Configuration spaces
Configuration spaces
Configuration spaces
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Introduction Homotopy type of S(P) Example<br />
On the <strong>Configuration</strong> Space of a Certain<br />
n-arms Machine in the Euclidean Space<br />
Y. Kamiyama S. Tsukuda<br />
University of the Ryukyus<br />
19 December 2008 / National University of Singapore
Introduction Homotopy type of S(P) Example<br />
Introduction<br />
An n-arms Machine and the <strong>Configuration</strong> Space M(P, a)<br />
Results<br />
Homotopy type of S(P)<br />
Short Arms Case<br />
Example<br />
Octahedron
Introduction Homotopy type of S(P) Example<br />
Introduction<br />
An n-arms Machine and the <strong>Configuration</strong> Space M(P, a)<br />
Results<br />
Homotopy type of S(P)<br />
Short Arms Case<br />
Example<br />
Octahedron
Introduction Homotopy type of S(P) Example<br />
Introduction<br />
An n-arms Machine and the <strong>Configuration</strong> Space M(P, a)<br />
Results<br />
Homotopy type of S(P)<br />
Short Arms Case<br />
Example<br />
Octahedron
Introduction Homotopy type of S(P) Example<br />
Introduction<br />
An n-arms Machine and the <strong>Configuration</strong> Space M(P, a)<br />
Results<br />
Homotopy type of S(P)<br />
Short Arms Case<br />
Example<br />
Octahedron
Introduction Homotopy type of S(P) Example<br />
An n-arms Machine<br />
How much freedom does a trapped bug have?<br />
× <br />
× <br />
• •<br />
<br />
<br />
∩ • <br />
×<br />
• ○ <br />
×<br />
•<br />
<br />
• <br />
<br />
× ×
Introduction Homotopy type of S(P) Example<br />
An n-arms Machine<br />
How much freedom does a trapped bug have?<br />
× × <br />
• •<br />
<br />
<br />
∩ × ○ ×<br />
•<br />
<br />
<br />
• <br />
•<br />
<br />
<br />
• × ×
Introduction Homotopy type of S(P) Example<br />
An n-arms Machine<br />
a > 0, v 1 , . . . , v n ∈ R d .<br />
v 3<br />
v 2<br />
v 4<br />
v 5 v 6<br />
v 1
Introduction Homotopy type of S(P) Example<br />
<br />
An n-arms Machine<br />
a > 0, v 1 , . . . , v n ∈ R d .<br />
We consider an n-arms machine,<br />
which have n two-joined arms of length a,<br />
v 3<br />
v 2<br />
<br />
•<br />
<br />
• <br />
<br />
• <br />
<br />
v 4 • v 1<br />
•<br />
<br />
<br />
<br />
• <br />
•<br />
<br />
<br />
<br />
v 5 v 6
Introduction Homotopy type of S(P) Example<br />
An n-arms Machine<br />
a > 0, v 1 , . . . , v n ∈ R d .<br />
We consider an n-arms machine,<br />
which have n two-joined arms of length a,<br />
with the end of the i-th arm fixed at v i .<br />
v 3<br />
v 2<br />
<br />
•<br />
<br />
• <br />
• <br />
v • <br />
4 • <br />
v 1<br />
•<br />
<br />
• <br />
<br />
v 5 v 6
Introduction Homotopy type of S(P) Example<br />
The <strong>Configuration</strong> Space M(P, a)<br />
Denote the configuration space of the n-arms machine by<br />
M(v 1 , . . . , v n , a).
