T OPOLOGY OF LINE ARRANGEMENTS Alex Suciu Northeastern University Workshop on Configuration Spaces Il Palazzone di Cortona September 1, 2014 A LEX S UCIU (N ORTHEASTERN ) T OPOLOGY OF LINE ARRANGEMENTS C ORTONA , S EPT . 2014 1 / 24
O UTLINE I NTRODUCTION 1 Plane algebraic curves Line arrangements Residual properties Milnor fibration Techniques R ESIDUAL PROPERTIES 2 The RFR p property Boundary manifolds Towers of congruence covers M ILNOR FIBRATION 3 Resonance varieties and multinets Modular inequalities Combinatorics and monodromy A LEX S UCIU (N ORTHEASTERN ) T OPOLOGY OF LINE ARRANGEMENTS C ORTONA , S EPT . 2014 2 / 24
I NTRODUCTION P LANE ALGEBRAIC CURVES P LANE ALGEBRAIC CURVES Let C Ă CP 2 be a plane algebraic curve, defined by a homogeneous polynomial f P C [ z 1 , z 2 , z 3 ] . In the 1930s, Zariski studied the topology of the complement, U = CP 2 z C . He commissioned Van Kampen to find a presentation for the fundamental group, π = π 1 ( U ) . Zariski noticed that π is not determined by the combinatorics of C , but depends on the position of its singularities. He asked whether π is residually finite , i.e., whether the map to its π = : π alg , is injective. profinite completion, π Ñ p A LEX S UCIU (N ORTHEASTERN ) T OPOLOGY OF LINE ARRANGEMENTS C ORTONA , S EPT . 2014 3 / 24
I NTRODUCTION L INE ARRANGEMENTS L INE ARRANGEMENTS Let A be an arrangement of lines in CP 2 , defined by a polynomial ź f = f H P C [ z 1 , z 2 , z 3 ] , H P A with f H linear forms so that H = P ker ( f H ) for each H P A . Let L ( A ) be the intersection lattice of A , with L 1 ( A ) = t lines u and L 2 ( A ) = t intersection points u . Let U ( A ) = CP 2 z Ť H P A H be the complement of A . A LEX S UCIU (N ORTHEASTERN ) T OPOLOGY OF LINE ARRANGEMENTS C ORTONA , S EPT . 2014 4 / 24
I NTRODUCTION R ESIDUAL PROPERTIES R ESIDUAL PROPERTIES OF ARRANGEMENT GROUPS T HEOREM (T HOMAS K OBERDA –A.S. 2014) Let A be a complexified real line arrangement, and let π = π 1 ( U ( A )) . Then π is residually finite. 1 π is residually nilpotent. 2 π is torsion-free. 3 A LEX S UCIU (N ORTHEASTERN ) T OPOLOGY OF LINE ARRANGEMENTS C ORTONA , S EPT . 2014 5 / 24
I NTRODUCTION M ILNOR FIBRATION M ILNOR FIBRATION Let f P C [ z 1 , z 2 , z 3 ] be a homogeneous polynomial of degree n . The map f : C 3 zt f = 0 u Ñ C ˚ is a smooth fibration (Milnor), with fiber F = f ´ 1 ( 1 ) , and monodromy h : F Ñ F , z ÞÑ e 2 π i / n z . The Milnor fiber F is a regular, Z n -cover of U = CP 2 zt f = 0 u . C OROLLARY (T.K.–A.S.) Let A be an arrangement defined by a polynomial f P R [ z 1 , z 2 , z 3 ] , let F = F ( A ) be its Milnor fiber, and let π = π 1 ( F ) . Then π is residually finite. 1 π is residually nilpotent. 2 π is torsion-free. 3 A LEX S UCIU (N ORTHEASTERN ) T OPOLOGY OF LINE ARRANGEMENTS C ORTONA , S EPT . 2014 6 / 24
