

Geometric Topology Seminar
April 19, 2018 14:00–16:50, Moscow, Math Department of the Higher School of Economics, Room 108






Hexagon relations, their cohomologies, and invariants of 4dimensional PL manifolds
N. M. Sadykov^{} 
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Abstract:
Pachner moves, also called bistellar moves, are elementary rebuildings 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 fourdimensional Pachner moves. It can be said that hexagon relations are in the same relationship with 4manifolds and Pachner moves as quandles are with knots and Reidemeister moves, or as the same quandles are with 2knots 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 4manifolds, with a clear correspondence between such colorings before and after a Pachner move. We then define a “rough” invariant of a 4manifold, 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 4manifolds.

