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On the topology of matrix configuration spaces

Daniel C. Cohen

Department of MathematicsLouisiana State University

Daniel C. Cohen (LSU) Matrix configuration spaces Summer 2013 1

Configuration spaces

M topological space (e.g., a manifold, a graph. . . ) Mn = M × · · · ×M

F (M,n) = {(x1, . . . , xn) ∈ Mn : xi 6= xj if i 6= j}the configuration space of n distinct ordered points in M

symmetric group Σn acts freely on F (M,n)

F (M,n)/Σn the configuration space of unordered points in M

ExampleM = C

F (C,n)/Σn is a K (G,1)-space for G = Bn the Artin full braid group

F (C,n) is a K (G,1)-space for G = Pn the Artin pure braid group

1→ Pn → Bn → Σn → 1


Generalize. . .

X (n,1) = F (C,n) = {(x1, . . . , xn) ∈ Cn : xi 6= xj if i 6= j}space of n-tuples of points in C no two equal

X (n,2) space of n-tuples of points in C2 no three colinear

(xi , yi), (xj , yj), (xk , yk ) colinear ⇐⇒

∣∣∣∣∣∣1 1 1xi xj xkyi yj yk

∣∣∣∣∣∣ = 0

X (n,2) =

1 1 · · · 1

x1 x2 · · · xny1 y2 · · · yn

:xi , yj ∈ C

all 3× 3 minors nonzero

Note:

X (n,1) = F (C,n) =

{[1 1 · · · 1x1 x2 · · · xn

]:

xi ∈ Call 2× 2 minors nonzero

}Daniel C. Cohen (LSU) Matrix configuration spaces Summer 2013 3

Matrix configuration space

X (n, k) =

1 . . . 1x1,1 · · · x1,n

......

xk ,1 · · · xk ,n

:xi,j ∈ C

all (k + 1)× (k + 1) minors nonzero

X (n, k) space of n-tuples of points in Ck

no k + 1 of which lie on an affine (k − 1)-plane

related space:

G(n, k) =

x1,1 · · · x1,n

......

xk ,1 · · · xk ,n

:xi,j ∈ C

all k × k minors nonzero

G(n, k)/Σn is the generic stratum in the matroid “stratification” of theGrassmannian due to Gelfand-Goresky-MacPherson-Serganova

topology of such spaces?Daniel C. Cohen (LSU) Matrix configuration spaces Summer 2013 4

Some observations

Do nice features of X (n,1) = F (C,n) extend to X (n, k) for k ≥ 2?

Focus on X (n,2)

Theorem (Fadell-Neuwirth)

forgetful map p : F (C,n)→ F (C,n − 1),(x1, . . . , xn−1, xn) 7→ (x1, . . . , xn−1), is a bundle, fiber Cr {n − 1 points}

forgetful map P : X (n,2)→ X (n − 1,2) forget last columnis not in general a bundle

Example

n = 5 P : X (5,2)→ X (4,2) is not a bundle

fiber over x ∈ X (4,2) is C2 r {arrangement of 6 lines}

combinatorics (i.e., intersection pattern) of these lines varies with x

=⇒ topology of P−1(x) is not constant


not bundle

fibers over x =

1 1 1 10 1 0 10 0 1 1

and y =

1 1 1 10 1 0 20 0 1 1

in X (4,2)


cohomology

cohomology with C coefficients throughout H∗(X ) = H∗(X ;C)

Theorem (Arnol’d, Cohen)

H∗(X (n,1)) = H∗(F (C,n)) is generated in degree one by

ωi,j =d(xi−xj )

xi−xj1 ≤ i < j ≤ n

relations: ωi,jωi,k − ωi,jωj,k + ωi,kωj,k = 0 1 ≤ i < j < k ≤ nand their consequences

H∗(X (n,2)) is not generated in degree one

the map X (n,2)→ GL2(C),

1 1 · · · 1x1 x2 · · · xny1 y2 · · · yn

7→ [x2 − x1 x3 − x1y2 − y1 y3 − y1

]induces a monomorphism H∗(GL2(C)) ↪→ H∗(X (n,2)) in cohomology


higher homotopy


X (n,1) = F (C,n) is a K (G,1)-spaceπk (F (C,n)) = 0 for k ≥ 2

X (n,2) is not a K (G,1)-space

X (n,2) ' GL2(C)× Y (n,2)

