Geodesic
On relations and homology of the Dehn quandle
Zablow, Joel1
1Department of Mathematics, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester NY 14623, USA
Algebraic and Geometric Topology, Tome 8 (2008) no. 1, pp. 19-51
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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.

DOI : 10.2140/agt.2008.8.19
Keywords: quandle homology, Dehn twist, Lefschetz fibration

Zablow, Joel  1

1 Department of Mathematics, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester NY 14623, USA 
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Zablow, Joel. On relations and homology of the Dehn quandle. Algebraic and Geometric Topology, Tome 8 (2008) no. 1, pp. 19-51. doi: 10.2140/agt.2008.8.19

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