Simply-connected manifolds with large homotopy stable classes

2021/09/02 by Anthony Conway, Diarmuid Crowley, Conway, Anthony +5
Mathematics · #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2109.00654

Abstract

For every k ≥ 2 and n ≥ 2 we construct n pairwise homotopically inequivalent simply-connected, closed 4k-dimensional manifolds, all of which are stably diffeomorphic to one another. Each of these manifolds has hyperbolic intersection form and is stably parallelisable. In dimension 4, we exhibit an analogous phenomenon for spin^c structures on S² × S². For m≥ 1, we also provide similar (4m-1)-connected 8m-dimensional examples, where the number of homotopy types in a stable diffeomorphism class is related to the order of the image of the stable J-homomorphism π_4m-1(SO) → π^s_4m-1.

Related

• Stable equivalence relations on 4-manifolds
• The stable mapping class group of simply connected 4-manifolds
• Extended surgery theory for simply-connected 4k-manifolds
• Four-manifolds, two-complexes and the quadratic bias invariant
• Stably diffeomorphic manifolds and the realisation of modified surgery obstructions