearity as follows: assign to e l 2i±1 the vector eh 2i Vxi if eh 2i is an adjacent intersection point to the points el 2i±1 on the spiral. If eh 2i does not exist, e l 2i±1 are assigned zero. If el 2i±1 do not go to zero, h has to be l, l 1, or l + 1. The construction above provides a representation on G 2m which is indecomposable. One can also think these representations as the bar codes [s i, s j + 2kπ], (s i, s j + 2kπ], [s i, s j + 2kπ), and (s i, s j + 2kπ). ox 2j ox 2i V x3 s j o s i e 2 2i e 3 2i e 1 2i 1 ox 2i 1 e 2 2i 1 e 2 2i V x4 e 3 2i V x2 V x1 e 1 2i 1 e2 2i 1 V x2m V x2m 1 Figure 3: The spiral for [s i, s j + 4π). Figure: The spiral for [i, j + 2m). For any Jordan cell (λ, k) we associate a representation ρ J (λ, k) defined as follows. Assign the vector space with base e 1, e 2,, e k to each x i and take all linear maps α i but one (say α 1) and β i the identity. The map α 1 is given by the Jordan cell matrix (λ, k). Again this representation is indecomposable. It follows from the work of [10, 14, 15] that bar codes and Jordan cells as constructed above are all and only indecomposable representations of the quiver G 2m. Observation 4.1 If a representation ρ does not contain any indecomposable representations of type ρ I in its decomposition, all linear maps α is and β is are isomorphisms. For such a representation, starting with an index i, consider the linear isomorphism T i = Author, βi 1 αanother i βi 1 1 αi 1 Short β 1 2 Paper α 2 βtitle 1 1 α 1 βm 1 αm β 1 m 1 αm 1 β 1 i+1 αi+1.
Notations: For a Z or G 2m representation ρ denote by B c (ρ) closed bar codes B o (ρ) open bar codes B co (ρ) left closed right open bar codes B oc (ρ) left open right closed bar codes and B(ρ) = B c B o B co B oc all bar codes For a G 2m representation ρ denote by J (ρ) the collection of all Jordan cells.
Invariants for a tame map Let f be a tame map. Consider (real valued) the critical values < θ 1 < θ 2 < < θ m < (angle valued) the critical angles 0 < θ 1 < < θ m 2π. Choose t i, i = 1, 2,..., m, with θ 1 < t 1 < θ 2 < < t m 1 < θ m < t m, and for f angle valued 2π < t m < θ 1 + 2π. The tameness of f - when real valued induces the diagram : b i 1 X ti 1 a i b X θi i X ti a i+1 b X θi+1 i+1
- when angle valued induces the diagram: b 1 X t1 X θ1 a 1 X tm X θ2 a 2 X tm 2 b m X θm a m X tm 1 a m 1 b m 1 X θm 1
For r dim X let ρ r (f ) be the Z- resp. G 2m -representation defined by: V 2i = H r (X θi ), V 2i+1 = H r (X ti ) α i : V 2i 1 V 2i, β i : V 2i+1 V 2i induced by the continuous maps a i and b i. Consider the decomposition of ρ r in indecomposable components. Convert the intervals {i, j} into {θ i, θ j } (for f real valued) resp. the intervals {i, j + km}, 1 i, j m, into {θ i, θ j + 2πk} (for f angle valued). Denote the set of these intervals whose ends are critical values/angles by B r (f ) and the sets of Jordan cells by J r (f ).
The r invariants. Definition 1. For f : X R the sets B r (f ), r = 1, 2, dim X are the r-invariants of the map f. 2. For f : X S 1 the sets B r (f ), and J r (f ), r = 1, 2, dim X are the r-invariants of the map f. Call the pair (V r (f ), T r (f )) = (λ,k) J r (f ) (κk, T (λ, k)) the r-monodromy of the angle valued f. Note One has two types of monodromy : associated to an angle valued map derived from Jordan cells, associated to ξ H 1 (X; Z) via algebraic topology.
The main results Theorem If f : X R is a tame map and X t = f 1 (t) then Definition 1. β r (X t ) = { I B r (f ) I t } 2. dim im ( H r (X t ) H r (X) ) = { I B c r (f ) I t } 3. β r (X) = B c r (f ) + B o r 1 (f ) For an interval I R and an angle θ (0, 2π] denote by n θ (I) = {k Z θ + 2πk I}. For J J r write J = (λ(j), k(j)
Theorem If f : X S 1 is a tame map, ξ f H 1 (X; Z) represents f and X θ = f 1 (θ) then: 1. β r (X θ ) = I B r (f ) n θ (I) + J J r (f ) k(j) 2. dim im ( H r (X θ ) H r (X) ) = { I Br c (f ) } θ I + Br c (f ) + Br 1 o (f )+ 3. β r (X) = { (λ, k) J r (f ) λ = 1 } + { (λ, k) J r 1 (f ) λ = 1 } 4. βn r (X; ξ f ) = Br c (f ) + Br 1 o (f ).
