A new approach to the equivariant topological complexity

Transkrypt

Topological complexity
A new approach to the equivariant topological
complexity
Wojciech Lubawski
(Jagiellonian University)
Applied Topology, June 23, 2013
Wojciech Lubawski(Jagiellonian University)
Basic definition
Lusternik-Schnirelmann category
Whitehead definition
Picture 1 : A machanical robot arm
Basic definition
Basic definition
configuration space X := {(X1 , . . . , X6 ) | Xi ∈ S1 ⊆ C}.
Basic definition
Definition
For PX := {γ : [0, 1] → X } we define π : PX → X × X as
π(γ) := (γ(0), γ(1)).
Basic definition
Definition
For PX := {γ : [0, 1] → X } we define π : PX → X × X as
π(γ) := (γ(0), γ(1)).
Definition (Motion planning algorithm)
By a motion planning algorithm for a topological space X we mean
a continuous map
s : U → PX
where U ⊆ X × X is open and π ◦ s = idU .
Basic definition
Definition (Farber)
Topological complexity of X is the minimal number n such that
there exists a covering U1 , . . . , Un of X × X such that for each i
there exists a motion planning algorithm si : Ui → PX .
Basic definition
Definition (Farber)
We denote the topological complexity by TC (X ).
Basic definition
Definition (Farber)
We denote the topological complexity by TC (X ).
Example (Farber)
For the sphere Sn we have that
(
2
TC (Sn ) =
3
for n odd
for n even.
Basic definition
How can we modify the topological complexity if the mechanical
robot arm admits a symmetry?
Basic definition
In our case there is a G := Z /2 = {1, t} symmetry.
Picture 3 : The action of t
Basic definition
In our case there is a G := Z /2 = {1, t} symmetry.
Picture 3 : The action of t
Using the angle notation we have
t : X 3 (X1 , X2 , X3 , X4 , X5 , X6 ) 7→ (X1 , X2 , X4 , X3 , X6 , X5 ) ∈ X .
Basic definition
Let us consider an arbitrary path γ ∈ PX between states x and y .
Since X admits a symmetry we would like to exploit that
symmetry in the definition of the motion planning algorithm.
Basic definition
Whenever we choose a path γ between points x and y in X we
would like it to determine:
Basic definition
a path tγ between tx and ty in X ;
Basic definition
this approach leads us to the definition of equivariant
topological complexity of Colman and Grant
Basic definition
: States x and tx.
Basic definition
a ”path” η between tx and y ;
a ”path” tη between x and ty ;
Basic definition
a ”path” η between tx and y ;
a ”path” tη between x and ty ;
this approach leads us to a completely new definition of
symmetric topological complexity.
Basic definition
Let X be a G -space. The map
π : PX → X × X
is G -equivariant.
Basic definition
π : PX → X × X
is G -equivariant.
Definition
Let X be a G -space. An equivariant motion planning algorithm is
a G -map
s : U → PX
for an open invariant U ⊆ X × X which satisfies π ◦ s = idU .
Basic definition
π : PX → X × X
is G -equivariant.
Definition
a G -map
s : U → PX
Definition (Colman, Grant)
Equivariant topological complexity of X is the smallest n such that
there exists invariant open covering U1 , . . . , Un of X × X such that
for each i there exists an equivariant motion planning algorithm
si : Ui → PX .
Basic definition
π : PX → X × X
is G -equivariant.
Definition
a G -map
s : U → PX
Definition (Colman, Grant)
Equivariant topological complexity of X is the smallest n such that
there exists invariant open covering U1 , . . . , Un of X × X such that
for each i there exists an equivariant motion planning algorithm
si : Ui → PX .
We denote the equiv. topological complexity of X by TCG (X ).
Basic definition
The main problem with the second approach is that in the
definition of the motion planning algorithm s : U → PX for an
open U ⊆ X × X the space PX is a G -space whereas X × X is a
