Configuration spaces

Introduction Homotopy type of S(P) Example
On the Configuration Space of a Certain
n-arms Machine in the Euclidean Space
Y. Kamiyama S. Tsukuda
University of the Ryukyus
19 December 2008 / National University of Singapore

Introduction Homotopy type of S(P) Example
Introduction
An n-arms Machine and the Configuration Space M(P, a)
Results
Homotopy type of S(P)
Short Arms Case
Example
Octahedron

Introduction Homotopy type of S(P) Example
Introduction
An n-arms Machine and the Configuration Space M(P, a)
Results
Homotopy type of S(P)
Short Arms Case
Example
Octahedron

Introduction Homotopy type of S(P) Example
Introduction
An n-arms Machine and the Configuration Space M(P, a)
Results
Homotopy type of S(P)
Short Arms Case
Example
Octahedron

Introduction Homotopy type of S(P) Example
Introduction
An n-arms Machine and the Configuration Space M(P, a)
Results
Homotopy type of S(P)
Short Arms Case
Example
Octahedron

Introduction Homotopy type of S(P) Example
An n-arms Machine
How much freedom does a trapped bug have?
×
×
• •
∩ •
×
• ○
×
•
•
× ×

Introduction Homotopy type of S(P) Example
An n-arms Machine
How much freedom does a trapped bug have?
× ×
• •
∩ × ○ ×
•
•
•
• × ×

Introduction Homotopy type of S(P) Example
An n-arms Machine
a > 0, v 1 , . . . , v n ∈ R d .
v 3
v 2
v 4
v 5 v 6
v 1

Introduction Homotopy type of S(P) Example
An n-arms Machine
a > 0, v 1 , . . . , v n ∈ R d .
We consider an n-arms machine,
which have n two-joined arms of length a,
v 3
v 2
•
•
•
v 4 • v 1
•
•
•
v 5 v 6

Introduction Homotopy type of S(P) Example
An n-arms Machine
a > 0, v 1 , . . . , v n ∈ R d .
We consider an n-arms machine,
which have n two-joined arms of length a,
with the end of the i-th arm fixed at v i .
v 3
v 2
•
•
•
v •
4 •
v 1
•
•
v 5 v 6

Introduction Homotopy type of S(P) Example
The Configuration Space M(P, a)
Denote the configuration space of the n-arms machine by
M(v 1 , . . . , v n , a).

Introduction Homotopy type of S(P) Example
The Configuration Space M(P, a)
Definition
M(v 1 , . . . , v n , a) ⊂ ( R d) n+1

Introduction Homotopy type of S(P) Example
The Configuration Space M(P, a)
Definition
M(v 1 , . . . , v n , a) ⊂ ( R d) n+1
:= { (p 1 , . . . , p n , q) ‖p i − v i ‖ = ‖q − p i ‖ = a 2 , ∀i}

Introduction Homotopy type of S(P) Example
The Configuration Space M(P, a)
Definition
M(v 1 , . . . , v n , a) ⊂ ( R d) n+1
:= { (p 1 , . . . , p n , q) ‖p i − v i ‖ = ‖q − p i ‖ = a 2 , ∀i}
v 3
v 2
p
p 2 3
p
v p 4
4 q 1
v 1
p 5 p 6
v 5 v 6

Introduction Homotopy type of S(P) Example
The Configuration Space M(P, a)
Definition
When {v 1 , . . . , v n } is the set of vertices of a polyhedron P ⊂ R d ,
we denote M(v 1 , . . . , v n , a) by M(P, a).
v 3
v 2
p
p 2 3
p
v p 4
4 q 1
v 1
p 5 p 6
v 5 v 6

Introduction Homotopy type of S(P) Example
Known Results (d = 2)
When d = 2, namely, in R 2 ,
M(v 1 , . . . , v n , a) is an orientable closed surface in many cases.
• D. Eldar, Maps and Machines, Hebrew Univ. MSc project
http://www.math.toronto.edu/~drorbn/People/Eldar/thesis
• [1] D. Blanc, M. Shoham, N. Shvalb
• [3] N. Shvalb, G. Liu, M. Shoham, J. C. Trinkle

Introduction Homotopy type of S(P) Example
Known Results (d = 2)
When d = 2, namely, in R 2 ,
M(v 1 , . . . , v n , a) is an orientable closed surface in many cases.
• D. Eldar, Maps and Machines, Hebrew Univ. MSc project
http://www.math.toronto.edu/~drorbn/People/Eldar/thesis
• [1] D. Blanc, M. Shoham, N. Shvalb
• [3] N. Shvalb, G. Liu, M. Shoham, J. C. Trinkle

Introduction Homotopy type of S(P) Example
Known Results (d ≥ 3)
When d ≥ 3, namely, in R d ,
we determined the homotopy type of M(P, a) in the case when
P ⊂ R 2 ⊂ R d is a regular n-gon and the arms are short ([2]
Y. Kamiyama, T).

