The Topology of Configuration Spaces of Coverings

Shuchi Agrawal, Daniel Barg, Derek Levinson

Summer@ICERM

November 5, 2015

Agrawal, Barg, Levinson (ICERM) Topology of Coverings November 5, 2015 1 / 30

Overview

1 Introduction

2 k-coverings of the unit interval

3 Excess 0 coverings of S1

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Introduction

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General Question

Question

Given a metric space Y , a radius r and n closed balls of this radius, whatis the topology of the configuration space of the balls (i.e. their centers)such that every point in Y is covered by (at least) one ball?

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Configuration Spaces

Given n balls, label these balls 1, 2, . . . , n. Suppose Y ⊂ Rd , and considerall vectors in Y n ⊂ Rdn of the form ~x = (~x1, ~x2, . . . , ~xn), where ball i hascenter ~xi .

Definition (Configuration Space)

The configuration space of coverings of Y is all ~x ∈ Y n such that Y iscovered, i.e.

Covn(r ,Y ) = {~x ∈ Y n | ∀y ∈ Y ∃ 1 ≤ i ≤ n s.t. d(y , xi ) ≤ r}

Agrawal, Barg, Levinson (ICERM) Topology of Coverings November 5, 2015 5 / 30

Single coverings of the interval

Suppose we now consider coverings of the unit interval I = [0, 1].

18

38

58

78

x1 x2 x3 x4

Figure: The configuration above corresponds to the point ( 18 ,

38 ,

58 ,

78 ) ∈ R4

18

38

58

78

x2, x5 x4, x6 x3 x1

Figure: The configuration above corresponds to the point ( 78 ,

18 ,

58 ,

38 ,

18 ,

38 ) ∈ R6

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“Excess”

Definition (Excess)

The excess given a radius r is defined as the largest number m for which itis still possible to cover the interval with (n −m) r -balls.

18

38

58

78

x1 x2 x3 x4

Figure: Excess 0, as with 4 balls of radius 18 , we can just cover I .

18

38

58

78

x2, x5 x4, x6 x3 x1

Figure: Excess 2, as we can cover the interval with 4 balls of radius 18 .

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Background and Goal

Theorem (Baryshnikov)

Covn(r , I ) ∼ Skelm(Pn), where m is the excess, Skelm is an m-skeleton,and Pn is the permutahedron on n vertices.

Example (n = 3; the 3-permutahedron is a 2-dimensional hexagon)

for 0 ≤ 2r < 13 , cannot cover, so Cov3(r , I ) ∼= ∅

for 13 ≤ 2r < 1

2 , m = 0, Cov3(r , I ) ∼ vertices of hexagon (0-sk.)

for 12 ≤ 2r < 1, m = 1, Cov3(r , I ) ∼ 1-sk. of hexagon ∼ S1

for 1 ≤ 2r , m = 2, contractible

Our Goal

Find an analogue for the case of k-covering I , where k is arbitrary.

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Indices

Definition

Define “indices” as points in the unit interval of the form ij = 2j−12n for

1 ≤ j ≤ n.

i1 = 18 i2 = 3

8 i3 = 58 i4 = 7

8

Suppose we have balls of radius r = 12n . Then if we are k-covering I , kn

balls will cover I , and then the excess m =(# of balls)−kn.

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k-coverings of the unit interval

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The Space of Double Coverings: Excess 1

Suppose now that we want to double-cover every point in I , so that

∀ y ∈ I , ∃ 1 ≤ j 6= k ≤ 2n + 1 s.t. max(d(y , xj), d(y , xk)) ≤ r

Definition

Let 2-Cov2n+1(r , I ) be the configuration space of double coverings of theinterval with 2n + 1 balls, with 1

2n ≤ r < 12(n−1) , so the excess is 1.

14

34

x1, x2, x3 x4, x50 1

14

12

34

x1x2 x3x4, x5

0 1

Figure: Two configurations with 5 balls of radius 14 which double-cover I ,

corresponding to the the points ( 14 ,

14 ,

14 ,

34 ,

34 ) and ( 1

4 ,18 ,

38 ,

34 ,

34 ), respectively, in

2-Cov5( 14 , I ).

