Morphisms between Discrete Vector Fields

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Discrete Vector Fields

Ana Romero, Universidad de La Rioja Francis Sergeraert, Institut Fourier, Grenoble Oberwolfach, May-2013

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Semantics of colours:

Blue = “Standard” Mathematics Red = Constructive, effective,

algorithm, machine object, . . . Violet = Problem, difficulty,

obstacle, disadvantage, . . . Green = Solution, essential point,

mathematicians, . . .

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→ • Introduction.

• Basics of Algebraic Discrete Vector Fields.

• Discrete Vector Field ⇒ Canonical Reduction.

• Guessing the reduction formulas.

• Vector Fields and Morphisms.

• Seeing complicated products.

• The Eilenberg-Zilber vector field.

• The Eilenberg-Zilber vector field is natural.

• EZ vector field ⇒ EZ reduction.

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Introduction.

Eilenberg-MacLane conjecture (1953):

G = simplicial group ⇒

BG = Classifying space of G

Bar(C_∗G) = Algebraic classifying object of C_∗G Then ∃ a reduction ρ : C_∗(BG) ⇒⇒ Bar(C_∗G)

Essential for effective methods computing π_nX.

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First proof = Pedro Real (1993). Never implemented.

Second “proof ” through discrete vector fields.

Easily implemented ⇒







Program code divided by ∼ 3

Computing times divided by ∼ 20

But proof quite complex not yet finished.

Main problem =

Compatibility: Vector fields ↔ Algebraic structures Typical problem = Naturality of reductions

coming from discrete vector fields.

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Basic Algebraic Topology:

Serre + Eilenberg-Moore spectral sequences Non-constructive

Making basic algebraic topology constructive:

1. Eilenberg-Zilber theorem.

2. Twisted Eilenberg-Zilber theorem.

3. Eilenberg-MacLane correspondence:

Topological Classifying space

⇔

Algebraic Classifying space

But constructive requires: with explicit homotopies.

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Example: Rubio-Morace homotopy for Eilenberg-Zilber:

RM : C_∗(X × Y ) → C_∗+1(X × Y )

RM(x_p , y_p) = X

0≤r≤p−1 0≤s≤p−r−1 (η,η⁰)∈Sh(s+1,r)

(−1)^p−r−sε(η, η⁰). . .

. . .(↑^p−r−s(η⁰)η_{p−r−s−1}∂_p−r+1· · ·∂_px_p , ↑^p−r−s(η)∂_p−r−s· · ·∂_p−r−1y_p)

with Sh(p, q) = {(p, q)-shuffles} = {(η_i_p−1· · ·η_i₀, η_j_q−1· · ·η_j₀)}

for 0 ≤ i₀ < · · · < i_p−1 ≤ p +q − 1 and 0 ≤ j₀ < · · · < j_q−1 ≤ p +q − 1 and {i₀, . . . , i_p−1} ∩ {j₀, . . . , j_q−1} = ∅.

and ↑^k (η_αη_β· · ·) = η_α+kη_β+k · · · (↑^k = k-shift operator.)

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Simpler:

•

⇒ ⇒

•

once the notion of discrete vector field is understood.

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X • Introduction.

→ • Basics of Algebraic Discrete Vector Fields.

• Discrete Vector Field ⇒ Canonical Reduction.

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Basics of Algebraic Discrete Vector Fields.

Definition: R = unitary commutative ring.

A cellular R-chain complex is an indexed triple:

(C_p,β_p, d_p)_p∈_Z with:

• C_p = free R-module (non necessarily of finite type);

• β_p = distinguished R-basis of C_p;

• d_p : C_p → C_p−1 = differential.

The elements of β_p are the p-cells of the cellular complex.

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Main examples:

• Geometrical cellular complexes (R = Z):

– Cubical complexes (digital images);

– Simplicial complexes ; – Simplicial sets ;

– CW-complexes ;

• Algebraic cellular complexes (R = k = field):

– Free resolutions;

– Koszul complexes;

– . . . .