Theorem 1. V r (ξ f ) := ker(h r ( X) Hr N (X; ξ f )) is a finite dimensional κ-vector space and (V r (ξ f ) κ, T r (ξ f )) κ) = (V r (f ), T r (f )) 2. H r ( X) = κ[t 1, t] N V r (ξ f ) as κ[t 1, t]-modules with N = βn r (f ) = Br c (f ) + Br 1 o (f ).
Remark: For f real valued the numbers Br c + Br 1 o homotopy invariants. For f circle valued te numbers B c r + B o r 1 collections J r (f ) are homotopy invariants. Define Cr r (f ) as a configuration of points in: C (for f : X R) consisting of r closed bar code [s, s ] z = s + s i C (r 1) open bar code (s, s ) z = s + is C (= βr (X)) are (= βnr (X) and the C \ 0 (for f : X S 1 ) consisting of r closed bar code [s, s ] z = e (s s)+2πsi C \ 0 (r 1) open bar code (s, s ) z = e (s s )+2πs i C \ 0
blue r closed barcodes, red (r 1) open bar codes
Cr(f ) can be regarded as points in the symmetric product S βr (X) (R 2 ) resp. S βnr (X;ξ f ) (C \ 0). N {}}{ S N (M) = ( M M M)/Σ N, Σ N - the N symmetric group. C 0 tame(m; R) resp.c 0 tame(m; S 1 ) the space of tame maps dense in the space C 0 (M; R) resp.c 0 (M; S 1 ) of all continuous maps. Theorem The assignments f Cr r (f ) is a continuous map on C 0 tame(m; R) resp. C 0 tame(m; S 1 ) hence has a continuous extension to the entire C 0 (M; R) resp. C 0 tame(m; S 1 ).
As a consequence The closed and open bar codes as well as the Jordan cells can be defined for any continuous maps. The monic polynomials P r (f )(z) whose roots are the points of Cr(f )are well defined for any continuous map and the assignment f P r (f )(z) is continuous. The collection J (f ) r remains constant on a connected component of C 0 (M; S 1 ).
About proof Start with f : X S 1 X θ1 X t1 b 1 a 1 X tm X θ2 a 2 X tm 2 Consider R = 1 i m X ti, X = 1 i m X si. a m 1 X θm 1 b m 1 b m X θm a m X tm 1
Derive the long exact sequence (the canonical sequence ) H r (R) M(ρr ) H r (X ) H r (X) H r 1 (R) M(ρ r 1) H r 1 (X ) with α1 r β r 1 0 0 0 α2 r β r... 2. M(ρ r ) =.......... 0 0 0 αm 1 r βm 1 r βm r 0 0 αm r. α r i : H r (R i ) H r (X i ) and β r i : H r (R i+1 ) H r (X i ) induced by the maps a i and b i.
Examples Spaces : Y map ϕ : Y 0 Y 1 X := Y ϕ. 3 0 0 Maps: p : Y [0, 2π] R, f : X S 1, ϕ = 1 2 1. 0 0 2 φ 1 circle 1 2 3 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 r-invariants
r-invariants for p dimension bar codes 0 [0, 2π] [0, 2π] [0, 2π] 1 [0, θ 1 ] [θ 2, θ 3 ] (θ 3, θ 5 ) (θ 6, 2π] Table 1: r-invariants of f dimension bar codes Jordan cells 0 (1, 1) (θ 6, θ 1 + 2π] (2, 2) 1 [θ 2, θ 3 ] (θ 4, θ 5 ) Table 2.
Calculations: Input Record X as a square N N matrix, N the total number of simplicee of X with entries the numbers I(τ, σ), and record the values of f on vertices as an additional row. Output 1: ρ r = {α r, β r } Output 2 A table of boxes with two columns ( bar codes, Jordan cells) and dim X rows : r = 1, 2, dim X.
The meaning of bar codes 0 x H r (X t ) : dead (right) at t > t if image in H r (X [t,t ]) vanishes, dead (left) at t < t if image in H r (X [t,t]) vanishes, observable at t if image in H r (X [t,t ]) is not trivial and lies in the image of H r (X t ). N{s i, s j } = { maximal number of linearly independent elements in H r (X t ) which are dead/ observable at s i /s j but not before/after s i /s j.}
REFERENCES 1. D. Burghelea and T. K. Dey, Persistence for circle valued maps. (arxiv:1104.5646), 2011. 2. D. Burghelea and S. Haller, Graph representations and the topology of real and angle valued maps, (arxiv:1204.5646), 2012. 3. D. Burghelea, On the bar codes of continuous real and angle valued maps. (in preparation) 4. G. Carlsson, V. de Silva and D. Morozov, Zigzag persistent homology and real-valued functions, Proc. of the 25th Annual Symposium on Computational Geometry 2009, 247 256.