Introduction Homotopy type of S(P) Example<br />
The <strong>Configuration</strong> Space M(P, a)<br />
Definition<br />
M(v 1 , . . . , v n , a) ⊂ ( R d) n+1
Introduction Homotopy type of S(P) Example<br />
The <strong>Configuration</strong> Space M(P, a)<br />
Definition<br />
M(v 1 , . . . , v n , a) ⊂ ( R d) n+1<br />
:= { (p 1 , . . . , p n , q) ‖p i − v i ‖ = ‖q − p i ‖ = a 2 , ∀i}
Introduction Homotopy type of S(P) Example<br />
The <strong>Configuration</strong> Space M(P, a)<br />
Definition<br />
M(v 1 , . . . , v n , a) ⊂ ( R d) n+1<br />
:= { (p 1 , . . . , p n , q) ‖p i − v i ‖ = ‖q − p i ‖ = a 2 , ∀i}<br />
v 3<br />
v 2<br />
<br />
p<br />
p 2 3 <br />
p<br />
v p 4 <br />
<br />
4 q 1 <br />
v 1<br />
<br />
<br />
p 5 p 6<br />
<br />
<br />
<br />
v 5 v 6
Introduction Homotopy type of S(P) Example<br />
The <strong>Configuration</strong> Space M(P, a)<br />
Definition<br />
When {v 1 , . . . , v n } is the set of vertices of a polyhedron P ⊂ R d ,<br />
we denote M(v 1 , . . . , v n , a) by M(P, a).<br />
v 3<br />
v 2<br />
<br />
p<br />
p 2 3 <br />
p<br />
v p 4 <br />
<br />
4 q 1 <br />
v 1<br />
<br />
<br />
p 5 p 6<br />
<br />
<br />
<br />
v 5 v 6
Introduction Homotopy type of S(P) Example<br />
Known Results (d = 2)<br />
When d = 2, namely, in R 2 ,<br />
M(v 1 , . . . , v n , a) is an orientable closed surface in many cases.<br />
• D. Eldar, Maps and Machines, Hebrew Univ. MSc project<br />
http://www.math.toronto.edu/~drorbn/People/Eldar/thesis<br />
• [1] D. Blanc, M. Shoham, N. Shvalb<br />
• [3] N. Shvalb, G. Liu, M. Shoham, J. C. Trinkle
Introduction Homotopy type of S(P) Example<br />
Known Results (d = 2)<br />
When d = 2, namely, in R 2 ,<br />
M(v 1 , . . . , v n , a) is an orientable closed surface in many cases.<br />
• D. Eldar, Maps and Machines, Hebrew Univ. MSc project<br />
http://www.math.toronto.edu/~drorbn/People/Eldar/thesis<br />
• [1] D. Blanc, M. Shoham, N. Shvalb<br />
• [3] N. Shvalb, G. Liu, M. Shoham, J. C. Trinkle
Introduction Homotopy type of S(P) Example<br />
Known Results (d ≥ 3)<br />
When d ≥ 3, namely, in R d ,<br />
we determined the homotopy type of M(P, a) in the case when<br />
P ⊂ R 2 ⊂ R d is a regular n-gon and the arms are short ([2]<br />
Y. Kamiyama, T).
Introduction Homotopy type of S(P) Example<br />
Introduction<br />
An n-arms Machine and the <strong>Configuration</strong> Space M(P, a)<br />
Results<br />
Homotopy type of S(P)<br />
Short Arms Case<br />
Example<br />
Octahedron
Introduction Homotopy type of S(P) Example<br />
In This Talk<br />
We consider the case when d = 3 and<br />
v 3<br />
v 2<br />
<br />
p<br />
p 2 3 <br />
p<br />
v p 4 <br />
<br />
4 q 1 <br />
v 1<br />
<br />
<br />
p 5 p 6<br />
<br />
<br />
<br />
v 5 v 6
Introduction Homotopy type of S(P) Example<br />
In This Talk<br />
We consider the case when d = 3 and<br />
P is a regular polyhedron P ⊂ R 3 or<br />
a regular n-gon P ⊂ R 2 × {0} ⊂ R 3 .<br />
v 3<br />
v 2<br />
v 4<br />
v 5 v 6<br />
v 1
Introduction Homotopy type of S(P) Example<br />
In This Talk<br />
We consider the case when d = 3 and<br />
P is a regular polyhedron P ⊂ R 3 or<br />
a regular n-gon P ⊂ R 2 × {0} ⊂ R 3 .<br />
We assume that the center of P is located at the origin of R d .<br />
v 3<br />
v 2<br />
v 4<br />
v 5 v 6<br />
o v 1
Introduction Homotopy type of S(P) Example<br />
