## Algebraic & Geometric Topology

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

### On relations and homology of the Dehn quandle

#### 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