Topological Complexity of Braid Groups (Joint with G

Home , Borromean rings

Topological complexity of braid groups (joint with G. Lupton and J. Oprea)

Mark Grant

24th July 2013 Overview

1 Topological complexity of groups

2 A new lower bound for TC(G)

3 Examples

4 Further questions Let X be a space, and let

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

be the free path ﬁbration on X .

Topological complexity of groups Deﬁnitions

Recall that topological complexity is a numerical homotopy invariant, which quantifies the complexity of navigation in a configuration space. Topological complexity of groups Definitions

Recall that topological complexity is a numerical homotopy invariant, which quantiﬁes the complexity of navigation in a conﬁguration space.

Let X be a space, and let

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

be the free path ﬁbration on X .

Deﬁnition (Farber) TC(X ) is the minimum k such that X × X admits a cover by open sets I U0, U1,..., Uk , each of which admits a partial section si : Ui → X of πX (meaning πX ◦ si equals the inclusion Ui ,→ X × X ). Another important special case of sectional category is (Lusternik–Schnirelmann) category of a space.

Deﬁnition Let X be a space. The category of X , denoted cat(X ), is the minimum k such that X admits a cover by open sets U0, U1,..., Uk such that each inclusion Ui ,→ X is null-homotopic.

Topological complexity of groups

This is a special case of

Defnition Let p : E → B be a ﬁbration. The sectional category of p, denoted secat(p), is the minimum k such that B admits a cover by open sets U0, U1,..., Uk , each of which admits a partial section si : Ui → E of p. Topological complexity of groups

This is a special case of

Another important special case of sectional category is (Lusternik–Schnirelmann) category of a space.

Deﬁnition Let X be a space. The category of X , denoted cat(X ), is the minimum k such that X admits a cover by open sets U0, U1,..., Uk such that each inclusion Ui ,→ X is null-homotopic. Proposition (Svarc)ˇ If the square Y E q p

A B

is a pullback, then secat(q) ≤ secat(p).

Topological complexity of groups

Proposition (Svarc)ˇ Let p : E → B be a ﬁbration. 1 If p is surjective, then secat(p) ≤ cat(B). 2 If p is null-homotopic, then secat(p) ≥ cat(B). Topological complexity of groups

Proposition (Svarc)ˇ Let p : E → B be a ﬁbration. 1 If p is surjective, then secat(p) ≤ cat(B). 2 If p is null-homotopic, then secat(p) ≥ cat(B).

Proposition (Svarc)ˇ If the square Y E q p

A B is a pullback, then secat(q) ≤ secat(p). Topological complexity of groups

Corollary For any path-connected space X we have

cat(X ) ≤ TC(X ) ≤ cat(X × X ). This construction is functorial, so K(G, 1) is unique up to homotopy equivalence.

Problem (Farber) Describe TC(G) := TCK(G, 1) in terms of algebraic properties of the group G.

Topological complexity of groups Topological complexity of groups

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

G (i = 1), π K(G, 1) = i 0 (i > 1). Problem (Farber) Describe TC(G) := TCK(G, 1) in terms of algebraic properties of the group G.

Topological complexity of groups Topological complexity of groups

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

G (i = 1), π K(G, 1) = i 0 (i > 1).

This construction is functorial, so K(G, 1) is unique up to homotopy equivalence. Topological complexity of groups Topological complexity of groups

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

G (i = 1), π K(G, 1) = i 0 (i > 1).

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

Problem (Farber) Describe TC(G) := TCK(G, 1) in terms of algebraic properties of the group G. Theorem (Eilenberg–Ganea, Stallings, Swan) For any group G we have

cat(G) := catK(G, 1) = cd(G).

Topological complexity of groups Category of groups

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

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

Theorem (Eilenberg–Ganea, Stallings, Swan) For any group G we have

cat(G) := catK(G, 1) = cd(G). Topological complexity of groups Topological complexity of groups: a survey

Note that the inequalities

cd(G) ≤ TC(G) ≤ cd(G × G)

show that TC(G) = ∞ if G has torsion. So the problem is interesting mainly for torsion-free groups (of ﬁnite cohomological dimension). n Free abelian groups Z (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 plane Pn,m = ker(Pn → Pm) = π1 Fn(C \ m points) (Farber–G.–Yuzvinsky 2006)

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

Basis-conjugating automorphism groups PΣn and upper-triangular + McCool groups PΣn (Cohen–Pruidze 2008) Pure braid groups of surfaces π1 Fn(Σg ) (Cohen–Farber 2011)

Topological complexity of groups Topological complexity of groups: a survey

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

Groups for which the exact value of TC(G) is known include:

n Free abelian groups Z (Farber 2003) Orientable surface groups π1(Σg ), g ≥ 1 (Farber 2003)

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

Basis-conjugating automorphism groups PΣn and upper-triangular + McCool groups PΣn (Cohen–Pruidze 2008) Pure braid groups of surfaces π1 Fn(Σg ) (Cohen–Farber 2011) ∗ ∗ 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.