In This Talk<br />
Let l(P) and L(P) be the radius and the diameter of P, respectively.<br />
Definition<br />
l(P) := ‖v 1 ‖ (= ‖v i ‖, ∀i)<br />
L(P) := max 1≤i≤n ‖v i − v 1 ‖<br />
v 3<br />
L(P)<br />
v 2<br />
v o v 1<br />
4<br />
l(P)<br />
v 5 v 6
Introduction Homotopy type of S(P) Example<br />
Easy Facts<br />
• M(P, a) = ∅ if a < l(P)<br />
• M(P, a) = {∗} if a = l(P)<br />
v 1<br />
•<br />
v 2 v 3
Introduction Homotopy type of S(P) Example<br />
Easy Facts<br />
• M(P, a) = ∅ if a < l(P)<br />
• M(P, a) = {∗} if a = l(P)<br />
v 1<br />
•<br />
v 2 v 3
Introduction Homotopy type of S(P) Example<br />
Short and Long Arms (Changing a for a Fixed P)<br />
Proposition<br />
1. The topological type of M(P, a) is constant for<br />
l(P) < a < L(P).<br />
2. The topological type of M(P, a) is constant for a > L(P).<br />
We denote M(P, a) by S(P) if l(P) < a < L(P) and by L(P) if<br />
a > L(P).<br />
v 1<br />
•<br />
v 2 v 3
Introduction Homotopy type of S(P) Example<br />
Short and Long Arms (Changing a for a Fixed P)<br />
Proposition<br />
1. The topological type of M(P, a) is constant for<br />
l(P) < a < L(P).<br />
2. The topological type of M(P, a) is constant for a > L(P).<br />
We denote M(P, a) by S(P) if l(P) < a < L(P) and by L(P) if<br />
a > L(P).<br />
v 1<br />
•<br />
v 2 v 3
Introduction Homotopy type of S(P) Example<br />
Homology<br />
Theorem<br />
The integral homology groups H ∗ (S(P)), H ∗ (L(P)) are torsion<br />
free and the Poincaré polynomials are given by<br />
1. Regular n-gon<br />
PS(S(P)) =1 +<br />
n∑<br />
i=4<br />
{<br />
n<br />
( n − 2<br />
i − 3<br />
)<br />
−<br />
( )} n<br />
t i + t n+3<br />
i − 2<br />
PS(L(P)) =1 + t 2 + nt 3 (1 + t) n−2<br />
∑n+1<br />
( ) n<br />
+ (2n + 3 − 2i) t i + t n+3<br />
i − 2<br />
i=2
Introduction Homotopy type of S(P) Example<br />
Homology<br />
Theorem<br />
The integral homology groups H ∗ (S(P)), H ∗ (L(P)) are torsion<br />
free and the Poincaré polynomials are given by<br />
2. Tetrahedron<br />
PS(S(P)) =1 + t 7<br />
PS(L(P)) =1 + 8t 2 + 24t 3 + 24t 4 + 8t 5 + t 7
Introduction Homotopy type of S(P) Example<br />
Homology<br />
Theorem<br />
The integral homology groups H ∗ (S(P)), H ∗ (L(P)) are torsion<br />
free and the Poincaré polynomials are given by<br />
3. Octahedron<br />
PS(S(P)) =1 + 3t 3 + 3t 6 + t 9<br />
PS(L(P)) =1 + 12t 2 + 63t 3 + 120t 4<br />
+ 120t 5 + 63t 6 + 12t 7 + t 9
Introduction Homotopy type of S(P) Example<br />
Homology<br />
Theorem<br />
The integral homology groups H ∗ (S(P)), H ∗ (L(P)) are torsion<br />
free and the Poincaré polynomials are given by<br />
4. Cube<br />
PS(S(P)) =1 + 4t 3 + 8t 5 + 36t 6 + 40t 7 + 16t 8 + t 11<br />
PS(L(P)) =1 + 16t 2 + 116t 3 + 336t 4 + 568t 5<br />
+ 596t 6 + 376t 7 + 128t 8 + 16t 9 + t 11
Introduction Homotopy type of S(P) Example<br />
Homology<br />
Theorem<br />
The integral homology groups H ∗ (S(P)), H ∗ (L(P)) are torsion<br />
free and the Poincaré polynomials are given by<br />
5. Icosahedron and Dodecahedron: omitted
Introduction Homotopy type of S(P) Example<br />
Manifold<br />
Remark<br />
If P is a regular n-gon (n = 2, 3) or a simplicial regular<br />
polyhedron, namely, a tetrahedron, an octahedron or an<br />
icosahedron, then M(P, a) is a smooth closed orientable<br />
manifold. ( D. Blanc, M. Shoham, N. Shvalb [1, Theorem 2.1] )
