1. LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
ABSTRACT
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
Petar Pavešić, University of Ljubljana
The topological complexity, denoted TC(X), is a homotopy invariant of the space X (often interpreted as the configuration space of some complex mechanical system) that has important applications in topological robotics. The determination of TC(X) is usually based on the computation of suitable upper and lower estimates (e.g., dimension, Lusternik-Schnirelmann category, cup-length, category weight,...). In this talk I will describe the results of a joint work with Aleksandra Franc on a general framework for the study of lower estimates for TC(X).
Rijeka 1

2. BREAKING NEWS: Algebraic topology has gone applied!
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
INTRODUCTION
BREAKING NEWS: Algebraic topology has gone applied!
KEYWORDS:
persistent homology
topological robotics
statistical topology
directed topology, concurrency
Algebraic topology enters robotics through the notion of configuration space.
The configuration space of a mechanical device is the space of all its possible states.
These states are usually described by finitely many parameters, so the configuration space may be viewed as a subspace of some n. 2

3. EXAMPLES mechanical device configuration space
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
CONFIGURATION SPACES
EXAMPLES
mechanical device
configuration space 3

4. How do we get configuration spaces other than tori?
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
CONFIGURATION SPACES
S1×S1×S1×S1×S1×S1
How do we get configuration spaces other than tori?
By restricting the movement of the components.
OBSTACLES 2π 4

5. The configuration space of the 3-bar linkage is S1.
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
CONFIGURATION SPACES x
LINKAGES
For each admissible angle of the left rod there are two possible admissible angles of the right rod...
...except at the two extreme admissible values, where there is only one admissible angle on the right.
The configuration space of the 3-bar linkage is S1. 5

6. The configuration space of the 4-bar linkage is S2.
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
CONFIGURATION SPACES x
The configuration space of the 4-bar linkage is S2.
Theorem (M. Kapovich, J.L.Millson, 2002)
For every smooth compact manifold M there is a linkage whose configuration space is homeomorphic to a disjoint union of a finite number of copies of M. 6

7. Let X be the configuration space of some device.
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
NAVIGATION PLANS
Let X be the configuration space of some device.
Continuous motion of the device is represented by a path α: IX.
We will always assume that X is path-connected.
A navigation plan for X is a rule that takes as input a pair of points x,y in X, and returns as output a path α in X starting at x and ending at y. x y XI
X×X p
Formally, a navigation plan is a section of the evaluation fibration p: XIX×X, p(α)=(α(0),α(1)). 7

8. We are interested in continuous navigation plans s: X×X  XI .
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
NAVIGATION PLANS
We are interested in continuous navigation plans s: X×X  XI .
However, a continuous navigation plan for X exists if and only if X is contractible.
In fact, given a continuous s: X×X  XI, the map x s(x, x0) determines a contraction of X to x0. x0
Conversely, given a contraction of X to x a canonical path from x to y is obtained by joining the given path from x to x0 to the inverse of the path from y to x0. x0 x y 8

9. For non-contractible spaces we may consider partial
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
TOPOLOGICAL COMPLEXITY x y
On a non-contractible space it is impossible to find a navigation plan depending continuously on the input data. XI
X×X p U sU
For non-contractible spaces we may consider partial
navigation plans, i.e. continuous sections
M. Farber (2003)
The topological complexity of X is the minimal number of continuous partial navigation plans needed to describe all possible navigation plans for X.
Formally: TC(X) is the minimal n such that there is an open cover U1,...,Un, for X×X admitting partial continuous sections si:Ui  XI for the projection p: XIX×X. 9

10. TC(X) is a homotopy invariant of X.
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
TOPOLOGICAL COMPLEXITY
OBSERVATIONS:
Topological complexity of X is the Schwartz genus of the fibration p: XIX×X.
Open covers are required for convenience. For nice spaces (manifolds, polyhedra) closed covers or covers by arbitrary ENRs yield the same result.
TC(X) is a homotopy invariant of X.
TC(X)=1 if and only if X is contractible.
If the inclusion U X×X is null-homotopic, then p: XIX×X admits a section over U.
cat(X)  TC(X)  cat(X×X),
(where cat(X) denotes the Lusternik-Schnirelmann category of X =
minimal number of open subspaces that are contractible in X needed to cover X)
cat(X)  TC(X)  2cat(X)1 10

11. Hence TC(S1)=2. Similarly TC(S2n+1)=2.
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
TOPOLOGICAL COMPLEXITY
EXAMPLES x
cat(Sn)=2  2  TC(Sn)  3.
2 or 3?
For TC(S1) divide S1× S1 into:
U={(x,y) | x+ y  0},
sU (x,y):= shortest path on S1 from x to y
V={(x,y) | x+ y =0},
sV (x,y):= path on S1 from x to y in the positive sense
Hence TC(S1)=2. Similarly TC(S2n+1)=2. x
For TC(S2) divide S2× S2 into:
U={(x,y) | x+ y  0}, sU (x,y):= shortest path on S2 from x to y
V={(x,y) | x+ y =0}, sV (x,y):= path on the great circle on S2 from x through north pole N to y
Doesn’t work for the pair (N,-N), so we need
W={(N,-N)}, sW (N,-N):= any path on S2 connecting N to –N
Can we do better? 11

