Universal quadratic forms and Whitney tower intersection invariants

TL;DR

This paper introduces a geometric description of the Kirby-Siebenmann invariant using quadratic forms and develops a universal theory of intersection invariants for Whitney towers in 4-manifolds, advancing the understanding of their topology.

Contribution

It provides a new geometric interpretation of the Kirby-Siebenmann invariant and establishes a universal symmetric intersection invariant for Whitney towers, connecting quadratic forms to 4-manifold topology.

Findings

Geometric description of Kirby-Siebenmann invariant via quadratic refinement

Development of a universal symmetric intersection invariant for Whitney towers

Short exact sequence elucidating Whitney tower structure in the 4-ball

Abstract

The first part of this paper exposits a simple geometric description of the Kirby-Siebenmann invariant of a 4--manifold in terms of a quadratic refinement of its intersection form. This is the first in a sequence of higher-order intersection invariants of Whitney towers studied by the authors, particularly for the 4--ball. In the second part of this paper, a general theory of quadratic forms is developed and then specialized from the non-commutative to the commutative to finally, the symmetric settings. The intersection invariant for twisted Whitney towers is shown to be the universal symmetric refinement of the framed intersection invariant. As a corollary we obtain a short exact sequence that has been essential in the understanding of Whitney towers in the 4--ball.