Theorem (Farber) We have TC(X ) ≥ nil ker(∪).

(The nilpotency nil I of an ideal I C R is the minimum k such that I k+1 = 0.)

Topological complexity of groups Cohomological lower bounds In all of these cases, sharp lower bounds are given by zero-divisors cup-length. Theorem (Farber) We have TC(X ) ≥ nil ker(∪).

(The nilpotency nil I of an ideal I C R is the minimum k such that I k+1 = 0.)

Topological complexity of groups Cohomological lower bounds In all of these cases, sharp lower bounds are given by zero-divisors cup-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. Topological complexity of groups Cohomological lower bounds In all of these cases, sharp lower bounds are given by zero-divisors cup-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.

Theorem (Farber) We have TC(X ) ≥ nil ker(∪).

(The nilpotency nil I of an ideal I C R is the minimum k such that I k+1 = 0.) Under certain conditions, secat(q) may be readily computable.

In particular, if q is null-homotopic, then secat(q) = cat(Y ) ≤ TC(X ).

A new lower bound for TC(G) Main idea

Given any map f : Y → X × X we obtain a pullback diagram

E X I

q πX f Y X × X ,

in which secat(q) ≤ secat(πX ) = TC(X ). In particular, if q is null-homotopic, then secat(q) = cat(Y ) ≤ TC(X ).

A new lower bound for TC(G) Main idea

Given any map f : Y → X × X we obtain a pullback diagram

E X I

q πX f Y X × X ,

in which secat(q) ≤ secat(πX ) = TC(X ).

Under certain conditions, secat(q) may be readily computable. A new lower bound for TC(G) Main idea

Given any map f : Y → X × X we obtain a pullback diagram

E X I

q πX f Y X × X ,

in which secat(q) ≤ secat(πX ) = TC(X ).

Under certain conditions, secat(q) may be readily computable.

In particular, if q is null-homotopic, then secat(q) = cat(Y ) ≤ TC(X ). If f = (φ, ψ): Y → X × X , the connecting homomorphism is given by

∂ : πi+1(Y ) → πi (ΩX ) = πi+1(X )

−1 ∂(y) = φ](y) · ψ](y) .

A new lower bound for TC(G)

Since ﬁber(πX ) = ΩX , the pullback is a ﬁbration

q ΩX E Y with long exact homotopy sequence

∂ q] ··· πi+1(Y ) πi (ΩX ) πi (E) πi (Y ) ··· . A new lower bound for TC(G)

Since ﬁber(πX ) = ΩX , the pullback is a ﬁbration

q ΩX E Y with long exact homotopy sequence

∂ q] ··· πi+1(Y ) πi (ΩX ) πi (E) πi (Y ) ··· .

If f = (φ, ψ): Y → X × X , the connecting homomorphism is given by

∂ : πi+1(Y ) → πi (ΩX ) = πi+1(X )

−1 ∂(y) = φ](y) · ψ](y) . Theorem 1 (G.–Lupton–Oprea) Let A and B be subgroups of G such that gAg −1 ∩ B = {1} for every g ∈ G. Then cd(A × B) ≤ TC(G).

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

Theorem 2 (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) If X is a K(G, 1) we can let Y be a K(A × B, 1) where A and B are subgroups of G. The map f : Y → X × X is induced by the inclusion of subgroups A × B ,→ G × G. Recall that A and B are complementary in G if A ∩ B = {1} and AB = G.

Theorem 2 (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) If X is a K(G, 1) we can let Y be a K(A × B, 1) where A and B are subgroups of G. The map f : Y → X × X is induced by the inclusion of subgroups A × B ,→ G × G.

Theorem 1 (G.–Lupton–Oprea) Let A and B be subgroups of G such that gAg −1 ∩ B = {1} for every g ∈ G. Then cd(A × B) ≤ TC(G). A new lower bound for TC(G) If X is a K(G, 1) we can let Y be a K(A × B, 1) where A and B are subgroups of G. The map f : Y → X × X is induced by the inclusion of subgroups A × B ,→ G × G.

Theorem 1 (G.–Lupton–Oprea) Let A and B be subgroups of G such that gAg −1 ∩ B = {1} for every g ∈ G. Then cd(A × B) ≤ TC(G).

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

Theorem 2 (G.–Lupton–Oprea) Let A and B be complementary subgroups of G. Then

cd(A × B) ≤ TC(G). In Theorem 1, the space E may be disconnected; the assumption gAg −1 ∩ B = {1} for all g ensures that each path-component is contractible.

