映射的次数与部分旗流形的上同调刚性
Degrees of Maps and Cohomological Rigidity of Partial Flag Manifolds
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中文总结 AI 辅助
该研究证明不同复或四元数部分旗流形间若上域非Grassmann流形则连续映射Brouwer次数为零，推出这类流形具上同调刚性，并猜想其具同调刚性。
中文摘要 AI 辅助
我们研究部分旗流形之间非零Brouwer次数连续映射的存在性。证明：两个不同复（或四元数）部分旗流形之间，若上域不是Grassmann流形，则所有连续映射的Brouwer次数均为零。这为这类部分旗流形建立了与Ramani-Sankaran、Sankaran-Sarkar针对复和四元数Grassmann流形所得结果，以及Paranjape和Srinivas针对复Grassmann流形的代数几何结果的类似结论。由此证明复和四元数部分旗流形是上同调刚性的，即它们的有理上同调环决定了自身的同胚类型。我们猜想复和四元数部分旗流形是同调刚性的。
英文摘要
We study the existence of continuous maps with nonzero Brouwer degree between partial flag manifolds. We prove that every continuous map between distinct complex or quaternionic partial flag manifolds has degree zero if either the domain or the codomain is not a Grassmannian. This establishes, for such partial flag manifolds, the analogue of the results of Ramani-Sankaran and Sankaran-Sarkar for complex and quaternionic Grassmannians, as well as an algebraic-geometric result of Paranjape and Srinivas for complex Grassmannians. As a consequence, we prove that complex and quaternionic partial flag manifolds are cohomologically rigid; that is, their rational cohomology rings determine their homeomorphism types. We conjecture that complex and quaternionic partial flag manifolds are homologically rigid.
