SIMPLY CONNECTED MANIFOLDS WITH LARGE HOMOTOPY STABLE CLASSES
SIMPLY CONNECTED MANIFOLDS WITH LARGE HOMOTOPY STABLE CLASSES
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DOI：
10.1017/s1446788722000167
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发表时间：
2021-09
影响因子：
0.7
通讯作者：
Anthony Conway;D. Crowley;Mark Powell;Joerg Sixt
中科院分区：
文献类型：
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作者：
Anthony Conway;D. Crowley;Mark Powell;Joerg Sixt
对任意的$k \geq 2$和$n \geq 2$，我们构造了n个两两同伦不等价的单连通闭$4k$维流形，它们都是彼此稳定同构的.这些流形中的每一个都具有双曲相交形式，并且是稳定可平行的。在四维空间中，我们展示了类似的现象，自旋$^{c}$结构在$S^2 \times S^2$上。对于$m\geq 1$，我们也提供了类似的$（4 m-1）$ -连通的$8m$ -维例子，其中一个稳定同态类中同伦类型的数量与稳定J-同态$\pi _{4 m-1}（SO）\to \pi ^s_{4 m-1}$的像的阶有关。
Abstract 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 $\pi _{4m-1}(SO) \to \pi ^s_{4m-1}$ .