REFINMENTS OF HOMOLOGY (homological spectral package). Dan Burghelea Department of Mathematics Ohio State University, Columbus, OH
In analogy with linear algebra over C we propose REFINEMENTS of the simplest topological invariants HOMOLOGY of X NOVIKOV HOMOLOY / L 2 HOMOLOGY of (X, ξ H 1 (X, Z)), associated to a continuous REAL or ANGLE VALUED MAP This work is : implicit in joint work with S.Haller (Vienna) influenced by joint work with Tamal Dey (OSU- Columbus) and Du Dong (Shanghai)
HOMOLOGY for a fixed field Homology : Space X : (1.) H r (X), Novikov homology : Pair (X; ξ H 1 (X; Z)) : β r (X) = dim H r (X) (2.) H N r (X, ξ), β N r (X, ξ) = rank H N r (X, ξ) (X, ξ) X X infinite cyclic cover associated to ξ. X compact ANR H r ( X) a f.g. κ[t 1, t] module X compact ANR TH r ( X) a κ[t 1, t] module which is a f.d. κ vector space. X compact ANR H N (X; ξ) := H r ( X)/TH r ( X) a f.g. free κ[t 1, t] module
L 2 homology If κ = C by "completion" C[t 1, t] L (S 1 ) a von Neumann algebra. Hr N (X : ξ) H L 2 r ( X) finite type L (S 1 ) Hilbert module and β N (X; ξ) = dim vn (H L 2 r ( X)).
New invariants - configurations Configurations of points in Y with multiplicities in N Conf n (Y ) := {δ : Y N 0 (supp δ) < } = Y n /Σ n Conf n (Y ) := {δ : Y N 0 y δ(y) = n} = Y n /Σ n Configurations of points in Y indexing subspaces of V Conf vect (Y ) := {ˆδ : Y Vect (supp ˆδ) < } Conf V (Y ) := {ˆδ : Y S(V ) y ˆδ(y) = V } Y = C Conf n (Y ) = C n and identifies with degree n monic polynomials. Y = C \ 0 Conf n (Y ) = C n 1 (C \ 0) and identifies with degree n monic polynomials with nonzero free coefficient.
SPECTRAL PACKAGE in LINEAR ALGEBRA SYSTHEM (V, T : V V ) { V f.d. complex vector space. T : V V linear map = SPECTRAL PACKAGE dim V = n z 1, z 2,, z k 1, z k n 1, n 2,, n k 1, n k V 1, V 2,, V k 1, V k N C; eigenvalues N; multiplicities V ; generalized eigenspaces PROPERTIES : dim V = n i, dim V i = n i, V = V i
geometrization δ T : { z1, z 2,, z k 1, z k n 1, n 2,, n k 1, n k finite configurations of points with multiplicities s.t. δ T (z i ) = dim V i δ T P T (z) = (z z 1 ) n 1(z z 2 ) n 2 (z z k ) n k the characteristic polynomial, n = dim V. ˆδ T : { z1, z 2,, z k 1, z k V 1, V 2,, V k 1, V k finite configurations of points indexinig disjoint subspaces of V s.t. i ˆδ T (z i ) = V δ T Conf n (C), ˆδ T Conf V (C)
basic properties 1 (Stability) L(V, V ) T δ T (z) = P T (z) C n is continuous 2 (Duality) δ T = δ T 3 For an open and dense set of T L(V, V ), δ T (z) = 0 or 1 4 (Computability) P T (z) can be calculated with arbitrary accuracy. One regards δ T P T (z) as a refinement of dim V, One regards ˆδ T as an implementation of the refinement δ T.