We may also prove Theorem 1 by employing the one-dimensional category cat1 of Fox.

A new lower bound for TC(G) Remarks on the proofs

In the proof of Theorem 2, the assumption AB = G ensures that E is path-connected. The assumption A ∩ B = {1} ensures that π∗(E) = 0. We may also prove Theorem 1 by employing the one-dimensional category cat1 of Fox.

A new lower bound for TC(G) Remarks on the proofs

In the proof of Theorem 2, the assumption AB = G ensures that E is path-connected. The assumption A ∩ B = {1} ensures that π∗(E) = 0.

In Theorem 1, the space E may be disconnected; the assumption gAg −1 ∩ B = {1} for all g ensures that each path-component is contractible. A new lower bound for TC(G) Remarks on the proofs

In the proof of Theorem 2, the assumption AB = G ensures that E is path-connected. The assumption A ∩ B = {1} ensures that π∗(E) = 0.

In Theorem 1, the space E may be disconnected; the assumption gAg −1 ∩ B = {1} for all g ensures that each path-component is contractible.

We may also prove Theorem 1 by employing the one-dimensional category cat1 of Fox. Theorem (Cohen–Pruidze) We have TC(GΓ) = z(Γ) := max |VK ∪ VL| , K,L the maximum number of vertices covered by two cliques in Γ.

Examples Right-angled Artin groups

Given a ﬁnite graph Γ, one gets a ﬁnite group presentation with a generator xv for each vertex v ∈ VΓ and a relation [xv , xw ] for each edge {v, w} ∈ EΓ.

The resulting group GΓ is the Right-angled Artin group associated to Γ. Examples Right-angled Artin groups

Given a ﬁnite graph Γ, one gets a ﬁnite group presentation with a generator xv for each vertex v ∈ VΓ and a relation [xv , xw ] for each edge {v, w} ∈ EΓ.

The resulting group GΓ is the Right-angled Artin group associated to Γ.

Theorem (Cohen–Pruidze) We have TC(GΓ) = z(Γ) := max |VK ∪ VL| , K,L the maximum number of vertices covered by two cliques in Γ. Let K and L be disjoint cliques realizing z(Γ). Let A and B be the free abelian subgroups of GΓ generated by the vertices of K and L, respectively.

−1 It is easily veriﬁed that gAg ∩ B = {1} for all g ∈ GΓ.

Hence by Theorem 1,

TC(GΓ) ≥ cd(A × B) = cd(A) + cd(B) = |K| + |L| = z(Γ).

Examples Right-angled Artin groups

Here we observe that the lower bound z(Γ) ≤ TC(GΓ) follows easily from our Theorem 1. −1 It is easily veriﬁed that gAg ∩ B = {1} for all g ∈ GΓ.

Hence by Theorem 1,

TC(GΓ) ≥ cd(A × B) = cd(A) + cd(B) = |K| + |L| = z(Γ).

Examples Right-angled Artin groups

Here we observe that the lower bound z(Γ) ≤ TC(GΓ) follows easily from our Theorem 1.

Let K and L be disjoint cliques realizing z(Γ). Let A and B be the free abelian subgroups of GΓ generated by the vertices of K and L, respectively. Hence by Theorem 1,

TC(GΓ) ≥ cd(A × B) = cd(A) + cd(B) = |K| + |L| = z(Γ).

Examples Right-angled Artin groups

Here we observe that the lower bound z(Γ) ≤ TC(GΓ) follows easily from our Theorem 1.

Let K and L be disjoint cliques realizing z(Γ). Let A and B be the free abelian subgroups of GΓ generated by the vertices of K and L, respectively.

−1 It is easily veriﬁed that gAg ∩ B = {1} for all g ∈ GΓ. Examples Right-angled Artin groups

Here we observe that the lower bound z(Γ) ≤ TC(GΓ) follows easily from our Theorem 1.

Let K and L be disjoint cliques realizing z(Γ). Let A and B be the free abelian subgroups of GΓ generated by the vertices of K and L, respectively.

−1 It is easily veriﬁed that gAg ∩ B = {1} for all g ∈ GΓ.

Hence by Theorem 1,

TC(GΓ) ≥ cd(A × B) = cd(A) + cd(B) = |K| + |L| = z(Γ). It has cd(Pn) = n − 1.

Theorem (Farber–Yuzvinsky) We have TC(Pn) = 2n − 3 for all n ≥ 2.

Examples Pure braid groups

The pure braid group on n strands can be deﬁned as Pn = π1 Fn(C) ,

n where Fn(C) = {(z1,..., zn) ∈ C | i 6= j =⇒ zi 6= zj } is the classical conﬁguration space. Theorem (Farber–Yuzvinsky) We have TC(Pn) = 2n − 3 for all n ≥ 2.