G -space as well as a G × G -space.
Basic definition
Here we suggest the following solution.
Basic definition
Definition
Let k(X ) := (G × G ) · ∆(X ). We define a G × G space
PX ×k(X ) PX := {(γ, δ) ∈ PX × PX | G · γ(1) = G · δ(0)}
Basic definition
Definition
Let k(X ) := (G × G ) · ∆(X ). We define a G × G space
PX ×k(X ) PX := {(γ, δ) ∈ PX × PX | G · γ(1) = G · δ(0)}
and a G × G -map
p : PX ×k(X ) PX 3 (γ, δ) 7→ (γ(0), δ(1)) ∈ X × X .
Basic definition
Definition
The symmetric motion planning algorithm is a G × G -map
s : U → PX ×k(X ) PX
for an open invariant subset U ⊆ X × X such that p ◦ s = idU .
Basic definition
Definition
Definition (L., Marzantowicz)
The symmetric topological complexity of X is the smallest n such
that there exists an open invariant cover U1 , . . . , Un of X × X
such that for each i there exists si : Ui → PX ×k(X ) PX a
symmetric motion planning algorithm.
Basic definition
Definition
Definition (L., Marzantowicz)
The symmetric topological complexity of X is the smallest n such
that there exists an open invariant cover U1 , . . . , Un of X × X
such that for each i there exists si : Ui → PX ×k(X ) PX a
symmetric motion planning algorithm.
We denote the symmetric topological complexity by STCG (X ).
Basic definition
Definition
Let A ⊆ X be a closed G -subset of a G -space X . An open
G -subset U ⊆ X will be called G -compressible into A whenever the
inclusion map ιU : U ⊆ X is G -homotopic to a G -map c : U → X
such that c(U) ⊆ A.
Basic definition
Definition
If G = ∗ and A = ∗ then A-compressable sets are called
contractible in X .
Basic definition
Definition
contractible in X .
Definition (Clapp, Puppe in nonequivariant case)
For a given G -subset A ⊆ X the A-Lusternik-Schnirelmann
G -category of a G space X is the smallest n such that X can be
covered by U1 , . . . , Un open G -subsets of X each G -compressible
into A.
Basic definition
Definition
contractible in X .
Definition (Clapp, Puppe in nonequivariant case)
For a given G -subset A ⊆ X the A-Lusternik-Schnirelmann
G -category of a G space X is the smallest n such that X can be
covered by U1 , . . . , Un open G -subsets of X each G -compressible
into A.
We denote the A-LS G -category of X by A catG (X ).
Basic definition
Remark
If X is path connected and the action of G on X is trivial
then {∗} catG (X ) = cat(X ) for every ∗ ∈ X .
Basic definition
Remark
If X is G -connected and ∗ ∈ X G then {∗} catG (X ) = catG (X )
where catG (X ) denotes the equivariant
Lusternik-Schnirelmann category of X .
Basic definition
Remark
If X is G -connected and ∗ ∈ X G then {∗} catG (X ) = catG (X )
where catG (X ) denotes the equivariant
Lusternik-Schnirelmann category of X .
A catG (X )
is an invariant of G -homotopy type of (X , A).
Basic definition
Proposition (Colman, Grant)
For a G -space X the following statements are equivalent:
1) TCG (X ) 6 n;
2) there exist n invariant open sets U1 , . . . , Un which cover X × X
and s¯i : Ui → PX such that p ◦ s¯i is G -homotopic to
Ui → X × X ;
Basic definition
Proposition (Colman, Grant)
1) TCG (X ) 6 n;
2) there exist n invariant open sets U1 , . . . , Un which cover X × X
and s¯i : Ui → PX such that p ◦ s¯i is G -homotopic to
Ui → X × X ;
3)
∆(X ) catG (X
× X ) 6 n.
Basic definition
Proposition
1) STCG (X ) 6 n;
2) there exist n G × G invariant open sets U1 , . . . , Un which cover
X × X and G × G maps s¯i : Ui → PX ×k(X ) PX such that
p ◦ s¯i is G × G -homotopic to id (as maps Ui → X × X );
Basic definition
Proposition
1) STCG (X ) 6 n;
2) there exist n G × G invariant open sets U1 , . . . , Un which cover
X × X and G × G maps s¯i : Ui → PX ×k(X ) PX such that
p ◦ s¯i is G × G -homotopic to id (as maps Ui → X × X );
3)
k(X ) catG ×G (X
× X ) 6 n.
