Spin(7) 和 Spin(8) 列群同调类的几何表示

arXiv - MATH - Algebraic Topology Pub Date : 2024-09-10 DOI:arxiv-2409.06491

Eiolf Kaspersen, Gereon Quick

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引用次数: 0

摘要

通过构建具体的面向复数的映射，我们证明了自旋群 Spin(7) 和 Spin(8) 的第三积分同调的生成器的八叠层在从复数共线性到奇异同调的 Thom 形态的映像中，而生成器本身不在映像中。因此，我们给出了霍普金斯（Hopkins）和辛格（Singer）针对 Lie 群 Spin(7) 和 Spin(8) 的微分托姆态内核中一个非小类的年龄计量构造。该构造利用了八元数的特殊对称性。

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Geometric representation of cohomology classes for the Lie groups Spin(7) and Spin(8)

By constructing concrete complex-oriented maps we show that the eight-fold of the generator of the third integral cohomology of the spin groups Spin(7) and Spin(8) is in the image of the Thom morphism from complex cobordism to singular cohomology, while the generator itself is not in the image. We thereby give a geometric construction for a nontrivial class in the kernel of the differential Thom morphism of Hopkins and Singer for the Lie groups Spin(7) and Spin(8). The construction exploits the special symmetries of the octonions.

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arXiv - MATH - Algebraic Topology

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