H YPERPLANE ARRANGEMENTS AND M ILNOR FIBRATIONS Alex Suciu Northeastern University Workshop on Computational Geometric Topology in Arrangement Theory ICERM, Brown University July 8, 2015 A LEX S UCIU A RRANGEMENTS AND M ILNOR FIBRATIONS ICERM, J ULY 8, 2015 1 / 16
T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT T HE M ILNOR FIBRATION ( S ) OF AN ARRANGEMENT Let A be a (central) hyperplane arrangement in C ℓ . For each H P A , let f H : C ℓ Ñ C be a linear form with kernel H . For each choice of multiplicities m “ p m H q H P A with m H P N , let ź f m H Q m : “ Q m p A q “ H , H P A a homogeneous polynomial of degree N “ ř H P A m H . The map Q m : C ℓ Ñ C restricts to a map Q m : M p A q Ñ C ˚ . This is the projection of a smooth, locally trivial bundle, known as the Milnor fibration of the multi-arrangement p A , m q , Q m � M p A q � C ˚ . F m p A q A LEX S UCIU A RRANGEMENTS AND M ILNOR FIBRATIONS ICERM, J ULY 8, 2015 2 / 16
T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT T HE M ILNOR FIBRATION ( S ) OF AN ARRANGEMENT Let A be a (central) hyperplane arrangement in C ℓ . For each H P A , let f H : C ℓ Ñ C be a linear form with kernel H . For each choice of multiplicities m “ p m H q H P A with m H P N , let ź f m H Q m : “ Q m p A q “ H , H P A a homogeneous polynomial of degree N “ ř H P A m H . The map Q m : C ℓ Ñ C restricts to a map Q m : M p A q Ñ C ˚ . This is the projection of a smooth, locally trivial bundle, known as the Milnor fibration of the multi-arrangement p A , m q , Q m � M p A q � C ˚ . F m p A q A LEX S UCIU A RRANGEMENTS AND M ILNOR FIBRATIONS ICERM, J ULY 8, 2015 2 / 16
T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT The typical fiber, F m p A q “ Q ´ 1 m p 1 q , is called the Milnor fiber of the multi-arrangement. F m p A q is a Stein manifold. It has the homotopy type of a finite cell complex, with gcd p m q connected components, of dim ℓ ´ 1. The (geometric) monodromy is the diffeomorphism z ÞÑ e 2 π i { N z . h : F m p A q Ñ F m p A q , If all m H “ 1, the polynomial Q “ Q p A q is the usual defining polynomial, and F p A q is the usual Milnor fiber of A . A LEX S UCIU A RRANGEMENTS AND M ILNOR FIBRATIONS ICERM, J ULY 8, 2015 3 / 16
T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT The typical fiber, F m p A q “ Q ´ 1 m p 1 q , is called the Milnor fiber of the multi-arrangement. F m p A q is a Stein manifold. It has the homotopy type of a finite cell complex, with gcd p m q connected components, of dim ℓ ´ 1. The (geometric) monodromy is the diffeomorphism z ÞÑ e 2 π i { N z . h : F m p A q Ñ F m p A q , If all m H “ 1, the polynomial Q “ Q p A q is the usual defining polynomial, and F p A q is the usual Milnor fiber of A . A LEX S UCIU A RRANGEMENTS AND M ILNOR FIBRATIONS ICERM, J ULY 8, 2015 3 / 16
T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT The typical fiber, F m p A q “ Q ´ 1 m p 1 q , is called the Milnor fiber of the multi-arrangement. F m p A q is a Stein manifold. It has the homotopy type of a finite cell complex, with gcd p m q connected components, of dim ℓ ´ 1. The (geometric) monodromy is the diffeomorphism z ÞÑ e 2 π i { N z . h : F m p A q Ñ F m p A q , If all m H “ 1, the polynomial Q “ Q p A q is the usual defining polynomial, and F p A q is the usual Milnor fiber of A . A LEX S UCIU A RRANGEMENTS AND M ILNOR FIBRATIONS ICERM, J ULY 8, 2015 3 / 16
T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT E XAMPLE Let A be the single hyperplane t 0 u inside C . Then M p A q “ C ˚ , Q m p A q “ z m , and F m p A q “ m -roots of 1. E XAMPLE Let A be a pencil of 3 lines through the origin of C 2 . Then F p A q is a thrice-punctured torus, and h is an automorphism of order 3: h A F p A q F p A q More generally, if A is a pencil of n lines in C 2 , then F p A q is a Riemann ` n ´ 1 ˘ surface of genus , with n punctures. 2 A LEX S UCIU A RRANGEMENTS AND M ILNOR FIBRATIONS ICERM, J ULY 8, 2015 4 / 16
T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT E XAMPLE Let A be the single hyperplane t 0 u inside C . Then M p A q “ C ˚ , Q m p A q “ z m , and F m p A q “ m -roots of 1. E XAMPLE Let A be a pencil of 3 lines through the origin of C 2 . Then F p A q is a thrice-punctured torus, and h is an automorphism of order 3: h A F p A q F p A q More generally, if A is a pencil of n lines in C 2 , then F p A q is a Riemann ` n ´ 1 ˘ surface of genus , with n punctures. 2 A LEX S UCIU A RRANGEMENTS AND M ILNOR FIBRATIONS ICERM, J ULY 8, 2015 4 / 16
