The homotopy type of the complement of a coordinate subspace - PDF document

I NTERNATIONAL C ONFERENCE ON T ORIC T OPOLOGY O SAKA C ITY U NIVERSITY 29 M AY - 3 J UNE O SAKA 2006 The homotopy type of the complement of a coordinate subspace arrangement Jelena Grbi´ c University of Aberdeen

Problem COORDINATE SUBSPACE ARRANGEMENT is a finite set CA = { L 1 , . . . , L r } ⊂ C n of coordinate subspaces, that is, � ( z 1 , . . . , z n ) ∈ C n : z i 1 = . . . = z i k = 0 � L ω = , where ω = { i 1 , . . . , i k } ⊂ [ n ] and its complement U( CA ) is de- fined as r � U( CA ) := C n \ L i . i =1 G OAL : The homotopy type of U( CA ) . Toric topology-main definitions and constructions S IMPLICIAL COMPLEXES V = { v 1 , . . . , v n } = [ n ] set of vertices K := { σ 1 , . . . , σ s : σ i ⊂ V } ( ∅ ∈ K ) – abstract simplicial complex closed under formation of subsets σ ∈ K – simplex dim( K ) = d if ♯σ ≤ d + 1 for all σ ∈ K S TANLEY -R EISNER FACE RING R – commutative ring with unit; deg( v i ) = 2 – topological grading R [ V ] = R [ v 1 , . . . , v n ] graded polynomial algebra on V over R Given σ ⊂ [ n ] , set � v σ := v σ = v i 1 . . . v i r for σ = { i 1 , . . . , i r } . v i , i ∈ σ The Stanley-Reisner algebra (or face ring) of K is R [ K ] := R [ v 1 , . . . , v n ] / ( v σ : σ / ∈ K ) . 1

‘‘Topological models" for the algebraic objects D AVIS –J ANUSZKIEWICZ SPACE DJ( K ) –topological realisation of the Stanley–Reisner ring R [ K ] , that is, H ∗ (DJ( K ); R ) = R [ K ] ( for R = Z or R = Z / 2) . DJ( K ) = ET n × T n Z K Davis–Januszkiewicz Buchstaber–Panov through a simple colimit of nice blocks Assume R = Z . Denote C P ∞ = BS 1 , thus BT n = ( C P ∞ ) n For ω ⊂ [ n ] , define BT ω := � ( x 1 , . . . , x n ) ∈ BT n : x i = ∗ if i / � ∈ ω . For K on [ n ] , the Davis-Januszkiewicz space of K is given by � BT σ ⊂ BT n . DJ( K ) := σ ∈ K M OMENT – ANGLE COMPLEX Z K � � Torus T n ⊂ ( D 2 ) n = ( z 1 , . . . , z n ) ∈ C n : | z i | ≤ 1 , ∀ i For arbitrary σ ⊂ [ n ] , define � � ( z 1 , . . . , z n ) ∈ ( D 2 ) n : | z i | = 1 B σ := i / ∈ σ . = ( D 2 ) | σ | × T n −| σ | B σ ∼ For K on [ n ] , define the moment–angle complex Z K by � B σ ⊂ ( D 2 ) n . Z K := σ ∈ K T n acts on Z K B σ invariant under the action of T n � 2

Proposition. The moment–angle complex Z K is the homotopy fibre of the inclusion → BT n . i : DJ( K ) − H ∗ T n ( Z K ) = Z [ K ] Proposition. Arrangements and their complements For K on set [ n ] , define the complex coordinate subspace arrangement as � � CA ( K ) := L σ : σ / ∈ K and its complement in C n by � U( K ) := C n \ L σ . σ / ∈ K If L ⊂ K is a subcomplex, then U( L ) ⊂ U( K ) . Proposition. The assignment K �→ U( K ) defines a one–to–one order preserving correspondence   complements of � �   simplicial coordinate subspace  . � complexes on [ n ] arrangements in C n  3

CONNECTION BETWEEN Z K AND U( K ) Theorem (Buchstaber–Panov) . There is an equivariant deformation retraction ≃ U( K ) − → Z K . COHOMOLOGY OF U( K ) Theorem (Buchstaber–Panov) . The following isomorphism of graded algebra holds � � H ∗ (U( K ); k ) ∼ = Tor k [ v 1 ,...,v n ] ( k [ K ] , k ) ∼ = H Λ[ u 1 , . . . , u n ] ⊗ k [ K ] , d . hints from ALGEBRA and COMBINATORICS Definition. The Stanley-Reisner ring k [ K ] is Golod if all Massey products in Tor k [ v 1 ,...,v n ] ( k [ K ] , k ) vanish. Definition. A simplicial complex K is shifted if there is an ordering σ ∈ K, v ′ < v ⇒ ( σ − v ) ∪ v ′ ∈ K . Proposition. If K is shifted, then its face ring k [ K ] is Golod. THE MAIN THEOREM (G., Theriault) Let K be a shifted complex. Then Z K is a wedge of spheres. 4

