In this expository article we describe some recent results on the existence and general properties of complex homology 2-cells and contractible surfaces. The motivation for these can be directly traced back to C P Ramanujam’s important paper, ‘A topological characterization of the affine plane as an algebraic variety’. We also discuss the related results of M Miyanishi, T Sugie and T Fujita.

We prove the following resultTheorem.Let R be an affine normal 2-dimensional subring of ℂ[X, Y] generated by homogeneous polynomials, say, F₁,…,F_n. If the g.c.d. (F₁,…,F_n) has at most two distinct linear factors (possibly occurring with multiplicities), then R is isomorphic to a ring of invariants ℂ[v, Y]^Wfor some finite group W.

This generalizes a result of D Anderson on rings generated by monomials. As a corollary of the theorem, we prove some special cases of a conjecture of CTC wall. We also prove some results on general graded affine subrings of ℂ[X, Y].

We give a different proof of a formula of Suzuki and its strengthening by Zaidenberg for the topological Euler characteristic of an affine surface fibered over a curve. We deduce this formula using the ideas for the proof of an analogous formula for a proper morphism.