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1 Computational Homology in Topological Dynamics ACAT School, Bologna, Italy MAY 26, 2012 Marian Mrozek Jagiellonian University, Krak ow Outline 2 Dynamical systems Rigorous numerics of dynamical systems Homological invariants of

  1. Cubical Homology 17 • The kernel of ∂ X q is called the group of q -cycles of X and denoted by Z q ( X ) . • The image of ∂ X q +1 is called the group of q -boundaries of X and denoted by B q ( X ) . • One can verify that B q ( X ) ⊂ Z q ( X ) , which allows us to define the q th homology group of X by H q ( X ) := Z q ( X ) /B q ( X ) • By homology of X we mean the collection of all homology groups H ( X ) := { H q ( X ) } .

  2. Standard approach 18 Immediate algebraization:

  3. Standard approach 19 Immediate algebraization: • generate the faces

  4. Standard approach 20 Immediate algebraization: • generate the faces • construct the boundary maps   − 1 0 0 ... 1   0 0 ... 1 1     D k = − 1 0 0 ... 1     0 0 0 ... 1 . . . . .

  5. Standard approach 21 Immediate algebraization: • generate the faces • construct the boundary maps • find Smith diagonalization and read Betti numbers   − 1 0 0 ... 1   0 0 ... 1 1     D k = − 1 0 0 ... 1     0 0 0 ... 1 . . . . . B k = Q − 1 D k R   2 0 0 0 ...   ... 0 1 0 0     0 0 0 ... B k = 1     0 0 0 0 ... . . . . .

  6. Standard approach 21 Immediate algebraization: • generate the faces • construct the boundary maps • find Smith diagonalization and read Betti numbers Advantages:   − 1 0 0 ... 1 • standard linear algebra   0 0 ... 1 1   • may be easily adapted to ho-   D k = − 1 0 0 ... 1     mology generators 0 0 0 ... 1 . . . . . B k = Q − 1 D k R   2 0 0 0 ...   ... 0 1 0 0     0 0 0 ... B k = 1     0 0 0 0 ... . . . . .

  7. Standard approach 21 Immediate algebraization: • generate the faces • construct the boundary maps • find Smith diagonalization and read Betti numbers Advantages:   − 1 0 0 ... 1 • standard linear algebra   0 0 ... 1 1   • may be easily adapted to ho-   D k = − 1 0 0 ... 1     mology generators 0 0 0 ... 1 . . . . . Problems: • constructing faces immediately B k = Q − 1 D k R   may increase data size 2 0 0 0 ... • complexity: Cn 3   ... 0 1 0 0   • sparseness of matrices may not   0 0 0 ... B k = 1     help (fill-in process) 0 0 0 0 ... . . . . . • C large for sparse matrices (dynamic storage allocation)

  8. Geometric reduction algorithms 22 Geometric Reductions • Reduce the set so that – the representation used is preserved – the homology is not changed • build chain complex • compute homology

  9. Shaving 23 • If X = � X is cubical and Q ∈ X is an elementary cube such that Q ∩ X is acyclic and X ′ = � ( X \ { Q } ) then H ( X ) ∼ = H ( X ′ ) • full cubes representation is used! • acyclicity tests via lookup ta- bles: – 2 3 d − 1 entries – extremely fast in dimension 2 and 3

  10. Shaving 23 • If X = � X is cubical and Q ∈ X is an elementary cube such that Q ∩ X is acyclic and X ′ = � ( X \ { Q } ) then H ( X ) ∼ = H ( X ′ ) • full cubes representation is used! • acyclicity tests via lookup ta- bles: – 2 3 d − 1 entries – extremely fast in dimension 2 and 3 – not enough memory for di- mension above 3 • partial acyclicity tests in higher dimensions

  11. Acyclic subspace 24 • If X is cubical and A ⊂ X is acyclic then � for n ≥ 1 H n ( X, A ) , H n ( X ) ∼ = Z ⊕ H n ( X, A ) , for n = 0 • breadth-first search construc- tion

