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 . 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 ). 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 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. 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) = 1 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 n 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) 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 n 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) ∗ ∗ Let H (−) = H (−; |) with | 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 (−; |) with | 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 (−; |) with | 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, ): Y ! X × X , the connecting homomorphism is given by @ : πi+1(Y ) ! πi (ΩX ) = πi+1(X ) −1 @(y) = φ](y) · ](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 ) ··· .