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The n = 5 case and the Desargues Graph

Theorem

For 14 ≤ r < 1

2 (excess 1),2-Cov5(r , I ) ∼=h G (10, 3), the“Desargues Graph”- the bipartitedouble cover of the PetersenGraph G (5, 2).

1 42 53

1 32 45

1 32 54

1 23 45

1 32 45

2 13 45

1 23 54

1 24 35

1 23 45

2 14 35

2 13 54

1 52 34

2 13 54

2 15 34

1 24 35

3 14 25

2 14 35

3 15 24

3 14 25

4 51 23

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Theorem 1: Our space is homotopic to a graph

Theorem

For r = 12n (excess 1), 2-Cov2n+1(r , I ) ∼ G , for G a graph, i.e. a

1-dimensional simplicial complex.

Definition

Let G2,n ⊂ 2-Cov2n+1(r , I ) be the following graph. For ~x ∈-Cov2n+1(r , I )to be in G2,n we first of all require that ∀ 1 ≤ j ≤ n, ∃ 1 ≤ p 6= q ≤ 2n + 1s.t. ij = xp = xq. Thus, any point on this graph has at least 2 ballscentered at each index ij . A vertex of this graph also has one index with 3balls centered at it. An edge of this graph has exactly 2 balls centered ateach index, and one ball centered in an interval of the form(ij , ij+1) = (2j−12n , 2(j+1)−1

2n ) for 1 ≤ j ≤ n − 1.

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Some Lemmas

Theorem

For any double-covering in 2-Cov2n+1( 12n , I ) ⊂ I 2n+1, every index must

have at least 1 ball centered at it, that is:

∀ 1 ≤ j ≤ n, ∃ 1 ≤ k ≤ 2n + 1 s.t. ij = xk .

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Some Lemmas

Theorem

For any double covering, at most 1 ball can be centered in any interval ofthe form (ij , ij+1) for 1 ≤ j ≤ n − 1.

ij−1 ij ij+1 ij+2

xi−1 xi xi+1 xi+20 1

Figure: The above cannot happen.

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Some Lemmas

Theorem

Suppose a double-covering has no balls centered in (0, i1) or (in, 1).Suppose the balls with centers xI1 , xI2 , . . . , xIp (re-labeled in ascendingorder) are not centered at indices, for 1 ≤ Ij ≤ 2n + 1 and 1 ≤ p ≤ n + 1.Then xI1 ∈ (ij , ij+1), . . . , xIp ∈ (ij+p−1, ij+p) for 1 ≤ j ≤ n.

ij−1 ij ij+1 ij+2

xI1xI2 xI3 xI4

0 1

Figure: The above cannot happen.

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Flow for the Space of Double Coveringsn = 3, excess 1; 7 balls of radius 1

6

16

12

56

x3, x4 x5 x6, x7x1 x20 1

x1

x2

16

12

56

16

12

56

Figure: 16 ≤ x1 ≤ 1

2 ∩12 ≤ x2 ≤ 5

6 ∩ x2 − x1 ≤ 13

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Flow for the Space of Double Coveringsn = 4, excess 1; 9 balls of radius 1

8

18

38

58

78

x4, x5 x6 x7 x8, x9x1 x2 x30 1

x1

x2

x3( 18, 38, 58) ( 3

8, 38, 58)

( 38, 58, 58)

( 38, 58, 78)

( 12, 12, 12)

Figure: 18 ≤ x1 ≤ 3

8 ∩38 ≤ x2 ≤ 5

8 ∩58 ≤ x3 ≤ 7

8 ∩ x2 − x1 ≤ 14 ∩ x3 − x2 ≤ 1

4

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Main Theorem

We calculated the vertex and edge counts of Gk,n:

V (Gk,n) = k( kn+1k k k k ... k+1

)= k(kn+1)!

(k!)n−1·(k+1)

E (Gk,n) =2n·V ·(k+1)+ n−2

n·V ·2(k+1)

2 = V (n−1)(k+1)n = k(kn+1)!(n−1)

n(k!)n−1

Theorem (See, for example: [Katok, 2006])

Suppose X and Y are 1-dimensional simplicial complexes, i.e. graphs.Then X ∼ Y ↔ χ(X ) = χ(Y ), where χ(X ) = V (X )− E (X ), where χ(X )is called the “Euler Characteristic” of X .

