Triangulations of manifolds and combinatorial bundle theory
by
Laura Anderson
Texas A&M University
Coauthors: Nikolai Mn"ev (Steklov Institute -- St Petersburg)

Let X be a PL manifold with boundary. A category of triangulations of X is a category of combinatorial manifold structures on X and poset maps induced by PL subdivision. We require the class of objects to be closed under stellar subdivision and the class of morphisms to include all poset maps induced by stellar subdivision. Interesting examples include the category of all combinatorial manifold structures on a given X and all PL subdivisions, the category of all combinatorial manifold structures on a given X and all stellar subdivisions, and the category of all Brouwer manifold structures on X and all stellar subdivisions.

Our main result is: Let C be a category of triangulations of X. The nerve of C is homotopic to BPL(X), the classifying space for PL X-bundles.

The result is proven by defining a category of combinatorial C-bundles, whose classifying space is the nerve of C, and showing this category is equivalent to the category of PL X-bundles.

Date received: June 3, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacy-36.