P OLYHEDRAL PRODUCTS , DUALITY PROPERTIES , AND C OHEN –M ACAULAY COMPLEXES Alex Suciu Northeastern University Special Session Geometry and Combinatorics of Cell Complexes Mathematical Congress of the Americas Montréal, Canada July 28, 2017 A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS AND DUALITY M ONTRÉAL , J ULY 28, 2017 1 / 22
P OLYHEDRAL PRODUCTS P OLYHEDRAL PRODUCTS Let p X , A q be a pair of topological spaces, and let L be a simplicial complex on vertex set r m s . The corresponding polyhedral product (or, generalized moment-angle complex ) is defined as ď p X , A q σ Ă X ˆ m , Z L p X , A q “ σ P L where p X , A q σ “ t x P X ˆ m | x i P A if i R σ u . Homotopy invariance: p X , A q » p X 1 , A 1 q ù ñ Z L p X , A q » Z L p X 1 , A 1 q . Converts simplicial joins to direct products: Z K ˚ L p X , A q – Z K p X , A q ˆ Z L p X , A q . Takes a cellular pair p X , A q to a cellular subcomplex of X ˆ m . A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS AND DUALITY M ONTRÉAL , J ULY 28, 2017 2 / 22
P OLYHEDRAL PRODUCTS The usual moment-angle complexes (which play an important role in toric topology) are: Complex moment-angle complex, Z L p D 2 , S 1 q . π 1 “ π 2 “ t 1 u . Real moment-angle complex, Z L p D 1 , S 0 q . π 1 “ W 1 L , the derived subgroup of W Γ , the right-angled Coxeter group associated to Γ “ L p 1 q . E XAMPLE Let L “ two points. Then: Z L p D 2 , S 1 q “ D 2 ˆ S 1 Y S 1 ˆ D 2 “ S 3 Z L p D 1 , S 0 q “ D 1 ˆ S 0 Y S 0 ˆ D 1 “ S 1 D 1 × S 0 S 0 × D 1 D 1 Z L ( D 1 , S 0 ) S 0 S 0 × S 0 A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS AND DUALITY M ONTRÉAL , J ULY 28, 2017 3 / 22
P OLYHEDRAL PRODUCTS E XAMPLE Let L be a circuit on 4 vertices. Then: Z L p D 2 , S 1 q “ S 3 ˆ S 3 Z L p D 1 , S 0 q “ S 1 ˆ S 1 E XAMPLE More generally, let L be an m -gon. Then: ˆ m ´ 2 ˙ S r ` 2 ˆ S m ´ r . m ´ 3 Z L p D 2 , S 1 q “ # r “ 1 r ¨ r ` 1 (McGavran 1979) Z L p D 1 , S 0 q “ an orientable surface of genus 1 ` 2 m ´ 3 p m ´ 4 q . (Coxeter 1937) A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS AND DUALITY M ONTRÉAL , J ULY 28, 2017 4 / 22
P OLYHEDRAL PRODUCTS If p M , B M q is a compact manifold of dimension d , and L is a PL-triangulation of S m on n vertices, then Z L p M , B M q is a compact manifold of dimension p d ´ 1 q n ` m ` 1. (Bosio–Meersseman 2006) If K is a polytopal triangulation of S m , then Z L p D 2 , S 1 q if n ` m ` 1 is even, or Z L p D 2 , S 1 q ˆ S 1 if n ` m ` 1 is odd is a complex manifold. This construction generalizes the classical constructions of complex structures on S 2 p ´ 1 ˆ S 1 (Hopf) and S 2 p ´ 1 ˆ S 2 q ´ 1 (Calabi–Eckmann). In general, the resulting complex manifolds are not symplectic, thus, not Kähler. In fact, they may even be non-formal (Denham–Suciu 2007). A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS AND DUALITY M ONTRÉAL , J ULY 28, 2017 5 / 22