Introduction Homotopy type of S(P) Example
Introduction
An n-arms Machine and the Configuration Space M(P, a)
Results
Homotopy type of S(P)
Short Arms Case
Example
Octahedron

Introduction Homotopy type of S(P) Example
In This Talk
We consider the case when d = 3 and
v 3
v 2
p
p 2 3
p
v p 4
4 q 1
v 1
p 5 p 6
v 5 v 6

Introduction Homotopy type of S(P) Example
In This Talk
We consider the case when d = 3 and
P is a regular polyhedron P ⊂ R 3 or
a regular n-gon P ⊂ R 2 × {0} ⊂ R 3 .
v 3
v 2
v 4
v 5 v 6
v 1

Introduction Homotopy type of S(P) Example
In This Talk
We consider the case when d = 3 and
P is a regular polyhedron P ⊂ R 3 or
a regular n-gon P ⊂ R 2 × {0} ⊂ R 3 .
We assume that the center of P is located at the origin of R d .
v 3
v 2
v 4
v 5 v 6
o v 1

Introduction Homotopy type of S(P) Example
In This Talk
Let l(P) and L(P) be the radius and the diameter of P, respectively.
Definition
l(P) := ‖v 1 ‖ (= ‖v i ‖, ∀i)
L(P) := max 1≤i≤n ‖v i − v 1 ‖
v 3
L(P)
v 2
v o v 1
4
l(P)
v 5 v 6

Introduction Homotopy type of S(P) Example
Easy Facts
• M(P, a) = ∅ if a < l(P)
• M(P, a) = {∗} if a = l(P)
v 1
•
v 2 v 3

Introduction Homotopy type of S(P) Example
Easy Facts
• M(P, a) = ∅ if a < l(P)
• M(P, a) = {∗} if a = l(P)
v 1
•
v 2 v 3

Introduction Homotopy type of S(P) Example
Short and Long Arms (Changing a for a Fixed P)
Proposition
1. The topological type of M(P, a) is constant for
l(P) < a < L(P).
2. The topological type of M(P, a) is constant for a > L(P).
We denote M(P, a) by S(P) if l(P) < a < L(P) and by L(P) if
a > L(P).
v 1
•
v 2 v 3

Introduction Homotopy type of S(P) Example
Short and Long Arms (Changing a for a Fixed P)
Proposition
1. The topological type of M(P, a) is constant for
l(P) < a < L(P).
2. The topological type of M(P, a) is constant for a > L(P).
We denote M(P, a) by S(P) if l(P) < a < L(P) and by L(P) if
a > L(P).
v 1
•
v 2 v 3

Introduction Homotopy type of S(P) Example
Homology
Theorem
The integral homology groups H ∗ (S(P)), H ∗ (L(P)) are torsion
free and the Poincaré polynomials are given by
1. Regular n-gon
PS(S(P)) =1 +
n∑
i=4
{
n
( n − 2
i − 3
)
−
( )} n
t i + t n+3
i − 2
PS(L(P)) =1 + t 2 + nt 3 (1 + t) n−2
∑n+1
( ) n
+ (2n + 3 − 2i) t i + t n+3
i − 2
i=2

Introduction Homotopy type of S(P) Example
Homology
Theorem
The integral homology groups H ∗ (S(P)), H ∗ (L(P)) are torsion
free and the Poincaré polynomials are given by
2. Tetrahedron
PS(S(P)) =1 + t 7
PS(L(P)) =1 + 8t 2 + 24t 3 + 24t 4 + 8t 5 + t 7

Introduction Homotopy type of S(P) Example
Homology
Theorem
The integral homology groups H ∗ (S(P)), H ∗ (L(P)) are torsion
free and the Poincaré polynomials are given by
3. Octahedron
PS(S(P)) =1 + 3t 3 + 3t 6 + t 9
PS(L(P)) =1 + 12t 2 + 63t 3 + 120t 4
+ 120t 5 + 63t 6 + 12t 7 + t 9