Theorem (Main Theorem)

k-Covkn+1( 12n , I )

∼=h Gk,n, with the above vertex and edge counts.

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Future Work: Extending to higher excesses

Conjecture

k-Cov2n+m( 12n , I ) ∼ m-dimensional simplicial complex.

Need extension of excess 1 flows.

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Future Work: Non-smooth Morse Theory

Theorem (Milnor, Classical Morse Theory)

Let f : M → R be smooth. Let a < b. If f −1[a, b] iscompact, and contains no points where 5f = 0, thenMa = f −1[−∞, a] is homotopy equivalent toMb = f −1[−∞, b].

Definition (Tautological Function)

Define τ : Covn(r ,Y )→ R by τ(~x = (x1, . . . , xn)) = maxy∈Y

min1≤i≤n

d(xi , y)

τ is only piece-wise smooth; must use techniques such as in[Agrachev, 1997].

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Future Work: Non-smooth Morse Theory

Definition (Tautological Function for Double-covering)

Define τ : 2-Covn(r ,Y )→ R by

τ(~x) = maxy∈Y

min1≤i<j≤n

max{d(xi , y), d(xj , y)}

Conjecture

The only critical points of τ occur when the excess changes.

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Excess 0 coverings of S1

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Space of Single Coverings of the Circlen balls of radius 1

2n, total length n · 1

n= 1

1

2

3

Permutations 123, 312, 231 equivalent up to rotation.

1

3

2

Permutations 132, 321, 213 equivalent up to rotation.

Theorem

Covn( 12n , S

1) ∼=(n−1)!⊔i=1

S1

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Space of Double Coverings of the Circle, for odd nn balls of radius 1

n, total length n · 1

2n= 2

1

2

3

Theorem (same reasoning as single-covering case)

2-Covn( 1n ,S1) ∼=

(n−1)!⊔i=1

S1

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Space of Double Coverings for n = 2, r = 12

Both balls cover the entire circle, can be moved independently of eachother.

Theorem

2-Cov2(12 ,S1) ∼= S1 × S1 ∼= T 2

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Space of Double Coverings for n ≥ 4, n even

1, 23, 4 1

3

2

4

1

2

3

4

Theorem

2-Cov4(14 ,S1) ∼= 3 tori, glued as below. In general, for n even,

2-Covn( 1n ,S1) ∼= 2(n−1)!

n tori; each torus glued to n2 · (2

n−22 − 1) other tori.

1, 2

3, 4

1, 3

2, 4

1, 4

2, 3 π1(2− Cov4(14, S1)) =

Z× F3

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Generalization to k-covering, Future Work

Small Result: k-Covk(12 , S1) ∼= T k .

Conjecture

k-Covn(kn ,S1) ∼=

(n−1)!⊔i=1

S1 if k - n.

Generalize to higher k , and look at higher excess.

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References (selection)

John Milnor (1963)

Morse Theory

Princeton University Press.

Han Wang (2014)

On the Topology of the Spaces of Coverings

PhD Thesis University of Illinois at Urbana-Champaign.

Anatole Katok and Alexey Sossinsky (2006)

Introduction to Modern Topology and Geometry

Lecture Notes Penn State University 44-50.

Yuliy Baryshnikov, Peter Bubenik and Matthew Kahle (2014)

Min-Type Morse Theory for Configuration Spaces of Hard Spheres

University of Illinois at Urbana-Champaign.

A. A. Agrachev, D. Pallaschke and S. Scholtes (2014)

On Morse Theory For Piecewise Smooth Functions

Journal of Dynamical and Control Systems 449-469.

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Acknowledgements

Thank you to Yuliy for inventing the problem, and for being an incrediblyhelpful advisor.

Thank you to Tarik for many enlightening conversations.

Thank you to Stefan for his willingness to listen and help.

Thank you to ICERM for the opportunity, for the facilities, and for thesustenance (shout out to Danielle!).

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