Geometry & T opology 10 (2006) 2351–2381 2351 Duality for Legendrian contact homology J OSHUA MSABLOFF The main result of this paper is that, off of a “fundamental class” in degree 1, the linearized Legendrian contact homology obeys a version of Poincaré duality between homology groups in degrees k and k . Not only does the result itself simplify calculations, but its proof also establishes a framework for analyzing cohomology operations on the linearized Legendrian contact homology. 57R17; 53D12, 53D40, 57M25 1 Introduction 1.1 Legendrian contact homology As in smooth knot theory, a fundamental problem in Legendrian knot theory is to find effective invariants and to understand their structure and meaning. Bennequin [1] initiated the modern study of Legendrian knots by introducing two “classical” invariants: the Thurston–Bennequin number (which measures the difference between the framing of a knot coming from the contact planes and the Seifert surface framing) and the rotation number (which measures the twisting of the tangent to the knot inside the contact planes with respect to a suitable trivialization). These two invariants suffice to classify Legendrian knots in the standard contact structure on  3 when the underlying smooth knot type is the unknot (Eliashberg and Fraser [8]), a torus knot or the figure eight knot (Etnyre and Honda [11]) or a cable link (Ding and Geiges [5]). The first non-classical invariant of Legendrian knots was Legendrian contact homology, a Floer-type theory that comes from geometric ideas of Eliashberg and Hofer [9; 7] and was rendered combinatorially computable by Chekanov [4] for knots in the standard contact  3 . The Legendrian contact homology of a Legendrian knot is the homology of a freely-generated differential graded algebra (DGA) .A; @/ , which we shall refer to as the Chekanov–Eliashberg DGA, itself an invariant up to “stable tame isomorphism.” It is difficult to extract information from the stable tame isomorphism class of the Chekanov– Eliashberg DGA, but Chekanov defined a linearized version which was sufficient to distinguish the first examples of Legendrian knots with the same smooth knot type and classical invariants [4]. The homology of the linearized DGA is usually encoded in a Published: 8 December 2006 DOI: 10.2140/gt.2006.10.2351