Introduction Homotopy type of S(P) Example
Homology
Theorem
The integral homology groups H ∗ (S(P)), H ∗ (L(P)) are torsion
free and the Poincaré polynomials are given by
4. Cube
PS(S(P)) =1 + 4t 3 + 8t 5 + 36t 6 + 40t 7 + 16t 8 + t 11
PS(L(P)) =1 + 16t 2 + 116t 3 + 336t 4 + 568t 5
+ 596t 6 + 376t 7 + 128t 8 + 16t 9 + t 11

Introduction Homotopy type of S(P) Example
Homology
Theorem
The integral homology groups H ∗ (S(P)), H ∗ (L(P)) are torsion
free and the Poincaré polynomials are given by
5. Icosahedron and Dodecahedron: omitted

Introduction Homotopy type of S(P) Example
Manifold
Remark
If P is a regular n-gon (n = 2, 3) or a simplicial regular
polyhedron, namely, a tetrahedron, an octahedron or an
icosahedron, then M(P, a) is a smooth closed orientable
manifold. ( D. Blanc, M. Shoham, N. Shvalb [1, Theorem 2.1] )

Introduction Homotopy type of S(P) Example
Introduction
An n-arms Machine and the Configuration Space M(P, a)
Results
Homotopy type of S(P)
Short Arms Case
Example
Octahedron

Introduction Homotopy type of S(P) Example
Homotopy Type of S(P)
Theorem
1. Let P(n) ⊂ R 2 ⊂ R 3 be a regular n-gon. Then there exists
a homotopy equivalence
⎛ ⎞
n−2
∨
S(P(n)) ≃ ⎝ ∨
S i+2 ⎠ ∨ S n+3
i=2
where α(n, i) := n ( n−2
i−1)
−
( n
i
)
.
α(n,i)

Introduction Homotopy type of S(P) Example
Homotopy Type of S(P)
Theorem
2. If P ⊂ R 3 is a regular convex polyhedron, then
⎛
⎞
ΣS(P) ≃ Σ ⎝ ∨
S |I| ∧ S(I ∆ ) ⎠
∅≠I⊂V
where V : the set of verticex of P,
I ∆ : the union of the faces of P ∆ those correspond to
vertices in I

Introduction Homotopy type of S(P) Example
Workspace
Definition
Let D i := { x ∈ R 3 ‖x − v i ‖ ≤ a } ⊂ R 3 be the disc centered at
v i .
The space B = ⋂ n
i=1 D i is called the workspace.
We set F i = B ∩ ∂D i . ∂B is devided into spherical regions
F 1 , . . . , F n , and by this decomposition,
B ∼ = P ∆ if P is a regular polyhedron (P ∆ : the dual of P),
B ∼ = S(P ∆ ∩ R 2 ) if P is a regular n-gon.
F 3
v 1
F 2
v 2
B
v 3

Introduction Homotopy type of S(P) Example
Workspace
Definition
Let D i := { x ∈ R 3 ‖x − v i ‖ ≤ a } ⊂ R 3 be the disc centered at
v i .
The space B = ⋂ n
i=1 D i is called the workspace.
We set F i = B ∩ ∂D i . ∂B is devided into spherical regions
F 1 , . . . , F n , and by this decomposition,
B ∼ = P ∆ if P is a regular polyhedron (P ∆ : the dual of P),
B ∼ = S(P ∆ ∩ R 2 ) if P is a regular n-gon.
F 3
v 1
F 2
v 2
B
v 3

Introduction Homotopy type of S(P) Example
Workspace
Definition
Let D i := { x ∈ R 3 ‖x − v i ‖ ≤ a } ⊂ R 3 be the disc centered at
v i .
The space B = ⋂ n
i=1 D i is called the workspace.
We set F i = B ∩ ∂D i . ∂B is devided into spherical regions
F 1 , . . . , F n , and by this decomposition,
B ∼ = P ∆ if P is a regular polyhedron (P ∆ : the dual of P),
B ∼ = S(P ∆ ∩ R 2 ) if P is a regular n-gon.
F 3
v 1
F 2
v 2
B
v 3