TOPOLOGY ; HOMOLOGICAL SPECTRAL PACKAGE (real valued map) SYSTHEM (X, f : X R) X a compact ANR, f continuous, κ a field, H r (X) := H r (X; κ), r N 0. HOMOLOGICAL SPECTRAL PACKAGE dim H r (X) = β r (X) ; Betti number z 1, z 2,, z k 1, z k C; barcodes n 1, n 2,, n k 1, n k N; multiplicities V 1, V 2,, V k 1, V k ; quotients of subspaces of V. β r = i n i, dim V i = n i, V V i
specifications V i = L i /L i are quotients of subspaces of H r (X), L i L i H r (X), essentially disjoint If H r (X) has an inner product then V i is canonically realizable as a subspace of H r (X) with V i V j, i j.
geometrization δ f r : { z1, z 2,, z k 1, z k n 1, n 2,, n k 1, n k finite configurations of points with multiplicities s.t. δr(z f i ) = β r (X) δ f r P f r (z) = (z z 1 ) n 1(z z 2 ) n 2 (z z k ) n k the homological characteristic polynomial. ˆδ f r : { z1, z 2,, z k 1, z k V 1, V 2,, V k 1, V k δ f r Conf βr (X)(C), ˆδ f r Conf Hr (X)(C) finite configurations of points indexing subspaces of H r (X) s.t. ˆδ r f (z i ) = H r (X)
TOPOLOGY ; HOMOLOGICAL SPECTRAL PACKAGE (angle valued map) SYSTHEM (X, f : X S 1 ) (X, f : X S 1 ) ξ f H 1 (X; Z) X a compact ANR, f continuous, κ a field, Hr N (X; ξ f ) := H r ( X)/TH r ( X) r N 0. f : X R denotes the infinite cyclic cover of f
HOMOLOGICAL SPECTRAL PACKAGE rank H N r (X; ξ f ) = βr N (X; ξ f ) ; Novikov Betti number z 1, z 2,, z k 1, z k C \ 0; exponentiated barcodes n 1, n 2,, n k 1, n k N; multiplicities V 1, V 2,, V k 1, V k ; free κ[t 1, t] modules. V i = L i /L i quotients of free κ[t 1, t] submodules of H N r (X; ξ f ), L i L i H r (X), essentially disjoint β N r = i n i, rank V i = n i, H N r (X; ξ f ) V i
If κ = C and a C[t 1, t] inner product by completion Hr N (X; ξ) is replaced by the Hilbert module H L 2 r ( X) and the configuration of free C[t 1, t] by configurations of mutually orthogonal closed Hilbert L (S 1 ) submodules.
geometrization δ f r : { finite configurations of z1, z 2,, z k 1, z k points with multiplicities in n 1, n 2,, n k 1, n k C \ 0 s.t. δr(z f i ) = βr N (X; ξ) δ f r P f r (z) = (z z 1 ) n 1(z z 2 ) n 2 (z z k ) n k the homological characteristic polynomial of degree β N r (X; ξ). ˆδ f r : { z1, z 2,, z k 1, z k V 1, V 2,, V k 1, V k finite configurations of points indexing submodules of δr f Conf β N r (X;ξ f ) (C \ 0), ˆδ r f Conf H N r (X;ξ f )(C \ 0) H N r (X; ξ)s.t. ˆδ f r (z i ) = H N r (X; ξ)
Definitions of δ f r and ˆδ f r For f : X R a proper map a, b R, κ a field Denote: X a := f 1 ((, a]) sub-level X b := f 1 ([b, )) over-level Define: I a (r) := img(h r (X a ) H r (X)) I b (r) := img(h r (X b ) H r (X)) F r (a, b) := I a (r) I b (r)
Observe a a, b b imply F r (a, b ) F r (a, b). Prove F r (a, b) finite dimensional. Define F r (a, b, ɛ) := F r (a, b)/f r (a ɛ, b) + F r (a, b + ɛ) for ɛ > 0 Observe that ɛ > ɛ induces a surjective map F r (a, b; ɛ ) F r (a, b; ɛ ). Definition ˆd r f (a, b) := lim F r (a, b, ɛ) ɛ 0 d f r (a, b) := dim ˆd f r (a, b)
The case of f : X R, X compact. Define δ f r (z) = d f r (a, b), z = a + ib ˆδ f r (z) = ˆd f r (a, b), z = a + ib.
The case of f : X S 1, X compact. Consider f : X R an infinite cyclic cover of f. Observe that 1 t : ˆd f r (a, b) ˆd f r (a + 2π, b + 2π) is an isomorphism and 2 dr f (a, b) = d f r (a + 2π, b + 2π) Define δ f r (z) = d f r (a, b), z = e (b a)+ia ˆδ f r (z) = k ˆd f r (a + 2πk, b + 2πk), z = e (b a)+ia.