  12. Acyclic subspace 24 • If X is cubical and A ⊂ X is acyclic then � for n ≥ 1 H n ( X, A ) , H n ( X ) ∼ = Z ⊕ H n ( X, A ) , for n = 0 • breadth-first search construc- tion

  13. Free face reductions 25 • free face - a generator with foreach σ do exactly one generator in if cbd( σ ) = { τ } then coboundary remove( σ ); • a combinatorial counterpart of remove( τ ); deformation retraction endif ; • on algebraic level: endfor ;   1 1 1 1 1 ... 0 0 0   1 1 1 1 ... 0 0 0 0     1 ... 0 0 0 0 0 0 0    1 1 ...  0 0 0 0 0 0   ...  1 1  0 0 0 0 0 0   ...  0 0 1 0 0 0 1 0    ...  0 0 0 1 0 0 0 1    ... 0 0 0 1 0 0 0 0     ... 0 0 0 0 1 0 0 0     ... 0 0 0 0 1 0 0 0     . . . . . . . . ...     . . . . . . . . ... . . . . . . . . ...

  14. Dual reductions? 26   0 0 0 ... 1 1 1 1 1   0 0 0 0 ... 1 1 1 1     0 0 0 0 0 0 0 ... 1     ... 0 0 0 0 0 0 1 1    0 0 0 0 0 0 0 ...  1   0 0 0 0 0 0 ...  1 1    0 0 0 0 0 0 ... 1 1     0 0 0 0 0 0 0 ... 1     0 0 0 0 0 0 0 ... 1     0 0 0 0 0 0 0 ... 1     . . . . . . . . ...     . . . . . . . . ... . . . . . . . . ...

  15. Dual reductions? 26   0 0 0 ... 1 1 1 1 1   0 0 0 0 ... 1 1 1 1     0 0 0 0 0 0 0 ... 1     ... 0 0 0 0 0 0 1 1    0 0 0 0 0 0 0 ...  1   0 0 0 0 0 0 ...  1 1    0 0 0 0 0 0 ... 1 1     0 0 0 0 0 0 0 ... 1     0 0 0 0 0 0 0 ... 1     0 0 0 0 0 0 0 ... 1     . . . . . . . . ...     . . . . . . . . ... . . . . . . . . ... • free coface - a generator with exactly one generator in boundary

  16. Dual reductions? 26   0 0 0 ... 1 1 1 1 1   0 0 0 0 ... 1 1 1 1     0 0 0 0 0 0 0 ... 1     ... 0 0 0 0 0 0 1 1    0 0 0 0 0 0 0 ...  1   0 0 0 0 0 0 ...  1 1    0 0 0 0 0 0 ... 1 1     0 0 0 0 0 0 0 ... 1     0 0 0 0 0 0 0 ... 1     0 0 0 0 0 0 0 ... 1     . . . . . . . . ...     . . . . . . . . ... . . . . . . . . ... • free coface - a generator with exactly one generator in boundary • one space homology theory with compact supports for locally compact sets (Steenrod 1940, Massey 1978)

  17. Dual reductions? 26   0 0 0 ... 1 1 1 1 1   0 0 0 0 ... 1 1 1 1     0 0 0 0 0 0 0 ... 1     ... 0 0 0 0 0 0 1 1    0 0 0 0 0 0 0 ...  1   0 0 0 0 0 0 ...  1 1    0 0 0 0 0 0 ... 1 1     0 0 0 0 0 0 0 ... 1     0 0 0 0 0 0 0 ... 1     0 0 0 0 0 0 0 ... 1     . . . . . . . . ...     . . . . . . . . ... . . . . . . . . ... • free coface - a generator with exactly one generator in boundary • one space homology theory with compact supports for locally compact sets (Steenrod 1940, Massey 1978) • combinatorial version (MM, B. Batko, 2006)