P OLYHEDRAL PRODUCTS The GMAC construction enjoys nice functoriality properties in both arguments. E.g: Let f : p X , A q Ñ p Y , B q be a (cellular) map. Then f ˆ n : X ˆ n Ñ Y ˆ n restricts to a (cellular) map Z L p f q : Z L p X , A q Ñ Z L p Y , B q . Much is known from work of M. Davis about the fundamental group and the asphericity problem for Z L p X q “ Z L p X , ˚q . E.g.: π 1 p Z L p X , ˚qq is the graph product of G v “ π 1 p X , ˚q along the graph Γ “ L p 1 q “ p V , E q , where Prod Γ p G v q “ ˚ v P V G v {tr g v , g w s “ 1 if t v , w u P E, g v P G v , g w P G w u . Suppose X is aspherical. Then: Z L p X , ˚q is aspherical iff L is a flag complex. A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS AND DUALITY M ONTRÉAL , J ULY 28, 2017 6 / 22
T ORIC COMPLEXES AND RAAG S T ORIC COMPLEXES T ORIC COMPLEXES Let L be a simplicial complex on vertex set V “ t v 1 , . . . , v m u . Let T L “ Z L p S 1 , ˚q be the subcomplex of T m obtained by deleting the cells corresponding to the missing simplices of L . T L is a connected, minimal CW-complex, of dimension dim L ` 1. T L is formal (Notbohm–Ray 2005). (Kim–Roush 1980, Charney–Davis 1995) The cohomology algebra H ˚ p T L , k q is the exterior Stanley–Reisner ring k x L y “ Ź V ˚ {p v ˚ σ | σ R L q , where k “ Z or a field, V is the free k -module on V, and V ˚ “ Hom k p V , k q , while v ˚ σ “ v ˚ i 1 ¨ ¨ ¨ v ˚ i s for σ “ t i 1 , . . . , i s u . A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS AND DUALITY M ONTRÉAL , J ULY 28, 2017 7 / 22
T ORIC COMPLEXES AND RAAG S R IGHT ANGLED A RTIN GROUPS R IGHT ANGLED A RTIN GROUPS The fundamental group π Γ : “ π 1 p T L , ˚q is the RAAG associated to the graph Γ : “ L p 1 q “ p V , E q , π Γ “ x v P V | r v , w s “ 1 if t v , w u P E y . Moreover, K p π Γ , 1 q “ T ∆ Γ , where ∆ Γ is the flag complex of Γ . (Kim–Makar-Limanov–Neggers–Roush 1980, Droms 1987) Γ – Γ 1 ð ñ π Γ – π Γ 1 . (Papadima–S. 2006) The associated graded Lie algebra of π Γ has (quadratic) presentation gr p π Γ q “ L p V q{pr v , w s “ 0 if t v , w u P E q . (Duchamp–Krob 1992, PS06) The lower central series quotients of π Γ are torsion-free, with ranks φ k given by ź 8 k “ 1 p 1 ´ t k q φ k “ P Γ p´ t q , where P Γ p t q “ ř k ě 0 f k p ∆ Γ q t k is the clique polynomial of Γ . A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS AND DUALITY M ONTRÉAL , J ULY 28, 2017 8 / 22
T ORIC COMPLEXES AND RAAG S C HEN RANKS C HEN RANKS The Chen Lie algebra of a f.g. group π is the associated graded Lie algebra of its maximal metabelian quotient, gr p π { π 2 q . Write θ k p π q “ rank gr k p π { π 2 q for the Chen ranks. (K.-T. Chen 1951) gr p F n { F 2 n q is torsion-free, with ranks θ 1 “ n and ` n ` k ´ 2 ˘ θ k “ p k ´ 1 q for k ě 2. k (PS 06) gr p π Γ { π 2 Γ q is torsion-free, with ranks given by θ 1 “ | V | and ˆ ˙ 8 ÿ t θ k t k “ Q Γ . 1 ´ t k “ 2 Here Q Γ p t q “ ř j ě 2 c j p Γ q t j is the “cut polynomial" of Γ , with ÿ ˜ c j p Γ q “ b 0 p Γ W q . W Ă V : | W |“ j A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS AND DUALITY M ONTRÉAL , J ULY 28, 2017 9 / 22
