Introduction Coh.op. Lens spaces TC(X × Y )
Motion planning algorithsms, TC(X ) and Schwartz
genus (II)
Albert Ruiz Cirera
http://mat.uab.cat/∼albert
Universitat Autònoma de Barcelona
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Introduction
Yesterday Aleksandra Franc gave us several equivalent definitions
of TC(X ). Let us recall this one:
Definition
The Topological complexity of X is the minimal k such that there
exist open subsets:
X × X = V1 ∪ V2 ∪ · · · ∪ Vk = X × X
and continuous sections si : Vi → PX of the map PX → X × X
defined by γ 7→ (γ(0), γ(1)).
Aleksandra also gave us tools to compute TC(X ):
zero-divisors(-cup-length), weight of a cohomology class, . . . and
computed the topological complexity of spheres, a graph and
oriented surfaces.
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Today
We will use some of the tools and results that Aleksandra
explained us with the following table of contents:
1
Cohomology operations and weight.
2
Application: Topological complexity of lens spaces.
3
Topological complexity and products.
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Cohomology operations
Definition
A stable cohomology operation in degree i is a family of natural
transformations
θ : H n (−; R) → H n+i (−; S)
which commute with suspensions.
Natural transformation
if f : X → Y is a continuous map of topological spaces, the
following diagram commute:
H n (Y ; R)
θ
H n+i (Y ; S)
A. Ruiz
fn
f n+i
/ H n (X ; R)
θ
/ H n+i (X ; S)
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Cohomology operations
Definition
A stable cohomology operation in degree i is a family of natural
transformations
θ : H n (−; R) → H n+i (−; S)
which commute with suspensions.
Commuting with suspensions
Consider S ∗ the isomorphism induced by the cohomology long
exact sequence of a pair:
S∗
H̃ p (X )
δ∗ /
∼
=
H̃ p+1 (CX , X )
∼
=
,
/ H̃ p+1 (SX )
then S ∗ θ = θS ∗ . In fact S ∗ δ ∗ = δ ∗ S ∗ .
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Excess
Definition
The excess of a stable cohomology operation θ, denoted as e(θ), is
the largest integer n such that θ(u) = 0 for all u such that |u| < n.
Examples
e(β) = 1, where β is the Bockstein operation.
e(Sqi ) = i, where Sqi is the degree i Steenrod operation for
p = 2.
e(P i ) = 2i, where P i is the Steenrod power for odd p.
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Recall
Recall from yesterday’s talk:
Definition
Let u ∈ H ∗ (X × X ; R).
We say that u has weight k if k is the largest integer such
that for any open subset A ⊂ X × X with TCX (A) ≤ k one
has u|A = 0.
We say that u is a zero divisor if
u|∆(X ) = 0 ∈ H ∗ (X ; R|∆(X )), where ∆(X ) ⊂ X × X is the
diagonal.
Proposition
If u, v ∈ H ∗ (X × X ; R) and uv is the cup product then
wgt(uv ) ≥ wgt(u) + wgt(v ) .
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Recall
Proposition
If there exists a nonzero cohomology class u ∈ H ∗ (X × X ; R) with
wgt(u) ≥ k, then TC(X ) > k.
Notation
If u ∈ H j (X ; R), we denote
u = 1 × u − u × 1 ∈ H j (X × X ; R)
Remark
If θ is a stable cohomology operation, then
θ(u) = θ(u) .
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Weight and excess
Theorem
Let θ be stable cohomology operation in degree i and excess
e(θ) ≥ n. Then for any cohomology class u ∈ H n (X ; R),
wgt(θ(u)) ≥ 2.
Proof
Assume A ⊂ X × X is an open subset with TCX (A) ≤ 2. Then
A = B ∪ C , with B and C open sets and such that the projections
p1B : B → X ' p2B : B → X and p1C : C → X ' p2C : C → X .
Consider:
u|A = (p2A )∗ (u) − (p1A )∗ (u) ∈ H n (A; R)
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Proof (cont.)
Proof.