Introduction Homotopy type of S(P) Example
Workspace
Definition
Let D i := { x ∈ R 3 ‖x − v i ‖ ≤ a } ⊂ R 3 be the disc centered at
v i .
The space B = ⋂ n
i=1 D i is called the workspace.
We set F i = B ∩ ∂D i . ∂B is devided into spherical regions
F 1 , . . . , F n , and by this decomposition,
B ∼ = P ∆ if P is a regular polyhedron (P ∆ : the dual of P),
B ∼ = S(P ∆ ∩ R 2 ) if P is a regular n-gon.
F 3
v 1
F 2
v 2
B
v 3

Introduction Homotopy type of S(P) Example
S(P) as a Quotient Space
It is not hard to see that
S(P) ∼ =
(S 1) n /
× B
∼
(1)
where the equivalence relation ∼ is generated by the following
relations:
(x 1 , . . . , x n , q) ∼ (x 1 , . . . , i ∗, . . . , x n , q) if q ∈ F i .

Introduction Homotopy type of S(P) Example
S(P) as a Homotopy Colimit (Source and Target
Categories)
We consider the case when P is a regular convex polyhedron.
• V = {v 1 , . . . , v n } : the set of vertices of P
• P : the face poset of ∂P,namely,
P ⊂ P(V ) : subposet
I ∈ P ⇔ I spans a face of ∂P
¯P = P ∪ {∅} ⊂ P(V ).
• T ∗ : the category of pointed compactly generated
topological spaces

Introduction Homotopy type of S(P) Example
S(P) as a Homotopy Colimit (Source and Target
Categories)
We consider the case when P is a regular convex polyhedron.
• V = {v 1 , . . . , v n } : the set of vertices of P
• P : the face poset of ∂P,namely,
P ⊂ P(V ) : subposet
I ∈ P ⇔ I spans a face of ∂P
¯P = P ∪ {∅} ⊂ P(V ).
• T ∗ : the category of pointed compactly generated
topological spaces

Introduction Homotopy type of S(P) Example
S(P) as a Homotopy Colimit (Source and Target
Categories)
We consider the case when P is a regular convex polyhedron.
• V = {v 1 , . . . , v n } : the set of vertices of P
• P : the face poset of ∂P,namely,
P ⊂ P(V ) : subposet
I ∈ P ⇔ I spans a face of ∂P
¯P = P ∪ {∅} ⊂ P(V ).
• T ∗ : the category of pointed compactly generated
topological spaces

Introduction Homotopy type of S(P) Example
S(P) as a Homotopy Colimit (Source and Target
Categories)
We consider the case when P is a regular convex polyhedron.
• V = {v 1 , . . . , v n } : the set of vertices of P
• P : the face poset of ∂P,namely,
P ⊂ P(V ) : subposet
I ∈ P ⇔ I spans a face of ∂P
¯P = P ∪ {∅} ⊂ P(V ).
• T ∗ : the category of pointed compactly generated
topological spaces

Introduction Homotopy type of S(P) Example
S(P) as a Homotopy Colimit (Source and Target
Categories)
We consider the case when P is a regular convex polyhedron.
• V = {v 1 , . . . , v n } : the set of vertices of P
• P : the face poset of ∂P,namely,
P ⊂ P(V ) : subposet
I ∈ P ⇔ I spans a face of ∂P
¯P = P ∪ {∅} ⊂ P(V ).
• T ∗ : the category of pointed compactly generated
topological spaces

Introduction Homotopy type of S(P) Example
S(P) as a Homotopy Colimit (Source and Target
Categories)
We consider the case when P is a regular convex polyhedron.
• V = {v 1 , . . . , v n } : the set of vertices of P
• P : the face poset of ∂P,namely,
P ⊂ P(V ) : subposet
I ∈ P ⇔ I spans a face of ∂P
¯P = P ∪ {∅} ⊂ P(V ).
• T ∗ : the category of pointed compactly generated
topological spaces

Introduction Homotopy type of S(P) Example
S(P) as a Homotopy Colimit (Functor)
We define a functor ¯X: ¯P → T ∗ as follows:
• For each I ∈ ¯P,
¯X(I) =
(
S 1) n−|I|
= {(x1 , . . . , x n )
x i = ∗ if v i ∈ I} ⊂
(S 1) n
• For each I ⊂ J ∈ ¯P, the canonical projection
¯X(I ⊂ J): ¯X(I) =
(S 1) n−|I|
→
(
S 1) n−|J|
= ¯X(J)

