Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, February 21, 2008, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Uli Wagner

Van Kampen's Obstruction and Embeddability of Skeleta of Spheres

Consider the following statement: If G is a planar simple graph all of whose faces are triangles, then G is maximally planar, i.e., adding a single edge to G makes G nonplanar.

For graphs, this is an immediate consequence of Euler's formula. I will present a different proof based on the so-called van Kampen obstruction. This proof has the disadvantage that it is more complicated than mere counting, but on the other hand, it also works in much more generality to prove the following statement, which generalizes a theorem of van Kampen:

If S is the d-skeleton of a 2d-dimensional piecewise linear sphere then adding a missing d-face to S makes S nonembeddable in R^{2d}.

I will try to explain all the relevant notions in the course of the talk, which means that most likely, there will not be much time to explain the motivation, i.e., how the more general statement is related to a conjecture by Kalai and Sarkaria and the so-called g-Conjecture for simplicial spheres, but one can always be optimistic.

Joint work with Eran Nevo (Cornell).

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