Basic definition
Proposition
Let X be a free G space and A its closed invariant subset. Then
A catG (X )
= A/G cat(X /G ).
Basic definition
Proposition
A catG (X )
= A/G cat(X /G ).
Corollary
Let X be a free G -space. Note that for ∆(X ) ⊆ X × X we have
∆(X )/G = ∆(X /G ) and k(X )/G × G = ∆(X /G ).
Basic definition
Proposition
A catG (X )
= A/G cat(X /G ).
Corollary
∆(X )/G = ∆(X /G ) and k(X )/G × G = ∆(X /G ).
STCG (X ) = TC (X /G )
Basic definition
Proposition
A catG (X )
= A/G cat(X /G ).
Corollary
∆(X )/G = ∆(X /G ) and k(X )/G × G = ∆(X /G ).
STCG (X ) = TC (X /G )
while usually
TCG (X ) 6= TC (X /G ).
Basic definition
The A catG (X ) has its Whitehead definition.
Basic definition
The A catG (X ) has its Whitehead definition.
Definition
Let A ⊆ X be a closed G -cofibration. By a fat A-wedge we mean
for every n ∈ N a G -space FAn (X ) ⊆ X n := X × . . . × X defined as
FAn (X ) = {(x1 , . . . , xn ) ∈ X n | ∃i : xi ∈ A}.
Basic definition
For A = {0} and X = [−1, 1] we have
FA1 (X ) = {0};
Basic definition
FA1 (X ) = {0};
FA2 (X ) = [−1, 1] × {0} ∪ {0} × [−1, 1];
Basic definition
FA1 (X ) = {0};
FA2 (X ) = [−1, 1] × {0} ∪ {0} × [−1, 1];
FA3 (X ) =
Basic definition
Definition
We say that the G -Whitehead A-category is less or equal n if and
only if there is a G -map ξn : X → FAn (X ) such that the following
diagram is G -homotopy commutative:
ξn
/ F n (X )
X OOO
A
OOO
OO∆On
⊆
OOO
OO' Xn
where ∆n : X → X n is the diagonal mapping.
Basic definition
Definition
We say that the G -Whitehead A-category is less or equal n if and
only if there is a G -map ξn : X → FAn (X ) such that the following
diagram is G -homotopy commutative:
ξn
/ F n (X )
X OOO
A
OOO
OO∆On
⊆
OOO
OO' Xn
where ∆n : X → X n is the diagonal mapping.
We denote the G -Whitehead A-category by A catGWh (X ).
Basic definition
Theorem
Let X be a G -space and A ⊆ X closed G -cofibration. Then
Wh
A catG (X ) = A catG (X ).
Basic definition
Theorem
Let X be a G -space and A ⊆ X closed G -cofibration. Then
Wh
A catG (X ) = A catG (X ).
Theorem
Let X be a G -CW complex. The inclusion
∆(X ) ⊆ X × X
is a closed G -cofibration.
Corollary
We have equality
TCG (X ) = ∆(X ) catGWh (X × X ).
Basic definition
Theorem (L., Marzantowicz)
Let X be a G -CW complex for G finite. The inclusion
k(X ) ⊆ X × X
is a closed G × G -cofibration.
Corollary
We have equality
STCG (X ) = k(X ) catGWh
×G (X × X ).
Basic definition
Let X , Y be two G -spaces, A ⊆ X , B ⊆ Y their closed G -subsets.
Then
Wh
A×B catG (X
× Y ) 6 A catGWh (X ) + B catGWh (Y ) − 1.
Basic definition
Let X , Y be two G -spaces, A ⊆ X , B ⊆ Y their closed G -subsets.
Then
Wh
A×B catG (X
× Y ) 6 A catGWh (X ) + B catGWh (Y ) − 1.
Corollary
Let X , Y be two G -CW complexes. Then
TCG (X × Y ) 6 TCG (X ) + TCG (Y ) − 1.
If in addition G is finite then
STCG ×G (X × Y ) 6 STCG (X ) + STCG (Y ) − 1.
Basic definition
Example
Let G act on itself by left translations then
TCG (G ) = cat(G ),
Basic definition
Example
TCG (G ) = cat(G ),
STCG (G ) = TC (G /G ) = 1.
Basic definition
Example
TCG (G ) = cat(G ),
STCG (G ) = TC (G /G ) = 1.
Example
Let G := Z /2 acts on Sn , n > 1 by reflecting the last coordinate.
Then
Basic definition
Example
TCG (G ) = cat(G ),
STCG (G ) = TC (G /G ) = 1.
Example
Then
TCG (Sn ) = 3 for n > 1,
Basic definition
Example
TCG (G ) = cat(G ),
STCG (G ) = TC (G /G ) = 1.
Example
Then
TCG (Sn ) = 3 for n > 1,
(
3 for n odd,
STCG (Sn ) =
2 for n even.
Basic definition
Example
TCG (G ) = cat(G ),
STCG (G ) = TC (G /G ) = 1.
Example
Then
TCG (Sn ) = 3 for n > 1,
(
3 for n odd,
STCG (Sn ) =
= TC (Sn−1 ).
2 for n even.

A new approach to the equivariant topological complexity

Transkrypt

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