Examples Pure braid groups

The pure braid group on n strands can be deﬁned as Pn = π1 Fn(C) ,

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

It has cd(Pn) = n − 1. Examples Pure braid groups

The pure braid group on n strands can be deﬁned as Pn = π1 Fn(C) ,

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

It has cd(Pn) = n − 1.

Theorem (Farber–Yuzvinsky) We have TC(Pn) = 2n − 3 for all n ≥ 2. There is an inclusion Pn−1 ,→ Pn given by intro- ducing an n-th non-interacting strand after the other strands.

Examples Pure braid groups

Recall that elements of Pn can also be described geometrically as isotopy classes of braids, with the group operation given by concatenation. Examples Pure braid groups

Recall that elements of Pn can also be described geometrically as isotopy classes of braids, with the group operation given by concatenation.

There is an inclusion Pn−1 ,→ Pn given by intro- ducing an n-th non-interacting strand after the other strands. =

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

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

So Theorem 1 gives

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

Examples Pure braid groups

For j = 1,..., n−1, let αj be the braid which runs the j-th strand in front of the last n − j strands, then passes behind the last n − j strands to its original position. Since conjugate braids close to isotopic links, one checks using linking −1 numbers with the last strand that gAg ∩ Pn−1 = {1} for all g ∈ Pn.

So Theorem 1 gives

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

Examples Pure braid groups

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

The αj ’s commute pairwise, so they generate a free abelian subgroup A of rank (n − 1). So Theorem 1 gives

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

Examples Pure braid groups

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

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

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

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

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

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

So Theorem 1 gives

TC(Pn) ≥ cd(A × Pn−1) = (n − 1) + (n − 2) = 2n − 3. ∗ All cup-products vanish in He (X ; k) for any ﬁeld k, so the zero-divisors cup-length is 2.

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

Examples Borromean rings

The link complement X of the Borromean rings is a compact aspherical 3-manifold with fundamental group

G =∼ ha, b, c | [a, [b−1, c]], [b, [c−1, a]]i. ∗ Using Massey products in H (X ; Q) and sectional category weight, one can show that TC(X ) ≥ 3 (G., 2009).

Examples Borromean rings

The link complement X of the Borromean rings is a compact aspherical 3-manifold with fundamental group

G =∼ ha, b, c | [a, [b−1, c]], [b, [c−1, a]]i.

∗ All cup-products vanish in He (X ; k) for any ﬁeld k, so the zero-divisors cup-length is 2. Examples Borromean rings

The link complement X of the Borromean rings is a compact aspherical 3-manifold with fundamental group

G =∼ ha, b, c | [a, [b−1, c]], [b, [c−1, a]]i.

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

∗ Using Massey products in H (X ; Q) and sectional category weight, one can show that TC(X ) ≥ 3 (G., 2009). There results a split extension

p K G F2hα, βi,

  a 7→ α p : b 7→ β  c 7→ 1

Letting A = hai and B = p−1hβi, one can show algebraically that gAg −1 ∩ B = {1} for all g in G.

Since B is not free, Theorem 1 gives

TC(G) ≥ cd(A × B) = 1 + 2 = 3.

Examples Borromean rings Removing one component gives an unlink. Letting A = hai and B = p−1hβi, one can show algebraically that gAg −1 ∩ B = {1} for all g in G.

Since B is not free, Theorem 1 gives

TC(G) ≥ cd(A × B) = 1 + 2 = 3.

Examples Borromean rings Removing one component gives an unlink. There results a split extension

p K G F2hα, βi,

  a 7→ α p : b 7→ β  c 7→ 1 Since B is not free, Theorem 1 gives

TC(G) ≥ cd(A × B) = 1 + 2 = 3.

Examples Borromean rings Removing one component gives an unlink. There results a split extension

p K G F2hα, βi,

  a 7→ α p : b 7→ β  c 7→ 1

Letting A = hai and B = p−1hβi, one can show algebraically that gAg −1 ∩ B = {1} for all g in G. Examples Borromean rings Removing one component gives an unlink. There results a split extension

p K G F2hα, βi,

  a 7→ α p : b 7→ β  c 7→ 1

Letting A = hai and B = p−1hβi, one can show algebraically that gAg −1 ∩ B = {1} for all g in G.

Since B is not free, Theorem 1 gives

TC(G) ≥ cd(A × B) = 1 + 2 = 3. Further questions Further work

1 There are analogous results for non-aspherical spaces, and in the rational setting. 2 Can we recover all zero-divisors lower bounds using these methods? 3 Can the result be generalized to give lower bounds when gAg 1 ∩ B is “small”? 4 Can we give new calculations of TC(G) using these results? Further questions Thank you for listening!