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Topological Methods in Nonlinear Analysis

Borsuk-Ulam theorems for products of spheres and Stiefel manifolds revisited
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Borsuk-Ulam theorems for products of spheres and Stiefel manifolds revisited

Authors

  • Yu Hin Chan
  • Shujian Chen
  • Florian Frick https://orcid.org/0000-0002-7635-744X
  • J. Tristan Hull

Keywords

Borsuk-Ulam theorem, Stiefel manifold, equivariant map

Abstract

We give a different and possibly more accessible proof of a general Borsuk-Ulam theorem for a product of spheres, originally due to Ramos. That is, we show the non-existence of certain $(\mathbb Z/2)^k$-equivariant maps from a product of $k$ spheres to the unit sphere in a real $(\mathbb Z/2)^k$-representation of the same dimension. Our proof method allows us to derive Borsuk-Ulam theorems for certain equivariant maps from Stiefel manifolds, from the corresponding results about products of spheres, leading to alternative proofs and extensions of some results of Fadell and Husseini.

References

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P.V.M. Blagojević, F. Frick, A. Haase and G.M. Ziegler, Topology of the Grünbaum–Hadwiger–Ramos hyperplane mass partition problem, Trans. Amer. Math. Soc. 370 (2018), no. 10, 6795–6824.

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Published

2020-05-30

How to Cite

1.
CHAN, Yu Hin, CHEN, Shujian, FRICK, Florian and HULL, J. Tristan. Borsuk-Ulam theorems for products of spheres and Stiefel manifolds revisited. Topological Methods in Nonlinear Analysis. Online. 30 May 2020. Vol. 55, no. 2, pp. 553 - 564. [Accessed 4 July 2025].
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