Every A1−bundle over the complex affine plane punctured at the
origin, is trivial in the differentiable category but there are infinitely many
distinct isomorphy classes of algebraic bundles. Isomorphy types of total
spaces of such algebraic bundles are considered; in particular, the complex
affine 3-sphere admitts such a structure with an additional homogeneity
property. Total spaces of nontrivial homogeneous A1-bundles over
the punctured plane are classified up to Gm​-equivariant algebraic
isomorphism and a criterion for nonisomorphy is given. In fact the affine
3-sphere is not isomorphic as an abstract variety to the total space of any
A1-bundle over the punctured plane of different homogeneous
degree, which gives rise to the existence of exotic spheres, a phenomenon that
first arises in dimension three. As a by product, an example is given of two
biholomorphic but not algebraically isomorphic threefolds, both with a trivial
Makar-Limanov invariant, and with isomorphic cylinders