On the topology of matrix configuration spaces

Transcription

1 On the topology of matrix configuration spaces Daniel C. Cohen Department of Mathematics Louisiana State University Daniel C. Cohen (LSU) Matrix configuration spaces Summer

2 Configuration spaces M topological space (e.g., a manifold, a graph... ) M n = M M F(M, n) = {(x 1,..., x n ) M n : x i x j if i j} the configuration space of n distinct ordered points in M symmetric group Σ n acts freely on F(M, n) F(M, n)/σ n the configuration space of unordered points in M Example M = C F(C, n)/σ n is a K (G, 1)-space for G = B n the Artin full braid group F(C, n) is a K (G, 1)-space for G = P n the Artin pure braid group 1 P n B n Σ n 1 Daniel C. Cohen (LSU) Matrix configuration spaces Summer

3 Generalize... X(n, 1) = F (C, n) = {(x 1,..., x n ) C n : x i x j if i j} space of n-tuples of points in C no two equal X(n, 2) space of n-tuples of points in C 2 no three colinear (x i, y i ), (x j, y j ), (x k, y k ) colinear x i x j x k y i y j y k = X(n, 2) = x 1 x 2 x n x : i, y j C all 3 3 minors nonzero y 1 y 2 y n Note: {[ ] } x X(n, 1) = F(C, n) = : i C x 1 x 2 x n all 2 2 minors nonzero Daniel C. Cohen (LSU) Matrix configuration spaces Summer

4 Matrix configuration space x 1,1 x 1,n X(n, k) =.. : x i,j C all (k + 1) (k + 1) minors nonzero x k,1 x k,n X(n, k) space of n-tuples of points in C k no k + 1 of which lie on an affine (k 1)-plane related space: x 1,1 x 1,n x G(n, k) =.. : i,j C all k k minors nonzero x k,1 x k,n G(n, k)/σ n is the generic stratum in the matroid stratification of the Grassmannian due to Gelfand-Goresky-MacPherson-Serganova topology of such spaces? Daniel C. Cohen (LSU) Matrix configuration spaces Summer

5 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), (x 1,..., x n 1, x n ) (x 1,..., x n 1 ), is a bundle, fiber C {n 1 points} forgetful map P : X(n, 2) X(n 1, 2) is not in general a bundle forget last column Example n = 5 P : X(5, 2) X(4, 2) is not a bundle fiber over x X(4, 2) is C 2 {arrangement of 6 lines} combinatorics (i.e., intersection pattern) of these lines varies with x = topology of P 1 (x) is not constant Daniel C. Cohen (LSU) Matrix configuration spaces Summer

6 not bundle fibers over x = and y = in X(4, 2) Daniel C. Cohen (LSU) Matrix configuration spaces Summer

7 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 ω i,j = d(x i x j ) x i x j 1 i < j n relations: ω i,j ω i,k ω i,j ω j,k + ω i,k ω j,k = 0 1 i < j < k n and their consequences H (X(n, 2)) is not generated in degree one [ ] the map X(n, 2) GL 2 (C), x 1 x 2 x n x2 x 1 x 3 x 1 y y 1 y 2 y 2 y 1 y 3 y 1 n induces a monomorphism H (GL 2 (C)) H (X(n, 2)) in cohomology Daniel C. Cohen (LSU) Matrix configuration spaces Summer

8 higher homotopy Theorem (Fadell-Neuwirth) X(n, 1) = F (C, n) is a K (G, 1)-space π k (F(C, n)) = 0 for k 2 X(n, 2) is not a K (G, 1)-space X(n, 2) GL 2 (C) Y (n, 2) Y (n, 2) = x 4 x n x : i, y j C all 3 3 minors nonzero y 4 y n Y (n, 2) is not a K (G, 1)-space either Example n = 4 Y (4, 2) = {(x, y) C 2 : xy(1 x y) 0} [Hattori] Y (4, 2) S 1 S 1 S 1 {point} = π 2 (Y (4, 2)) nontrivial Daniel C. Cohen (LSU) Matrix configuration spaces Summer

9 section Theorem (Fadell-Neuwirth) 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: π 2 (Y (n, 2)) is nontrivial for all n 4 idea: x X(n 1, 2) collection of lines in C 2 produce new line in C 2 in general position with respect to these Daniel C. Cohen (LSU) Matrix configuration spaces Summer

10 TC(X(n, 2)) X topological space PX space of all continuous paths γ : [0, 1] X π : PX X X, γ (γ(0), γ(1)), is a fibration Farber (2003) topological approach to the motion planning problem from 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 k elements, over each of which π : PX X X has a continuous section X X = U 1 U k s i : U i PX continuous π s i = id Ui Theorem 3 if n = 3 TC(X(n, 2)) = 6 if n = 4 4n 9 if n 5 Daniel C. Cohen (LSU) Matrix configuration spaces Summer

