Stably diffeomorphic manifolds and modified surgery obstructions

Conway, Anthony; Crowley, Diarmuid; Powell, Mark D.; Sixt, Joerg

doi:10.48550/arxiv.2109.05632

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2022

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“…However, as far as we know, our construction gives the first simply connected examples and the first for which the homotopy stable class has been shown to have arbitrary cardinality. In a companion paper [CCPS21], we investigate the homotopy stable class in more detail, also for manifolds with nontrivial fundamental group, and we relate the homotopy stable class to computations of the -monoid from [CS11].…”

Section: Introductionmentioning

confidence: 99%

Simply Connected Manifolds With Large Homotopy Stable Classes

Conway¹,

Crowley²,

Powell

et al. 2022

J. Aust. Math. Soc.

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For every $k \geq 2$ and $n \geq 2$ , 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 four, we exhibit an analogous phenomenon for spin $^{c}$ structures on $S^2 \times S^2$ . For $m\geq 1$ , 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 $\pi _{4m-1}(SO) \to \pi ^s_{4m-1}$ .

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Section: Introductionmentioning

confidence: 99%