Natural distinguished basis ⇒ Cellular complex.

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C_∗ = {C_p, β_p, d_p}_p∈_Z = Cellular chain complex.

Definition: A p-cell is an element of β_p. Definition: If τ ∈ β_p and σ ∈ β_p−1,

then ε(σ, τ) := coefficient of σ in dτ is called the incidence number between σ and τ. Definition: σ is a face of τ if ε(σ, τ) 6= 0.

Definition: σ is a regular face of τ if ε(σ, τ) is R-invertible.

(⇔ ε(σ, τ) = ±1 if R = Z)

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•

• • • • • •

• • •

Definition:

(C_∗, β_∗, d_∗) = Cellular complex

A Discrete Vector Field is a pairing:

V = {(σ_i, τ_i)}_i∈I satisfying:

• ∀i ∈ I, τ_i = some k_i-cell and σ_i = some (k_i − 1)-cell.

• ∀i ∈ I, σ_i is a regular face of τ_i.

• ∀i 6= j ∈ I, {σ_i, τ_i} ∩ {σ_j, τ_j} = ∅.

The vector field V is admissible or not.

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V = {(σ_i, τ_i)}_i∈I = Vector field.

Definition: V -path = sequence (σ_i₁, τ_i₁, σ_i₂, τ_i₂, . . . , σ_i_n, τ_i_n) satisfying: 1. (σ_i_j, τ_i_j) ∈ V .

2. σ_i_j face of τ_i_j−1. 3. σ_i_j 6= σ_i_j−1.

Remark: σ_i_j not necessarily regular face of τ_i_j−1. Examples:

•

V -path of length 6

V -path of length 3

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Definition: A vector field is admissible if for every source cell σ,

the length of any path starting from σ

is bounded by a fixed integer λ(σ).

Example of two different paths with the same starting cell.

•

Remark: The paths from a starting cell

are not necessarily organized as a tree.

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Typical examples of non-admissible vector fields.

• • •

•

• •

•

• •

•

∼

???!!!

∅

•

R × I

∼

???!!!

∅

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V = {(σ_i, τ_i)}_i∈I = Vector field.

Definition: A critical p-cell is an element of β_p

which does not occur in V . Other cells divided in source cells and target cells.

Example:

•

Critical cells Source cells

Target cells

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Plan.

X • Introduction.

X • Basics of Algebraic Discrete Vector Fields.

→ • Discrete Vector Field ⇒ Canonical Reduction.

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Discrete Vector Field ⇒ Canonical Reduction.

Fundamental Theorem:

Given: C_∗ = (C_p, β_p, d_p)_p∈_Z = Cellular chain complex.

V = (σ_i, τ_i)_i∈I = Admissible Discrete Vector Field.

⇒

A canonical process constructs: d^c_∗ + f + g + h

defining a canonical reduction:

(C_p, β_p, d_p)_p∈_Z (C_p^c, β_p^c,d^c_p)_p∈_Z

f h g

ρ_V =

Initial Complex ⇒^ρ⇒^V Critical complex

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In (C_∗, β_∗, d_∗) (C_∗^c, β_∗^c,d^c_∗)

f g

ρ = (f,g,h) = h ,

the most important component is h.

To be understood as a contraction C_∗ ⇒⇒ C_∗^c.

•

^C^∗

•

C_∗^c

h

The homotopy h reduces the “cherry” C_∗

onto the “stone” C_∗^c.

⇒ h = the “flow” generated by the vector field.

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Toy Example.

•

Critical cells Source cells Target cells C_∗ =

Fundamental Reduction Theorem ⇒

•

d^c₁

ρ : C_∗ ⇒⇒ C_∗^c = = Z

d^c₁=0

←− Z = Circle

Rank(C_∗) = (16,24, 8) vs Rank(C_∗^c) = (1, 1,0)

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Plan.

X • Introduction.

X • Discrete Vector Field ⇒ Canonical Reduction.

