Fake ℝ⁴

Abstract

In this chapter we give an account of the topological ideas leading to the existence of a “fake R⁴” What distinguishes R ⁴_fake is its differentiable structure. After first reviewing the notion of a differentiable structure on a manifold, we describe the algebraic invariants used to classify topological 4-manifolds. Not all 4-manifolds admit a smooth structure, and specific nonexistence results, including Donaldson’s Theorem, are stated. Finally, all of this is tied together by a sketch of the proof that an exotic differentiable structure exists on R⁴. We refer the reader to [Fr²] for another expository account of this material.