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

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.

45 pages. v3: Minor corrections. Final version, to appear in Advances in Mathematics

References in corpus (10)

• Stable classification of 4-manifolds with 3-manifold fundamental groups
• Two remarks on Wall's D2 problem
• On CW-complexes over groups with periodic cohomology
• Counterexamples in 4-manifold topology
• Topological 4-manifolds with right-angled Artin fundamental groups
• Projective modules and the homotopy classification of $(G,n)$-complexes
• ${\mathbb{CP}}^2$-stable classification of $4$-manifolds with finite fundamental group
• Homotopy classification of 4-manifolds whose fundamental group is dihedral
• Algebraic criteria for stable diffeomorphism of spin 4-manifolds
• Homotopy classification of 4-manifolds with finite abelian 2-generator fundamental groups