I NTRODUCTION M ILNOR FIBRATION Let ∆ ( t ) = det ( tI ´ h ˚ ) be the characteristic polynomial of the algebraic monodromy, h ˚ : H 1 ( F , C ) Ñ H 1 ( F , C ) . P ROBLEM When f is the defining polynomial of an arrangement A , is ∆ = ∆ A determined solely by L ( A ) ? T HEOREM (S TEFAN P APADIMA –A.S. 2014) Suppose A has only double and triple points. Then ∆ A ( t ) = ( t ´ 1 ) | A |´ 1 ¨ ( t 2 + t + 1 ) β 3 ( A ) , where β 3 ( A ) is an integer between 0 and 2 that depends only on L ( A ) . A LEX S UCIU (N ORTHEASTERN ) T OPOLOGY OF LINE ARRANGEMENTS C ORTONA , S EPT . 2014 7 / 24
I NTRODUCTION T ECHNIQUES T ECHNIQUES Common themes: Homology with coefficients in rank 1 local systems. Homology of finite abelian covers. Specific techniques for residual properties: Boundary manifold of line arrangement. Towers of congruence covers. The RFRp property. Specific techniques for Milnor fibration: Nets, multinets, and pencils. Cohomology jump loci (in characteristic 0 and p ). Modular bounds for twisted Betti numbers. A LEX S UCIU (N ORTHEASTERN ) T OPOLOGY OF LINE ARRANGEMENTS C ORTONA , S EPT . 2014 8 / 24
R ESIDUAL PROPERTIES T HE RFR p PROPERTY T HE RFR p PROPERTY Let G be a finitely generated group and let p be a prime. We say that G is residually finite rationally p if there exists a sequence of subgroups G = G 0 ą ¨ ¨ ¨ ą G i ą G i + 1 ą ¨ ¨ ¨ such that G i + 1 Ÿ G i . 1 Ş i ě 0 G i = t 1 u . 2 G i / G i + 1 is an elementary abelian p -group. 3 ker ( G i Ñ H 1 ( G i , Q )) ă G i + 1 . 4 Remarks: May assume each G i Ÿ G . Compare with Agol’s RFRS property, where G i / G i + 1 only finite. G RFR p ñ residually p ñ residually finite and residually nilpotent. G RFR p ñ G RFRS ñ torsion-free. A LEX S UCIU (N ORTHEASTERN ) T OPOLOGY OF LINE ARRANGEMENTS C ORTONA , S EPT . 2014 9 / 24
R ESIDUAL PROPERTIES T HE RFR p PROPERTY The class of RFR p groups is closed under the following operations: Taking subgroups. 1 Finite direct products. 2 Finite free products. 3 The following groups are RFR p : Finitely generated free groups. 1 Closed, orientable surface groups. 2 Right-angled Artin groups. 3 A LEX S UCIU (N ORTHEASTERN ) T OPOLOGY OF LINE ARRANGEMENTS C ORTONA , S EPT . 2014 10 / 24
R ESIDUAL PROPERTIES B OUNDARY MANIFOLDS B OUNDARY MANIFOLDS Let N be a regular neighborhood of Ť H P A H inside CP 2 . Let U = CP 2 z int ( N ) be the exterior of A . The boundary manifold of A is M = B U = B N , a compact, orientable, smooth manifold of dimension 3. E XAMPLE Let A be a pencil of n hyperplanes in C 2 , defined by f = z n 1 ´ z n 2 . If n = 1, then M = S 3 . If n ą 1, then M = 7 n ´ 1 S 1 ˆ S 2 . E XAMPLE Let A be a near-pencil of n planes in CP 2 , defined by ) . Then M = S 1 ˆ Σ n ´ 2 , where Σ g = 7 g S 1 ˆ S 1 . f = z 1 ( z n ´ 1 ´ z n ´ 1 2 3 A LEX S UCIU (N ORTHEASTERN ) T OPOLOGY OF LINE ARRANGEMENTS C ORTONA , S EPT . 2014 11 / 24
R ESIDUAL PROPERTIES B OUNDARY MANIFOLDS Work of Hirzebruch, Jiang–Yau, and E. Hironaka shows that M = M Γ is a graph-manifold. The graph Γ is the incidence graph of A , with vertex set V ( Γ ) = L 1 ( A ) Y L 2 ( A ) and edge set E ( Γ ) = t ( H , P ) | P P H u . For each v P V ( Γ ) , there is a vertex manifold M v = S 1 ˆ S v , with ď S v = S 2 z D 2 v , w , t v , w uP E ( Γ ) a sphere with deg v disjoint open disks removed. For each e P E ( Γ ) , there is an edge manifold M e = S 1 ˆ S 1 . Vertex manifolds are glued along edge manifolds via flips. A LEX S UCIU (N ORTHEASTERN ) T OPOLOGY OF LINE ARRANGEMENTS C ORTONA , S EPT . 2014 12 / 24
