(Joint with Greg Lupton and John Oprea) Mark...

Lower bounds for the topological complexity of groups(Joint with Greg Lupton and John Oprea)

Mark Grant

School of Mathematics & StatisticsNewcastle University

University of Leicester Pure Mathematics seminar19th November 2013

1 Topological complexity of robot motion planning

2 Topological complexity of groups

3 A new lower bound for TC(G )

4 ExamplesPure braid groupsThe Borromean ringsHigman’s groupFurther work

Topological complexity of robot motion planning

Topological complexity is a numerical homotopy invariant defined byMichael Farber in the early 2000s.

Its definition is motivated by the motion planning problem in Robotics.

Configuration spaces

Any mechanical system is parameterized by a topological space X , theconfiguration space of the system.

Points in X correspond to states or configurations of the system.

Paths in X correspond to motions of the system.

The motion planning problem

Find an algorithm which, given states A and B of the system, outputs amotion from A to B.

In terms of configuration spaces, the input is a point (A,B) ∈ X × X , andthe output is a path γ ∈ X I = {paths in X} with γ(0) = A and γ(1) = B.

More formally, consider the endpoint map

πX : X I → X × X , πX (γ) =(γ(0), γ(1)

A motion planning algorithm is a section of πX , that is, a functions : X × X → X I such that πX ◦ s = IdX×X .

πX : X I → X × X , πX (γ) =(γ(0), γ(1)

When X I is given the compact-open topology, the map πX : X I → X × Xis continuous (in fact a fibration).

Observation

There exists a continuous section s : X ×X → X I of πX if and only if X iscontractible.

So motion planning algorithms in X often have essential discontinuities,due to the topology of X .

Observation

Topological complexity

Premise

It is desirable to find motion planning algorithms with fewest domains ofcontinuity, since these will be optimally robust to changes in the input.

Definition (Farber)

The topological complexity of a space X , denoted TC(X ), is the leastinteger k such that X × X admits a cover by open sets U0,U1, . . . ,Uk , oneach of which πX admits a local section (a continuous map si : Ui → X I

such that πX ◦ si = incl : Ui ⊆ X × X ). If no such integer exist, we setTC(X ) =∞.

Note that TC(X ) is one less than the number of sets in the cover!

Premise

Definition (Farber)

Premise

Definition (Farber)

Topological complexity: basic properties

If X ' Y then TC(X ) = TC(Y ) (homotopy invariance).

TC(X ) = 0 if and only if X is contractible.

Example

The topological complexity of the n-sphere (n ≥ 1) is given by

TC(Sn) =

{1 if n is odd2 if n is even.

Topological complexity: basic properties

If X ' Y then TC(X ) = TC(Y ) (homotopy invariance).

TC(X ) = 0 if and only if X is contractible.

Example

The topological complexity of the n-sphere (n ≥ 1) is given by

TC(Sn) =

{1 if n is odd2 if n is even.

Cohomological lower bounds

Lower bounds are given by cohomology, in particular the zero-divisorscup-length.

Let H∗(−) = H∗(−;k) with k a field. Recall that

∪ : H∗(X )⊗ H∗(X )→ H∗(X )

is a ring homomorphism. Its kernel ker(∪) is the ideal of zero-divisors.

Define the nilpotency nil I of an ideal I C R to be the least integer k suchthat I k+1 = 0.

Theorem (Farber)

For any space X ,TC(X ) ≥ nil ker(∪).

∪ : H∗(X )⊗ H∗(X )→ H∗(X )

Theorem (Farber)

∪ : H∗(X )⊗ H∗(X )→ H∗(X )

Theorem (Farber)

∪ : H∗(X )⊗ H∗(X )→ H∗(X )

Theorem (Farber)

Lusternik–Schnirelmann category

Definition

The (Lusternik–Schnirelmann) category of a space X , denoted cat(X ), isthe least integer k such that X admits a cover by open setsU0,U1, . . . ,Uk , with each inclusion Ui ↪→ X null-homotopic.

Example

The category of the n-sphere (n ≥ 1) is cat(Sn) = 1.

Proposition

For any path-connected space X we have

cat(X ) ≤ TC(X ) ≤ cat(X × X ).

