通过拓扑模形式与高阶实K理论研究酉群同伦中的高秩挠
High-corank torsion in homotopy of unitary groups via topological modular forms and higher real $K$-theories
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中文总结 AI 辅助
该研究利用Weiss的酉微积分将向量丛问题转化为稳定同伦计算，借助色与等变同伦理论中的广义上同调理论，推导出球面与复射影空间上非平凡稳定平凡向量丛的存在性。
中文摘要 AI 辅助
户田的工作将偶维球面上的亚稳向量丛群与某些截断射影谱的稳定同伦群对应起来。第二名作者利用Weiss的酉微积分，将该对应推广为：偶维胞腔复形上的亚稳、稳定平凡向量丛可自然对应到到移位截断射影谱的映射的稳定同伦类。因此，关于向量丛（或酉群同伦）的某些经典问题可重述为稳定同伦计算。本文中，我们证明，色同伦理论与等变同伦理论中产生的某些广义上同调理论，可用于推导出球面与复射影空间上非平凡稳定平凡向量丛的存在性。
英文摘要
Work of Toda identifies groups of metastable vector bundles on even-dimensional spheres with stable homotopy groups of certain stunted projective spectra. Using Weiss' unitary calculus, the second-named author generalized this identification to show that metastable, stably trivial vector bundles on even cell complexes can naturally be identified with stable homotopy classes of maps into a shifted stunted projective spectrum. Thus, certain classical questions about vector bundles (or homotopy of unitary groups) can be rephrased as stable computations. In this note, we show that certain generalized cohomology theories arising in chromatic and equivariant homotopy theory can be used to deduce the existence of non-trivial, stably trivial vector bundles on spheres and complex projective spaces.