Introduction Homotopy type of S(P) Example<br />
Introduction<br />
An n-arms Machine and the <strong>Configuration</strong> Space M(P, a)<br />
Results<br />
Homotopy type of S(P)<br />
Short Arms Case<br />
Example<br />
Octahedron
Introduction Homotopy type of S(P) Example<br />
Homotopy Type of S(P)<br />
Theorem<br />
1. Let P(n) ⊂ R 2 ⊂ R 3 be a regular n-gon. Then there exists<br />
a homotopy equivalence<br />
⎛ ⎞<br />
n−2<br />
∨<br />
S(P(n)) ≃ ⎝ ∨<br />
S i+2 ⎠ ∨ S n+3<br />
i=2<br />
where α(n, i) := n ( n−2<br />
i−1)<br />
−<br />
( n<br />
i<br />
)<br />
.<br />
α(n,i)
Introduction Homotopy type of S(P) Example<br />
Homotopy Type of S(P)<br />
Theorem<br />
2. If P ⊂ R 3 is a regular convex polyhedron, then<br />
⎛<br />
⎞<br />
ΣS(P) ≃ Σ ⎝ ∨<br />
S |I| ∧ S(I ∆ ) ⎠<br />
∅≠I⊂V<br />
where V : the set of verticex of P,<br />
I ∆ : the union of the faces of P ∆ those correspond to<br />
vertices in I
Introduction Homotopy type of S(P) Example<br />
Workspace<br />
Definition<br />
Let D i := { x ∈ R 3 ‖x − v i ‖ ≤ a } ⊂ R 3 be the disc centered at<br />
v i .<br />
The space B = ⋂ n<br />
i=1 D i is called the workspace.<br />
We set F i = B ∩ ∂D i . ∂B is devided into spherical regions<br />
F 1 , . . . , F n , and by this decomposition,<br />
B ∼ = P ∆ if P is a regular polyhedron (P ∆ : the dual of P),<br />
B ∼ = S(P ∆ ∩ R 2 ) if P is a regular n-gon.<br />
F 3<br />
v 1<br />
F 2<br />
v 2<br />
B<br />
v 3
Introduction Homotopy type of S(P) Example<br />
Workspace<br />
Definition<br />
Let D i := { x ∈ R 3 ‖x − v i ‖ ≤ a } ⊂ R 3 be the disc centered at<br />
v i .<br />
The space B = ⋂ n<br />
i=1 D i is called the workspace.<br />
We set F i = B ∩ ∂D i . ∂B is devided into spherical regions<br />
F 1 , . . . , F n , and by this decomposition,<br />
B ∼ = P ∆ if P is a regular polyhedron (P ∆ : the dual of P),<br />
B ∼ = S(P ∆ ∩ R 2 ) if P is a regular n-gon.<br />
F 3<br />
v 1<br />
F 2<br />
v 2<br />
B<br />
v 3
Introduction Homotopy type of S(P) Example<br />
Workspace<br />
Definition<br />
Let D i := { x ∈ R 3 ‖x − v i ‖ ≤ a } ⊂ R 3 be the disc centered at<br />
v i .<br />
The space B = ⋂ n<br />
i=1 D i is called the workspace.<br />
We set F i = B ∩ ∂D i . ∂B is devided into spherical regions<br />
F 1 , . . . , F n , and by this decomposition,<br />
B ∼ = P ∆ if P is a regular polyhedron (P ∆ : the dual of P),<br />
B ∼ = S(P ∆ ∩ R 2 ) if P is a regular n-gon.<br />
F 3<br />
v 1<br />
F 2<br />
v 2<br />
B<br />
v 3
Introduction Homotopy type of S(P) Example<br />
Workspace<br />
Definition<br />
Let D i := { x ∈ R 3 ‖x − v i ‖ ≤ a } ⊂ R 3 be the disc centered at<br />
v i .<br />
The space B = ⋂ n<br />
i=1 D i is called the workspace.<br />
We set F i = B ∩ ∂D i . ∂B is devided into spherical regions<br />
F 1 , . . . , F n , and by this decomposition,<br />
B ∼ = P ∆ if P is a regular polyhedron (P ∆ : the dual of P),<br />
B ∼ = S(P ∆ ∩ R 2 ) if P is a regular n-gon.<br />
F 3<br />
v 1<br />
F 2<br />
v 2<br />
B<br />
v 3
Introduction Homotopy type of S(P) Example<br />
S(P) as a Quotient Space<br />
It is not hard to see that<br />
S(P) ∼ =<br />
(S 1) n /<br />
× B<br />
∼<br />
(1)<br />
where the equivalence relation ∼ is generated by the following<br />
relations:<br />
(x 1 , . . . , x n , q) ∼ (x 1 , . . . , i ∗, . . . , x n , q) if q ∈ F i .