T ORIC COMPLEXES AND RAAG S C HEN RANKS E XAMPLE Let Γ be a pentagon, and Γ 1 a square with an edge attached to a vertex. Then: P Γ “ P Γ 1 “ 1 ` 5 t ` 5 t 2 , and so φ k p π Γ q “ φ k p π Γ 1 q , for all k ě 1 . Q Γ “ 5 t 2 ` 5 t 3 but Q Γ 1 “ 5 t 2 ` 5 t 3 ` t 4 , and so θ k p π Γ q ‰ θ k p π Γ 1 q , for k ě 4 . A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS AND DUALITY M ONTRÉAL , J ULY 28, 2017 10 / 22
T ORIC COMPLEXES AND RAAG S C OHOMOLOGY JUMP LOCI C OHOMOLOGY JUMP LOCI Let X be a connected, finite CW-complex X with π : “ π 1 p X q . Fix a field k and set A “ H . p X , k q . If char p k q “ 2, assume H 1 p X , Z q is torsion-free. Then, for each a P A 1 , we have a 2 “ 0, and so we get a cochain complex, p A , ¨ a q : A 0 ¨ a ¨ a � A 2 � A 1 � ¨ ¨ ¨ . The resonance varieties of X are defined as s p X q “ t a P A 1 | dim H i p A , ¨ a q ě s u . R i They are Zariski closed, homogeneous subsets of A 1 “ H 1 p X , k q . The characteristic varieties of X are the jump loci for homology with coefficients in rank-1 local systems, V i s p X , k q “ t ρ P Hom p π, k ˚ q | dim H i p X , k ρ q ě s u . These loci are Zariski closed subsets of the character group. For i “ 1, they depend only on π { π 2 (and k ). A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS AND DUALITY M ONTRÉAL , J ULY 28, 2017 11 / 22
T ORIC COMPLEXES AND RAAG S J UMP LOCI OF TORIC COMPLEXES J UMP LOCI OF TORIC COMPLEXES For a field k , identify H 1 p T L , k q “ k V , the k -vector space with basis V. T HEOREM (P APADIMA –S. 2009) ď R i k W , s p T L , k q “ W Ă V ř σ P L V z W dim k r H i ´ 1 ´| σ | p lk L W p σ q , k qě s where L W is the subcomplex induced by L on W , and lk K p σ q is the link of a simplex σ in a subcomplex K Ď L. In particular, ď R 1 k W . 1 p π Γ q “ W Ď V Γ W disconnected Similar formulas hold for the characteristic varieties V i s p T L , k q . A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS AND DUALITY M ONTRÉAL , J ULY 28, 2017 12 / 22
T ORIC COMPLEXES AND RAAG S J UMP LOCI OF TORIC COMPLEXES 1 1 2 3 s s s s � ❅ ❅ ❅ � ❅ ❅ ❅ 2 3 s s ❅ � � ❅ ❅ ❅ � ❅ � ❅ ❅ ❅ s s s s s s 4 5 6 4 5 6 E XAMPLE Let Γ and Γ 1 be the two graphs above. Both have P p t q “ 1 ` 6 t ` 9 t 2 ` 4 t 3 , Q p t q “ t 2 p 6 ` 8 t ` 3 t 2 q . and Thus, π Γ and π Γ 1 have the same LCS and Chen ranks. Each resonance variety has 3 components, of codimension 2: R 1 p π Γ , k q “ k 23 Y k 25 Y k 35 , R 1 p π Γ 1 , k q “ k 15 Y k 25 Y k 26 . Yet the two varieties are not isomorphic, since dim p k 23 X k 25 X k 35 q “ 3 , dim p k 15 X k 25 X k 26 q “ 2 . but A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS AND DUALITY M ONTRÉAL , J ULY 28, 2017 13 / 22
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