  18. Coreduction algorithm 27 Q := empty queue; enqueue( Q , s ); while Q � = ∅ do s :=dequeue( Q ); if bd S s = { t } then remove( s ); remove( t ); enqueue( Q, cbd K t ); else if bd S s = ∅ then enqueue( Q, cbd K s ); endif ; endwhile ;

  19. Coreduction algorithm 27 Q := empty queue; enqueue( Q , s ); while Q � = ∅ do s :=dequeue( Q ); if bd S s = { t } then remove( s ); remove( t ); enqueue( Q, cbd K t ); else if bd S s = ∅ then enqueue( Q, cbd K s ); endif ; endwhile ;

  20. Coreduction algorithm 28

  21. Coreductions for S-complexes 29 • S -complex - a free chain complex with a fixed basis S which allows computation of incidence coefficients κ ( s, t ) directly from the coding of the basis

  22. Coreductions for S-complexes 29 • S -complex - a free chain complex with a fixed basis S which allows computation of incidence coefficients κ ( s, t ) directly from the coding of the basis • Examples: cubical complexes, simplicial complexes

  23. Coreductions for S-complexes 29 • S -complex - a free chain complex with a fixed basis S which allows computation of incidence coefficients κ ( s, t ) directly from the coding of the basis • Examples: cubical complexes, simplicial complexes • Rectangular CW-complexes (P. D� lotko, T. Kaczynski, MM, T. Wanner, 2010)

  24. Augmentible S -complexes 30 Definition. An S -complex is augmentible iff there exists ǫ : S 0 → R (augmentation) such that • ǫ ( t ) � = 0 for t ∈ S 0 • � t κ ( s, t ) ǫ ( t ) = 0 for s ∈ S 1 Coreductions may be applied to any augmentible S -complexes.

  25. Coreduction algorithm 31 Unlike torus, coreductions of Bing’s House result in a non-augmentible S - complex.

  26. CAPD::RedHom 32 • http://redhom.ii.uj.edu.pl

  27. CAPD::RedHom 32 • http://redhom.ii.uj.edu.pl • A subproject of CAPD (http://capd.ii.uj.edu.pl)

  28. CAPD::RedHom 32 • http://redhom.ii.uj.edu.pl • A subproject of CAPD (http://capd.ii.uj.edu.pl) • A sister project of CHomP (http://chomp.rutgers.edu)

  29. CAPD::RedHom 32 • http://redhom.ii.uj.edu.pl • A subproject of CAPD (http://capd.ii.uj.edu.pl) • A sister project of CHomP (http://chomp.rutgers.edu) Generic homology software based on geometric reductions

  30. CAPD::RedHom 32 • http://redhom.ii.uj.edu.pl • A subproject of CAPD (http://capd.ii.uj.edu.pl) • A sister project of CHomP (http://chomp.rutgers.edu) Generic homology software based on geometric reductions • AS, CR, DMT algorithms

  31. CAPD::RedHom 32 • http://redhom.ii.uj.edu.pl • A subproject of CAPD (http://capd.ii.uj.edu.pl) • A sister project of CHomP (http://chomp.rutgers.edu) Generic homology software based on geometric reductions • AS, CR, DMT algorithms • Betti and torsion numbers, homology generators, homology maps, per- sistence intervals

  32. CAPD::RedHom 32 • http://redhom.ii.uj.edu.pl • A subproject of CAPD (http://capd.ii.uj.edu.pl) • A sister project of CHomP (http://chomp.rutgers.edu) Generic homology software based on geometric reductions • AS, CR, DMT algorithms • Betti and torsion numbers, homology generators, homology maps, per- sistence intervals • Z and Z p coefficients

  33. CAPD::RedHom 32 • http://redhom.ii.uj.edu.pl • A subproject of CAPD (http://capd.ii.uj.edu.pl) • A sister project of CHomP (http://chomp.rutgers.edu) Generic homology software based on geometric reductions • AS, CR, DMT algorithms • Betti and torsion numbers, homology generators, homology maps, per- sistence intervals • Z and Z p coefficients • generic but efficient: for cubical sets, simplicial sets, cubical CW com- plexes, ...