Introduction Homotopy type of S(P) Example<br />
S(P) as a Homotopy Colimit (Source and Target<br />
Categories)<br />
We consider the case when P is a regular convex polyhedron.<br />
• V = {v 1 , . . . , v n } : the set of vertices of P<br />
• P : the face poset of ∂P,namely,<br />
P ⊂ P(V ) : subposet<br />
I ∈ P ⇔ I spans a face of ∂P<br />
¯P = P ∪ {∅} ⊂ P(V ).<br />
• T ∗ : the category of pointed compactly generated<br />
topological <strong>spaces</strong>
Introduction Homotopy type of S(P) Example<br />
S(P) as a Homotopy Colimit (Source and Target<br />
Categories)<br />
We consider the case when P is a regular convex polyhedron.<br />
• V = {v 1 , . . . , v n } : the set of vertices of P<br />
• P : the face poset of ∂P,namely,<br />
P ⊂ P(V ) : subposet<br />
I ∈ P ⇔ I spans a face of ∂P<br />
¯P = P ∪ {∅} ⊂ P(V ).<br />
• T ∗ : the category of pointed compactly generated<br />
topological <strong>spaces</strong>
Introduction Homotopy type of S(P) Example<br />
S(P) as a Homotopy Colimit (Source and Target<br />
Categories)<br />
We consider the case when P is a regular convex polyhedron.<br />
• V = {v 1 , . . . , v n } : the set of vertices of P<br />
• P : the face poset of ∂P,namely,<br />
P ⊂ P(V ) : subposet<br />
I ∈ P ⇔ I spans a face of ∂P<br />
¯P = P ∪ {∅} ⊂ P(V ).<br />
• T ∗ : the category of pointed compactly generated<br />
topological <strong>spaces</strong>
Introduction Homotopy type of S(P) Example<br />
S(P) as a Homotopy Colimit (Source and Target<br />
Categories)<br />
We consider the case when P is a regular convex polyhedron.<br />
• V = {v 1 , . . . , v n } : the set of vertices of P<br />
• P : the face poset of ∂P,namely,<br />
P ⊂ P(V ) : subposet<br />
I ∈ P ⇔ I spans a face of ∂P<br />
¯P = P ∪ {∅} ⊂ P(V ).<br />
• T ∗ : the category of pointed compactly generated<br />
topological <strong>spaces</strong>
Introduction Homotopy type of S(P) Example<br />
S(P) as a Homotopy Colimit (Source and Target<br />
Categories)<br />
We consider the case when P is a regular convex polyhedron.<br />
• V = {v 1 , . . . , v n } : the set of vertices of P<br />
• P : the face poset of ∂P,namely,<br />
P ⊂ P(V ) : subposet<br />
I ∈ P ⇔ I spans a face of ∂P<br />
¯P = P ∪ {∅} ⊂ P(V ).<br />
• T ∗ : the category of pointed compactly generated<br />
topological <strong>spaces</strong>
Introduction Homotopy type of S(P) Example<br />
S(P) as a Homotopy Colimit (Source and Target<br />
Categories)<br />
We consider the case when P is a regular convex polyhedron.<br />
• V = {v 1 , . . . , v n } : the set of vertices of P<br />
• P : the face poset of ∂P,namely,<br />
P ⊂ P(V ) : subposet<br />
I ∈ P ⇔ I spans a face of ∂P<br />
¯P = P ∪ {∅} ⊂ P(V ).<br />
• T ∗ : the category of pointed compactly generated<br />
topological <strong>spaces</strong>
Introduction Homotopy type of S(P) Example<br />
S(P) as a Homotopy Colimit (Functor)<br />
We define a functor ¯X: ¯P → T ∗ as follows:<br />
• For each I ∈ ¯P,<br />
¯X(I) =<br />
(<br />
S 1) n−|I|<br />
= {(x1 , . . . , x n )<br />
x i = ∗ if v i ∈ I} ⊂<br />
(S 1) n<br />
• For each I ⊂ J ∈ ¯P, the canonical projection<br />
¯X(I ⊂ J): ¯X(I) =<br />
(S 1) n−|I|<br />
→<br />
(<br />
S 1) n−|J|<br />
= ¯X(J)
Introduction Homotopy type of S(P) Example<br />
S(P) as a Homotopy Colimit (Functor)<br />
We define a functor ¯X: ¯P → T ∗ as follows:<br />
• For each I ∈ ¯P,<br />
¯X(I) =<br />
(<br />
S 1) n−|I|<br />
= {(x1 , . . . , x n )<br />
x i = ∗ if v i ∈ I} ⊂<br />
(S 1) n<br />
• For each I ⊂ J ∈ ¯P, the canonical projection<br />
¯X(I ⊂ J): ¯X(I) =<br />
(S 1) n−|I|<br />
→<br />
(<br />
S 1) n−|J|<br />
= ¯X(J)
Introduction Homotopy type of S(P) Example<br />
S(P) as a Homotopy Colimit (Functor)<br />
We define a functor ¯X: ¯P → T ∗ as follows:<br />
• For each I ∈ ¯P,<br />
¯X(I) =<br />
(<br />
S 1) n−|I|<br />
= {(x1 , . . . , x n )<br />
x i = ∗ if v i ∈ I} ⊂<br />
(S 1) n<br />
• For each I ⊂ J ∈ ¯P, the canonical projection<br />
¯X(I ⊂ J): ¯X(I) =<br />
(S 1) n−|I|<br />
→<br />
(<br />
S 1) n−|J|<br />
= ¯X(J)
Introduction Homotopy type of S(P) Example<br />
S(P) as a Homotopy Colimit<br />
Then, we have a homeomorphism<br />
hocolim ¯X ∼ = S(P)<br />
where hocolim ¯X is the unpointed homotopy colimit.
Introduction Homotopy type of S(P) Example<br />
Proof of the Theorem<br />
Since B ¯P is contractible,<br />
hocolim ¯X ≃ hocolim ∗ ¯X<br />
( hocolim ∗ ¯X is the pointed homotopy colimit.)
