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

TL;DR

The authors extend Kreck–Schafer’s doubling framework to define a quadratic bias invariant that detects stable diffeomorphism classes of doubles of finite $(G,n)$–complexes beyond homotopy equivalence. They show the quadratic bias is governed by the quadratic 2–type and develop a higher–dimensional theory with an obstruction group $B_Q(G,n)$ computable via unitary groups in appropriate cases. The main achievements include: (i) a homotopy–invariance of $eta_Q$ for minimal doubles, (ii) an explicit description of $B_Q(G,n)$ when $H_n(G)\cong (b Z/m)^d$, with $n$ even, and (iii) numerous examples of families of manifolds which are stably diffeomorphic yet pairwise not homotopy equivalent, including new nonabelian cases such as $G=Q_8 imes(b Z/p)^3$. The results provide concrete tools to distinguish finite–fundamental group 4–manifolds (and higher even–dimensional doubles) via their quadratic bias, with implications for the Andrews–Curtis landscape and related surgery–theoretic questions.

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.

Theorems & Definitions (161)

• Theorem 1
• Theorem 2
• Remark 1.1
• Theorem 3
• Remark 1.2
• Theorem 4
• Theorem 1.4
• Theorem 1.6
• Remark 1.7
• Definition 2.1