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Gysin Sequences in Supergeometry

Primarily human-written textHuman understanding: some partsmath.DG — Differential Geometry

Contributed by Sebastian Denton ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We study spectral sequences in supergeometry and construct the Gysin sequence within the supermanifold framework. First, the foundational definitions are established and utilized to develop differential forms and de Rham cohomology on supermanifolds. We then apply a standard de Rham-to-Cech cohomology isomorphism to assemble a spectral sequence that accounts for the ˇ Z2-graded nature of supermanifolds. Finally, we explore additional properties arising from this construction, leading directly into Gysin sequences.

Author comments: Comments and suggestions are welcome. Please direct any correspondence to the author.

Provenance statement

Following correspondence with researchers in supergeometry, I incorporated several suggested improvements to the manuscript and addressed potential errors. While the researchers offered differing perspectives on the construction of the Gysin sequence, I have carefully reviewed and revised the work accordingly.

References

  1. Raoul Bott and Loring W. Tu (1982), Differential Forms in Algebraic Topology , Springer. Available at: https://link.springer.com/book/10.1007/978-1-4757-3951-0DOI
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  8. Elizaveta Vishnyakova, (2011) A Classification Theorem and a Spectral Sequence for a Locally Free Sheaf Cohomology of a Supermanifold , arXiv preprint. Available at: https://arxiv.org/pdf/1110.3912link

Version history

  1. v1 · 8 pagesSubmitted by Sebastian DentonComments and suggestions are welcome. Please direct any correspondence to the author.CurrentNov 10, 2026