The Topology of Configuration Spaces of Coverings
Shuchi Agrawal, Daniel Barg, Derek Levinson
Summer@ICERM
November 5, 2015
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Overview
1
Introduction
2
k-coverings of the unit interval
3
Excess 0 coverings of S 1
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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, what
is 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 consider
all vectors in Y n ⊂ Rdn of the form ~x = (x~1 , x~2 , . . . , x~n ), where ball i has
center x~i .
Definition (Configuration Space)
The configuration space of coverings of Y is all ~x ∈ Y n such that Y is
covered, i.e.
Covn (r , Y ) = {~x ∈ Y n | ∀y ∈ Y ∃ 1 ≤ i ≤ n s.t. d(y , xi ) ≤ r }
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Single coverings of the interval
Suppose we now consider coverings of the unit interval I = [0, 1].
x1
x2
x3
x4
1
8
3
8
5
8
7
8
Figure: The configuration above corresponds to the point ( 18 , 38 , 58 , 78 ) ∈ R4
x2 , x5
x4 , x6
x3
x1
1
8
3
8
5
8
7
8
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 it
is still possible to cover the interval with (n − m) r -balls.
x1
x2
x3
x4
1
8
3
8
5
8
7
8
Figure: Excess 0, as with 4 balls of radius 18 , we can just cover I .
x2 , x5
x4 , x6
x3
x1
1
8
3
8
5
8
7
8
Figure: Excess 2, as we can cover the interval with 4 balls of radius 81 .
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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
for
1
3
1
2
≤ 2r < 12 , m = 0, Cov3 (r , I ) ∼ vertices of hexagon (0-sk.)
≤ 2r < 1, m = 1, Cov3 (r , I ) ∼ 1-sk. of hexagon ∼ S 1
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 =
1 ≤ j ≤ n.
i1 =
1
8
i2 =
3
8
i3 =
5
8
2j−1
2n
i4 =
for
7
8
1
Suppose we have balls of radius r = 2n
. 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 the
1
1
≤ r < 2(n−1)
, so the excess is 1.
interval with 2n + 1 balls, with 2n
x1 , x2 , x3
x4 , x5
1
4
3
4
0
1 0
x2
x1
1
4
x4 , x5
x3
1
1
2
3
4
Figure: Two configurations with 5 balls of radius 14 which double-cover I ,
corresponding to the the points ( 41 , 14 , 14 , 34 , 34 ) and ( 41 , 18 , 38 , 43 , 34 ), respectively, in
2-Cov5 ( 14 , I ).
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The n = 5 case and the Desargues Graph
Theorem
For 14 ≤ r < 12 (excess 1),
2-Cov5 (r , I ) ∼
=h G (10, 3), the
“Desargues Graph”- the bipartite
double cover of the Petersen
Graph G (5, 2).
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1 4
2 5
3
1 3
2 4
5
1 3
2 5
4
1 2
3 4
5
1 3
2 4
5
2 1
3 4
5
1 2
3 5
4
1 2
4 3
5
1 2
3 4
5
2 1
4 3
5
2 1
3 5
4
1 5
2 3
4
2 1
3 5
4
2 1
5 3
4
1 2
4 3
5
3 1
4 2
5
2 1
4 3
5
3 1
5 2
4
3 1
4 2
5
4 5
1 2
3
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Theorem 1: Our space is homotopic to a graph
Theorem
1
For r = 2n
(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 + 1
s.t. ij = xp = xq . Thus, any point on this graph has at least 2 balls
centered at each index ij . A vertex of this graph also has one index with 3
balls centered at it. An edge of this graph has exactly 2 balls centered at
each index, and one ball centered in an interval of the form
2(j+1)−1
) for 1 ≤ j ≤ n − 1.
(ij , ij+1 ) = ( 2j−1
2n ,
2n
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Some Lemmas
Theorem
1
For any double-covering in 2-Cov2n+1 ( 2n
, 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 of
the form (ij , ij+1 ) for 1 ≤ j ≤ n − 1.
xi−1
0
ij−1
xi+2
xi xi+1
ij
ij+1
1
ij+2
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 ascending
order) 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.
xI1
0
ij−1
x I2
x I3
ij
ij+1
x I4
1
ij+2
Figure: The above cannot happen.
