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I

N T E R N AT I O N A L

C

ONFERENCE ON TY

T

ORIC

T

OPOLOGY

OSAKA CI 2 9 M AY - 3 J

U

NIVERSITY

UNE

O

SAKA

2006

The homotopy type of the complement of a coordinate subspace arrangement
´ Jelena Grbic University of Aberdeen


Problem is a finite set CA = {L1 , . . . , Lr } Cn of coordinate subspaces, that is, L = (z1 , . . . , zn) Cn : zi1 = . . . = zik = 0 , where = {i1 , . . . , ik } [n] and its complement U(CA) is defined as r U(CA) := Cn\ Li.
i=1

C O O R D I N AT E S U B S PAC E A R R A N G E M E N T

G OA L: The homotopy type of U(CA). Toric topology-main definitions and constructions S
IMPLICIAL COMPLEXES

V = {v1 , . . . , vn} = [n] set of ver tices K := {1 , . . . , s : i V } ( K ) ­ abstract simplicial complex closed under formation of subsets K ­ simplex S
TA N L E Y

dim(K ) = d if d + 1 for all K

-R

E I S N E R FAC E R I N G

R ­ commutative ring with unit; deg(vi) = 2 ­ topological grading R[V ] = R[v1 , . . . , vn] graded polynomial algebra on V over R Given [n], set v :=
i

vi ,

v = vi1 . . . v

ir

for = {i1 , . . . , ir }.

The Stanley-Reisner algebra (or face ring) of K is R[K ] := R[v1 , . . . , vn]/(v : K ). /
1


``Topological models" for the algebraic objects D
AV I S

­J

A N U S Z K I E W I C Z S PAC E

DJ(K )

­topological realisation of the Stanley­Reisner ring R[K ], that is, H (DJ(K ); R) = R[K ] (for R = Z or R = Z/2). Davis­Januszkiewicz DJ(K ) = E T n âT n ZK Buchstaber­Panov through a simple colimit of nice blocks Assume R = Z. Denote CP For [n], define B T := (x1 , . . . , xn) B T n : xi = if i . / For K on [n], the Davis-Januszkiewicz space of K is given by DJ(K ) :=
K

= B S 1 , thus B T n = (CP )

n

B T B T n.

M

O M E N T­ A N G L E C O M P L E X

ZK

Torus T n (D2 )n = (z1 , . . . , zn) Cn : |zi| 1, i For arbitrary [n], define B := (z1 , . . . , zn) (D2 )n : |zi| = 1 i . / B = (D2 )|| â T n-|| For K on [n], define the moment­angle complex ZK by

ZK :=
K

B (D 2 ) n .
n

B invariant under the action of T

T n acts on ZK
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Proposition. The moment­angle complex ZK is the homotopy fibre of the inclusion i : DJ(K ) - B T n. Proposition.
HT n (ZK ) = Z[K ]

Arrangements and their complements For K on set [n], define the complex coordinate subspace arrangement as

CA(K ) := L : K / and its complement in Cn by
U(K ) := Cn\
K /

L .

If L K is a subcomplex, then U(L) U(K ). Proposition. The assignment K U(K ) defines a one­to­one order preserving correspondence complements of simplicial coordinate subspace . complexes on [n] arrangements in Cn

3


CONNECTION BETWEEN

ZK

AND

U(K )

Theorem (Buchstaber­Panov). There is an equivariant deformation retraction U(K ) - ZK .
COHOMOLOGY OF

U(K )

Theorem (Buchstaber­Panov). The following isomorphism of graded algebra holds H (U(K ); k) = Tor
k[v1 ,...,vn ]

(k[K ], k) = H [u1 , . . . , un]k[K ], d .

hints from ALGEBRA and COMBINATORICS Definition. The Stanley-Reisner ring k[K ] is Golod if all Massey products in Tork[v1 ,...,vn] (k[K ], k) vanish. Definition. A simplicial complex K is shifted if there is an ordering K , v < v ( - v ) v K . Proposition. If K is shifted, then its face ring k[K ] is Golod.
THE MAIN

T H E O R E M (G., Theriault)

Let K be a shifted complex. Then ZK is a wedge of spheres.