→ • Guessing the reduction formulas.

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Guessing the reduction formulas.

Transcription for discrete vector fields:

Elementary example of :

•0

•1

•2

f g

h •3

•3

f(0) = f(1) = f(2) = f(3) = 3

f(01) = f(12) = f(23) = 0 g(3) = 3

h(0) = −01 − 12− 23 h(1) = −12 − 23

h(2) = −23 h(3) = 0

id = f g

id = gf +dh + hd f h = 0

hg = 0 hh = 0

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Just a little more complicated:

•1

•2

•3

•4

•5

•6

•

• •

•1

•2

•3

•4

•5

•6 f

g = inc.

h

f(25) = 24 + 45 = −dh(25) + 25

h(25) = −245 = −τ(25) (25 = negative face of 245) f(23) = 24 + 45 + 56− 36

h(23) = −245 + 235 −356 = τ(23) + h(25)− h(35) f(13) = 12 + 24 + 45 + 56− 36

h(13) = −123− 245 + 235 − 356 = −τ(13) + h(23)

⇒ General formula for h(source cells):

h(σ) = v(σ) − h(d_stv(σ) −σ)

to be explained.

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(C_∗, β_∗, d_∗) + V = {(σ_i, τ_i)} = vector field defines a splitting:

C_n−1^c

· · ·

C_n−1^s

· · ·

C_n−1^t

· · ·

C_n^c · · · C_n^s · · · C_n^t · · ·

dst

d

C_n−1^c

· · ·

C_n−1^s

· · ·

C_n−1^t

· · ·

C_n^c · · · C_n^s · · · C_n^t · · ·

v

V defines a codifferential: v(σ) = ε(σ, τ)τ if (σ, τ) ∈ V

d =







d_tt d_ts d_tc d_st d_ss d_sc d_ct d_cs d_cc







v =







0 v 0 0 0 0 0 0 0







h(σ) = v(σ) −h(d_stv(σ) − σ) = [v −h(dv −1)](σ)

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Deciding also h(σ) = 0 if σ = target or critical cell

+ using some elementary facts, gives the general formula:

h(σ) = v(σ) − h(dv(σ) − σ)

valid even for σ target or critical cell.

In particular: v − hdv = 0 Then:

f = pr_c(1 − dh) d^c_∗ = d_cc − d_cthd_sc g = (1 − hd)

defines the searched reduction:

(C_∗, β_∗, d_∗) (C_∗^c, β_∗^c,d^c_∗)

f g

ρ = (f,g,h) = h

,

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Understanding: h ⇒ f and g and d^c:

h = v − h(dv − 1) : C_∗^s → C_∗+1^t

C_n^c C_n^s C_n^t

C_n+1^c C_n+1^s C_n+1^t

h

d_ct

f = pr_c(1 - dh)

C_n−1^c C_n−1^s C_n−1^t

C_n^c C_n^s C_n^t

dsc

h

g = (1 − hd)

C_n−1^c C_n−1^s C_n−1^t

C_n^c C_n^s C_n^t

d_sc h

d_ct

dcc

d^c = d_cc − d_cthd_sc

0•

1•

2•

•

0•

1•

2• f(01) = 01 f(02) = 01 + 12 f(012) =∅

•

0

1

2 3

• •

•

0

1

2 3

f(013) = 013 f(123) =∅

g(013) = 013 + 123

d^c(013) = 01 + 12 + 23−03

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Plan.

X • Introduction.

X • Guessing the reduction formulas.

→ • Vector Fields and Morphisms.

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Vector Fields and Morphisms.

Problem: Let C_∗ and C_∗⁰ be two cellular chain complexes respectively provided with vector fields V and V ⁰. Question: Right notion

of morphism φ : (C_∗, V ) → (C_∗⁰, V ⁰) ???

