Accelariting Applied Algebraic Topology String Topological ...myismail.net/docs/doctorat/exposes/2016/AalborgTalk.pdf · Our Main Goal String Topology (1999) Topological Robotics

String Topological Robotics

Accelariting Applied Algebraic Topology

Aalborg Univ., Denmark

Associate Professor (Habilite)CRMEF Rabat, Moroccohttp://[email protected]

My Ismail Mamouni

My Ismail Mamouni Aalborg Univ., Denmark April 14, 2016 1 / 32

Joint work with

Derfoufi YounesFac. Sc. Meknes, Morocco

Joint work with


Our Main Goal

String Topology (1999) Topological Robotics (2003)

M. Chas D. Sullivan M. Farber

Marrying


1 Topological Robotics (brief recalling)2 Intersection Product3 String Topology (brief recalling)4 Strings through Topological Robotics5 Acknowledgements

Content


Topological Robotics (brief recalling)

Topological Robotics

X = path-connected topological space.

Viewed as a configuration space of all states of a mechanicalsystem (a robot for example).

Context

How the topology of X interfers through the complexity ofthe motion of the robot.

Motivation




X = path-connected topological space.

Viewed as a configuration space of all states of a mechanicalsystem (a robot for example).

Context

How the topology of X interfers through the complexity ofthe motion of the robot.

Motivation




MPA = s : X 2 −→ PX a continuous sectionof the bi-evaluation at times t = 0 and t = 1ev0,1 : PX −→ X 2

γ 7−→ (γ(0), γ(1)).

Motion Planner algorithm (MPA)

Input = (x , y) a pair of points in X ;

Output = s(x , y) a path=motion from x to y .

Interpretation




MPA = s : X 2 −→ PX a continuous sectionof the bi-evaluation at times t = 0 and t = 1ev0,1 : PX −→ X 2

γ 7−→ (γ(0), γ(1)).

Motion Planner algorithm (MPA)


Output = s(x , y) a path=motion from x to y .

Interpretation




MPA exist iff X is contractible

M. Farber (2003)

Continuity of MPA = close initial-final pairs produceclose motions s(x , y) and s(x ′, y ′).

From discontinuity will result an instability of therobot’s motion.

Interpretation




MPA exist iff X is contractible

M. Farber (2003)

Continuity of MPA = close initial-final pairs produceclose motions s(x , y) and s(x ′, y ′).

From discontinuity will result an instability of therobot’s motion.

Interpretation


Intersection Product


Y ,Z some embedded orientable submanifolds of X .

dimY = i , dimZ = j with Y ∩ Z 6= ∅.

Context

TxY + TxZ = TxX for all x ∈ Y ∩ Z .

Intersection is transverse :






Context








Context






If Y and Z intersect transversally then, Y ∩Z is an orientablesubmanifold of X of with dimY ∩ Z = i + j − n.

Remark:

: Hi (X )⊗ Hj(X ) −→ Hi+j−n(X )[Y ]⊗ [Z ] 7−→ [Y ] · [Z ] := [Y ∩ Z ]

Definition:

Here [−], denotes the homological fundamental class that represents thenamed submanifold.




If Y and Z intersect transversally then, Y ∩Z is an orientablesubmanifold of X of with dimY ∩ Z = i + j − n.

Remark:

: Hi (X )⊗ Hj(X ) −→ Hi+j−n(X )[Y ]⊗ [Z ] 7−→ [Y ] · [Z ] := [Y ∩ Z ]

Definition:

Here [−], denotes the homological fundamental class that represents thenamed submanifold.



Mapping Intersection Product

Two smooth maps f : Y −→ X , g : Z −→ X are called transversein X when Im(Df (y)) + Im(Dg(z)) = TxM for all y , z , x such thatf (y) = g(z) = x . Therefore

Y ×X Z := {(y , z) ∈ Y × Z , f (y) = g(z)}

is naturally a smooth manifold .

f ∩ g : Y ×X Z −→ X(y , z) 7−→ f (y) = g(z)

is the intersection product of the named maps.

Definition:




Y × Zp1

++p2

""

Y ×X Zi

hh

//

��

f ∩g

##

Z

g

��Y

f// X

[f ].[g ] := [f ∩ g ] is well defined at level of homology.

Remark:




Y × Zp1

++p2

""

Y ×X Zi

hh

//

��

f ∩g

##

Z

g

��Y

f// X

[f ].[g ] := [f ∩ g ] is well defined at level of homology.

Remark:


String Topology (brief recalling)

String Topology

X = closed and orientable n-manifold.

Context

Study algebraic structures on the homology of the loop spaceof a manifold.

Motivation



String Topology

X = closed and orientable n-manifold.

Context

Study algebraic structures on the homology of the loop spaceof a manifold.

