We define quantum lens spaces as ‘direct sums of line bundles’ and exhibit them as ‘total spaces’ of certain principal bundles over quantum projective spaces. For each of these quantum lens spaces we construct an analogue of the classical Gysin sequence in K-theory. We use the sequence to compute the K-theory of the quantum lens spaces, in particular to give explicit geometric representatives of their K-theory classes. These representatives are interpreted as ‘line bundles’ over quantum lens spaces and generically define ‘torsion classes’. We work out explicit examples of these classes.

VL - 9 ER - TY - JOUR T1 - Moduli spaces of noncommutative instantons: gauging away noncommutative parameters JF - Quarterly Journal of Mathematics (2012) 63 (1): 41-86 Y1 - 2012 A1 - Simon Brain A1 - Giovanni Landi AB - Using the theory of noncommutative geometry in a braided monoidal category, we improve upon a previous construction of noncommutative families of instantons of arbitrary charge on the deformed sphere S^4_\\\\theta. We formulate a notion of noncommutative parameter spaces for families of instantons and we explore what it means for such families to be gauge equivalent, as well as showing how to remove gauge parameters using a noncommutative quotient construction. Although the parameter spaces are a priori noncommutative, we show that one may always recover a classical parameter space by making an appropriate choice of gauge transformation. PB - Oxford University Press UR - http://hdl.handle.net/1963/3777 U1 - 548 U2 - Mathematics U3 - Mathematical Physics ER - TY - RPRT T1 - Families of Monads and Instantons from a Noncommutative ADHM Construction Y1 - 2009 A1 - Simon Brain A1 - Giovanni Landi AB - We give a \\\\theta-deformed version of the ADHM construction of SU(2) instantons with arbitrary topological charge on the sphere S^4. Classically the instanton gauge fields are constructed from suitable monad data; we show that in the deformed case the set of monads is itself a noncommutative space. We use these monads to construct noncommutative `families\\\' of SU(2) instantons on the deformed sphere S^4_\\\\theta. We also compute the topological charge of each of the families. Finally we discuss what it means for such families to be gauge equivalent. UR - http://hdl.handle.net/1963/3478 U1 - 786 U2 - Mathematics U3 - Mathematical Physics ER -