Surgery and the Generalized Kervaire Invariant, I

Abstract

(i) Synopsis. The discovery, around 1960, of the 'Kervaire Invariant' for almost framed manifolds of dimension 4k+ 2 (see [12]) was an important stimulant for the development of surgery theory; but it also led to the theory of the 'generalized Kervaire Invariant' of Browder and Brown [2, 3]. The present paper is an attempt at uniting these two theories, by constructing a non-simply-connected and in other respects updated version of the generalized Kervaire Invariant. The construction has three surprising aspects. Firstly, it is conceptually satisfying and, in the simply-connected case, clarifies Brown's original theory; for instance, the 'product formula problem' (see [4]) evaporates. Most of the new concepts are borrowed from the 'algebraic theory of surgery'; see [15]. Secondly, it is computationally satisfying. Thirdly, it has applications to classical surgery theory, especially to the calculation of the symmetric L-groups of [13] and [15]; and therefore to anything which involves product formulae for surgery obstructions. A black box description of the theory has been given in [22]; in this introduction I shall concentrate on the concepts inside the box.