流形的连通和与直积的迷向指数
Isotropy index for the connected sum and the direct product of manifolds
摘要：
如果双线性映射对一个子空间或子群的约束是平凡的，则该子空间或子群在双线性映射下是迷向的。本文研究了反对称映射下的极大迷向子空间或极大迷向子空间的子群，特别是研究了迷向子空间的最大维数或极大迷向子空间的最大秩--迷向指数.对光滑闭可定向流形$M$，刻画了第一上同调群在上积下不同系数迷向子群的几何意义.我们计算了流形的连通和与直积的各向同性指数，以及极大各向同性子群的秩集。最后，我们研究了各向同性指数与基本群的第一贝蒂数和余秩的关系。我们还讨论了应用这些结果的拓扑结构的叶理。
英文摘要：
A subspace or subgroup is isotropic under a bilinear map if the restriction of the map on it is trivial. We study maximal isotropic subspaces or subgroups under skew-symmetric maps, and in particular the isotropy index---the maximum dimension of an isotropic subspace or maximum rank of an isotropic subgroup. For a smooth closed orientable manifold $M$, we describe the geometric meaning of the isotropic subgroups of the first cohomology group with different coefficients under the cup product. We calculate the corresponding isotropy index, as well as the set of ranks of all maximal isotropic subgroups, for the connected sum and the direct product of manifolds. Finally, we study the relationship of the isotropy index with the first Betti number and the co-rank of the fundamental group. We also discuss applications of these results to the topology of foliations.
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