# A note on 3-manifolds and complex surface singularities

## TL;DR

This paper introduces an integral invariant for certain 3-manifolds linked to complex surface singularities, connecting the Milnor number with the Adams e-invariant and providing new insights into Gorenstein singularities.

## Contribution

It defines the -invariant for 3-manifolds arising from Gorenstein singularities, linking it to the Milnor number and offering a novel perspective on surface singularities.

## Key findings

- The -invariant modulo 24 equals the Adams e-invariant.
- The -invariant for the canonical frame equals the Milnor number plus 1.
- Provides a new viewpoint on the Milnor number of smoothable Gorenstein 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.