1. Not trivial.

2. Essential to master the Eilenberg-Zilber vector fields.

3. Quite amazing !!

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Definition: A cellular morphism:

φ : (C_∗, d_∗, β_∗) → (C_∗⁰, d⁰_∗, β_∗⁰)

is a chain complex morphism φ : (C_∗, d_∗) → (C_∗⁰, d⁰_∗)

satisfying the extra condition:

For every p-cell σ ∈ β_p,

φ(σ) is null or ∈ β_p⁰.

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(C_∗, d, β, V ) and (C_∗⁰, d⁰, β⁰, V ⁰)

= cellular chain complexes

with respective admissible discrete vector fields V and V ⁰. Definition: A vectorious morphism:

φ : (C_∗, d, β, V ) → (C_∗⁰, d⁰, β⁰, V ⁰)

is a cellular morphism φ := (C_∗, d, β) → (C_∗⁰, d⁰, β⁰) satisfying the extra conditions:

1. For every critical cell χ ∈ β_p^c, φ(χ) is null or ∈ β_p^0c. 2. For every target cell τ ∈ β^t_p, φ(τ) is null or ∈ β^0t_p. 3. No condition at all for the source cells !!

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Theorem: φ : (C_∗, d, β, V ) → (C_∗⁰, d⁰, β⁰, V ⁰)

= vectorious morphism.

Then φ defines a morphism (φ, φ^c)

between the corresponding reductions:

C_∗^c C_∗^0c

C_∗ C_∗⁰ with:

d^0cφ^c = φ^cd^c f⁰φ = φ^cf g⁰φ^c = φg

h⁰φ = φh

φ^c φ

f g f⁰ g⁰

h h⁰

Note: φ^c := φ|C_∗^c

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Proof:

Definition:

λ_σ =











0 for target and critical cells, maximal length of a V -path

starting from the source cell σ.

Remember: Recursive formula:

h(σ) = v(σ) − h(dv(σ) − σ)

⇒ hdv(σ) = v(σ)

⇒ hdτ = τ for every target cell τ

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1. h⁰φσ = φhσ ??

Obvious for σ target or critical cell.

Assumed known for λ_σ < k.

Let σ be a source cell with λ_σ = k.

φhσ = φvσ − φh(dvσ − σ) OK for (dvσ − σ) ⇒ = φvσ − h⁰φ(dvσ − σ)

φd = d⁰φ ⇒ = φvσ − h⁰d⁰φvσ + h⁰φσ φvσ = target cell ⇒ = h⁰φσ

QED

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2. g⁰φ^c = φg ??

•

φ^c g⁰ g

φ

d^c d^0c

d d⁰

h h⁰

For a critical cell χ: gχ = χ − hdχ = (1 − hd)χ

⇒ φgχ = φ(1 − hd)χ φhd = h⁰d⁰φ ⇒ = (1 − h⁰d⁰)φχ

φχ = φ^cχ ⇒ = g⁰φ^cχ

QED

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3. φ^cd^c = d^0cφ^c ??

•

φ^c g⁰ g

φ

d^c d^0c

d d⁰

h h⁰

g⁰φ^c = φg ⇒ g⁰φ^cd^c = φgd^c gd^c = dg ⇒ = φdg φd = d⁰φ ⇒ = d⁰φg φg = g⁰φ^c ⇒ = d⁰g⁰φ^c d⁰g⁰ = g⁰d^0c ⇒ = g⁰d^0cφ^c g⁰ injective ⇒ φ^cd^c = d^0cφ^c

QED

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•

φ^c f⁰ f

φ

g⁰

d^c d^0c

d d⁰

h h⁰

4. f⁰φ = φ^cf ??

g⁰ injective ⇒ [(f⁰φ = φ^cf) ⇔ (g⁰f⁰φ = g⁰φ^cf)]

g⁰f⁰φ = (1 − d⁰h⁰ − h⁰d⁰)φ (d⁰φ = φd) + (h⁰φ = φh) ⇒ = φ(1 − dh − hd)

= φgf

= g⁰φ^cf

QED

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Plan.