11 The motion planning problem A motion planning algorithm for a mechanical system is a rule which assigns to a pair of states (A, B) of the system a continuous motion of the 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 path space 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 Daniel C. Cohen (LSU) Matrix configuration spaces Summer

12 Solving the motion planning problem Theorem (Farber) If X is a Euclidean Neighborhood Retract, then TC(X) is equal to the smallest integer k so that there is a section s : X X PX of the path space fibration and a decomposition X X = F 1 F 2 F k, F i F j =, with F i locally compact and s F i : F i PX continuous for each i This gives a motion planning algorithm: If (A, B) X X,! F i with (A, B) F i, and the path s(a, B) is a continuous motion of the system starting at A and ending at B Daniel C. Cohen (LSU) Matrix configuration spaces Summer

13 Spheres Example (X = S 1 ) F 1 = {(x, x) x X} X X F 2 = X X F 1 s F 1 : F 1 PX counterclockwise path from x to x s F 2 : F 2 PX shortest geodesic arc from x to y TC(S 1 ) = 2 Example (X = S 2 ) fix e X, ν a nowhere zero tangent vector field on X e F 1 = {(e, e)} F 2 = {(x, x) x e} F 3 = {(x, y) x y} s F 1 : F 1 PX any fixed path from e to e s F 2 : F 2 PX path x to x along semicircle tangent to ν(x) s F 3 : F 3 PX shortest geodesic arc from x to y TC(S 2 ) 3 Daniel C. Cohen (LSU) Matrix configuration spaces Summer

14 TC properties (Farber) TC(X) depends only on the homotopy type of X bounds in terms of the Lusternik-Schnirelman category cat(x): smallest integer k for which X has an open cover with k elements, each contractible in X Proposition (Weinberger) cat(x) TC(X) cat(x X) If G is a connected Lie group, then TC(G) = cat(g). dimensional upper bound dim(x): the covering dimension of X product inequality TC(X) 2 dim(x) + 1 TC(X Y ) TC(X) + TC(Y ) 1 Daniel C. Cohen (LSU) Matrix configuration spaces Summer

15 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 = S 2 continued recall TC(S 2 ) 3) If 0 x H 2 (S 2 ), then (x 1 1 x) 2 = 2x x 0 zcl H (S 2 ) 2 = TC(S 2 ) 3 so TC(S 2 ) = 3 zcl properties A, 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

16 Toric complexes T = T n = S 1 S 1 = {z = (z 1,..., z n ) C n : z i = 1} n-torus equipped with the standard minimal CW-decomposition k-dim l cell C K subset K of [n] = {1, 2,..., n} of cardinality k for K [n] C K = {z T : z i = 1 if i / K, z i 1 if i K } let T K = {z T : z i = 1 if i / K } T K = T K K -torus X T n a subcomplex z(x) = max{ J + K : J K = and T J T K is a subcomplex of X} Theorem (C-Pruidze) TC(X) = z(x) + 1 Daniel C. Cohen (LSU) Matrix configuration spaces Summer

17 Aside Right-angled Artin groups Γ finite graph on vertices [n] = {1, 2,..., n} no loops or multiple edges X Γ subcomplex of T n delete cells corresponding to noncliques of Γ X Γ 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 G Γ = x 1,..., x n x i x j = x j x i if {i, j} is an edge of Γ right-angled Artin group associated to Γ etc. Theorem (Charney-Davis, Meier-Van Wyk, Papadima-Suciu) X Γ is a K (G Γ, 1)-space P Γ (t) = dim H k (X Γ )t k = c k (Γ)t k c k (Γ) = #{k cliques in Γ} k 0 k 0 P Γ ( t) = k 1(1 t k ) φ k φ k = rank of k th LCS quotient of G Γ Daniel C. Cohen (LSU) Matrix configuration spaces Summer

18 TC(X Γ ) z(γ) = largest number of vertices of Γ covered by precisely two cliques Theorem (C-Pruidze) TC(X Γ ) = z(γ) Γ 1 Γ Example X i = X Γi G i = G Γi P Γ1 (t) = 1 + 6t + 9t 2 + 4t 3 = P Γ2 (t) X i (resp. G i ) have same homology, G i have same LCS quotients distinguished by TC: TC(X 1 ) = 6 TC(X 2 ) = 7 Daniel C. Cohen (LSU) Matrix configuration spaces Summer

