L. Paris. Intersection subgroups of complex hyperplane arrangements. preprint, 1998. 24 Home/Search Document Details and Download
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On the Homotopy Theory of Arrangements, II - Falk, Randell (1998) (Correct)
.... was also made on the problem of find ing sufficient conditions for an arrangement to be K(r, 1) The main results are the weight test of [31] and its application to factored arrangements by Paris [69] A new technique involving modular flats was recently discovered and presented at the conference [70, 35]. The complement M of a 2 dimensional affine arrangement .A is built up out of K(r, 1) spaces, specifically (r, r) torus link complements, in a relatively simple way, as is reflected in the Randell Salvetti Arvola presentations (see Section 4.1) In fact this structure mirrors precisely ....
....at least in rank three. Problem 3.12. Show that factored arrangements of arbitrary rank are K(r, 1) A flat X of a matroid G is modular if rk(X V Y) rk(X A Y) rk(X) rk(Y) for every flat Y. The following result was discovered independently by Paris and Falk Proudfoot Theorem 3.13. [70, 73, 35] [f X is a modular flat o arbitrary rank in G( A) then there is a topological fibration M( A) 4 M( Ax) whose fiber is the complement of a projectire arrangement. This generalizes the corank one case, which gives rise to fiber type arrangements, established in [87] The new result can be used to ....
L. Paris. Intersection subgroups of complex hyperplane arrangements. preprint, 1998. 24
On the Homotopy Theory of Arrangements, II - Falk, Randell (1998) (Correct)
.... was also made on the problem of finding sufficient conditions for an arrangement to be K( 1) The main results are the weight test of [31] and its application to factored arrangements by Paris [69] A new technique involving modular flats was recently discovered and presented at the conference [70, 35]. The complement M of a 2 dimensional affine arrangement A is built up out of K( 1) spaces, specifically (r; r) torus link complements, in a relatively simple way, as is reflected in the Randell Salvetti Arvola presentations (see Section 4.1) In fact this structure mirrors precisely ....
....at least in rank three. Problem 3.12. Show that factored arrangements of arbitrary rank are K( 1) A flat X of a matroid G is modular if rk(X Y ) rk(X Y ) rk(X) rk(Y ) for every flat Y . The following result was discovered independently by Paris and Falk Proudfoot Theorem 3.13. [70, 73, 35] If X is a modular flat of arbitrary rank in G(A) then there is a topological fibration M (A) M (AX ) whose fiber is the complement of a projective arrangement. This generalizes the corank one case, which gives rise to fiber type arrangements, established in [87] The new result can be used to ....
L. Paris. Intersection subgroups of complex hyperplane arrangements. preprint, 1998. 24 M. FALK AND R. RANDELL
Parallel Connections and Bundles of Arrangements - Falk, Proudfoot (2 citations) (Correct)
....elementary cone decone construction [22] The restriction OEj M(A) M (A) Gamma C is in fact a trivial fibration, with fiber isomorphic to the complement in C Gamma1 of an affine arrangement, the decone of A. The modular fibration theorem was independently discovered by L. Paris [24, 25], who also gave a proof using the Thom Isotopy Lemma. Our original 2 proof of local triviality incorrectly dealt with an important technical condition, so we rely here on Paris argument on this point. In [31] Terao establishes the result for modular copoints, and proves that for general modular ....
....3.3 The map j M(A) M (A) Gamma M (AX ) is a fiber bundle projection. proof: The complement M (A) coincides with the open stratum S 0L of C ae PX . So Lemma 3. 2 implies that the restriction of b to M (A) is a fiber bundle projection, by the Thom Isotopy Lemma [15, 20, 33] 2 Paris [24] uses the same proof, along with a result of [31] to prove a more general form of Corollary 3.3. As should be clear, the result is in essence an immediate consequence of modularity, specifically Lemma 2.3, given the heavy artillery of singularity theory. We proceed to generalize the properties ....
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L. Paris. Intersection subgroups of complex hyperplane arrangements. Topology and Its Applications. to appear.
On the Fundamental Group of the Complement of a Complex Hyperplane .. - Paris (1999) (1 citation) Self-citation (Paris) (Correct)
....a hyperplane H of A is parallel to X if either H X = or X H. Consider the projection pX : V V=X. If H is parallel to X, then pX (H) is a hyperplane of V=X. Let A=X = fpX (H) H 2 A and H parallel to Xg. Then the projection pX induces a projection pX : M(A) M(A=X) Proposition 3. 1 (Paris [Pa3]) The projection pX : M(A) M(A=X) admits a crosssection s X : M(A=X) M(A) Definition. Call Y 2 L(A) horizontal with respect to X if pX (Y ) V=X. Let HorX denote the set of horizontal elements of L(A) The bad set of M(A=X) is BX = fpX (Y ) M(A=X) Y 2 L(A) n HorX g : Theorem 3.2 ....
....M(A=X) admits a crosssection s X : M(A=X) M(A) Definition. Call Y 2 L(A) horizontal with respect to X if pX (Y ) V=X. Let HorX denote the set of horizontal elements of L(A) The bad set of M(A=X) is BX = fpX (Y ) M(A=X) Y 2 L(A) n HorX g : Theorem 3. 2 (Falk and Proudfoot [FP] Paris [Pa3]) Let NX = M(A=X) n BX ; MX = p Gamma1 X (NX ) M(A) 7 Then the restriction pX : MX NX of pX to MX is a locally trivial C 1 fibration. Remark. i) The restriction of s X to NX determines a cross section s X : NX MX of the fibration. ii) Let y 0 2 NX , and let z 0 = s X (y 0 ) 2 MX ....
[Article contains additional citation context not shown here]
L. Paris, Intersection subgroups of complex hyperplane arrangements, Topology and its Applications, to appear.
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