A PDF file should load here. If you do not see its contents
the file may be temporarily unavailable at the journal website
or you do not have a PDF plug-in installed and enabled in your browser.
Alternatively, you can download the file locally and open with any standalone PDF reader:
http://link.springer.com/content/pdf/10.1007%2FBF02573977.pdf
Preface
Discrete Comput Geom
-
This special issue collects six papers on oriented matroids that (except for the first
one) were written at the Institut Mittag-Leffter in Djursholm (Sweden), during its
program in "Combinatorics" (1991/92) directed by Anders Bj6rner.
Oriented matroids, a concept that was created 20 years ago, has since then grown
into a powerful theory. Oriented matroids form a combinatorial model for point
configurations and for real hyperplane arrangements (which are realizable
oriented matroids), but they contain more general objects; the precise geometric
meaning of general oriented matroids is given by the "Topological Representation
Theorem" of Lawrence, which identifies oriented matroids with arrangements of
possibly deformed hyperplanes (known as "pseudoarrangements").
All six papers are concerned with geometric and topological properties of
oriented matroids, and in particular with the "new effects" that appear if the
condition of realizability is relaxed or dropped. It seems that some of the most
interesting phenomena occur at the boundaries of realizability.
The first paper shows how the Topological Representation Theorem can be
used to represent an oriented matroid and its dual simultaneously in a single
pseudoarrangement.
The second paper, "Combinatorial models for the finite-dimensional
Grassmannians," presents a common framework for several situations where spaces
of oriented matroids are dealt with. We consider the spac~ of all extensions
of a fixed pseudoarrangement by a lower-dimensional pseudosphere (for instance,
the space of all pseudolines that can be inserted into an arrangement of
hyperplanes). It is a basic conjecture that this space should be a good model for a real
Grassmannian if the underlying pseudoarrangement is realizable.
In " T w o constructions of oriented matroids with disconnected extension space"
the basic conjecture is shown to fail if the realizability assumption is dropped for
the oriented matroid used as an ambient space. In "Oriented matroids with few
mutations" even more drastic nonlinear effects in oriented matroids are
constructed, which--for example--disprove an 18-year-old conjecture of Las Vergnas.
Moreover, the constructions presented there yield oriented matroids with
considerably fewer simplicial regions than hyperplanes, an effect which arises only in
the nonrealizable case.
There are other geometric applications of this line of investigation. Thus, in
"Combinatorial obstructions to the lifting of weaving diagrams" oriented matroids
are used to model the "cycles" in incorrect planar drawings of spatial
configurations of lines.
Finally, in "What is a complex matroid?" we propose a combinatorial model
for the study of discrete structures in complex space: we show how the
corresponding theory can in part be reduced to oriented matroid theory.
We are extremely grateful to Anders Bjrrner and the Institut Mittag-Leffler
for their support, help, and hospitality during our time there, and for the wonderful
working 'atmosphere they provided. In particular, we gratefully acknowledge
valuable discussions with our coauthor Nicolai E. Mn~v, and with Eric Babson,
Anders Bjfrner, Henry Crapo, Andreas Dress, Jacob E. Goodman, Robert D.
MacPherson, Richard Pollack, Grigori U Rybnikov, Bernd Sturmfels, and Walter
Whiteley on topics related to these papers. Thanks also to the referees for their
helpful comments and corrections. (...truncated)