Introduction Homotopy type of S(P) Example
S(P) as a Homotopy Colimit (Functor)
We define a functor ¯X: ¯P → T ∗ as follows:
• For each I ∈ ¯P,
¯X(I) =
(
S 1) n−|I|
= {(x1 , . . . , x n )
x i = ∗ if v i ∈ I} ⊂
(S 1) n
• For each I ⊂ J ∈ ¯P, the canonical projection
¯X(I ⊂ J): ¯X(I) =
(S 1) n−|I|
→
(
S 1) n−|J|
= ¯X(J)

Introduction Homotopy type of S(P) Example
S(P) as a Homotopy Colimit (Functor)
We define a functor ¯X: ¯P → T ∗ as follows:
• For each I ∈ ¯P,
¯X(I) =
(
S 1) n−|I|
= {(x1 , . . . , x n )
x i = ∗ if v i ∈ I} ⊂
(S 1) n
• For each I ⊂ J ∈ ¯P, the canonical projection
¯X(I ⊂ J): ¯X(I) =
(S 1) n−|I|
→
(
S 1) n−|J|
= ¯X(J)

Introduction Homotopy type of S(P) Example
S(P) as a Homotopy Colimit
Then, we have a homeomorphism
hocolim ¯X ∼ = S(P)
where hocolim ¯X is the unpointed homotopy colimit.

Introduction Homotopy type of S(P) Example
Proof of the Theorem
Since B ¯P is contractible,
hocolim ¯X ≃ hocolim ∗ ¯X
( hocolim ∗ ¯X is the pointed homotopy colimit.)

Introduction Homotopy type of S(P) Example
Proof of the Theorem
In T ∗ ,
Σ¯X ≃
∨
ΣȲI
I∈P(V )
where ȲI : ¯P → T ∗ is given as follows:
If I ≠ ∅,
• For each J ∈ ¯P,
Ȳ I (J) =
{
∗, I ∩ J ≠ ∅
S |I| ,
I ∩ J = ∅
(2)
• Arrows are collapsing maps.
If I = ∅,
Ȳ ∅ (J) = ∗,
∀J ∈ ¯P.

Introduction Homotopy type of S(P) Example
Proof of the Theorem
In T ∗ ,
Σ¯X ≃
∨
ΣȲI
I∈P(V )
where ȲI : ¯P → T ∗ is given as follows:
If I ≠ ∅,
• For each J ∈ ¯P,
Ȳ I (J) =
{
∗, I ∩ J ≠ ∅
S |I| ,
I ∩ J = ∅
(2)
• Arrows are collapsing maps.
If I = ∅,
Ȳ ∅ (J) = ∗,
∀J ∈ ¯P.

Introduction Homotopy type of S(P) Example
Proof of the Theorem
In T ∗ ,
Σ¯X ≃
∨
ΣȲI
I∈P(V )
where ȲI : ¯P → T ∗ is given as follows:
If I ≠ ∅,
• For each J ∈ ¯P,
Ȳ I (J) =
{
∗, I ∩ J ≠ ∅
S |I| ,
I ∩ J = ∅
(2)
• Arrows are collapsing maps.
If I = ∅,
Ȳ ∅ (J) = ∗,
∀J ∈ ¯P.

Introduction Homotopy type of S(P) Example
Proof of the Theorem
In T ∗ ,
Σ¯X ≃
∨
ΣȲI
I∈P(V )
where ȲI : ¯P → T ∗ is given as follows:
If I ≠ ∅,
• For each J ∈ ¯P,
Ȳ I (J) =
{
∗, I ∩ J ≠ ∅
S |I| ,
I ∩ J = ∅
(2)
• Arrows are collapsing maps.
If I = ∅,
Ȳ ∅ (J) = ∗,
∀J ∈ ¯P.