Definition

Example

Proposition

cat(X ) ≤ TC(X ) ≤ cat(X × X ).

Definition

Example

Proposition

cat(X ) ≤ TC(X ) ≤ cat(X × X ).

Topological complexity of groups

Recall that for any group G , one can construct a path-connected complexK (G , 1) which has

πi(K (G , 1)

{G (i = 1),0 (i > 1).

This construction is functorial up to homotopy, and so K (G , 1) is uniqueup to homotopy equivalence.

Problem (Farber)

Describe TC(G ) := TC(K (G , 1)

)in terms of known algebraic invariants of

the group G .

πi(K (G , 1)

{G (i = 1),0 (i > 1).

Problem (Farber)

the group G .

πi(K (G , 1)

{G (i = 1),0 (i > 1).

Problem (Farber)

the group G .

Cohomological dimension

Definition

The cohomological dimension of a group G , denoted cd(G ), is theminimum k such that H i (G ;M) = 0 for all i > k and all Z[G ]-modules M.

Theorem (Eilenberg–Ganea)

If cd(G ) ≥ 3 then cd(G ) = gd(G ), where gd(G ) denotes the smallestpossible dimension of a K (G , 1) complex.

Theorem (Stallings, Swan)

If cd(G ) = 1 then G is a free group (and hence cd(G ) = gd(G )).

The remaining question, of whether cd(G ) = 2 implies gd(G ) = 2, isknown as the Eilenberg–Ganea conjecture.

Definition

Category of groups

Theorem (Eilenberg–Ganea, Stallings, Swan)

For any group G we have

cat(G ) := cat(K (G , 1)

)= cd(G ).

Examples

If G is free then cat(G ) = 1.

If G is an infinite surface group then cat(G ) = 2.

If G ∼= Zn then cat(G ) = n.

If G has torsion then cat(G ) =∞.

Category of groups

Theorem (Eilenberg–Ganea, Stallings, Swan)

For any group G we have

cat(G ) := cat(K (G , 1)

)= cd(G ).

Examples

If G is free then cat(G ) = 1.

If G is an infinite surface group then cat(G ) = 2.

If G ∼= Zn then cat(G ) = n.

If G has torsion then cat(G ) =∞.

Topological complexity of groups: a survey

Note that the inequalities

cd(G ) = cat(G ) ≤ TC(G ) ≤ cat(G × G ) = cd(G × G )

show that TC(G ) =∞ if G has torsion. So the problem is interestingmainly for torsion-free groups (of finite cohomological dimension).

Topological complexity of groups: a surveyGroups for which the exact value of TC(G ) is known include:

Free abelian groups Zn (Farber 2003)

Orientable surface groups π1(Σg ), g ≥ 1 (Farber 2003)

Free groups Fn (Farber 2004)

Pure braid groups Pn = π1

(Fn(C)

)(Farber–Yuzvinsky 2004)

Pure braid groups of the punctured planePn,m = ker(Pn → Pm) = π1

(Fn(C \m points)

)(Farber–G.–Yuzvinsky

Right-angled Artin groups GΓ (Cohen–Pruidze 2008)

Basis-conjugating automorphism groups PΣn and upper-triangularMcCool groups PΣ+

n (Cohen–Pruidze 2008)

Almost-direct products of free groups (Cohen 2010)

Pure braid groups of surfaces π1

(Fn(Σg )

)(Cohen–Farber 2011)

Topological complexity of groups: a surveyGroups for which the exact value of TC(G ) is known include:

Free abelian groups Zn (Farber 2003)

Orientable surface groups π1(Σg ), g ≥ 1 (Farber 2003)

Free groups Fn (Farber 2004)

Pure braid groups Pn = π1

(Fn(C)

)(Farber–Yuzvinsky 2004)

Pure braid groups of the punctured planePn,m = ker(Pn → Pm) = π1

(Fn(C \m points)

)(Farber–G.–Yuzvinsky

Right-angled Artin groups GΓ (Cohen–Pruidze 2008)

Basis-conjugating automorphism groups PΣn and upper-triangularMcCool groups PΣ+

n (Cohen–Pruidze 2008)

Almost-direct products of free groups (Cohen 2010)

Pure braid groups of surfaces π1

(Fn(Σg )

)(Cohen–Farber 2011)

Topological complexity of groups: a survey

Groups conspicuously missing from this list include:

Finitely generated torsion-free nilpotent groups

Non-orientable surface groups

A new lower bound for TC(G)

A new lower bound for TC(G )

Theorem (G.–Lupton–Oprea)

Let A and B be subgroups of G such that gAg−1 ∩ B = {1} for everyg ∈ G . Then

cd(A× B) ≤ TC(G ).