Y (n,2) =

1 1 1 1 · · · 1

0 1 0 x4 · · · xn0 0 1 y4 · · · yn

:xi , yj ∈ C


Y (n,2) is not a K (G,1)-space either

Example

n = 4 Y (4,2) = {(x , y) ∈ C2 : xy(1− x − y) 6= 0}

[Hattori] Y (4,2) ' S1×S1×S1 r {point} =⇒ π2(Y (4,2)) nontrivial


“section”


bundle p : F (C,n)→ F (C,n − 1) admits a section∃ s : F (C,n − 1)→ F (C,n) with p ◦ s = id|F (C,n−1)

there is a map S : X (n − 1,2)→ X (n,2) with P ◦ S = id|X(n−1,2)

consequence: π2(Y (n,2)) is nontrivial for all n ≥ 4

idea: x ∈ X (n − 1,2)←→ collection of lines in C2

produce new line in C2 in general position with respect to these


TC(X (n,2))

X topological space PX space of all continuous paths γ : [0,1]→ X

π : PX → X × X , γ 7→ (γ(0), γ(1)), is a fibration

Farber (2003) – topological approach to the motion planning problemfrom robotics: The topological complexity of X is the Schwarz genus,or sectional category, of the fibration π : PX → X × X

TC(X ) = genus(π : PX → X × X )

TC(X ): smallest integer k for which X × X has an open cover with kelements, over each of which π : PX → X ×X has a continuous section

X × X = U1 ∪ · · · ∪ Uk si : Ui → PX continuous π ◦ si = id|Ui

Theorem

TC(X (n,2)) =

3 if n = 36 if n = 44n − 9 if n ≥ 5


The motion planning problem

A motion planning algorithm for a mechanical system is a rule whichassigns to a pair of states (A,B) of the system a continuous motion ofthe system starting at A and ending at B

X the configuration space of the system

PX the space of all continuous paths γ : [0,1]→ X as before

A motion planning algorithm is a section s : X × X → PX of the pathspace fibration π : PX → X × X

(not necessarily continuous)

Proposition∃ a globally continuous section s : X × X → PX of π : PX → X × X

⇐⇒ X is contractible

that is, TC(X ) = 1 ⇐⇒ X is contractible


Solving the motion planning problem

Theorem (Farber)

If X is a Euclidean Neighborhood Retract, then TC(X ) is equal to thesmallest integer k so that there is a section s : X × X → PX of the pathspace fibration and a decomposition

X × X = F1 ∪ F2 ∪ · · · ∪ Fk , Fi ∩ Fj = ∅,

with Fi locally compact and s|Fi : Fi → PX continuous for each i

This gives a motion planning algorithm:

If (A,B) ∈ X × X , ∃! Fi with (A,B) ∈ Fi , and the path s(A,B) is acontinuous motion of the system starting at A and ending at B


Spheres

Example (X = S1)

F1 = {(x ,−x) | x ∈ X} ⊂ X × X F2 = X × X r F1

s|F1 : F1 → PX counterclockwise path from x to −xs|F2 : F2 → PX shortest geodesic arc from x to y

TC(S1) = 2

Example (X = S2)fix e ∈ X , ν a nowhere zero tangent vector field on X r eF1 = {(e,−e)} F2 = {(x ,−x) | x 6= e} F3 = {(x , y) | x 6= −y}s|F1 : F1 → PX any fixed path from e to −es|F2 : F2 → PX path x to −x along semicircle tangent to ν(x)

s|F3 : F3 → PX shortest geodesic arc from x to yTC(S2) ≤ 3


TC properties (Farber)

• TC(X ) depends only on the homotopy type of X

• bounds in terms of the Lusternik-Schnirelman categorycat(X ): smallest integer k for which X has an open cover with kelements, each contractible in X

cat(X ) ≤ TC(X ) ≤ cat(X × X )

Proposition (Weinberger)

If G is a connected Lie group, then TC(G) = cat(G).

