Algebraic & Geometric Topology

On relations and homology of the Dehn quandle

Joel Zablow

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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.

Article information

Source
Algebr. Geom. Topol., Volume 8, Number 1 (2008), 19-51.

Dates
Received: 4 October 2007
Accepted: 22 October 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796806

Digital Object Identifier
doi:10.2140/agt.2008.8.19

Mathematical Reviews number (MathSciNet)
MR2377276

Zentralblatt MATH identifier
1146.57031

Subjects
Primary: 18G60: Other (co)homology theories [See also 19D55, 46L80, 58J20, 58J22] 57T99: None of the above, but in this section

Keywords
quandle homology Dehn twist Lefschetz fibration

Citation

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


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