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Flow for the Space of Double Coverings
n = 3, excess 1; 7 balls of radius
1
6
x2
x3 , x4
5
6
0
x5
x1
x2
x6 , x7
1
2
1
6
1
5
6
1
2
1
6
x1
1
6
Figure:
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1
2
1
6
5
6
≤ x1 ≤
1
2
∩
1
2
≤ x2 ≤
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5
6
∩ x2 − x1 ≤
1
3
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Flow for the Space of Double Coverings
n = 4, excess 1; 9 balls of radius
1
8
( 38 , 58 , 78 )
0
x4 , x5 x1
1
8
x6
x7
x2
3
8
5
8
x3 x8 , x9
1
7
8
( 83 , 58 , 85 )
( 18 , 83 , 58 )
x3
( 83 , 38 , 58 )
( 12 , 12 , 12 )
x2
x1
Figure:
1
8
≤ x1 ≤
3
8
∩
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3
8
≤ x2 ≤
5
8
∩
5
8
≤ x3 ≤
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7
8
∩ x2 − x1 ≤
1
4
∩ x3 − x2 ≤
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Main Theorem
We calculated the vertex and edge counts of Gk,n :
k(kn+1)!
V (Gk,n ) = k k k kkn+1
k ... k+1 = (k!)n−1 ·(k+1)
E (Gk,n ) =
2
·V ·(k+1)+ n−2
·V ·2(k+1)
n
n
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)
1
k-Covkn+1 ( 2n
,I) ∼
=h Gk,n , with the above vertex and edge counts.
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Future Work: Extending to higher excesses
Conjecture
1
k-Cov2n+m ( 2n
, 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] is
compact, and contains no points where 5f = 0, then
M a = f −1 [−∞, a] is homotopy equivalent to
M b = f −1 [−∞, b].
Definition (Tautological Function)
Define τ : Covn (r , Y ) → R by τ (~x = (x1 , . . . , xn )) = max min d(xi , y )
y ∈Y 1≤i≤n
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 ) = max min max{d(xi , y ), d(xj , y )}
y ∈Y 1≤i<j≤n
Conjecture
The only critical points of τ occur when the excess changes.
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Excess 0 coverings of S 1
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Space of Single Coverings of the Circle
n balls of radius
1
,
2n
total length n ·
1
n
=1
2
1 Permutations 123, 312, 231 equivalent up to rotation.
3
3
1 Permutations 132, 321, 213 equivalent up to rotation.
2
Theorem
(n−1)!
F
1
Covn ( 2n
, S 1) ∼
S1
=
i=1
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Space of Double Coverings of the Circle, for odd n
n balls of radius n1 , total length n ·
1
2n
=2
2
1
3
Theorem (same reasoning as single-covering case)
(n−1)!
F
2-Covn ( n1 , S 1 ) ∼
S1
=
i=1
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Space of Double Coverings for n = 2, r =
1
2
Both balls cover the entire circle, can be moved independently of each
other.
Theorem
2-Cov2 ( 12 , S 1 ) ∼
= T2
= S1 × S1 ∼
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Space of Double Coverings for n ≥ 4, n even
3
3, 4
1, 2
2
2
1
3
1
4
4
Theorem
2-Cov4 ( 14 , S 1 ) ∼
= 3 tori, glued as below. In general, for n even,
n−2
2(n−1)!
1
1
2-Covn ( n , S ) ∼
tori; each torus glued to n2 · (2 2 − 1) other tori.
=
n
3, 4
2, 4
π1 (2 − Cov4 ( 14 , S 1 )) =
2, 3
Z × F3
1, 2
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1, 3
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1, 4
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Generalization to k-covering, Future Work
Small Result: k-Covk ( 12 , S 1 ) ∼
= Tk.
Conjecture
(n−1)!
F
k-Covn ( kn , S 1 ) ∼
S 1 if k - n.
=
i=1
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 incredibly
helpful 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 the
sustenance (shout out to Danielle!).
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