Geometric Topology Seminar: N. Sadykov

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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

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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

Abstract:

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

4-manifolds.