arXiv:1312.5690v1 [math.QA] 19 Dec 2013 REAL SPECTRAL TRIPLES ON 3-DIMENSIONAL NONCOMMUTATIVE LENS SPACES. ANDRZEJ SITARZ 1 Institute of Physics, Jagiellonian University, Reymonta 4, Krak´ ow, 30-059 Poland, Institute of Mathematics of the Polish Academy of Sciences, ´ Sniadeckich 8, Warszawa, 00-950 Poland. JAN JITSE VENSELAAR Mathematics Department, Mail Code 253-37, Caltech, 1200 E. California Blvd. Pasadena, CA 91125, USA Abstract. We study almost real spectral triples on quantum lens spaces, as orbit spaces of free actions of cyclic groups on the spectral geome- try on the quantum group SUq (2). These spectral triples are given by weakening some of the conditions of a real spectral triple. We classify the irreducible almost real spectral triples on quantum lens spaces and we study unitary equivalences of such quantum lens spaces. We show that if r is coprime to p, the C ∗ -algebras corresponding to the quantum lens spaces Lq (p, r) and Lq (p, 1) are isomorphic. Also, we show that all such quantum lens spaces are Seifert fibrations over quantum teardrops, and calculate the Dirac spectrum on the base space coming from this fibration. Lens spaces, orbit spaces of free actions of cyclic groups on odd-dimensional spheres, were first introduced in 1884 by Walther Dyck [15]. Lens spaces are interesting because they are some of the simplest manifolds exhibiting the difference between homotopy type and homeomorphism type. In this article we study almost real spectral triples on quantum lens spaces, as orbit spaces of free actions of cyclic groups on the spectral geometry on the quantum group SU q (2), as constructed in [11]. These spectral triples are given by weakening some of the conditions of a real spectral triple, just like in [11]. We classify the irreducible spectral geometries on quantum lens spaces and we study unitary equivalences of such quantum lens spaces. E-mail addresses: andrzej.sitarz@uj.edu.pl, janjitse@caltech.edu . 1 Partially supported by NCN grant 2011/01/B/ST1/06474. 1