On the Complements of Affine Subspace Arrangements
نویسندگان
چکیده
Let V be an l−dimensional real vector space. A subspace arrangement A is a finite collection of affine subspaces in V . There is no assumption on the dimension of the elements of A. Let M(A) = V −∪A∈AA be the complement of A. A method of calculating the additive structure of H(M(A)) was given in [G-MP] using stratified Morse theory, proving that H(M(A)) depends only on the set of all intersections of elements of A partially ordered by inclusion. An alternate method of calculating H(M(A)) was obtained in [J] using the generalized Mayer–Vietoris spectral sequence, one-point compactifications, and the nerve poset. In this paper, we present an explicit isomorphism between the two results, offer an interpretation of the coincidence of the two methods and obtain a simplification of the method of calculation in [J].
منابع مشابه
The Cohomology Rings of Complements of Subspace Arrangements
The ring structure of the integral cohomology of complements of real linear subspace arrangements is considered. While the additive structure of the cohomology is given in terms of the intersection poset and dimension function by a theorem of Goresky and MacPherson, we describe the multiplicative structure in terms of the intersection poset, the dimension function and orientations of the partic...
متن کاملSubspace Arrangements of Curve Singularities and q-Analogs of the Alexander Polynomial
This paper introduces q-series for a curve singularity (C, 0) in the affine space (Cn, 0) via subspace arrangements. These q-series are certain multivariable generating functions whose coefficients are the characteristic polynomials of subspace arrangements associated with the singularity (C, 0) at various orders; the q is the variable in the characteristic polynomials of the subspace arrangeme...
متن کاملThe maximum likelihood degree of a very affine variety
We show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao’s solution to Varchenko’s conjecture on complements of hyperplane arrangements to smooth very affine varieties. For very affine varieties satisfying a genericity condition at infinity, the result is further strengthened to relate th...
متن کاملHigher Homotopy Groups of Complements of Complex Hyperplane Arrangements
We generalize results of Hattori on the topology of complements of hyperplane arrangements, from the class of generic arrangements, to the much broader class of hypersolvable arrangements. We show that the higher homotopy groups of the complement vanish in a certain combinatorially determined range, and we give an explicit Zπ1-module presentation of πp, the first non-vanishing higher homotopy g...
متن کاملAffine Spaces within Projective Spaces
We endow the set of complements of a fixed subspace of a projective space with the structure of an affine space, and show that certain lines of such an affine space are affine reguli or cones over affine reguli. Moreover, we apply our concepts to the problem of describing dual spreads. We do not assume that the projective space is finitedimensional or pappian. Mathematics Subject Classification...
متن کامل