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Back to COMBINATORICS P R O B L E M: Determine the homotopy type of the complement of arbitrar y codimension coordinate subspace arrangements. ST 1 2 3
R AT E G Y:

determine the simplicial complex K which corresponds to a codimenison­i coordinate subspace arrangement, U(K ); associate to the determined simplicial complex K its Davis­ Januszkiewicz space, i.e, DJ(K ); looking at the fibration

ZK - DJ(K ) - B T n, describe the homotopy type of ZK . 1 Look at an i + 2­codimension coordinate subspace in Cn, that is,
L = (z1 , . . . , zn) Cn : zj1 = . . . = z Then K = ski(n-1 ). Hence,
ji+2

= 0 , = {j1 , . . . , ji+2 }.

Cn\CAi+2 = U ski(n-1) .

2 A colimit model of the Davis-Januszkiewicz space for K is given by
DJ(K ) :=
K

B T B T n,

ver tices in K.

Then we have
n DJ(K ) = Tn- 1- i

= (z1 , . . . , zn) : at least n - 1 - i coordinates are (CP )n.
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3 Determine the homotopy fibre ZK of the fibration sequence
n (ZK )n - Tk - (CP )n for 1 k n - 1. k

Let X1 , . . . , Xn be path-connected spaces. There is a filtration of X1 â . . . â Xn given by
n n n Tn - Tn-1 - · · · - T0 n were Tk = (x1 , . . . , xn) X1 â . . . â Xn : at least k of xi`s are .

Theorem (Por ter; G., Theriault). For n 1, and k such that n 1 k n - 1, the homotopy fibre Fk of the inclusion n i : Tk - X1 â . . . â Xn decomposes as
n

F

n k j +n-k+1 1i1 <...
j-1 n-k Xi1 . . . Xij . n-k

Take for X1 = . . . = Xn = CP . Then we have the inclusion
n i : Tk (CP ) - (CP )n.

It follows that
n

F

n k j =n-k+1

n j
n

j-1 n-k CP n-k n j j-1 S n-k



. . . CP
j



n+j -k

.

j =n-k+1

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Family

Ft = K - simplicial complex|tZK a wedge of spheres
Notice that F0 F1 . . . Ft . . . F. We have shown if K -shifted, then K F
0

(ski(n-1 ) F0 )

Want: make simplicial complexes out of our building blocks W
H AT C A N H O M OTO P Y S E E

?

DISJOINT UNION OF SIMPLICIAL COMPLEXES

Let K1 Ft and K2 Fs . Then K

1 i

K2 Fm , m = max{t, s}. S 1) (
j

ZK

(

i

S1

j

S 1 ) (ZK

1

S

1

ZK2 )

G L U I N G A L O N G A C O M M O N FAC E

Let K = K1



K2 . If K1 , K2 F0 , then K F0 . S 1 ) (ZK
1

ZK

(

S1

S 1 ) (ZK

2

S 1)

JOIN OF SIMPLICIAL COMPLEXES

K1 , K2 simplicial complexes on sets S1 and S2 , belonging to Ft and Fs . The join K1 K2 := { S1 S2 : = 1 2 , 1 K1 , 2 K1 } Notice k[K1 K2 ] = k[K1 ] k[K2 ] Therefore for the join of K1 and K2 we get a product fibration DJ (K1 ) â DJ (K2 ) - B T hence ZK
1

m1

â BT

m2

K

2

ZK1 â ZK2 and K1 K2 Fm, m = max{t, s} + 1.
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Our contribution to ALGEBRA Let A be a polynomial ring on n variables k[x1 , . . . , xn] over a field k and S = A/I , where I is a homogeneous ideal, i.e, S = k[K ] for some simplicial complex K . P
RO B L E M

: The nature of TorS (k, k).

´ The Poincare series


P (S ) =
i=0

biti

where bi = dimk TorS (k, k) i

P

RO B L E M

: The rationality of P (S ).
n

Theorem. (Golod) There exist non-negative integers n, c1 , . . . , c such that (1 + t)n P (S ) . 1 - n citi+1 i=1

Theorem. (G.,Theriault) There is a topological proof of Golod's inequality.

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Theorem. (Buchstaber-Panov-Ray) Tor
k [K ]

(k, k) = H (DJ (K ); k).

Looking at the split fibration ZK - DJ(K ) - T Tor
k [K ] n

(k, k) = H (DJ(K )) = H (T n) H (ZK )

Using the bar resolution, P (H (ZK )) P (T (-1 H (ZK ))). Therefore P (k[K ]) = (1 + t)nP (H (ZK )) (1 + t)nP (T (-1 H (ZK ))) (1 + t)n = . 1 - P (-1 H (ZK )) Equality is obtained when H (ZK ) is Golod. Corollary. When K F0 , then P (k[K ]) is rational.

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