• dimensional upper bounddim(X ): the covering dimension of X

TC(X ) ≤ 2 dim(X ) + 1

• product inequalityTC(X × Y ) ≤ TC(X ) + TC(Y )− 1


more TC properties

• cohomological lower bound

TC(X ) ≥ zcl(H∗(X )) + 1

zcl(H∗(X )) the zero-divisor cup length of H∗(X )

the cup length of ker[H∗(X )⊗ H∗(X )

∪−−→ H∗(X )]

Example (X = S2 continued recall TC(S2) ≤ 3)

If 0 6= x ∈ H2(S2), then (x ⊗ 1− 1⊗ x)2 = −2x ⊗ x 6= 0zcl H∗(S2) ≥ 2 =⇒ TC(S2) ≥ 3 so TC(S2) = 3

• zcl propertiesA, B graded, graded commutative, connected, unital algebras/C

B a subalgebra of A =⇒ zcl(A) ≥ zcl(B)

B an epimorphic image of A =⇒ zcl(A) ≥ zcl(B)

zcl(A⊗ B) ≥ zcl(A) + zcl(B)Daniel C. Cohen (LSU) Matrix configuration spaces Summer 2013 15

Toric complexes

T = T n = S1 × · · · × S1 = {z = (z1, . . . , zn) ∈ Cn : |zi | = 1} n-torus

equipped with the standard minimal CW-decomposition

k -dim’l cell CK ←→ subset K of [n] = {1,2, . . . ,n} of cardinality k

for K ⊆ [n] CK = {z ∈ T : zi = 1 if i /∈ K , zi 6= 1 if i ∈ K}

let TK = {z ∈ T : zi = 1 if i /∈ K} TK∼= T |K | |K |-torus

X ⊂ T n a subcomplex

z(X ) = max{|J|+ |K | : J ∩ K = ∅ and TJ ∨ TK is a subcomplex of X}

Theorem (C-Pruidze)

TC(X ) = z(X ) + 1


Aside – Right-angled Artin groups

Γ finite graph on vertices [n] = {1,2, . . . ,n} no loops or multiple edges

XΓ subcomplex of T n delete cells corresponding to noncliques of Γ

XΓ has a 0-cell a 1-cell for each vertex of Γ

a 2-cell for each edge of Γ

a 3-cell for each triangle in Γ etc.

GΓ = 〈x1, . . . , xn | xixj = xjxi if {i , j} is an edge of Γ〉right-angled Artin group associated to Γ

Theorem (Charney-Davis, Meier-Van Wyk, Papadima-Suciu)

• XΓ is a K (GΓ,1)-space

• PΓ(t) =∑k≥0

dim Hk (XΓ)tk =∑k≥0

ck (Γ)tk ck (Γ) = #{k cliques in Γ}

• PΓ(−t) =∏k≥1

(1− tk )φk φk = rank of k th LCS quotient of GΓ


TC(XΓ)

z(Γ) = largest number of vertices of Γ covered by precisely two cliques

Theorem (C-Pruidze)

TC(XΓ) = z(Γ) + 1

• • •

•

• •

• • •

• • •Γ1 Γ2

1 2 3 1 2 3

4 5 64 5

6

Example

Xi = XΓi Gi = GΓi PΓ1(t) = 1 + 6t + 9t2 + 4t3 = PΓ2(t)

Xi (resp. Gi ) have same homology, Gi have same LCS quotients

distinguished by TC: TC(X1) = 6 TC(X2) = 7


TC of right-angled Artin groups

G a discrete groupcat(G) := cat(X ) TC(G) := TC(X ) where X is a K (G,1)-space

Theorem (Eilenberg-Ganea)

cat(G) = 1 + geometric dimension of G (for most G)

Example

G = GΓ a right-angled Artin group TC(GΓ) = TC(XΓ) = z(Γ) + 1for instance TC(Zn) = n + 1 TC(Fn) = 3 (for n ≥ 2)Γ = Γ1 q Γ2 (disjoint ∪) =⇒ TC(GΓ) = TC(GΓ1 ∗GΓ2) = TC(XΓ1 ∨ XΓ2)

Theorem (Rudyak)For each natural number k and each natural number ` with k ≤ ` ≤ 2k,there is a discrete group G with cat(G) = k + 1 and TC(G) = `+ 1

can take G = Zk ∗ Z`−k in the theoremDaniel C. Cohen (LSU) Matrix configuration spaces Summer 2013 19

Hattori theorem

G = {H1,H2, . . . ,Hn} n hyperplanes in general position in C` (n ≥ `)

XG = C` r⋃n

i=1 Hi the complement

Example

` = 1 XG = Cr {n points} '∨

n S1

` = n XG = (C∗)n ' T n

` = 2 G = {{x = 0}, {y = 0}, {x + y = 1}} XG = Y (4,2)

Theorem (Hattori)

