发表机构
University of Buea（布埃亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究建立了保体积微分同胚的伊斯马吉洛夫定理的$C^0$类似物，证明了保体积同胚群单位分支的一阶连续上同调与一阶德拉姆上同调同构，解决了保体积$C^0$通量猜想。
AI 中文摘要
我们建立了关于保体积微分同胚的一阶连续上同调的伊斯马吉洛夫定理的$C^0$类似物。对于一个闭定向流形，我们证明：保体积同胚群的单位分支，在以连续零均值函数构成的巴拿赫空间为系数时，其一阶连续上同调同构于一阶德拉姆上同调。该证明引入了针对保体积同痕的拓扑输运理论，得到了一个连续体积通量同态及相关的输运上闭链。作为应用，我们解决了保体积$C^0$通量猜想。
英文摘要
We establish a topological analogue of Ismagilov's theorem concerning the first continuous cohomology of volume-preserving diffeomorphisms. For a closed oriented manifold \(M\) with volume form \(Ω\), we consider the group \(\mathbb{G}^Ω(M)\) of homeomorphisms obtained as uniform limits of smooth volume-preserving isotopies. We show that its first continuous cohomology with values in the Banach space of zero-mean continuous functions is isomorphic to the first de Rham cohomology of \(M\). The proof develops a theory of transport for volume-preserving isotopies, producing a topological volume flux homomorphism and its associated transport cocycle. Under the additional hypothesis that \((M,Ω)\) satisfies the local \(C^0\)-generation property, and assuming the Müller-Sikorav approximation theorem (known for \(n\neq 4\)), the isomorphism extends to the full identity component \(\operatorname{Homeo}_0^Ω(M)\) of the group of volume-preserving homeomorphisms, and the topological flux conjecture follows in that setting.
