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Geometric Topology Seminar: N. Sadykov

Event ended

Geometric Topology Seminar
Math Department of the Higher School of Economics (Usachyova, 6)
Room 108

[For external visitors, you may be required to present identification
(such as a passport) at security. You are welcome to call Ash
Lightfoot (+7 925 8897129) in the case of any difficulties.]

Thursday, April 19, 14:00

Speaker: N. M. Sadykov

Title: Hexagon relations, their cohomologies, and invariants of
4-dimensional PL manifolds


Pachner moves, also called bistellar moves, are elementary
re-buildings of a manifold triangulation. A triangulation of a given
piecewise linear (PL) manifold can be transformed into any other one
by a finite sequence of Pachner moves. Hexagon relations are algebraic
realizations of four-dimensional Pachner moves. It can be said that
hexagon relations are in the same relationship with 4-manifolds and
Pachner moves as quandles are with knots and Reidemeister moves, or as
the same quandles are with 2-knots and Roseman moves.

We present an explicit hexagon relation in terms of vector spaces over
a finite field. This allows us to define "permitted colorings" on
triangulations of 4-manifolds, with a clear correspondence between
such colorings before and after a Pachner move. We then define a
"rough" invariant of a 4-manifold, based on the total number of
permitted triangulation colorings.

And like in the quandle case, there are cohomologies that can be
introduced for hexagon relations. Remarkably, nontrivial cohomologies
do exist, and they give much more interesting invariants of PL