Introduction Homotopy type of S(P) Example<br />
Proof of the Theorem<br />
In T ∗ ,<br />
Σ¯X ≃<br />
∨<br />
ΣȲI<br />
I∈P(V )<br />
where ȲI : ¯P → T ∗ is given as follows:<br />
If I ≠ ∅,<br />
• For each J ∈ ¯P,<br />
Ȳ I (J) =<br />
{<br />
∗, I ∩ J ≠ ∅<br />
S |I| ,<br />
I ∩ J = ∅<br />
(2)<br />
• Arrows are collapsing maps.<br />
If I = ∅,<br />
Ȳ ∅ (J) = ∗,<br />
∀J ∈ ¯P.
Introduction Homotopy type of S(P) Example<br />
Proof of the Theorem<br />
In T ∗ ,<br />
Σ¯X ≃<br />
∨<br />
ΣȲI<br />
I∈P(V )<br />
where ȲI : ¯P → T ∗ is given as follows:<br />
If I ≠ ∅,<br />
• For each J ∈ ¯P,<br />
Ȳ I (J) =<br />
{<br />
∗, I ∩ J ≠ ∅<br />
S |I| ,<br />
I ∩ J = ∅<br />
(2)<br />
• Arrows are collapsing maps.<br />
If I = ∅,<br />
Ȳ ∅ (J) = ∗,<br />
∀J ∈ ¯P.
Introduction Homotopy type of S(P) Example<br />
Proof of the Theorem<br />
In T ∗ ,<br />
Σ¯X ≃<br />
∨<br />
ΣȲI<br />
I∈P(V )<br />
where ȲI : ¯P → T ∗ is given as follows:<br />
If I ≠ ∅,<br />
• For each J ∈ ¯P,<br />
Ȳ I (J) =<br />
{<br />
∗, I ∩ J ≠ ∅<br />
S |I| ,<br />
I ∩ J = ∅<br />
(2)<br />
• Arrows are collapsing maps.<br />
If I = ∅,<br />
Ȳ ∅ (J) = ∗,<br />
∀J ∈ ¯P.
Introduction Homotopy type of S(P) Example<br />
Proof of the Theorem<br />
In T ∗ ,<br />
Σ¯X ≃<br />
∨<br />
ΣȲI<br />
I∈P(V )<br />
where ȲI : ¯P → T ∗ is given as follows:<br />
If I ≠ ∅,<br />
• For each J ∈ ¯P,<br />
Ȳ I (J) =<br />
{<br />
∗, I ∩ J ≠ ∅<br />
S |I| ,<br />
I ∩ J = ∅<br />
(2)<br />
• Arrows are collapsing maps.<br />
If I = ∅,<br />
Ȳ ∅ (J) = ∗,<br />
∀J ∈ ¯P.
Introduction Homotopy type of S(P) Example<br />
Proof of the Theorem<br />
ΣS(P) ≃ Σ hocolim ¯X<br />
(S(P) ∼ = hocolim ¯X)<br />
≃ Σ hocolim ∗ ¯X<br />
(hocolim ¯X ≃ hocolim∗ ¯X)<br />
≃ hocolim ∗ Σ¯X (Σ hocolim ∗ ≃ hocolim ∗ Σ)<br />
∨<br />
≃ hocolim ∗ ΣȲI<br />
(Σ¯X ≃ ∨ΣȲI)<br />
≃ Σ<br />
∨<br />
I∈P(V )<br />
I∈P(V )<br />
hocolim ∗ Ȳ I (hocolim ∗ ∨Σ ≃ Σ ∨ hocolim ∗ )
Introduction Homotopy type of S(P) Example<br />
Proof of the Theorem<br />
ΣS(P) ≃ Σ hocolim ¯X<br />
(S(P) ∼ = hocolim ¯X)<br />
≃ Σ hocolim ∗ ¯X<br />
(hocolim ¯X ≃ hocolim∗ ¯X)<br />
≃ hocolim ∗ Σ¯X (Σ hocolim ∗ ≃ hocolim ∗ Σ)<br />
∨<br />
≃ hocolim ∗ ΣȲI<br />
(Σ¯X ≃ ∨ΣȲI)<br />
≃ Σ<br />
∨<br />
I∈P(V )<br />
I∈P(V )<br />
hocolim ∗ Ȳ I (hocolim ∗ ∨Σ ≃ Σ ∨ hocolim ∗ )
Introduction Homotopy type of S(P) Example<br />
Proof of the Theorem<br />
ΣS(P) ≃ Σ hocolim ¯X<br />
(S(P) ∼ = hocolim ¯X)<br />
≃ Σ hocolim ∗ ¯X<br />
(hocolim ¯X ≃ hocolim∗ ¯X)<br />
≃ hocolim ∗ Σ¯X (Σ hocolim ∗ ≃ hocolim ∗ Σ)<br />
∨<br />
≃ hocolim ∗ ΣȲI<br />
(Σ¯X ≃ ∨ΣȲI)<br />
≃ Σ<br />
∨<br />
I∈P(V )<br />
I∈P(V )<br />
hocolim ∗ Ȳ I (hocolim ∗ ∨Σ ≃ Σ ∨ hocolim ∗ )
Introduction Homotopy type of S(P) Example<br />
