Page 255 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 255
APPLIED COMBINATORIAL AND GEOMETRIC TOPOLOGY (MS-34)
Partitioning the projective plane and the dunce hat
Andrés David Santamaría-Galvis, andres.santamaria@famnit.upr.si
University of Primorska, Slovenia
The faces of a simplicial complex induce a partial order by inclusion in a natural way. We say
that the complex is partitionable if its poset can be partitioned into boolean intervals, with a
maximal face at the top of each.
In this work we show that all the triangulations of the real projective plane, the dunce hat,
and the open Möbius strip are partitionable. To prove that, we introduce simple yet useful
gluing tools that allow us to abridge the discussion about partitionability of a given complex in
terms of smaller constituent relative subcomplexes. The gluing process generates partitioning
schemes with a distinctive shelling-like flavor.
Shellings from relative shellings
Russ Woodroofe, russ.woodroofe@famnit.upr.si
University of Primorska, Slovenia
Coauthor: Andrés David Santamaría-Galvis
It is frequently helpful to build complicated mathematical objects by breaking down into sub-
objects with simpler properties. In joint work with Andrés Santamaría-Galvis, we have shown
how to usefully glue together shellings of relative simplicial complexes to construct a shelling
of a large simplicial complex. Indeed, one of the main other ways of using the "divide and
conquer" approach to build a shelling, vertex-decomposability, can be viewed as a consequence
of our approach. One useful consequence is that our approach makes it relatively straightfor-
ward to find a shellable simplicial complex satisfying any of a variety of conditions on its facets
that contain a free face. Another is an improved proof that the shellability decision problem is
NP-complete.
253
Partitioning the projective plane and the dunce hat
Andrés David Santamaría-Galvis, andres.santamaria@famnit.upr.si
University of Primorska, Slovenia
The faces of a simplicial complex induce a partial order by inclusion in a natural way. We say
that the complex is partitionable if its poset can be partitioned into boolean intervals, with a
maximal face at the top of each.
In this work we show that all the triangulations of the real projective plane, the dunce hat,
and the open Möbius strip are partitionable. To prove that, we introduce simple yet useful
gluing tools that allow us to abridge the discussion about partitionability of a given complex in
terms of smaller constituent relative subcomplexes. The gluing process generates partitioning
schemes with a distinctive shelling-like flavor.
Shellings from relative shellings
Russ Woodroofe, russ.woodroofe@famnit.upr.si
University of Primorska, Slovenia
Coauthor: Andrés David Santamaría-Galvis
It is frequently helpful to build complicated mathematical objects by breaking down into sub-
objects with simpler properties. In joint work with Andrés Santamaría-Galvis, we have shown
how to usefully glue together shellings of relative simplicial complexes to construct a shelling
of a large simplicial complex. Indeed, one of the main other ways of using the "divide and
conquer" approach to build a shelling, vertex-decomposability, can be viewed as a consequence
of our approach. One useful consequence is that our approach makes it relatively straightfor-
ward to find a shellable simplicial complex satisfying any of a variety of conditions on its facets
that contain a free face. Another is an improved proof that the shellability decision problem is
NP-complete.
253