  34. CAPD::RedHom 32 • http://redhom.ii.uj.edu.pl • A subproject of CAPD (http://capd.ii.uj.edu.pl) • A sister project of CHomP (http://chomp.rutgers.edu) Generic homology software based on geometric reductions • AS, CR, DMT algorithms • Betti and torsion numbers, homology generators, homology maps, per- sistence intervals • Z and Z p coefficients • generic but efficient: for cubical sets, simplicial sets, cubical CW com- plexes, ... • written in C++, based on C++ templates and generic programming • Authors: P. D� lotko, M. Juda, A. Krajniak, MM, H. Wagner, ...

  35. Rectangular CW-complexes 33

  36. Rectangular CW-complexes 34 Theorem. (P. D� lotko, T. Kaczynski, MM, T. Wanner, 2010) Consider a rectangular CW-complex given by a rectangu- lar structure Q . Let P and Q denote two arbitrary rectangles in Q with dim Q = 1 + dim P , and define the number α QP as follows. For d = 1 and Q = [ a, b ] let  − 1  if P = [ a ] , α QP := 1 if P = [ b ] ,  0 otherwise , and for d > 1 set (1)  � j − 1  i =1 dim Q i α Q j P j ( − 1) if P < Q and P j < Q j , α QP :=  0 otherwise . Then the numbers α QP are incidence numbers for the given rectangular CW-complex.

  37. Rectangular CW complex versus cubical approach 35

  38. Numerical examples - manifolds 36 T × S 1 ( S 1 ) 3 S 1 × K T × T P × K dim 5 6 6 6 8 size in millions 0.07 0.10 0.40 2.36 32.05 Z Z Z Z Z H 0 Z 4 Z + Z 2 0 Z 2 + Z 2 Z 3 H 1 2 Z 3 Z 6 Z 2 Z + Z 2 H 2 0 2 Z 4 Z Z Z 2 H 3 Z H 4 130 350 > 600 > 600 - Linbox::Smith CHomP ::homcubes 1 . 3 1 . 7 10 56 17370 RedHom ::CR 0 . 03 0 . 04 0 . 26 2 . 5 34 RedHom ::CR+DMT 0 . 02 0 . 08 0 . 5 1 . 1 -

  39. Numerical examples - Cahn-Hillard 37 P0001 P0050 P0100 dim 3 3 3 size in millions 75.56 73.36 71.64 Z 7 Z 2 Z H 0 Z 6554 Z 2962 Z 1057 H 1 Z 2 H 2 > 600 > 600 > 600 Linbox::Smith CHomP ::homcubes 400 360 310 RedHom ::CR 18 16 15 RedHom ::CR+DMT 8 7 6 RedHom ::AS 10 5 3 . 5

  40. Numerical examples - random sets 38 d4s8f50 d4s12f50 d4s16f50 d4s20f50 dim 4 4 4 4 size in millions 0.07 0.34 1.04 2.48 Z 2 Z 2 Z 2 Z 2 H 0 Z 2 Z 17 Z 30 Z 51 H 1 Z 174 Z 1389 Z 5510 Z 15401 H 2 Z 2 Z 15 Z 71 Z 179 H 3 120 > 600 > 600 > 600 Linbox::Smith CHomP ::homcubes 1 8 . 3 41 170 RedHom ::CR 0 . 08 1 . 4 15 140 RedHom ::CR+DMT 0 . 03 0 . 16 0 . 58 2 . 9

  41. Numerical examples - simplicial sets 39 S 2 S 5 random set dim 4 2 5 size in millions 4.8 1.9 4.3 Z Z Z H 0 Z 39 H 1 0 0 Z 84 H 2 Z 0 H 3 0 H 4 Z CHomP ::homcubes 830 310 2100 RedHom ::CR+DMT 65 11 100

