instituto de matemáticas universidad de sevilla
Antonio de Castro Brzezicki
Geometric embeddings and immersions of pyramidal triangulations of surfaces in 3-space
Seminario de Matemática Discreta

Historically, it is a corollary of Steinitz's theorem (1922) that  every combinatorial triangulation  of  the  sphere  is  geometrically  embeddable  in  (Euclidean)  3-space. Grünbaum’s conjecture  (1967)  stated  that  every  triangulation  on  any  closed  orientable  surface  would  be geometrically embeddable in 3-space, but was disproved by Bokowski and Guedes de Oliveira in 2000.

The term “embedding” means that no two (geometric) faces may intersect in their interior. The term “immersion” means “locally embedding”—that is, two general faces may overlap or intersect along a line that is not an edge of either face, but no two faces with a common vertex may intersect in their interior. The focus of the lecture will be on geometric embeddability and immersability of “pyramidal triangulations”— that is, triangulations having a vertex connected to all other vertices. For pyramidal triangulations, an algorithm has been designed (Sabitov, 2012) to find a simpler form for Sabitov polynomials for the volumes of geometric embeddings of such triangulations.  

Pyramidal  simplicial  orientable  polyhedra  with  arbitrary  genus  have  been  constructed  as geometrically embedded  in  3-space  by Sabitov  and  the  speaker.  Sabitov  has  also constructed geometric  immersions  of  pyramidal  nonorientable  triangulations  in  3-space  for  arbitrary even nonorientable genus. However, for the case of odd nonorientable genus, especially for the case of the projective plane, the pyramidal immersion problem has turned out to be not so easy. The speaker conjectures that no pyramidal triangulation of the projective plane can be simplicially immersed in 3-space.

Possible approaches to the pyramidal projective plane immersion problem will be discussed. The lecture may  be  interesting  both  for  theoretical  as  well  as  for  computational  geometers  and topologists.