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SIMPLY CONNECTED MANIFOLDS WITH LARGE HOMOTOPY STABLE CLASSES
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 115 / Issue 2 / October 2023
- Published online by Cambridge University Press:
- 26 September 2022, pp. 172-203
- Print publication:
- October 2023
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- Article
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- You have access
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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 
$\pi _{4m-1}(SO) \to \pi ^s_{4m-1}$.