Introduction Homotopy type of S(P) Example
Proof of the Theorem
ΣS(P) ≃ Σ hocolim ¯X
(S(P) ∼ = hocolim ¯X)
≃ Σ hocolim ∗ ¯X
(hocolim ¯X ≃ hocolim∗ ¯X)
≃ hocolim ∗ Σ¯X (Σ hocolim ∗ ≃ hocolim ∗ Σ)
∨
≃ hocolim ∗ ΣȲI
(Σ¯X ≃ ∨ΣȲI)
≃ Σ
∨
I∈P(V )
I∈P(V )
hocolim ∗ Ȳ I (hocolim ∗ ∨Σ ≃ Σ ∨ hocolim ∗ )

Introduction Homotopy type of S(P) Example
Proof of the Theorem
ΣS(P) ≃ Σ hocolim ¯X
(S(P) ∼ = hocolim ¯X)
≃ Σ hocolim ∗ ¯X
(hocolim ¯X ≃ hocolim∗ ¯X)
≃ hocolim ∗ Σ¯X (Σ hocolim ∗ ≃ hocolim ∗ Σ)
∨
≃ hocolim ∗ ΣȲI
(Σ¯X ≃ ∨ΣȲI)
≃ Σ
∨
I∈P(V )
I∈P(V )
hocolim ∗ Ȳ I (hocolim ∗ ∨Σ ≃ Σ ∨ hocolim ∗ )

Introduction Homotopy type of S(P) Example
Proof of the Theorem
ΣS(P) ≃ Σ hocolim ¯X
(S(P) ∼ = hocolim ¯X)
≃ Σ hocolim ∗ ¯X
(hocolim ¯X ≃ hocolim∗ ¯X)
≃ hocolim ∗ Σ¯X (Σ hocolim ∗ ≃ hocolim ∗ Σ)
∨
≃ hocolim ∗ ΣȲI
(Σ¯X ≃ ∨ΣȲI)
≃ Σ
∨
I∈P(V )
I∈P(V )
hocolim ∗ Ȳ I (hocolim ∗ ∨Σ ≃ Σ ∨ hocolim ∗ )

Introduction Homotopy type of S(P) Example
Proof of the Theorem
ΣS(P) ≃ Σ hocolim ¯X
(S(P) ∼ = hocolim ¯X)
≃ Σ hocolim ∗ ¯X
(hocolim ¯X ≃ hocolim∗ ¯X)
≃ hocolim ∗ Σ¯X (Σ hocolim ∗ ≃ hocolim ∗ Σ)
∨
≃ hocolim ∗ ΣȲI
(Σ¯X ≃ ∨ΣȲI)
≃ Σ
∨
I∈P(V )
I∈P(V )
hocolim ∗ Ȳ I (hocolim ∗ ∨Σ ≃ Σ ∨ hocolim ∗ )

Introduction Homotopy type of S(P) Example
Proof of the Theorem
ΣS(P) ≃ Σ hocolim ¯X
(S(P) ∼ = hocolim ¯X)
≃ Σ hocolim ∗ ¯X
(hocolim ¯X ≃ hocolim∗ ¯X)
≃ hocolim ∗ Σ¯X (Σ hocolim ∗ ≃ hocolim ∗ Σ)
∨
≃ hocolim ∗ ΣȲI
(Σ¯X ≃ ∨ΣȲI)
≃ Σ
∨
I∈P(V )
I∈P(V )
hocolim ∗ Ȳ I (hocolim ∗ ∨Σ ≃ Σ ∨ hocolim ∗ )

Introduction Homotopy type of S(P) Example
Proof of the Theorem (hocolim ∗ Ȳ)
Let St(I) be the union of the closed star neghbourhoods in BP
of vertices in I.
St(I) ∼ = I ∆
C(St(I)): the cone on St(I). By the definition of Ȳ (2), if I ≠ ∅,
hocolim ∗ Ȳ I
∼ =
≃
S |I| × B¯P
S |I| × St(I) ∪ ∗ × B ¯P
S |I| × C(St(I))
S |I| × St(I) ∪ ∗ × C(St(I))
S ∼=
|I| × S(St(I))
S |I| × ∗ ∪ ∗ × S(St(I))
∼= S |I| ∧ S(St(I))
∼= S |I| ∧ S(I ∆ )

Introduction Homotopy type of S(P) Example
Proof of the Theorem (hocolim ∗ Ȳ)
Let St(I) be the union of the closed star neghbourhoods in BP
of vertices in I.
St(I) ∼ = I ∆
C(St(I)): the cone on St(I). By the definition of Ȳ (2), if I ≠ ∅,
hocolim ∗ Ȳ I
∼ =
≃
S |I| × B¯P
S |I| × St(I) ∪ ∗ × B ¯P
S |I| × C(St(I))
S |I| × St(I) ∪ ∗ × C(St(I))
S ∼=
|I| × S(St(I))
S |I| × ∗ ∪ ∗ × S(St(I))
∼= S |I| ∧ S(St(I))
∼= S |I| ∧ S(I ∆ )