X • Introduction.

X • Vector Fields and Morphisms.

→ • Seeing complicated products.

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Seeing complicated products.

Triangulating prisms ∆^p × ∆^q:

•

(0,0) (0,1)

(1,0) (1,1)

∆¹ × ∆¹

•

(0,0) (0,1)

(1,0) (1,1)

(2,0) (2,1)

∆² × ∆¹

Two ∆² in ∆¹ × ∆¹: (0,0) < (0,1) < (1,1) (0,0) < (1,0) < (1,1)

Three ∆³ in ∆² ×∆¹: (0,0) < (0,1) < (1,1) < (2,1) (0,0) < (1,0) < (1,1) < (2,1) (0,0) < (1,0) < (2,0) < (2,1)

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Rewriting the triangulation of ∆² × ∆¹.

•

(0,0) (0,1)

(1,0) (1,1)

(2,0) (2,1)

(0,0)<(0,1)<(1,1)<(2,1)

•

0 1 2

0 1

•

(0,0) (0,1)

(1,0) (1,1)

(2,0) (2,1)

(0,0)<(1,0)<(1,1)<(2,1)

•

0 1 2

0 1

•

(0,0) (0,1)

(1,0) (1,1)

(2,0) (2,1)

(0,0)<(1,0)<(2,0)<(2,1)

•

0 1 2

0 1

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Planar representations of simplices of ∆^p × ∆^q: Example of 5-simplex :

•

0 1 2 3 4 5 0

1 2 3

= σ ∈ (∆⁵ × ∆³)₅

⇒ 6 faces:

•

0 1 2 3 4 5 0

1 2 3

∂₀σ

•

0 1 2 3 4 5 0

1 2 3

∂₁σ

•

0 1 2 3 4 5 0

1 2 3

∂₂σ

•

0 1 2 3 4 5 0

1 2 3

∂₃σ

•

0 1 2 3 4 5 0

1 2 3

∂₄σ

•

0 1 2 3 4 5 0

1 2 3

∂₅σ

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Plan.

X • Introduction.

X • Seeing complicated products.

→ • The Eilenberg-Zilber vector field.

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The Eilenberg-Zilber vector field.

Eilenberg-Zilber problem for ∆¹ × ∆¹: Cubical version of ∆¹ × ∆¹:

•

(0,0) (0,1)

(1,0) (1,1)

C₀ ←− C₁ ←− C₂

Z⁴ ←− Z⁴ ←− Z C_∗∆¹ ⊗ C_∗∆¹

C0∆¹ ⊗ C0∆¹ ←− (C0∆¹ ⊗ C1∆¹) ⊕ (C1∆¹ ⊗ C0∆¹) ←− C1∆¹⊗ C1∆1

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Simplicial version of ∆¹ × ∆¹:

•

(0,0) (0,1)

(1,0) (1,1)

C₀⁰ ←− C₁⁰ ←− C₂⁰

Z⁴ ←− Z⁵ ←− Z² To be compared with:

C₀ ←− C₁ ←− C₂

Z⁴ ←− Z⁴ ←− Z¹

Difference = a vector field with a unique “vector”

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Eilenberg-Zilber reduction as induced by a vector field:

•

(0,0) (0,1)

(1,0) (1,1)

•

⇒ ⇒

•

(0,0) (0,1)

(1,0) (1,1)

“=”

•

(0,0) (0,1)

(1,0) (1,1)

Translation in planar diagrams: V = {(σ, τ)}

σ = source cell = edge[(0,0),(1,1)] =

•

τ = target cell = triangle[(0,0),(0,1),(1,1)] =

•

∂1τ =σ

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Generalizing the idea ⇒

Canonical discrete vector field for ∆⁵ × ∆³.

•

0 1 2 3 4 5 0

1 2 3

Source =∂₂(

•

0 1 2 3 4 5 0

1 2 3

Target)

•

0 1 2 3 4 5 0

1 2 3

Source =∂₁(

•

0 1 2 3 4 5 0

1 2 3

Target)

Recipe: First “event” = Diagonal step = •

•

⇒ Source cell.

