Signature modulo 8 of fibre bundles

Abstract

Lay summary Topology studies the geometric properties of spaces that are preserved by continuous deformations. Manifolds are the main examples of topological spaces, with the local properties of Euclidean space in an arbitrary dimension n. They are the higher dimensional analogs of curves and surfaces. For example a circle is a one-dimensional manifold. Balloons and doughnuts are examples of two-dimensional manifolds. A balloon cannot be deformed continuously into a doughnut, so we see that there are essential topological differences between them. An "invariant" of a topological space is a number or an algebraic structure such that topologically equivalent spaces have the same invariant. For example the essential topological difference between the balloon and the doughnut is calculated by the "Euler characteristic", which is 2 for a balloon and 0 for a doughnut. In this thesis I investigate the relation between three different but related invariants of manifolds with dimension divisible by 4: the signature, the Brown-Kervaire invariant and the Arf invariant. The signature invariant takes values in the set (...;-3;-2;-1; 0; 1; 2; 3; ...) of integers. In this thesis we focus on the signature invariant modulo 8, that is its remainder after division by 8. The Brown-Kervaire invariant takes values in the set (0; 1; 2; 3; 4; 5; 6; 7). The Arf invariant takes values in the set (0; 1). The main result of the thesis uses the Brown-Kervaire invariant to prove that for a manifold with signature divisible by 4, the divisibility by 8 is decided by the Arf invariant. The thesis is entirely concerned with pure mathematics. However it is possible that it may have applications in mathematical physics, where the signature modulo 8 plays a significant role.