On relations and homology of the Dehn quandle

Zablow, Joel

doi:10.2140/agt.2008.8.19

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On relations and homology of the Dehn quandle

Joel Zablow

Algebraic & Geometric Topology 8 (2008) 19–51

DOI: 10.2140/agt.2008.8.19

Abstract

Isotopy classes of circles on an orientable surface $F$ of genus $g$ form a quandle $Q$ under the operation of Dehn twisting about such circles. We derive certain fundamental relations in the Dehn quandle and then consider a homology theory based on this quandle. We show how certain types of relations in the quandle translate into cycles and homology representatives in this homology theory, and characterize a large family of 2–cycles representing homology elements. Finally we draw connections to Lefschetz fibrations, showing isomorphism classes of such fibrations over a disk correspond to quandle homology classes in dimension 2, and discuss some further structures on the homology.

Keywords

quandle homology, Dehn twist, Lefschetz fibration

Mathematical Subject Classification 2000

Primary: 18G60, 57T99

References

Publication

Received: 4 October 2007

Accepted: 22 October 2007

Published: 8 February 2008

Authors

Joel Zablow
Department of Mathematics Rochester Institute of Technology 85 Lomb Memorial Drive Rochester NY 14623 USA

Volume 8, issue 1 (2008)