� � � R ESIDUAL PROPERTIES B OUNDARY MANIFOLDS The inclusion i : M Ñ U induces a surjection i 7 : π 1 ( M ) ։ π 1 ( U ) . By collapsing each vertex manifold of M = M Γ to a point, we obtain a map κ : M Ñ Γ . Using work of D. Cohen–A.S. (2006, 2008), we get a split exact sequence i ˚ κ ˚ � H 1 ( U , Z ) � H 1 ( M , Z ) � H 1 ( Γ , Z ) 0 0 . L EMMA Suppose A is an essential line arrangement in CP 2 . Then, for each v P V ( Γ ) and e P E ( Γ ) , the inclusions i v : M v ã Ñ M and i e : M e ã Ñ M induce split injections on H 1 , whose images are contained in ker ( κ ˚ ) . A LEX S UCIU (N ORTHEASTERN ) T OPOLOGY OF LINE ARRANGEMENTS C ORTONA , S EPT . 2014 13 / 24
R ESIDUAL PROPERTIES B OUNDARY MANIFOLDS Using work of E. Hironaka (2001), we obtain: L EMMA Suppose A is the complexification of a real arrangement. There is then a finite, simplicial graph G and an embedding j : G ã Ñ M such that: The graph G is homotopy equivalent to the incidence graph Γ . 1 We have an exact sequence, 2 j ˚ i ˚ � H 1 ( G , Z ) � H 1 ( M , Z ) � H 1 ( U , Z ) � 0 . 0 We have an exact sequence, 3 j 7 i 7 � π 1 ( G ) � π 1 ( M ) � π 1 ( U ) � 1 . 1 A LEX S UCIU (N ORTHEASTERN ) T OPOLOGY OF LINE ARRANGEMENTS C ORTONA , S EPT . 2014 14 / 24
� � � R ESIDUAL PROPERTIES T OWERS OF CONGRUENCE COVERS T OWERS OF CONGRUENCE COVERS For each prime p , we construct a tower of regular covers of M , q i + 1 � M i q i q 1 � M 0 = M . � M i + 1 � ¨ ¨ ¨ ¨ ¨ ¨ Each M i is a graph-manifold, modelled on a graph Γ i . The group of deck-transformations for q i + 1 is the elementary � ( H 1 ( M i , Z ) /tors ) / H 1 ( Γ i , Z ) � abelian p -group b Z p . The covering maps preserve the graph-manifold structures, e.g., M v , i M i q v q � M M v where M v , i is a connected component of q ´ 1 ( M v ) and q v = q | M v , i . A LEX S UCIU (N ORTHEASTERN ) T OPOLOGY OF LINE ARRANGEMENTS C ORTONA , S EPT . 2014 15 / 24
R ESIDUAL PROPERTIES T OWERS OF CONGRUENCE COVERS The inclusions M v , i ã Ñ M i and M e , i ã Ñ M i induce injections on H 1 , whose images are contained in ker (( κ i ) ˚ ) . If A is complexified real, the graph G ã Ñ M lifts to a graph Ñ M i so that G i ã The group H 1 ( M i , Z ) splits off H 1 ( G i , Z ) as a direct summand. H 1 ( G i , Z ) X H 1 ( M v , i , Z ) = 0, for all v P V ( Γ ) . Finally, For each v P V ( Γ ) , the group π 1 ( M v ) = Z ˆ π 1 ( S v ) is RFR p . From the construction of the tower, it follows that π 1 ( M ) is RFR p . If A is complexified real, the above properties of the lifts of G imply that π 1 ( U ) = π 1 ( M ) / x x j 7 ( π 1 ( G )) y y is also RFR p . A LEX S UCIU (N ORTHEASTERN ) T OPOLOGY OF LINE ARRANGEMENTS C ORTONA , S EPT . 2014 16 / 24
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