XG ' T n` the `-dimensional skeleton of T n

=⇒ π`(XG) nontrivial if n > `

Theorem (Yuzvinsky, C-Pruidze)

TC(XG) = min{n + 1,2`+ 1}


Hyperplane arrangements

A hyperplane arrangement is a finite collection of codimension 1 affinesubspaces in a complex vector space V

A = {H1, . . . ,Hn} Hi = {fi = 0} fi a linear polynomial

XA = V r⋃n

i=1 Hi the complement

ExampleG general position arrangement

A = {Hi,j 1 ≤ i < j ≤ n} in V = Cn Hi,j = {xi = xj} braid arrangement

XA = Cn r⋃

i<j Hi,j = {(x1, . . . , xn) ∈ Cn : xi 6= xj if i 6= j} = F (C,n)

Theorem (Brieskorn-Orlik-Solomon)

H∗(XA) ∼= E/I E exterior algebra generated by df1f1, . . . , dfn

fnI explicit two-sided ideal in E


TC of some arrangement complements

A is central if 0 ∈ Hi ∀ Hi ∈ A Hi = ker(fi) fi a linear form

{Hi1 , . . . ,Hik} ⊂ A independent ⇐⇒ {fi1 , . . . , fik} linearly independent

rank(A) = cardinality of a maximal independent set of hyperplanes in A

Theorem (Farber-Yuzvinsky)

A a central arrangement with rank(A) = r . If ∃ H1, . . . ,H2r−1 ∈ A with{H1, . . . ,Hr} independent and {Hj ,Hr+1, . . . ,H2r−1} independent foreach j, 1 ≤ j ≤ r , then zcl(H∗(XA)) = 2r − 1 and TC(XA) = 2r .

Theorem (Farber-Yuzvinsky)

TC(F (C,n)) = 2n − 2 TC(Pn) = 2n − 2 (Artin pure braid group)

Theorem (Farber-Grant-Yuzvinsky)

TC(F (Cr {m points},n)) =

{2n m = 12n + 1 m ≥ 2


back to TC(X (n,2))

Theorem

TC(X (n,2)) =

3 if n = 36 if n = 44n − 9 if n ≥ 5

X (n,2) ' GL2(C)× Y (n,2)

X (3,2) ' GL2(C)

X (4,2) ' GL2(C)× Y (4,2)

' GL2(C)× {(x , y) ∈ C2 : xy(1− x − y) 6= 0}' GL2(C)× T 3

2

' GL2(C)× (S1 × S1 × S1 r {point})

these cases covered by results discussed previouslyDaniel C. Cohen (LSU) Matrix configuration spaces Summer 2013 23

TC(X (n,2))

X (n,2) ' GL2(C)× Y (n,2)

for n ≥ 5 enough to show that TC((Y (n,2)) = 4n − 11

Y (n,2) =

1 1 1 1 · · · 1

0 1 0 x4 · · · xn0 0 1 y4 · · · yn

:xi , yj ∈ C


' CW-complex of dimension 2(n − 3)

so need to see that TC(Y (n,2)) = 2 dim(Y (n,2)) + 1 = 4n − 11

TC(Y (n,2)) ≥ zcl(H∗(Y (n,2))) + 1

so if zcl(H∗(Y (n,2))) = 2 dim(Y (n,2)) = 4(n − 3)

then TC(Y (n,2)) = 2 dim(Y (n,2)) + 1 = 4n − 11


TC(X (n,2))

{(x , y) ∈ C2 : xy(1− x − y) 6= 0} → {(u, v) ∈ C2 : uv(1 + u + v) 6= 0}

(x , y) 7→(

x1−x−y ,

y1−x−y

)induces g : Y (n,2)→ XA A arrangement in C2(n−3)

(· · · xi , yi · · · ) 7→(· · · xi

1−xi−yi, yi

1−xi−yi· · ·)

A defined by linear polynomials ui , vi , 1 + ui + vi , uj − uk , vj − vk

3 ≤ i ≤ n 3 ≤ j < k ≤ n

Farber-Yuzvinsky Theorem + some work =⇒ zcl(H∗(XA)) = 4(n − 3)

Proposition

g∗ : H∗(XA)→ H∗(Y (n,2)) is a monomorphism

hence zcl(H∗(Y (n,2))) ≥ zcl(H∗(XA)) = 4(n − 3) as neededthe end


THANKS FOR LISTENING !


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