First Betti numbers of orbits of Morse functions on surfaces

2019/08/08 by Kuznietsova, Iryna, Soroka, Yuliia · 1 citation
#20E22 #20F16 #57T15 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.1908.03014

Abstract

In this article we study algebraic properties of the specific class of groups G generated by direct products and wreath products. Such class of groups appears in calculation of fundamental groups of orbits of Morse functions on compact manifolds. We prove that for any group G\inG the ranks of the center Z(G) and the quotient by commutator subgroup G/[G,G] coincide. Moreover, this rank is a first Betti number of the orbit of Morse function.

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