Published
Deletion-Restriction for Subspace Arrangements
Accepted to Topology and its Applications: 12/12/2007Abstract: Let A be a subspace arrangement in V with designated affine subspace A0. Let A '= A\{A0} be the deletion of A0 from A. Let A '' = {AÇA0 |A Ç A0 ≠Æ } be the restriction of A to A0. Let M = V \ ÈAÎ A A be the complement of A in V. Similarly, let M be the complement of A ' in V, and M'' the complement of A '' in A0. If A is an arrangement of complex hyperplanes, then there is a split short exact sequence, 0 ® Hk(M') ® Hk(M) ® Hk+1-codim (A0)(M'') ® 0. In [OT], an example of a subspace arrangement was found where the above sequence was not a split short exact sequence, but it was not known why the above sequence forms a split short exact sequence for some subspace arrangements and not others. In this paper, we explore which conditions on the triple (A , A ', A '') are needed to yield a split short exact sequence. We then generalize the no-broken-circuit basis nbc of Hk(M) for hyperplane arrangements to deletion-restriction subspace arrangements.
Geometric Relationship Between Cohomology of the Complement of Real and Complexified Arrangements
Published in Topology and Its Applications 118 (2002) 113-129 Authored with Peter OrlikAbstract: Let AR be a real hyperplane arrangement and let AC be its complexification. Let MR and MC be the respective complements. Then MR is the disjoint union of convex chambers whose number is given by its only Betti number, b0(MR). A real arrangement and its complexification satisfy the M-property: b0 = åq bq(MC), the number of chambers in MR equals the sum of the betti numbers of MC. The no-broken-circuit set, nbc, is a field independent combinatorial object. It has been used to label a basis for H*(MC) but not to label the chambers of MR in a way that makes the M-property explicit. In this paper we use the nbc set to label a combinatorial object in the nerve of the arrangement, which is field independent. This allows for simulaneous choices of nbc bases in H*(MR) and H*(MC). We also explore the geometrical connections between these bases.
Complements of Sphere and Subspace Arrangements
Published in Topology and its Applications 56 (1994) 199-214Abstract: Let S be an l-dimensional sphere. A sphere arrangement B is a finite collection of spheres in S with each possible intersection being either a sphere or a single point. There is no assumption on the dimension of the elements of B. If B is the one-point compactification of a subspace arrangement A, then M(B) @ M(A). A method of calculating the additive structure of H*(M(A)) was given by Goresky and MacPherson using stratified Morse Theory, proving that Hn(M(A)) depends only on the set of all intersections of elements of A partially ordered by inclusion. In this paper, we present an alternative method of calculating Hn(M(B)) for a sphere arrangement B, using the generalized Mayer-Vietoris spectral sequence and the nerve poset. We include an example of a subspace arrangement whose complement has torsion in its cohomology.
On the Complements of Affine Subspace Arrangements
Published in Topology and its Applications 56 (1994) 215-233 Authored with Peter Orlik and Boris Z. ShapiroAbstract: 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 - È AÎA A be the complement of A. A method of calculating the additive structure of H*(M(A)) was given by Goresky and MacPherson using stratified Morse Theory, proving that Hn(M(A)) depends only on the set of all intersections of elements of A partially ordered by inclusion. An alternative method of calculating Hn(M(A)) was obtained by Jewell 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 by Jewell.
In Progress
The Homotopy Type of the Complement of Complexified Real Arrangements
Last Updated: 10/12/02Abstract: In [O], by embedding a subspace arrangement in a real, essential hyperplane arrangement and using the face poset of the hyperplane arrangement and order complexes, a simplicial complex was found which has the same homotopy type as the complement of the subspace arrangement. In this paper, we specialize to the case of complexified real hyperplane arrangements and find a simpler CW-complex which is homotopy equivalent to the complement of the arrangement and which is different from Salvetti's complex described in [S].
When Will a Poset Be the Intersection Poset of a Real or Complex Hyperplane Arrangement?
Last Updated: 12/4/03Abstract: In [J], a poset needed to satisfy four simple conditions in order to be the intersection poset of some subspace arrangement. In this paper, we examine the question of what additional conditions might be necessary to determine whether or not a poset will be the intersection poset of a real or complex hyperplane arrangement.
Catergorization of Essential Real Hyperplane Arrangements by the Intersection Poset
Last Updated: 12/5/03Abstract: Based upon the conditions presented in [J2], this paper provides a listing of all hyperplane arrangements having 6 or fewer hyerplanes, and indicates which posets with a unique minimal element and having six or fewer vertices (semiminimal elements) are or are not the intersection poset of a real hyperplane arrangement.
Hyperflat Arrangements
Last Updated: 10/15/97 Work with Peter Orlik
Presentations
Knots, Links and Twisted Strips
MYCAP Summer Program Edgewood College July 21, 2005Wigwametry: Geometry in a Cultural Context
Presented with Diane Benjamin, Horse Livingston, Joni Theobald, Karen Thomas
Wisconsin Math Council Green Lake May 8, 2005Mathematical Modeling in Teaching Algebraic Ideas (PowerPoint Presentation)
Modeling Examples
Wisconsin Math Council Green Lake May 6, 2004How to Get the Most Out of Your Pizza: Or How Best to Cut a Brick of Cheese
Mathematics Seminar Edgewood College November 12, 2003A Modeling Approach Towards College Algebra
Wisconsin Mathematical Association of Two-Year Colleges
Madison Area Technical College September 27, 2003Knots, Links and Twisted Strips
Math Club Seminar Madison Area Technical College January 30, 2003Hyperplane Arrangements
Mathematics Seminar Edgewood College November 19, 2002Knots, Links and Twisted Strips
Family Weekend Edgewood College October 19, 2001