Recall that A and B are complementary in G if A∩B = {1} and AB = G .

Corollary (G.–Lupton–Oprea)

Let A and B be complementary subgroups of G . Then

cd(A× B) ≤ TC(G ).

A new lower bound for TC(G )

Theorem (G.–Lupton–Oprea)

Let A and B be subgroups of G such that gAg−1 ∩ B = {1} for everyg ∈ G . Then

cd(A× B) ≤ TC(G ).

Recall that A and B are complementary in G if A∩B = {1} and AB = G .

Corollary (G.–Lupton–Oprea)

Let A and B be complementary subgroups of G . Then

cd(A× B) ≤ TC(G ).

Remarks

The proof uses elementary homotopy theory together with propertiesof the sectional category under pullbacks.

This lower bound does not require knowledge of the cohomology ringstructure of G , and can improve on the zero-divisors cup-length lowerbound.

It illustrates that TC(G ) is related to the subgroup structure of G .For instance, upper bounds on TC(G ) imply that certain pairs ofsubgroups have conjugate elements.

Remarks

Examples Pure braid groups

Pure braid groups

The pure braid group on n strands can be defined as

Pn = π1

(Fn(C)

where Fn(C) = {(z1, . . . , zn) ∈ Cn | i 6= j =⇒ zi 6= zj} is the classicalconfiguration space.

It has cd(Pn) = n − 1.

Theorem (Farber–Yuzvinsky)

We haveTC(Pn) = 2n − 3

for all n ≥ 2.

Pure braid groups

Pn = π1

(Fn(C)

for all n ≥ 2.

Pure braid groups

Pn = π1

(Fn(C)

for all n ≥ 2.

Pure braid groups

Recall that elements of Pn can also be describedgeometrically as isotopy classes of braids, withthe group operation given by concatenation.

There is an inclusion Pn−1 ↪→ Pn given by intro-ducing an n-th non-interacting strand after theother strands.

Pure braid groups

Recall that elements of Pn can also be describedgeometrically as isotopy classes of braids, withthe group operation given by concatenation.

There is an inclusion Pn−1 ↪→ Pn given by intro-ducing an n-th non-interacting strand after theother strands.

Pure braid groups

For j = 1, . . . , n−1, let αj be the braid which runsthe j-th strand in front of the last n − j strands,then passes behind the last n − j strands to itsoriginal position.

The αj ’s commute pairwise, so they generate afree abelian subgroup A of rank (n − 1).

Since conjugate braids close to isotopic links, one checks using linkingnumbers with the last strand that gAg−1 ∩ Pn−1 = {1} for all g ∈ Pn.

So the Theorem gives

TC(Pn) ≥ cd(A× Pn−1) = (n − 1) + (n − 2) = 2n − 3.

Pure braid groups

TC(Pn) ≥ cd(A× Pn−1) = (n − 1) + (n − 2) = 2n − 3.

Pure braid groups

TC(Pn) ≥ cd(A× Pn−1) = (n − 1) + (n − 2) = 2n − 3.

Pure braid groups

TC(Pn) ≥ cd(A× Pn−1) = (n − 1) + (n − 2) = 2n − 3.

Examples The Borromean rings

The Borromean rings

The link complement X of the Borromeanrings is a compact aspherical 3-manifoldwith fundamental group

G ∼=⟨a, b, c

∣∣ [a, [b−1, c]], [b, [c−1, a]]⟩.

All cup-products vanish in H̃∗(X ;k) for any field k, so the zero-divisorscup-length is 2.

Using Massey products in H∗(X ;Q) and sectional category weight, we canshow that TC(X ) ≥ 3 (G., 2009).

The Borromean rings

G ∼=⟨a, b, c

∣∣ [a, [b−1, c]], [b, [c−1, a]]⟩.