= (-90^◦)-bend = •

• •

⇒ Target cell.

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Critical cells ??

Critical cell = cell without any “event”

= without any diagonal or −90^◦-bend.

Examples.

•

0 1 2 3 4 5 0

1 2 3

•

∆⁰₂ ⊗∆⁰₁

•

0 1 2 3 4 5 0

1 2 3

• • •

∆²_2,3,5 ⊗∆⁰₁

•

0 1 2 3 4 5 0

1 2 3

•

∆⁰₂ ⊗∆²_1,2,3

•

0 1 2 3 4 5 0

1 2 3

• • •

•

∆²_2,3,4 ⊗∆²_1,2,3

•

0 1 2 3 4 5 0

1 2 3

• • •

•

∆²_1,4,5⊗∆²_0,2,3

•

0 1 2 3 4 5 0

1 2 3

• • • • • •

•

∆⁵0,1,2,3,4,5 ⊗∆³_0,1,2,3

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Conclusion:

C_∗^c = C_∗(∆⁵) ⊗ C_∗(∆³)

Fundamental theorem of vector fields ⇒

Canonical Homological Reductions:

ρ : C_∗(∆⁵ × ∆³) ⇒⇒ C_∗(∆⁵) ⊗ C_∗(∆³) ρ : C_∗(∆^p × ∆^q) ⇒⇒ C_∗(∆^p) ⊗ C_∗(∆^q)

p = q = 10 ⇒ 16,583,583,743 vs 4,190,209

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X • Introduction.

X • Seeing complicated products.

X • The Eilenberg-Zilber vector field.

→ • The Eilenberg-Zilber vector field is natural.

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The Eilenberg-Zilber vector field is natural.

Naturality of the Eilenberg-Zilber reduction

with respect to product morphisms.

Standard methods ⇒ (∆^p → ∆^p⁰) × (∆^q → ∆^q⁰) is enough.

Given: φ : ∆^p → ∆^p⁰ ψ : ∆^q → ∆^q⁰

simplicial.

Prove:

C_∗∆^p ⊗ C_∗∆^q C_∗(∆^p × ∆^q)

C_∗∆^p⁰ ⊗ C_∗∆^q⁰ C_∗(∆^p⁰ × ∆^q⁰)

φ×ψ

φ⊗ψ

f g f⁰ g⁰

h h⁰

com ?

is commutative ??

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Proof:

Representation of a simplex of ∆^p × ∆^q via an s-path.

•

α β

= subsimplex of α × β ⊂ (∆^p × ∆^q)₄

spanned by the vertices (0,0) – (0,1)– (2,2) – (3,3) – (4,3).

The game first event “diagonal •

• ”

or “right-angle bend •

• • ”

determines the nature source, target or critical.

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42/56

Examples:

•

α β •

•

source = σ

•

α β •

• •

target = τ

•

α β

critical = χ

• • •

•

Here:

∂₂(τ) = σ

⇒ v(σ) = τ

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Two maps

φ : ∆^p → ∆^p⁰ ψ : ∆^q → ∆^q⁰

= simplicial morphisms.

Claim:

τ target cell in ∆^p × ∆^q ⇒

(φ × ψ)(τ) target or degenerate cell in ∆^p⁰ × ∆^q⁰ χ critical cell in ∆^p × ∆^q ⇒

(φ × ψ)(χ) critical or degenerate cell in ∆^p⁰ × ∆^q⁰

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44/56

Typical accidents with source cells.

α = id : ∆² → ∆² and ψ : ∆² → ∆¹as below:

•

0

1 2

0 1

ψ =

1)

•

• ϕ×ψ •

Then (φ × ψ)(source) = source

but for reasons which do not match!

Compare corresponding target cells.

•

• ϕ×ψ •

•

• ϕ×ψ •

V