T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT E XAMPLE Let A be the single hyperplane t 0 u inside C . Then M p A q “ C ˚ , Q m p A q “ z m , and F m p A q “ m -roots of 1. E XAMPLE Let A be a pencil of 3 lines through the origin of C 2 . Then F p A q is a thrice-punctured torus, and h is an automorphism of order 3: h A F p A q F p A q More generally, if A is a pencil of n lines in C 2 , then F p A q is a Riemann ` n ´ 1 ˘ surface of genus , with n punctures. 2 A LEX S UCIU A RRANGEMENTS AND M ILNOR FIBRATIONS ICERM, J ULY 8, 2015 4 / 16
� � � T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT Let B n be the Boolean arrangement, with Q m p B n q “ z m 1 ¨ ¨ ¨ z m n n . 1 Then M p B n q “ p C ˚ q n and F m p B n q “ ker p Q m q – p C ˚ q n ´ 1 ˆ Z gcd p m q Let A “ t H 1 , . . . , H n u be an essential arrangement. The inclusion ι : M p A q Ñ M p B n q restricts to a bundle map Q m p A q � C ˚ F m p A q M p A q ι Q m p B n q � C ˚ � M p B n q F m p B n q Thus, F m p A q “ M p A q X F m p B n q A LEX S UCIU A RRANGEMENTS AND M ILNOR FIBRATIONS ICERM, J ULY 8, 2015 5 / 16
� � � T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT Let B n be the Boolean arrangement, with Q m p B n q “ z m 1 ¨ ¨ ¨ z m n n . 1 Then M p B n q “ p C ˚ q n and F m p B n q “ ker p Q m q – p C ˚ q n ´ 1 ˆ Z gcd p m q Let A “ t H 1 , . . . , H n u be an essential arrangement. The inclusion ι : M p A q Ñ M p B n q restricts to a bundle map Q m p A q � C ˚ F m p A q M p A q ι Q m p B n q � C ˚ � M p B n q F m p B n q Thus, F m p A q “ M p A q X F m p B n q A LEX S UCIU A RRANGEMENTS AND M ILNOR FIBRATIONS ICERM, J ULY 8, 2015 5 / 16
T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT T HE HOMOLOGY OF THE M ILNOR FIBER T HE HOMOLOGY OF THE M ILNOR FIBER Some basic questions about the topology of the Milnor fibration: (Q1) Are the homology groups H q p F m p A q , k q determined by L p A q ? If so, is the characteristic polynomial of the algebraic monodromy, h ˚ : H q p F m p A q , k q Ñ H q p F m p A q , k q , also determined by L p A q ? (Q2) Are the homology groups H q p F m p A q , Z q torsion-free? If so, does F m p A q admit a minimal cell structure? (Q3) Is F m p A q a (partially) formal space? A LEX S UCIU A RRANGEMENTS AND M ILNOR FIBRATIONS ICERM, J ULY 8, 2015 6 / 16
T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT T HE HOMOLOGY OF THE M ILNOR FIBER T HE HOMOLOGY OF THE M ILNOR FIBER Some basic questions about the topology of the Milnor fibration: (Q1) Are the homology groups H q p F m p A q , k q determined by L p A q ? If so, is the characteristic polynomial of the algebraic monodromy, h ˚ : H q p F m p A q , k q Ñ H q p F m p A q , k q , also determined by L p A q ? (Q2) Are the homology groups H q p F m p A q , Z q torsion-free? If so, does F m p A q admit a minimal cell structure? (Q3) Is F m p A q a (partially) formal space? A LEX S UCIU A RRANGEMENTS AND M ILNOR FIBRATIONS ICERM, J ULY 8, 2015 6 / 16
T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT T HE HOMOLOGY OF THE M ILNOR FIBER T HE HOMOLOGY OF THE M ILNOR FIBER Some basic questions about the topology of the Milnor fibration: (Q1) Are the homology groups H q p F m p A q , k q determined by L p A q ? If so, is the characteristic polynomial of the algebraic monodromy, h ˚ : H q p F m p A q , k q Ñ H q p F m p A q , k q , also determined by L p A q ? (Q2) Are the homology groups H q p F m p A q , Z q torsion-free? If so, does F m p A q admit a minimal cell structure? (Q3) Is F m p A q a (partially) formal space? A LEX S UCIU A RRANGEMENTS AND M ILNOR FIBRATIONS ICERM, J ULY 8, 2015 6 / 16
T HE M ILNOR FIBRATIONS OF AN ARRANGEMENT T HE HOMOLOGY OF THE M ILNOR FIBER Let p A , m q be a multi-arrangement with gcd t m H | H P A u “ 1. Set N “ ř H P A m H . The Milnor fiber F m p A q is a regular Z N -cover of U p A q “ P p M p A qq defined by the homomorphism δ m : π 1 p U p A qq ։ Z N , x H ÞÑ m H mod N Let x δ m : Hom p Z N , k ˚ q Ñ Hom p π 1 p U p A qq , k ˚ q . If char p k q ∤ N , then ˇ ˇ ÿ ˇ ˇ ˇ V q s p U p A q , k q X im p x dim k H q p F m p A q , k q “ δ m q ˇ . s ě 1 This gives a formula for the polynomial ∆ q p t q “ det p t ¨ id ´ h ˚ q in terms of the characteristic varieties of U p A q . A LEX S UCIU A RRANGEMENTS AND M ILNOR FIBRATIONS ICERM, J ULY 8, 2015 7 / 16
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