The Borromean rings

G ∼=⟨a, b, c

∣∣ [a, [b−1, c]], [b, [c−1, a]]⟩.

The Borromean ringsRemoving one component gives an unlink.

There results a split extension

K G F2〈α, β〉,p

a 7→ αb 7→ βc 7→ 1

Letting A = 〈a〉 and B = p−1〈β〉, one can show algebraically thatgAg−1 ∩ B = {1} for all g in G .

Since B is not free, the Theorem gives

TC(G ) ≥ cd(A× B) = 1 + 2 = 3.

The Borromean ringsRemoving one component gives an unlink.There results a split extension

a 7→ αb 7→ βc 7→ 1

TC(G ) ≥ cd(A× B) = 1 + 2 = 3.

a 7→ αb 7→ βc 7→ 1

TC(G ) ≥ cd(A× B) = 1 + 2 = 3.

a 7→ αb 7→ βc 7→ 1

TC(G ) ≥ cd(A× B) = 1 + 2 = 3.

Examples Higman’s group

Higman’s group

Higman’s group H is a finitely presented group with presentation⟨x , y , z ,w | xyx−1y−2, yzy−1z−2, zwz−1w−2,wxw−1x−2

It is acyclic, so H̃∗(H;k) = 0 and the zero-divisors cup-length is 0 for anyfield k.

It has no non-trivial finite quotients. It follows that H∗(H;M) is trivial forany Z[H]-module M which is finitely generated as a Z-module.

The above presentation is aspherical, and so cd(H) = 2.

Higman’s group

⟩It is acyclic, so H̃∗(H; k) = 0 and the zero-divisors cup-length is 0 for anyfield k.

Higman’s group

Proposition

We have TC(H) = 4.

Proof We recall the original construction of H as an iterated amalgam.

For symbols x and y let Hxy be the Baumslag–Solitar group withpresentation

⟨x , y | xyx−1y−2

⟩. Note that cd(Hxy ) = 2.

We now form the amalgams

〈y〉 Hyz

〈w〉 Hwx

Higman’s group

Proposition

We have TC(H) = 4.

⟨x , y | xyx−1y−2

〈y〉 Hyz

〈w〉 Hwx

Higman’s group

Proposition

We have TC(H) = 4.

⟨x , y | xyx−1y−2

〈y〉 Hyz

〈w〉 Hwx

Higman’s group

Proposition

We have TC(H) = 4.

⟨x , y | xyx−1y−2

〈y〉 Hyz

〈w〉 Hwx

Higman’s group

Proposition

We have TC(H) = 4.

⟨x , y | xyx−1y−2

〈y〉 Hyz

Hxy Hxyz

〈w〉 Hwx

Hzw Hzwx

Higman’s group

Finally we form the amalgam

〈x , z〉 Hxyz

We have subgroups

Hxy = 〈x , y〉 ≤ H, Hzw = 〈z ,w〉 ≤ H.

We claim that gHxyg−1 ∩ Hzw = {1} for all g ∈ H. Hence

TC(H) ≥ cd(Hxy × Hzw ) = 4.

Higman’s group

〈x , z〉 Hxyz

Hzwx H

We have subgroups

Hxy = 〈x , y〉 ≤ H, Hzw = 〈z ,w〉 ≤ H.

Higman’s group

〈x , z〉 Hxyz

Hzwx H

We have subgroups

Hxy = 〈x , y〉 ≤ H, Hzw = 〈z ,w〉 ≤ H.

Higman’s group

〈x , z〉 Hxyz

Hzwx H

We have subgroups

Hxy = 〈x , y〉 ≤ H, Hzw = 〈z ,w〉 ≤ H.

Higman’s group

The proof of the claim (communicated to us by Yves Cornulier) usesBass–Serre theory, and the following Lemmas:

In an amalgam G = A ∗C B, if an element of A is conjugate in G to anelement of B, then it is conjugate in G to an element of C .

In an amalgam G = A ∗C B, if an element of A is conjugate in G to anelement of C , then it is conjugate in A to an element of C .

Examples Further work

Further work

Obtain a more general result about TC(G ) for G = A ∗C B.

Can our result be extended to deal with non-orientable surfaces?

Examples Further work

Thanks for listening!

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