Motivation



String Topology

Use evaluation at time t = 0 to define MappingIntersection Product for simplicesΣ : ∆i −→ LX := X S1

.

extend it at level of the regraded homologyH∗(LX ) := H∗+n(LX ;Z) to get a structure ofassociative graded commutative algebra.

define at level of homology H∗(LX ) a loop bracket toget a structure of Gerstenhaber algebra.

define at level of homology H∗(LX ) an operator ∆ toget a structure of Batalin-Vilkovisky algebra.

Machinery


Strings through Topological Robotics

String Topological Robotics

X path-connected n-manifold, compact or not,orientable or not

Compact Lie group acting transitively on X .

Context


Strings through Topological Robotics Loop Motion Planning Algorithms (LMPA)

Loop Motion Planning Algorithms (LMPA)

LMPA = continuous section s : X × X −→ LX of the loopbi-evaluation

evLP : LX −→ X × Xγ 7−→ (γ(0), γ( 1

2 )).

Derfoufi, M. (2015)


Output = s(x , y) a goings and free comings

Interpretation

The motion of a drone like an unmanned warplane or aguided TV camera;

The famous NP-complete problem of vehicle routingwith pick-up and delivery

Applications Areas

LMPAs exist iff X is contractible.

Derfoufi, M. (2015)





evLP : LX −→ X × Xγ 7−→ (γ(0), γ( 1

2 )).

Derfoufi, M. (2015)



Interpretation



Applications Areas


Derfoufi, M. (2015)





evLP : LX −→ X × Xγ 7−→ (γ(0), γ( 1

2 )).

Derfoufi, M. (2015)



Interpretation



Applications Areas


Derfoufi, M. (2015)





evLP : LX −→ X × Xγ 7−→ (γ(0), γ( 1

2 )).

Derfoufi, M. (2015)



Interpretation



Applications Areas


Derfoufi, M. (2015)



Goal

MLP(X ) denotes the set of all LMPAs on X

Notation

Study algebraic structures of the homology of the loop motionplanners.

Motivation

First Obstacle = MLP(X ) is contractible.

Derfoufi, M. (2015)



Goal


Notation


Motivation


Derfoufi, M. (2015)



Goal


Notation


Motivation


Derfoufi, M. (2015)



Issue

PX is replaced by PX×X/G PX

Where PX ×X/G PX := {(γ, τ) ∈ PX × PX , G .γ(0) = G .τ(1)}.

Lubawski’s & Marzantowicz’s definition of MPA



String Topological Robotics : Step 1

LMPA = continuous section s : X × X −→ LX of the loopbi-evaluation evLP : LX ×X/G LX −→ X × X

(γ, τ) 7−→ (γ(0), τ( 12 ))

.

Definition 1

LMPA product of two LMPA s1, s2 is:

µ(s1, s2)(x , y)(t) = s1(x , y)(t) if 0 ≤ t ≤ 12

= s1(x , y)(3t − 1) if 12 ≤ t ≤ 2

3= s2(x , y)(3t − 2) if 2

3 ≤ t ≤ 1

(MLP(X ), µ) is a monoid (up to a homotopy).

Definition 2




LMPA = continuous section s : X × X −→ LX of the loopbi-evaluation evLP : LX ×X/G LX −→ X × X

(γ, τ) 7−→ (γ(0), τ( 12 ))

.

Definition 1

LMPA product of two LMPA s1, s2 is:

µ(s1, s2)(x , y)(t) = s1(x , y)(t) if 0 ≤ t ≤ 12

= s1(x , y)(3t − 1) if 12 ≤ t ≤ 2

3= s2(x , y)(3t − 2) if 2

3 ≤ t ≤ 1

(MLP(X ), µ) is a monoid (up to a homotopy).

Definition 2




the bi-evaluation

ev : MLP(X ) −→ X 2

s 7−→ (s(−,−)(0), s(−,−)(1/2)),

relates any i-simplex Σ : ∆i −→ MLP(X ) of MLP(X ) tothe i-simplex σ := ev(Σ) : ∆i −→ X 2 of X 2.

Link simplices of MLP(X ) to that of X 2




Equiped X 2 with an atlas A of charts.

A i-simplex σ : ∆i −→ X 2 is said to be small, when itspreimage is contained in a chart from A (chosen oncefor all and denoted U(Σ)).

A i-simplex Σ : ∆i −→MLP(X ) is said to be small ifthe associated i-simplex σ := ev(Σ) : ∆i −→ X 2.

A small (i , j)-bi-simplexΣ×Θ : ∆i ×∆j −→MLP(X )×MLP(X ) is said to betransverse in MLP(X ), when σ × θ and all its faces aretransverse in X 4 to the diagonal map

∆X 2 : X 2 −→ X 4

(x , y) 7−→ (x , y , x , y).

Transeversality at level of chains of MLP(X )




According to Whitehead’s theorems,

W := (∆X 2 ◦ (σ × θ))−1 (X 4).

is an sub-manifold of ∆i × ∆j with corners and of dimen-sion i + j − 2n. Thus W can be smoothly triangulated bya PL-triangulation chosen so that the triangulated faces aresubcomplexes. Moreover, the use of the charts U(Σ) andU(Σ) whose product contains the image of σ × θ yields acanonical orientation on W .

