Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 8, Number 1 (2008), 19-51.
On relations and homology of the Dehn quandle
Isotopy classes of circles on an orientable surface of genus form a quandle 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.
Algebr. Geom. Topol., Volume 8, Number 1 (2008), 19-51.
Received: 4 October 2007
Accepted: 22 October 2007
First available in Project Euclid: 20 December 2017
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Zablow, Joel. On relations and homology of the Dehn quandle. Algebr. Geom. Topol. 8 (2008), no. 1, 19--51. doi:10.2140/agt.2008.8.19. https://projecteuclid.org/euclid.agt/1513796806