From: mmcconn@littlewood.math.okstate.edu Newsgroups: sci.math.research Subject: Sheafhom announcement Date: Fri, 28 Feb 97 15:21:58 CST This is an announcement of Sheafhom, a system of programs I have been developing for studying algebraic topology, especially intersection homology. INTRODUCTION Sheafhom lets the user work with finite-dimensional vector spaces V over the rational numbers, linear maps V --> W, and chain complexes V_n --> ... --> V_1 --> V_0. It provides partially ordered sets, which model simplicial complexes X. One can define sheaves on X with sections in the category of chain complexes. For example, one can write programs to construct the intersection homology [IH] sheaves on X in any perversity, and to find the groups IH_*(X;Q). This includes the ordinary cohomology H^*(X;Q) and homology H_*(X;Q) as special cases. To find these groups, Sheafhom uses spectral sequences. Given the E^1 page, it computes E^2, E^3, E^4,... in turn until it knows the sequence has stopped. It works directly from the definition of a spectral sequence, maintaining a finite-dimensional model of the complex IC_* of all intersection chains on X. We do not need special conditions like "the sequence collapses at E^2". At present, Sheafhom does not work over finite fields, or with torsion modules like Z/nZ. It computes IH_*(X;Q) rather than the full IH_*(X;Z). TORIC VARIETIES Sheafhom currently can find IH_*(X) when X is a toric variety (over the complex numbers, of any dimension, complete, normal, not necessarily smooth or projective). This holds in any perversity, and includes the ordinary H^*(X;Q) and H_*(X;Q). Methods are already known for computing H^*(X), H_*(X) [see [T]], and the middle-perversity Betti numbers. My approach is different because I treat all perversities equally, and because I construct actual cycles on X that generate the IH groups. In effect, I carry out Deligne's construction of IH as in [GM], replacing the infinite-dimensional complexes of injective sheaves with the finite-dimensional model of IC_*. One can prove the model is "fine enough" for this purpose. People have felt that an algorithm like this should exist, but it has needed care to write it down precisely. Within the next few months, I will write a paper that translates Sheafhom's toric variety algorithms into English and proves they are correct. A goal for the future is to write an algorithm for the intersection product on IH of toric varieties [including the cup product on H^*(X) as a special case]. Stanley proved the necessity of McMullen's conditions on a simplicial convex polytope by using the cup product on the cohomology of the associated toric variety X, which is essentially smooth. Understanding the product on IH would help us investigate the generalization of Stanley's proof to all rational convex polytopes. "NON-RATIONAL" TORIC VARIETIES AND POLYTOPES While Sheafhom works over Q, its toric variety algorithms are also well-defined over the real numbers. Thus the algorithms define "Betti numbers" for the fictitious toric varieties coming from fans with irrational slopes. Equivalently, they give Betti numbers for non-rational convex polytopes (with fixed position relative to the origin). We do not know if these Betti numbers are non-negative, or whether they have properties like Poincare Duality and hard Lefschetz. MORE INFORMATION; OBTAINING SHEAFHOM The current version of Sheafhom can be downloaded over the Web from http://www.math.okstate.edu/~mmcconn/shh.html The Web page also contains tutorial files, an article about Sheafhom, an encyclopedia of all of Sheafhom's data types and functions, and previous versions of the code (including a fast implementation for H_*(X;Q) using [T]). At present, Sheafhom is written in Common Lisp. To run the code, you will need a Common Lisp interpreter or compiler that supports CLOS (the object-oriented extension of Lisp). You can use the toric variety routines without knowing any Lisp--you can create a text file that describes the fan data for X according to a simple recipe, and just load the text file into Lisp. See the toric-variety tutorial for how to do this. A version of Sheafhom without Lisp is a possibility for the future. PLEASE CONTACT ME at mmcconn@math.okstate.edu or (405) 744-8220 if you have any questions or comments. If you use Sheafhom, either briefly or for a specific project, I'd love to hear about what you're doing and whether Sheafhom proved to be useful to you. --Mark McConnell Assoc. Prof. of Mathematics Oklahoma State University --------- [GM] Mark Goresky and Robert MacPherson, "Intersection Homology II", Inventiones Math. 72 (1983), pp. 77-130. [T] Burt Totaro, "Chow groups, Chow cohomology, and linear varieties", to appear in J. Algebraic Geom.