Four-manifolds, two-complexes and the quadratic bias invariant

TL;DR

This paper introduces a new homotopy invariant for 4-manifolds constructed as doubles of 2-complexes, demonstrating the existence of multiple non-homotopy equivalent manifolds that are stably diffeomorphic.

Contribution

It extends Kreck and Schafer's methods to formulate a novel invariant distinguishing certain 4-manifolds, revealing new examples of stably diffeomorphic but non-homotopy equivalent manifolds.

Findings

Existence of families of 4-manifolds that are stably diffeomorphic but not homotopy equivalent

Development of a new homotopy invariant for doubles of 2-complexes

Extension of previous constructions to broader classes of 4-manifolds

Abstract

Kreck and Schafer produced the first examples of stably diffeomorphic closed smooth 4-manifolds which are not homotopy equivalent. They were constructed by applying the doubling construction to 2-complexes over certain finite abelian groups of odd order. By extending their methods, we formulate a new homotopy invariant on the class of 4-manifolds arising as doubles of 2-complexes with finite fundamental group. As an application we show that, for any $k \ge 2$, there exist a family of $k$ closed smooth 4-manifolds which are all stably diffeomorphic but are pairwise not homotopy equivalent.