Back to COMBINATORICS P ROBLEM : Determine the homotopy type of the complement of ar- bitrary codimension coordinate subspace arrangements. S TRATEGY : 1 � determine the simplicial complex K which corresponds to a codimenison– i coordinate subspace arrangement, U( K ) ; 2 � associate to the determined simplicial complex K its Davis– Januszkiewicz space, i.e, DJ( K ) ; 3 � looking at the fibration → BT n , Z K − → DJ( K ) − describe the homotopy type of Z K . 1 � Look at an i +2 –codimension coordinate subspace in C n , that is, � � ( z 1 , . . . , z n ) ∈ C n : z j 1 = . . . = z j i +2 = 0 L ω = , ω = { j 1 , . . . , j i +2 } . Then K = sk i (∆ n − 1 ) . C n \ CA i +2 = U � � sk i (∆ n − 1 ) Hence, . 2 � A colimit model of the Davis-Januszkiewicz space for K is given by � BT σ ⊂ BT n , DJ( K ) := ♯ vertices in K. σ ∈ K Then we have DJ( K ) = T n n − 1 − i � � ⊂ ( C P ∞ ) n . = ( z 1 , . . . , z n ) : at least n − 1 − i coordinates are ∗ 5

3 � Determine the homotopy fibre Z K of the fibration sequence → ( C P ∞ ) n for 1 ≤ k ≤ n − 1 . ( Z K ) n → T n k − k − Let X 1 , . . . , X n be path-connected spaces. There is a filtration of X 1 × . . . × X n given by T n → T n → T n n − n − 1 − → · · · − 0 � � were T n k = ( x 1 , . . . , x n ) ∈ X 1 × . . . × X n : at least k of x i ‘s are ∗ . Theorem (Porter; G., Theriault) . For n ≥ 1 , and k such that 1 ≤ k ≤ n − 1 , the homotopy fibre F n k of the inclusion i : T n k − → X 1 × . . . × X n decomposes as n � � � j − 1 � � � F n Σ n − k Ω X i 1 ∧ . . . ∧ Ω X i j k ≃ . n − k j + n − k +1 1 ≤ i 1 <...<i j ≤ n Take for X 1 = . . . = X n = C P ∞ . Then we have the inclusion i : T n k ( C P ∞ ) − → ( C P ∞ ) n . It follows that n �� n � �� j − 1 � � Σ n − k Ω C P ∞ ∧ . . . ∧ Ω C P ∞ F n k ≃ j n − k � �� � j j = n − k +1 n � n �� j − 1 � � S n + j − k . ≃ j n − k j = n − k +1 6

Family � � K − simplicial complex | Σ t Z K a wedge of spheres F t = Notice that F 0 ⊂ F 1 ⊂ . . . ⊂ F t ⊂ . . . ⊂ F ∞ . ( sk i (∆ n − 1 ) ∈ F 0 ) We have shown if K -shifted, then K ∈ F 0 Want: make simplicial complexes out of our building blocks W HAT CAN HOMOTOPY SEE ? DISJOINT UNION OF SIMPLICIAL COMPLEXES � K 2 ∈ F m , m = max { t, s } . Let K 1 ∈ F t and K 2 ∈ F s . Then K 1 Z K ≃ ( � i S 1 ∗ � j S 1 ) ∨ ( Z K 1 ⋊ � i S 1 ) ∨ ( � j S 1 ⋉ Z K 2 ) GLUING ALONG A COMMON FACE � Let K = K 1 σ K 2 . If K 1 , K 2 ∈ F 0 , then K ∈ F 0 . Z K ≃ ( � S 1 ∗ � S 1 ) ∨ ( Z K 1 ⋊ � S 1 ) ∨ ( Z K 2 ⋊ � S 1 ) JOIN OF SIMPLICIAL COMPLEXES K 1 , K 2 simplicial complexes on sets S 1 and S 2 , belonging to F t and F s . The join K 1 ∗ K 2 := { σ ⊂ S 1 ∪ S 2 : σ = σ 1 ∪ σ 2 , σ 1 ∈ K 1 , σ 2 ∈ K 1 } Notice k [ K 1 ∗ K 2 ] = k [ K 1 ] ⊗ k [ K 2 ] Therefore for the join of K 1 and K 2 we get a product fibration → BT m 1 × BT m 2 DJ ( K 1 ) × DJ ( K 2 ) − hence Z K 1 ∗ K 2 ≃ Z K 1 × Z K 2 and K 1 ∗ K 2 ∈ F m , m = max { t, s } + 1 . 7

Our contribution to ALGEBRA Let A be a polynomial ring on n variables k [ x 1 , . . . , x n ] 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 ROBLEM : The nature of Tor S ( k, k ) . The Poincar´ e series ∞ � b i t i where b i = dim k Tor S P ( S ) = i ( k, k ) i =0 P ROBLEM : The rationality of P ( S ) . Theorem. (Golod) There exist non-negative integers n, c 1 , . . . , c n such that (1 + t ) n P ( S ) ≤ i =1 c i t i +1 . 1 − � n Theorem. (G.,Theriault) There is a topological proof of Golod’s in- equality. 8

Theorem. (Buchstaber-Panov-Ray) ( k, k ) ∼ Tor k [ K ] = H ∗ (Ω DJ ( K ); k ) . ∗ Looking at the split fibration → T n Ω Z K − → ΩDJ( K ) − ( k, k ) ∼ Tor k [ K ] = H ∗ (ΩDJ( K )) = H ∗ ( T n ) ⊗ H ∗ (Ω Z K ) ∗ Using the bar resolution, P ( H ∗ (Ω Z K )) ≤ P ( T (Σ − 1 H ∗ ( Z K ))) . Therefore P ( k [ K ]) = (1 + t ) n P ( H ∗ (Ω Z K )) ≤ (1 + t ) n P ( T (Σ − 1 H ∗ ( Z K ))) (1 + t ) n = 1 − P (Σ − 1 H ∗ ( Z K )) . Equality is obtained when H ∗ ( Z K ) is Golod. Corollary. When K ∈ F 0 , then P ( k [ K ]) is rational. 9