Key Remark




(δ1, δ2) ∈W ⊂ ∆i ×∆j Σ×Θ //

σ×θ ++

(s1, s2) ∈MLP(X )×MLP(X )

ev×ev��

∆X 2 ⊂ X 2 × X 2

Illustrative diagram

Σ.Θ : W ' ∆i+j−2n −→ MLP(X )(δ1, δ2) 7−→ µ(s1, s2)

Intresection LMPA product




(δ1, δ2) ∈W ⊂ ∆i ×∆j Σ×Θ //

σ×θ ++

(s1, s2) ∈MLP(X )×MLP(X )

ev×ev��

∆X 2 ⊂ X 2 × X 2


Σ.Θ : W ' ∆i+j−2n −→ MLP(X )(δ1, δ2) 7−→ µ(s1, s2)

Intresection LMPA product




∆i+j−2n

Σ.Θ : interection LMP product

**Σ×Θ //

σ×θ**

MLP(X )×MLP(X )

ev×ev��

µ //MLP(X )

X 2


Σ.Θ := µ ◦ (Σ×Θ)|W is well defined.

Definition




∆i+j−2n

Σ.Θ : interection LMP product

**Σ×Θ //

σ×θ**

MLP(X )×MLP(X )

ev×ev��

µ //MLP(X )

X 2


Σ.Θ := µ ◦ (Σ×Θ)|W is well defined.

Definition




Boundary operator : Set ∂Σ :=∑i

k=0 εk(−1)kFkΣ.Here εk is the sign of the Jacobian of the coordinates changeU(ev(FkΣ)) −→ U(ev(Σ)) and Fk denotes the k-th face.

Set H∗(MLP) := H∗(MLP(X ), ∂), where coefficients aretaken in Z.

Put [Σ].[Γ] = [Σ.Γ].

Extend Intersection LMPA product at level of homology




Boundary operator : Set ∂Σ :=∑i

k=0 εk(−1)kFkΣ.Here εk is the sign of the Jacobian of the coordinates changeU(ev(FkΣ)) −→ U(ev(Σ)) and Fk denotes the k-th face.

Set H∗(MLP) := H∗(MLP(X ), ∂), where coefficients aretaken in Z.

Put [Σ].[Γ] = [Σ.Γ].

Extend Intersection LMPA product at level of homology




Any bi-cycle can be represented, up to a boundary preservinghomotopy, by a transverse bi-cycle, and that it does notdepend on the choice of the homological representants.

The homological class [Σ.Γ] does not depend on the choice ofthe homological representants.

String LMPA Product=Homology Intersection LMPA is well defined




The shifted homology H∗(MLP(X )) := H∗+2n(MLP(X )) equippedwith the the string loop motion product is an associative and com-mutative graded algebra

Derfoufi, M. (2016)

define at level of homology H∗(MLP(X )) a loop bracket toget a structure of Gerstenhaber algebra.

define at level of homology H∗(MLP(X )) an operator ∆ toget a structure of Batalin-Vilkovisky algebra.

Investigate examples and applications.

In progress




The shifted homology H∗(MLP(X )) := H∗+2n(MLP(X )) equippedwith the the string loop motion product is an associative and com-mutative graded algebra

Derfoufi, M. (2016)

define at level of homology H∗(MLP(X )) a loop bracket toget a structure of Gerstenhaber algebra.

define at level of homology H∗(MLP(X )) an operator ∆ toget a structure of Batalin-Vilkovisky algebra.

Investigate examples and applications.

In progress



M. Chas, D. Sullivan, String Topology,arXiv:math/9911159 [math.GT].

M. Farber, Topological complexity of motion planning,Discrete Comput. Geom., vol. 29 (2003), no. 2,211-221.

F. Laudenbach, A note on the Chas-Sullivan product,L’Enseignement Mathematique, Vol. 57, Issue 1-2(2011), 3-21.

W. Lubawski, W. Marzantowicz, Invariant topologicalcomplexity, Bull. London Math. Soc., vol. 47 (2015)101-117.


Acknowledgements

Acknowledgements

Organizers


Acknowledgements

For pointing out our attention to the Lubawski’s andMarzantowicz’s work

Zbigniew Baszczyk, Pozna, Poland


Acknowledgements

Acknowledgements

For pointing out our attention to the Laudenbach’s work

David Chataur, Lille, France


Acknowledgements

AcknowledgementsMAAT, Moroccan Research Group


Acknowledgements

MAAT: Moroccan Area of Algebraic Topology

Born: 2012;

Logo:

Home Page: http://algtop.net

Members: 4 professors, 1 PhD, 7 PhD students, 60 Master students;


Acknowledgements

Questions or Comments are accepted in

slowly formulated


Acknowledgements


Documents

Accelariting Applied Algebraic Topology String Topological ...myismail.net/docs/doctorat/exposes/2016/AalborgTalk.pdf · Our Main Goal String Topology (1999) Topological Robotics