Main results The values t R is CRITICAL if the homology of the levels of f changes at t. Cr(f ) = the set of critical values of f. Theorem 1. suppδ f r β r (X), z suppδ f r δf r (z) = β f r 2. For an open sense set of maps f, δ f r (z) = 0 or δ f r (z) = 1.
Proposition 1 If z = a + ib suppδr f then both a, b Cr(f ). 2 z = (a + ib) suppδr f above or on diagonal implies [a.b] is a closed r bar code. 3 z = (a + ib) suppδr f below diagonal implies (b, a) is an open (r 1) bar code.
Theorem The assignment C(X, R) f δ f r C βr is continuous. Theorem If M n is a closed κ orientable topological n dimensional manifold then δ f r (z) = δ f n r ( iz) The same remains true for the configuration ˆδ f r
Suppose f is an angle valued map Theorem 1. suppδr f βr N (X), z suppδr f δf r (z) = βr N (X; ξ) 2. For an open sense set of maps f, δr f (z) = 0 or δr f (z) = 1. Proposition 1 If z = e ia+(b a) suppδr f then both a, b Cr( f ) = {t e it Cr(f ). 2 z outside or on the unit circle implies [a, b] is a closed r bar code. 3 z inside the unit circle implies (b, a) is an open (r 1) bar code.
Theorem The assignment C(X, S 1 ) f δ f r C βn r 1 C \ 0 is continuous Theorem If M n is a closed κ orientable topological n dimensional manifold then δ f r (z) = δ f n r ( iz) The same remains true for the configuration ˆδ f r
The proofs are done in two steps. Step 1: Establish the results for X a finite simplicial complex and f a simplicial map Step 2. Use results about the topology of Hilbert cube manifolds and show that the statements are true for f iff are true for f Q = f p X, p X : X Q X, Q the Hilbert cube I.
Step 1 Introduce the concepts: 1 tame map, 2 homological critical value and for a tame map, 3 the number ɛ(f ) = inf c c c, c different homological critical values Verify that for f : X R proper tame map 1 For f proper tame map d f r (a, b) 0 a, b critical values, 2 For a box B = (a, b] [c, d) and f proper tame map F f r (B) = (a b ) B suppδ ˆd f r (a, b ) Let f a tame map, X compact and ɛ < ɛ(f ). Then g : X R proper map with g f < ɛ the support of δ g is in an 2ɛ neighborhood of the support of δ f and of equal cardinality as suppδ f.
Applications In view of effective computability of the configuration δ f r the result can be used to : Applications in topology: Calculation of Betti numbers, Novikov Betti numbers Refinements of Morse inequalities. Applications in data analysis: Homological recognition of shapes which can be manifolds. Homological differentiations of shapes. Applications in geometric analysis: Refinement of Hodge de Rham theorem on compact Riemannian maniflods, Canonical base in the space of Harmonic forms Applications in dynamics: for dynamics of flows which admit an action
References 1.D. Burghelea and T. K. Dey, Persistence for circle valued maps. Discrete and Computational Geometry, Vol 50 2013, pp 69-98 2.Dan Burghelea, Stefan Haller, Topology of angle valued maps, bar codes and Jordan blocks arxiv:1303.4328 3.Dan Burghelea, A refinement of Betti numbers in the presence of a continuous function ( I ), arxiv:1501.01012 4.Dan Burghelea, A refinement of Betti numbers in the presence of a continuous function ( II ), to be posted soon
Persistent homology PH r (X; f ) (Persistent homology) are If f real valued κ vector spaces If f angle valued κ[t 1, t] free modules / L (S 1 ) Hilbert modules. Refinments δ P,f r and ˆδ P,f r (Configurations) For f real valued configurations on C \ For f angle valued configurations on C \ {0 S 1 } Based on Fredholm cross ratio of α : A B, β : B C, γ : C D Fredholm maps.
Monodromy of f : X S 1 Isomorphism classes of pairs (T r, V r ) V r = TH r ( X), T r = t ) Invariants JORDAN CELLS J r (f ) := {(λ r i, nr i ), i = 1 k r λ r i κ, n r i N} Based on Linear relations.