Proof of the Theorem<br />
ΣS(P) ≃ Σ hocolim ¯X<br />
(S(P) ∼ = hocolim ¯X)<br />
≃ Σ hocolim ∗ ¯X<br />
(hocolim ¯X ≃ hocolim∗ ¯X)<br />
≃ hocolim ∗ Σ¯X (Σ hocolim ∗ ≃ hocolim ∗ Σ)<br />
∨<br />
≃ hocolim ∗ ΣȲI<br />
(Σ¯X ≃ ∨ΣȲI)<br />
≃ Σ<br />
∨<br />
I∈P(V )<br />
I∈P(V )<br />
hocolim ∗ Ȳ I (hocolim ∗ ∨Σ ≃ Σ ∨ hocolim ∗ )
Introduction Homotopy type of S(P) Example<br />
Proof of the Theorem<br />
ΣS(P) ≃ Σ hocolim ¯X<br />
(S(P) ∼ = hocolim ¯X)<br />
≃ Σ hocolim ∗ ¯X<br />
(hocolim ¯X ≃ hocolim∗ ¯X)<br />
≃ hocolim ∗ Σ¯X (Σ hocolim ∗ ≃ hocolim ∗ Σ)<br />
∨<br />
≃ hocolim ∗ ΣȲI<br />
(Σ¯X ≃ ∨ΣȲI)<br />
≃ Σ<br />
∨<br />
I∈P(V )<br />
I∈P(V )<br />
hocolim ∗ Ȳ I (hocolim ∗ ∨Σ ≃ Σ ∨ hocolim ∗ )
Introduction Homotopy type of S(P) Example<br />
Proof of the Theorem (hocolim ∗ Ȳ)<br />
Let St(I) be the union of the closed star neghbourhoods in BP<br />
of vertices in I.<br />
St(I) ∼ = I ∆<br />
C(St(I)): the cone on St(I). By the definition of Ȳ (2), if I ≠ ∅,<br />
hocolim ∗ Ȳ I<br />
∼ =<br />
≃<br />
S |I| × B¯P<br />
S |I| × St(I) ∪ ∗ × B ¯P<br />
S |I| × C(St(I))<br />
S |I| × St(I) ∪ ∗ × C(St(I))<br />
S ∼=<br />
|I| × S(St(I))<br />
S |I| × ∗ ∪ ∗ × S(St(I))<br />
∼= S |I| ∧ S(St(I))<br />
∼= S |I| ∧ S(I ∆ )
Introduction Homotopy type of S(P) Example<br />
Proof of the Theorem (hocolim ∗ Ȳ)<br />
Let St(I) be the union of the closed star neghbourhoods in BP<br />
of vertices in I.<br />
St(I) ∼ = I ∆<br />
C(St(I)): the cone on St(I). By the definition of Ȳ (2), if I ≠ ∅,<br />
hocolim ∗ Ȳ I<br />
∼ =<br />
≃<br />
S |I| × B¯P<br />
S |I| × St(I) ∪ ∗ × B ¯P<br />
S |I| × C(St(I))<br />
S |I| × St(I) ∪ ∗ × C(St(I))<br />
S ∼=<br />
|I| × S(St(I))<br />
S |I| × ∗ ∪ ∗ × S(St(I))<br />
∼= S |I| ∧ S(St(I))<br />
∼= S |I| ∧ S(I ∆ )
Introduction Homotopy type of S(P) Example<br />
Proof of the Theorem (hocolim ∗ Ȳ)<br />
Let St(I) be the union of the closed star neghbourhoods in BP<br />
of vertices in I.<br />
St(I) ∼ = I ∆<br />
C(St(I)): the cone on St(I). By the definition of Ȳ (2), if I ≠ ∅,<br />
hocolim ∗ Ȳ I<br />
∼ =<br />
≃<br />
S |I| × B¯P<br />
S |I| × St(I) ∪ ∗ × B ¯P<br />
S |I| × C(St(I))<br />
S |I| × St(I) ∪ ∗ × C(St(I))<br />
S ∼=<br />
|I| × S(St(I))<br />
S |I| × ∗ ∪ ∗ × S(St(I))<br />
∼= S |I| ∧ S(St(I))<br />
∼= S |I| ∧ S(I ∆ )
Introduction Homotopy type of S(P) Example<br />
Proof of the Theorem (hocolim ∗ Ȳ)<br />
Let St(I) be the union of the closed star neghbourhoods in BP<br />
of vertices in I.<br />
St(I) ∼ = I ∆<br />
C(St(I)): the cone on St(I). By the definition of Ȳ (2), if I ≠ ∅,<br />
hocolim ∗ Ȳ I<br />
∼ =<br />
≃<br />
S |I| × B¯P<br />
S |I| × St(I) ∪ ∗ × B ¯P<br />
S |I| × C(St(I))<br />
S |I| × St(I) ∪ ∗ × C(St(I))<br />
S ∼=<br />
|I| × S(St(I))<br />