The Mayer-Vietoris sequence for A = B ∪ C :
· · · → H n−1 (B ∩ C )
w
δ
→
7→
H n (A)
u|A
u|A
F
→
7→
H n (B) ⊕ H n (C ) → · · ·
0
Apply now that θ is a stable operation of excess ≥ n:
(θu)|A = θ(u|A ) = θδ(w ) = δθ(w ) = 0 .
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Example: lens spaces
Definition
Consider the cyclic group Zm = {1, ω, ω 2 , . . . , ω m−1 }, where ω is a
primitive m-root of unity, acting on S 2n+1 ⊂ Cn+1 by pointwise
multiplication. Define the lens space:
= S 2n+1 /Zm
L2n+1
m
Properties
1
There is a locally trivial fibration:
S 1 → L2n+1
→ CP n
m
2
If m is odd, H ∗ (L2n+1
; Zm ) = Zm [x, y ]/hx 2 , y n+1 i,
m
∗
2n+1
If m is even, H (Lm ; Zm ) = Zm [x, y ]/hx 2 − (m/2)y , y n+1 i,
In both cases |x| = 1, |y | = 2 and β(x) = y .
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
An upper bound
Theorem
TC(L2n+1
) ≤ 4n + 2 .
m
Proof.
Follows from results:
1
If F → E → B is a Hurewicz fibration, then:
TC(E ) ≤ TC (F ) · cat(B × B) .
2
cat(CP n × CP n ) = 2n + 1 .
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Lower bounds
Finding lower bounds is more difficult and we have to restrict to
some particular cases.
Theorem
For any pair of integers k, l such that m does not divide
0 ≤ k, l ≤ n:
TC(L2n+1
) ≥ 2(k + l) + 2 .
m
k+l
m
and
Corollary
Let p be an odd prime and n an integer such that its p-adic
expansion
n = n0 + n1 p + n2 p 2 + · · · + nk p k (with 0 ≤ ni < p)
satisfy that ni ≤ (p − 1)/2 for all i. Then:
TC(L2n+1
) = 4n + 2 .
p
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Proof of the Corollary
Proof.
We have just to prove TC(L2n+1
) ≥ 4n + 2.
p
As in the p-adic decomposition ni ≤ (p − 1)/2, then p doesn’t
divide 2n
n .
Using the Theorem:
) ≥ 2(2n) + 2 = 4n + 2
TC(L2n+1
p
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Proof of the Theorem
Proof.
Consider y = β(x) ∈ H ∗ (L2n+1
, Zm ). Then
m
wgt(β(x)) = wgt(β(x)) = wgt(y ) ≥ 2 .
where y = 1 ⊗ y − y ⊗ 1.
k+l 6= 0,
If for some 0 ≤ k, l ≤ n, m doesn’t divide k+l
k , then (y )
k+l
because contains the term (−1)k k y k ⊗ y l . Then the element
x(y )k+l 6= 0 and has weight at least (1 + 2(k + l)), so
TC(L2n+1
) ≥ 2(k + l) + 2 .
m
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Case m = 2r
Let α(n) be the number of 1’s in the 2-adic expansion of n.
Theorem
Assume that m = 2r . Then
) = 4n + 2
TC(L2n+1
m
for all n such that α(n) ≤ r − 1.
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Topological complexity and products: upper bound
Definition
A topological space X is a Euclidean Neighbourhood Retract
(ENR) if it can be embedded into an Euclidean space X ⊂ Rn such
that ∃U open with X ⊂ U ⊂ Rn and a retraction r : U → X (i.e.
r |X = IdX ).
Theorem
Let X and X 0 be ENRs. Then
TC(X × X 0 ) ≤ TC(X ) + TC(X 0 ) − 1 .
