Volume 8, issue 1 (2008)

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On relations and homology of the Dehn quandle

Joel Zablow

Algebraic & Geometric Topology 8 (2008) 19–51

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.

quandle homology, Dehn twist, Lefschetz fibration
Mathematical Subject Classification 2000
Primary: 18G60, 57T99
Received: 4 October 2007
Accepted: 22 October 2007
Published: 8 February 2008
Joel Zablow
Department of Mathematics
Rochester Institute of Technology
85 Lomb Memorial Drive
Rochester NY 14623