S |I| × ∗ ∪ ∗ × S(St(I))<br />
∼= S |I| ∧ S(St(I))<br />
∼= S |I| ∧ S(I ∆ )
Introduction Homotopy type of S(P) Example<br />
Proof of the Theorem (hocolim ∗ Ȳ)<br />
Let St(I) be the union of the closed star neghbourhoods in BP<br />
of vertices in I.<br />
St(I) ∼ = I ∆<br />
C(St(I)): the cone on St(I). By the definition of Ȳ (2), if I ≠ ∅,<br />
hocolim ∗ Ȳ I<br />
∼ =<br />
≃<br />
S |I| × B¯P<br />
S |I| × St(I) ∪ ∗ × B ¯P<br />
S |I| × C(St(I))<br />
S |I| × St(I) ∪ ∗ × C(St(I))<br />
S ∼=<br />
|I| × S(St(I))<br />
S |I| × ∗ ∪ ∗ × S(St(I))<br />
∼= S |I| ∧ S(St(I))<br />
∼= S |I| ∧ S(I ∆ )
Introduction Homotopy type of S(P) Example<br />
Proof of the Theorem (hocolim ∗ Ȳ)<br />
Let St(I) be the union of the closed star neghbourhoods in BP<br />
of vertices in I.<br />
St(I) ∼ = I ∆<br />
C(St(I)): the cone on St(I). By the definition of Ȳ (2), if I ≠ ∅,<br />
hocolim ∗ Ȳ I<br />
∼ =<br />
≃<br />
S |I| × B¯P<br />
S |I| × St(I) ∪ ∗ × B ¯P<br />
S |I| × C(St(I))<br />
S |I| × St(I) ∪ ∗ × C(St(I))<br />
S ∼=<br />
|I| × S(St(I))<br />
S |I| × ∗ ∪ ∗ × S(St(I))<br />
∼= S |I| ∧ S(St(I))<br />
∼= S |I| ∧ S(I ∆ )
Introduction Homotopy type of S(P) Example<br />
Introduction<br />
An n-arms Machine and the <strong>Configuration</strong> Space M(P, a)<br />
Results<br />
Homotopy type of S(P)<br />
Short Arms Case<br />
Example<br />
Octahedron
Introduction Homotopy type of S(P) Example<br />
Octahedron<br />
P =<br />
1 <br />
<br />
<br />
3<br />
<br />
<br />
4 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
5 <br />
2<br />
6
Introduction Homotopy type of S(P) Example<br />
Table<br />
|I| I ♯ S(I ∆ )<br />
1 [ 1 ] 6 ∗<br />
2 [ 1, 2 ] 12 ∗<br />
2 [ 1, 6 ] 3 S 1<br />
3 [ 1, 2, 3 ] 8 ∗<br />
3 [ 1, 2, 4 ] 12 ∗<br />
4 [ 1, 2, 3, 4 ] 12 ∗<br />
4 [ 1, 2, 4, 6 ] 3 S 2<br />
5 [ 1, 2, 3, 4, 5 ] 6 ∗
Introduction Homotopy type of S(P) Example<br />
Thank you very much
Introduction Homotopy type of S(P) Example<br />
Thank you very much<br />
for this nice conference!
Introduction Homotopy type of S(P) Example<br />
D. Blanc, M. Shoham and N. Shvalb, The configuration<br />
space of arachnoid mechanisms, Forum Math. 17 (2005),<br />
1033–1042.<br />
Y. Kamiyama and S. Tsukuda, The configuration space of<br />
the n-arms machine in the Euclidean space, Topology Appl.<br />
154 (2007), 1447–1464.<br />
N. Shvalb, G. Liu, M. Shoham and J. C. Trinkle, Motion<br />
planning for a class of planar closed-chain manipulators,<br />
Proceedings of the 2006 IEEE International Conference on<br />
Robotics and Automation, Orlando, Florida, (2006),<br />
133–138<br />
D. Eldar, Maps and Machines, Hebrew Univ. MSc project,<br />
http://www.math.toronto.edu/˜drorbn/People/Eldar/thesis