Introduction Homotopy type of S(P) Example
Proof of the Theorem (hocolim ∗ Ȳ)
Let St(I) be the union of the closed star neghbourhoods in BP
of vertices in I.
St(I) ∼ = I ∆
C(St(I)): the cone on St(I). By the definition of Ȳ (2), if I ≠ ∅,
hocolim ∗ Ȳ I
∼ =
≃
S |I| × B¯P
S |I| × St(I) ∪ ∗ × B ¯P
S |I| × C(St(I))
S |I| × St(I) ∪ ∗ × C(St(I))
S ∼=
|I| × S(St(I))
S |I| × ∗ ∪ ∗ × S(St(I))
∼= S |I| ∧ S(St(I))
∼= S |I| ∧ S(I ∆ )

Introduction Homotopy type of S(P) Example
Proof of the Theorem (hocolim ∗ Ȳ)
Let St(I) be the union of the closed star neghbourhoods in BP
of vertices in I.
St(I) ∼ = I ∆
C(St(I)): the cone on St(I). By the definition of Ȳ (2), if I ≠ ∅,
hocolim ∗ Ȳ I
∼ =
≃
S |I| × B¯P
S |I| × St(I) ∪ ∗ × B ¯P
S |I| × C(St(I))
S |I| × St(I) ∪ ∗ × C(St(I))
S ∼=
|I| × S(St(I))
S |I| × ∗ ∪ ∗ × S(St(I))
∼= S |I| ∧ S(St(I))
∼= S |I| ∧ S(I ∆ )

Introduction Homotopy type of S(P) Example
Proof of the Theorem (hocolim ∗ Ȳ)
Let St(I) be the union of the closed star neghbourhoods in BP
of vertices in I.
St(I) ∼ = I ∆
C(St(I)): the cone on St(I). By the definition of Ȳ (2), if I ≠ ∅,
hocolim ∗ Ȳ I
∼ =
≃
S |I| × B¯P
S |I| × St(I) ∪ ∗ × B ¯P
S |I| × C(St(I))
S |I| × St(I) ∪ ∗ × C(St(I))
S ∼=
|I| × S(St(I))
S |I| × ∗ ∪ ∗ × S(St(I))
∼= S |I| ∧ S(St(I))
∼= S |I| ∧ S(I ∆ )

Introduction Homotopy type of S(P) Example
Proof of the Theorem (hocolim ∗ Ȳ)
Let St(I) be the union of the closed star neghbourhoods in BP
of vertices in I.
St(I) ∼ = I ∆
C(St(I)): the cone on St(I). By the definition of Ȳ (2), if I ≠ ∅,
hocolim ∗ Ȳ I
∼ =
≃
S |I| × B¯P
S |I| × St(I) ∪ ∗ × B ¯P
S |I| × C(St(I))
S |I| × St(I) ∪ ∗ × C(St(I))
S ∼=
|I| × S(St(I))
S |I| × ∗ ∪ ∗ × S(St(I))
∼= S |I| ∧ S(St(I))
∼= S |I| ∧ S(I ∆ )

Introduction Homotopy type of S(P) Example
Introduction
An n-arms Machine and the Configuration Space M(P, a)
Results
Homotopy type of S(P)
Short Arms Case
Example
Octahedron

Introduction Homotopy type of S(P) Example
Octahedron
P =
1
3
4
5
2
6

Introduction Homotopy type of S(P) Example
Table
|I| I ♯ S(I ∆ )
1 [ 1 ] 6 ∗
2 [ 1, 2 ] 12 ∗
2 [ 1, 6 ] 3 S 1
3 [ 1, 2, 3 ] 8 ∗
3 [ 1, 2, 4 ] 12 ∗
4 [ 1, 2, 3, 4 ] 12 ∗
4 [ 1, 2, 4, 6 ] 3 S 2
5 [ 1, 2, 3, 4, 5 ] 6 ∗

Introduction Homotopy type of S(P) Example
Thank you very much

Introduction Homotopy type of S(P) Example
Thank you very much
for this nice conference!

Introduction Homotopy type of S(P) Example
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Y. Kamiyama and S. Tsukuda, The configuration space of
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N. Shvalb, G. Liu, M. Shoham and J. C. Trinkle, Motion
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http://www.math.toronto.edu/˜drorbn/People/Eldar/thesis