  42. Reduction equivalences 40 Assume S ′ is an S-complex resulting from Theorem. removing an coreduction pair ( a, b ) in an S -omplex S . Then the chain maps ψ ( a,b ) : R ( S ) → R ( S ′ ) and ι ( a,b ) : R ( S ′ ) → R ( S ) given by  c − � c,a �  for k = dim b − 1 , � ∂b,a � ∂b   ψ ( a,b ) ( c ) := c − � c, b � b for k = dim b , k    c otherwise , and � c − � ∂c,a � � ∂b,a � b for k = dim b , ι ( a,b ) ( c ) := k c otherwise , are mutually inverse chain equivalences.

  43. Reduction equivalences 40 Assume S ′ is an S-complex resulting from Theorem. removing an coreduction pair ( a, b ) in an S -omplex S . Then the chain maps ψ ( a,b ) : R ( S ) → R ( S ′ ) and ι ( a,b ) : R ( S ′ ) → R ( S ) given by  c − � c,a �  for k = dim b − 1 , � ∂b,a � ∂b   ψ ( a,b ) ( c ) := c − � c, b � b for k = dim b , k    c otherwise , and � c − � ∂c,a � � ∂b,a � b for k = dim b , ι ( a,b ) ( c ) := k c otherwise , are mutually inverse chain equivalences.

  44. Homology model 41 • the reduced S -complex S f — the homology model of S as a conve- nient model to solve the problems of decomposing homology classes on generators.

  45. Homology model 41 • the reduced S -complex S f — the homology model of S as a conve- nient model to solve the problems of decomposing homology classes on generators. • π f : R ( S ) → R ( S f ) and ι f : R ( S f ) → R ( S ) mutually inverse chain equivalences obtained by composing the maps π ( a,b ) and ι ( a,b ) .

  46. Homology model 41 • the reduced S -complex S f — the homology model of S as a conve- nient model to solve the problems of decomposing homology classes on generators. • π f : R ( S ) → R ( S f ) and ι f : R ( S f ) → R ( S ) mutually inverse chain equivalences obtained by composing the maps π ( a,b ) and ι ( a,b ) . • used to transport homology classes between H ∗ ( S ) and H ∗ ( S f )

  47. Homology model 41 • the reduced S -complex S f — the homology model of S as a conve- nient model to solve the problems of decomposing homology classes on generators. • π f : R ( S ) → R ( S f ) and ι f : R ( S f ) → R ( S ) mutually inverse chain equivalences obtained by composing the maps π ( a,b ) and ι ( a,b ) . • used to transport homology classes between H ∗ ( S ) and H ∗ ( S f ) The cost of transporting one generator through π f or ι f in general may be quadratic.

  48. Homology model 41 • the reduced S -complex S f — the homology model of S as a conve- nient model to solve the problems of decomposing homology classes on generators. • π f : R ( S ) → R ( S f ) and ι f : R ( S f ) → R ( S ) mutually inverse chain equivalences obtained by composing the maps π ( a,b ) and ι ( a,b ) . • used to transport homology classes between H ∗ ( S ) and H ∗ ( S f ) The cost of transporting one generator through π f or ι f in general may be quadratic. In the case of Free Face Reduction Algorithm and Free Coface Reduction Algorithm it is linear!

  49. Homology generators 42 • S q = { s q 1 , s q 2 , . . . s q r q } .

  50. Homology generators 42 • S q = { s q 1 , s q 2 , . . . s q r q } . • { [ u 1 ] , [ u 2 ] , . . . [ u n ] } – generators of the homology group H q ( S ) .