12. we see that Ker p*= Ker *.
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
TOPOLOGICAL COMPLEXITY
Enters cohomology: XI
X×X p U s i
H*( XI )
H*( X×X ) p*
H*(U) s* i*
H*( X×X,U ) j*
Ker p* Ker i*
= Im j*
If X×X has a cover U1,...,Un admitting local sections, then every product of n elements of Ker p* is in the image of an element in H*( X×X,U1...  Un )=0, therefore Ker p* is a nilpotent ideal of H*( X×X ) of order n. XI
X×X p X  
From the diagram
we see that Ker p*= Ker *.
nil(Ker *)  TC(X) 12

13. For x  H2(S2) we have x×11×x  Ker *  H*(S2× S2).
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
TOPOLOGICAL COMPLEXITY x
For x  H2(S2) we have x×11×x  Ker *  H*(S2× S2).
(x×11×x ) 2= 2 x×x  0 implies nil(Ker *)3, therefore TC(S2)=3.
Similarly TC(S2n)=3.
Further results (using the same methods):
TC(Mg)=
3, g =0,1
5, g > 1
Mg closed orientable surface of genus g
TC(Γ)=
2, Γ has one cycle
3, otherwise
1, Γ a tree
Γ finite connected graph
TC((S2n)k) =
k+1, n odd
2k+1, n even
TC(SO(3))= TC(P3)=3 13

14. ALTERNATIVE APPROACH (N.Iwase, S.Sakai 2008)
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
FIBREWISE CATEGORY
ALTERNATIVE APPROACH (N.Iwase, S.Sakai 2008) XI
X×X p U s x y
corresponds to a deformation of U to the diagonal X  X×X:
S:U×I X×X, S(x,y,t):= (y,s(x,y)(t))
i.e. U is a fibrewise categorical set in the fibrewise space pr2: X×X X, where the fibrewise basepoint is determined by the diagonal.
TC(X) is the fibrewise (Lusternik-Schnirelmann) category of the fibrewise pointed space
pr2 
X X×X X 14

15. Whitehead-Ganea framework for LS-category
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
FIBREWISE CATEGORY
Whitehead-Ganea framework for LS-category
U  X is categorical if U contracts to the base-point under a base-point preserving deformation of X
(i.e. there is a rel x0 homotopy H: X×IX , such that H0 =1X and H1(U)=x0 ).
Let
nX =X×...×X n-fold product
WnX={(x1 ,… , xn) nX | at least one xi is the base-point} n-fold “fat wedge”
Whitehead characterization
for nice spaces (normal, with non-degenerate base-point) cat(X) equals the minimal n such that there exists a lift in the diagram:
WnX
nX  X
 n 15

16. GnX = n-th Ganea space of X
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
FIBREWISE CATEGORY
GnX = n-th Ganea space of X
(n-th iterate of the Ganea construction of the path fibration PXX) X
GnX pn
Ganea characterization
for nice spaces cat(X) equals the minimal n such that there exists a section of the fibration:
WnX
nX X
 n
GnX pn
Whitehead and Ganea characterizations are related by the homotopy pull-back diagram:
Whitehead-Ganea framework forms the basis for all applications of homotopy theoretic methods to LS-category. Is there an analogous approach to the fibrewise category, and hence to the tooplogical complexity? 16

17. Assume U  X×X can be vertically deformed to the diagonal X.
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
FIBREWISE CATEGORY
Basic problem:
Assume U  X×X can be vertically deformed to the diagonal X.
Is there a vertical deformation of X×X that is stationary on X and compresses U to the diagonal?
In terms of navigation plans: given a partial navigation plan on U  X×X, we are looking for a global navigation plan s: X×X XI such that
s(x,y) is a path from x to y for all (x,y)U
s(x,x) is the constant path
Main difficulty is to achieve b). Iwase – Sakai gave a proof for polyhedra in 2008 but later retracted. Franc – P. proved the claim for finite polyhedra. 17

18. Whitehead-Ganea framework for fibrewise category
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
FIBREWISE CATEGORY
Whitehead-Ganea framework for fibrewise category
Define fibrewise spaces:
X X×X X
pr1 
XX:
XnX:
X X×nX X
pr1
1×n
XWnX:
XWnX={(x,y ) X×nX| y  WnX } x
X XWnX X
pr1
1×n
Whitehead characterization
for compact polyhedra, TC(X) equals the minimal n such that there exists a fibrewise lift in the diagram:
XWnX
XnX 
XX
1 n 18

19. Ganea characterization
LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
FIBREWISE CATEGORY
Similarly, we define the fibrewise pointed Ganea space XGnX and obtain the
XX
XGnX
1pn
Ganea characterization
for compact polyhedra TC(X) equals the minimal n such that there exists a fibrewise section of the fibration:
XWnX
XnX
XX
1 n
XGnX
1pn
Moreover, Whitehead and Ganea characterizations are related by the following fibrewise homotopy pull-back diagram:
This is the starting point for the application of obstruction theory methods to the computation of the topological complexity, as in
Franc – P. : Lower bounds for topological complexity, and
Spaces with high topological complexity 19

20. LOWER ESTIMATES FOR THE TOPOLOGICAL COMPLEXITY
THANK YOU! 20