19 TC of right-angled Artin groups G a discrete group cat(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 = G Γ a right-angled Artin group TC(G Γ ) = TC(X Γ ) = z(γ) + 1 for instance TC(Z n ) = n + 1 TC(F n ) = 3 (for n 2) Γ = Γ 1 Γ 2 (disjoint ) = TC(G Γ ) = TC(G Γ1 G Γ2 ) = TC(X Γ1 X Γ2 ) Theorem (Rudyak) For each natural number k and each natural number l with k l 2k, there is a discrete group G with cat(g) = k + 1 and TC(G) = l + 1 can take G = Z k Z l k in the theorem Daniel C. Cohen (LSU) Matrix configuration spaces Summer

20 Hattori theorem G = {H 1, H 2,..., H n } n hyperplanes in general position in C l (n l) X G = C l n i=1 H i the complement Example l = 1 l = n X G = C {n points} n S1 X G = (C ) n T n l = 2 G = {{x = 0}, {y = 0}, {x + y = 1}} X G = Y (4, 2) Theorem (Hattori) X G T n l the l-dimensional skeleton of T n Theorem (Yuzvinsky, C-Pruidze) TC(X G ) = min{n + 1, 2l + 1} = π l (X G ) nontrivial if n > l Daniel C. Cohen (LSU) Matrix configuration spaces Summer

21 Hyperplane arrangements A hyperplane arrangement is a finite collection of codimension 1 affine subspaces in a complex vector space V A = {H 1,..., H n } H i = {f i = 0} f i a linear polynomial X A = V n i=1 H i the complement Example G general position arrangement A = {H i,j 1 i < j n} in V = C n H i,j = {x i = x j } braid arrangement X A = C n i<j H i,j = {(x 1,..., x n ) C n : x i x j if i j} = F(C, n) Theorem (Brieskorn-Orlik-Solomon) H (X A ) = E/I E exterior algebra generated by df 1 f 1,..., dfn f n I explicit two-sided ideal in E Daniel C. Cohen (LSU) Matrix configuration spaces Summer

22 TC of some arrangement complements A is central if 0 H i H i A H i = ker(f i ) f i a linear form {H i1,..., H ik } A independent {f i1,..., f ik } 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 H 1,..., H 2r 1 A with {H 1,..., H r } independent and {H j, H r+1,..., H 2r 1 } independent for each j, 1 j r, then zcl(h (X A )) = 2r 1 and TC(X A ) = 2r. Theorem (Farber-Yuzvinsky) TC(F(C, n)) = 2n 2 TC(P n ) = 2n 2 (Artin pure braid group) Theorem (Farber-Grant-Yuzvinsky) { 2n m = 1 TC(F(C {m points}, n)) = 2n + 1 m 2 Daniel C. Cohen (LSU) Matrix configuration spaces Summer

23 back to TC(X(n, 2)) Theorem 3 if n = 3 TC(X(n, 2)) = 6 if n = 4 4n 9 if n 5 X(n, 2) GL 2 (C) Y (n, 2) X(3, 2) GL 2 (C) X(4, 2) GL 2 (C) Y (4, 2) GL 2 (C) {(x, y) C 2 : xy(1 x y) 0} GL 2 (C) T 3 2 GL 2 (C) (S 1 S 1 S 1 {point}) these cases covered by results discussed previously Daniel C. Cohen (LSU) Matrix configuration spaces Summer

24 TC(X(n, 2)) X(n, 2) GL 2 (C) Y (n, 2) for n 5 enough to show that TC((Y (n, 2)) = 4n Y (n, 2) = x 4 x n x : i, y j C all 3 3 minors nonzero y 4 y n 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 Daniel C. Cohen (LSU) Matrix configuration spaces Summer

25 TC(X(n, 2)) {(x, y) C 2 : xy(1 x y) 0} {(u, v) C 2 : uv(1 + u + v) 0} ( ) (x, y) x 1 x y, y 1 x y induces g : Y (n, 2) X A A arrangement in C 2(n 3) ( ) x ( x i, y i ) i y 1 x i y i, i 1 x i y i A defined by linear polynomials u i, v i, 1 + u i + v i, u j u k, v j v k 3 i n 3 j < k n Farber-Yuzvinsky Theorem + some work = zcl(h (X A )) = 4(n 3) Proposition g : H (X A ) H (Y (n, 2)) is a monomorphism hence zcl(h (Y (n, 2))) zcl(h (X A )) = 4(n 3) as needed the end Daniel C. Cohen (LSU) Matrix configuration spaces Summer

26 THANKS FOR LISTENING! Daniel C. Cohen (LSU) Matrix configuration spaces Summer