  51. Homology generators 42 • S q = { s q 1 , s q 2 , . . . s q r q } . • { [ u 1 ] , [ u 2 ] , . . . [ u n ] } – generators of the homology group H q ( S ) . • Task: decompose [ z ] ∈ H q ( X ) on homology generators n � [ z ] = x i [ u i ] i =1

  52. Homology generators 42 • S q = { s q 1 , s q 2 , . . . s q r q } . • { [ u 1 ] , [ u 2 ] , . . . [ u n ] } – generators of the homology group H q ( S ) . • Task: decompose [ z ] ∈ H q ( X ) on homology generators n � [ z ] = x i [ u i ] i =1 • Linear algebra problem n � z = x i u i + ∂c i =1 with unknown variables x 1 , x 2 , . . . , x n ∈ Z and c ∈ R q +1 ( S ) .

  53. Homology generators 43 r q r q r q +1 � � � z j s q u ij s q y k s q +1 z = j , u i = j , c = k j =1 j =1 k =1 r q � ∂s q +1 a kj s q = k j j =1 � r q +1 � r q � � s q ∂c = a kj y k j j =1 k =1

  54. Homology generators 43 r q r q r q +1 � � � z j s q u ij s q y k s q +1 z = j , u i = j , c = k j =1 j =1 k =1 r q � ∂s q +1 a kj s q = k j j =1 � r q +1 � r q � � s q ∂c = a kj y k j j =1 k =1 Thus, we get a system of r q linear equations with n + r q +1 unknowns r q +1 n � � z j = u ij x i + a kj y k for j = 1 , 2 , . . . r q . i =1 k =1 • In case of large S the cost is huge! transport the problem via π f to homology • Solution : model and solve it there

  55. Homology of cubical maps 44 • X, Y – cubical complexes

  56. Homology of cubical maps 44 • X, Y – cubical complexes • g : X → Y is cubical if maps elementary cubes to elementary cubes. • g induces chain map g # : R ( X ) → R ( Y )

  57. Homology of cubical maps 44 • X, Y – cubical complexes • g : X → Y is cubical if maps elementary cubes to elementary cubes. • g induces chain map g # : R ( X ) → R ( Y ) • examples of interest: inclusions and projections

  58. Homology of cubical maps 44 • X, Y – cubical complexes • g : X → Y is cubical if maps elementary cubes to elementary cubes. • g induces chain map g # : R ( X ) → R ( Y ) • examples of interest: inclusions and projections • U := { [ u 1 ] , [ u 2 ] , . . . [ u m ] } and W := { [ w 1 ] , [ w 2 ] , . . . [ w n ] } — bases of H q ( X ) and H q ( Y )

  59. Homology of cubical maps 44 • X, Y – cubical complexes • g : X → Y is cubical if maps elementary cubes to elementary cubes. • g induces chain map g # : R ( X ) → R ( Y ) • examples of interest: inclusions and projections • U := { [ u 1 ] , [ u 2 ] , . . . [ u m ] } and W := { [ w 1 ] , [ w 2 ] , . . . [ w n ] } — bases of H q ( X ) and H q ( Y ) • To find the matrix of g ∗ ecompose g # ( u i ) on generators in W

  60. Homology of cubical maps 44 • X, Y – cubical complexes • g : X → Y is cubical if maps elementary cubes to elementary cubes. • g induces chain map g # : R ( X ) → R ( Y ) • examples of interest: inclusions and projections • U := { [ u 1 ] , [ u 2 ] , . . . [ u m ] } and W := { [ w 1 ] , [ w 2 ] , . . . [ w n ] } — bases of H q ( X ) and H q ( Y ) • To find the matrix of g ∗ ecompose g # ( u i ) on generators in W • Using the diagram g # R ( X ) ✲ R ( Y ) ✻ π f ι f ❄ R ( Y f ) we can solve the problem in the homology model Y f , where it is much simpler.

  61. Computing Homology of Maps 45 In principle, computing homology of a map f : X → Y is a three step procedure: (1) Find a finite representation of f (2) Use it to build the chain map (3) Compute the map in homology from the chain map

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