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PreprintCC BY 4.0arXiv

C0-Analogue of Ismagilov's Theorem

  • Stéphane Tchuiaga
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Abstract

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 𝔾Ω(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 C0-generation property, and assuming the Müller-Sikorav approximation theorem (known for n≠ 4), the isomorphism extends to the full identity component Homeo0Ω(M) of the group of volume-preserving homeomorphisms, and the topological flux conjecture follows in that setting.

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Cite

BibTeXRIS
BibTeX
@misc{tchuiaga2026c0analogue,
  title = {{$C^0$-Analogue of Ismagilov's Theorem}},
  author = {Stéphane Tchuiaga},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.07802},
  eprint = {2608.07802},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.07802},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - C^0-Analogue of Ismagilov's Theorem
AU  - Stéphane Tchuiaga
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2608.07802
UR  - https://arxiv.org/abs/2608.07802
AB  - 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 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≠ 4), the isomorphism extends to the full identity component Homeo_0^Ω(M) of the group of volume-preserving homeomorphisms, and the topological flux conjecture follows in that setting.
ER  -