A note on 3-manifolds and complex surface singularities

Abstract

This article is motivated by the original Casson invariant regarded as an integral lifting of the Rochlin invariant. We aim to defining an integral lifting of the Adams e-invariant of stably framed 3-manifolds, perhaps endowed with some additional structure. We succeed in doing so for manifolds which are links of normal complex Gorenstein smoothable singularities. These manifolds are naturally equipped with a canonical $\SU$-frame. To start we notice that the set of homotopy classes of $\SU$-frames on the stable tangent bundle of every closed oriented 3-manifold is canonically a $\mathbb Z$-torsor. Then we define the $\widehat E$-invariant for the manifolds in question, an integer that modulo 24 is the Adams e-invariant. The $\widehat E$-invariant for the canonical frame equals the Milnor number plus 1, so this brings a new viewpoint on the Milnor number of the smoothable Gorenstein surface singularities.