A parametrized version of the Borsuk–Ulam–Bourgin–Yang–Volovikov theorem

Matschke, Benjamin

doi:10.1142/s1793525314500101

Cited by 3 publications

(1 citation statement)

References 38 publications

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“…index for a finitistic space with p-torus actions is finite. In 2014, Benjamin Matschke [13] defined ideal valued index using spectral sequences. We generalize Conner and Floyd's [1] index and co-index and related standard results to index and co-index for finitistic space X with free actions of G = S d , d = 1 or 3.…”

Section: Introductionmentioning

confidence: 99%

Fixed point Free Actions of Spheres and Equivariant maps

Kumari¹,

Singh²

2021

Preprint

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This paper generalizes the concept of index and co-index and some related results for free actions of G = S 0 on a paracompact Hausdorff space which were introduced by Conner and Floyd[1]. We define the index and co-index of a finitistic free G-space X, where G = S d , d = 1 or 3 and prove that the index of X is not more than the mod 2 cohomology index of X. We observe that the index and co-index of a (2n + 1)sphere S 2n+1 (resp. (4n+3)-sphere S 4n+3 ) for the action of componentwise multiplication of G = S 1 (resp. S 3 ) is n.We also determine the orbit spaces of free actions of G = S 3 on a finitistic space X with the mod 2 cohomology and the rational cohomology product of spheres S n × S m , 1 ≤ n ≤ m. The orbit spaces of circle actions on the mod 2 cohomology X is also discussed. Using these calculation, we obtain an upper bound of the index of X and the Borsuk-Ulam type results.

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Section: Introductionmentioning

confidence: 99%

Fixed point Free Actions of Spheres and Equivariant maps

Kumari¹,

Singh²

2021

Preprint

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Fixed point free actions of spheres and equivariant maps

Kumari

Singh

2022

Topology and its Applications

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Successive spectral sequences

Matschke¹

2022

Trans. Amer. Math. Soc.

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In this paper, we develop a structure theory for generalized spectral sequences, which are derived from chain complexes that are filtered over arbitrary partially ordered sets. Also, a more general construction method reminiscent of exact couples is studied, together with examples where they arise naturally. As for ordinary spectral sequences we will see differentials and group extensions, however the real power comes from the appearance of natural isomorphisms between pages of differing indices. The constructions reveal finer invariants than ordinary spectral sequences, and they connect to other fields such as Fary functors and perverse sheaves. They are based on a natural index scheme, which allows us to obtain new results even in the standard case of Z \mathbb {Z} -filtered chain complexes, e.g. a useful criterion for a product structure for Grothendieck’s spectral sequences, and new paths to connect the first or second page to the limit. This turns out to yield the right framework for unifying several spectral sequences that one would usually apply one after another. Examples that we work out are successive Leray–Serre spectral sequences, the Adams–Novikov spectral sequence following the chromatic spectral sequence, successive Grothendieck spectral sequences, and successive Eilenberg–Moore spectral sequences.

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A parametrized version of the Borsuk–Ulam–Bourgin–Yang–Volovikov theorem

Cited by 3 publications

References 38 publications

Fixed point Free Actions of Spheres and Equivariant maps

Fixed point Free Actions of Spheres and Equivariant maps

Fixed point free actions of spheres and equivariant maps

Successive spectral sequences

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