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Proof
Proof
Assume k = TC(X ), (resp. k 0 = TC(X 0 )), s : X × X → PX (resp.
s 0 : X 0 × X 0 → PX 0 ) section and
∅ = U0 ⊂ U1 ⊂ · · · ⊂ Uk = X × X
(resp. ∅ = U00 ⊂ U10 ⊂ · · · ⊂ Uk0 0 = X 0 × X 0 )
0 \U 0 ) is continuous.
such that s|Ui+1 \Ui (resp. s 0 |Ui+1
i
Define the product section:
s × s 0 : (X × X 0 ) × (X × X 0 ) → P(X × X 0 )
((x, x 0 ), (y , y 0 )) 7→
A. Ruiz
I
t
→ X × X0
7
→
(s(x, y )(t), s 0 (x 0 , y 0 )(t))
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Proof (cont.)
Proof.
Consider now for n ≤ k + k 0 − 1:
[
Wn =
Ui ×Uj0 ⊂ (X ×X )×(X 0 ×X 0 ) ≡ (X ×X 0 )×(X ×X 0 ) .
i+j=n+1
Then:
∅ = W0 ⊂ W1 ⊂ · · · ⊂ Wk+k 0 −1 = (X × X 0 ) × (X × X 0 )
and
Wn \ Wn−1 =
G
0
),
(Ui \ Ui−1 ) × (Uj0 \ Uj−1
i+j=n
so s × s 0 is continuous at Wn \ Wn−1 and this implies
TC(X × X 0 ) ≤ k + k 0 − 1 = TC(X ) + TC(X 0 ) − 1 .
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
˜
TC(X
)
Definition
˜
Define the reduced topological complexity TC(X
) as
˜
TC(X
) = TC(X ) − 1 .
Corollary
For ENR’s X1 , . . . , Xk one has:
˜ 1 × · · · × Xk ) ≤
TC(X
k
X
˜ i) .
TC(X
i=1
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Recall zcl(X )
Definition
Recall that the zero-divisor-cup-length of X , zcl(X ), is the length
of the longest nontrivial cup product of zero divisor elements in
H ∗ (X ; R).
And recall the following result:
Lemma
˜
TC(X
) ≥ zcl(X ).
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Topological complexity and products: lower bound
Consider H ∗ (X ; Q) and the zero-cup-length of X in this case
zclQ (X ) (rational coefficients).
Lemma
zclQ (X × Y ) ≥ zclQ (X ) + zclQ (Y ).
Proof.
Consider k = zclQ (X ) (resp. l = zclQ (Y )), and u1 , . . . , uk (resp.
v1 , . . . , vl ) zero divisors in H ∗ (X × X ; Q) (resp. H ∗ (Y × Y ; Q))
with nontrivial product. Consider the cohomology classes:
ũi = ui × 1 × 1 ∈ H ∗ (X ×X ×Y ×Y ; Q) ∼
= H ∗ (X ×Y ×X ×Y ; Q)
∗
v˜i = 1 × 1 × vi ∈ H (X ×X ×Y ×Y ; Q) ∼
= H ∗ (X ×Y ×X ×Y ; Q)
are zero divisors and
ũ1 · · · ũk ṽ1 · · · ṽl = ±(u1 . . . uk ) × (v1 . . . vl ) 6= 0 .
So zclQ (X × Y ) ≥ k + l = zclQ (X ) + zclQ (Y ) .
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Topological complexity and products: lower bound
Theorem
Suppose that one controls simultaneously k systems having
path-connected configurations spaces X1 , . . . , Xk . Assume that
H̃(Xi , Q) 6= 0 and there is a constant M such that for all i
˜ i ) ≤ M. Then
TC(X
˜ 1 × · · · × Xk ) ≤ kM .
k ≤ TC(X
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
References
M. Faber, Invitation to topological robtics. Zurich Lectures in
Advanced Mathematics. EMS, Zrich, 2008.
M. Farber and M. Grant, Robot motion planning, weights of
cohomology classes, and cohomology operations. Proceedings
of the AMS 136 (2008), 3339–3349.
M. Farber, S. Tabachnikov and S. Yuzvinsky, Topological
robotics: motion planning in projective spaces, International
Mathematics Research Notices 34 (2003), 1853–1870.
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)
Introduction Coh.op. Lens spaces TC(X × Y )
Thanks!
A. Ruiz
Motion planning algorithsms, TC(X ) and Schwartz genus (II)