Introduction to the First Edition

Abstract

Topologists study three types of manifolds—topological or continuous (TOP), piecewise linear (PL), differentiable (DIFF)—and the relation-ships among them. A basic problem is to ascertain when a topological manifold admits a PL structure and, if it does, whether there is also a compatible smooth structure. By the early 1950’s it was known that every topological manifold of dimension less than or equal to three admits a unique smooth structure. In 1968 Kirby and Siebenmann determined that for a topological manifold M of dimension at least five, there is a single obstruction α(M) ∈ H⁴(M;ℤ₂) to the existence of a PL structure. There are further discrete obstructions to lifting from PL to DIFF; these have coefficients in groups of homotopy spheres. Fortunately, a simplification in dimension four absolves us from having to consider the piecewise linear category again: Every PL 4-manifold carries a unique compatible differentiable structure. Now the Kirby-Siebenmann obstruction α(M), which lives on the 4-skeleton of an n-manifold M, relates in special cases to a result of Rohlin dating back to 1952. Rohlin’s Theorem states that the signature of a smooth spin 4-manifold is divisible by 16. The arithmetic of quadratic forms shows that the signature of a topological “spin” (= almost parallelizable) 4-manifold M is divisible by 8, and α(M) ∈₂ = 8ℤ/16ℤ is the signature mod 16. If M is not spinable, the Kirby-Siebenmann invariant is an extra piece of information not related to the intersection form.