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

Yu Hin Chan, Shujian Chen, Florian Frick, J. Tristan Hull

DOI: http://dx.doi.org/10.12775/TMNA.2019.103

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.

Keywords


Borsuk-Ulam theorem; Stiefel manifold; equivariant map

Full Text:

PREVIEW FULL TEXT

References


P.V.M. Blagojević, F. Frick, A. Haase and G.M. Ziegler, Hyperplane mass partitions via relative equivariant obstruction theory, Doc. Math. 21 (2016), 735–771.

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.

P.V.M. Blagojević, F. Frick and G.M. Ziegler, Tverberg plus constraints, Bull. Lond. Math. Soc. 46 (2014), no. 5, 953–967.

P.V.M. Blagojević and R. Karasev, Extensions of theorems of Rattray and Makeev, Topol. Methods Nonlinear Anal. 40 (2012), no. 1, 189–213.

P.V.M. Blagojević and G.M. Ziegler, The ideal-valued index for a dihedral group action, and mass partition by two hyperplanes, Topology Appl. 158 (2011), no. 12, 1326–1351.

P.V. M. Blagojević and G.M. Ziegler, Beyond the Borsuk–Ulam theorem: The topological Tverberg story, A Journey Through Discrete Mathematics, Springer, 2017, pp. 273–341.

Z. Dzedzej, A. Idzik and M. Izydorek, Borsuk–Ulam type theorems on product spaces II, Topol. Methods Nonlinear Anal. 14 (1999), no. 2, 345–352.

E. Fadell and S. Husseini, An ideal-valued cohomological index theory with applications to Borsuk–Ulam and Bourgin–Yang theorems, Ergodic Theory Dynam. Systems 8 (1988), no. 8*, 73–85.

A. Inoue, Borsuk–Ulam type theorems on Stiefel manifolds, Osaka J. Math. 43 (2006), no. 1, 183–191.

A.M. Kushkuley and Z.I. Balanov, Geometric Methods in Degree Theory for Equivariant Maps, Springer, 2006.

P. Mani-Levitska, S.T. Vrećica and R.T. Živaljević, Topology and combinatorics of partitions of masses by hyperplanes, Adv. Math. 207 (2006), no. 1, 266–296.

J. Matoušek, Using the Borsuk–Ulam Theorem. Lectures on Topological Methods in Combinatorics and Geometry, second ed., Universitext, Springer–Verlag, Heidelberg, 2008.

J. Matoušek and G.M. Ziegler, Topological lower bounds for the chromatic number : A hierarchy, Jahresber. Dtsch. Math.-Ver. 106 (2004), 71–90.

B. Matschke, A survey on the square peg problem, Notices Amer. Math. Soc. 61 (2014), no. 4, 346–352.

E. Outerelo and J.M. Ruiz, Mapping Degree Theory, Graduate Studies in Mathematics, vol. 108, Amer. Math. Soc., 2009.

E.A. Ramos, Equipartition of mass distributions by hyperplanes, Discrete Comput. Geom. 15 (1996), no. 2, 147–167.

S. Simon, Hyperplane equipartitions plus constraints, J. Combin. Theory, Ser. A 161 (2019), 29–50.

T. tom Dieck, Transformation Groups, vol. 8, Walter de Gruyter, 2011.

S.T. Vrećica and R.T. Živaljević, Hyperplane mass equipartition problem and the shielding functions of Ramos (2015), arXiv: 1508.01552.

R.T. Živaljević, Topological Methods, Handbook of Discrete and Computational Geometry (J. O’Rourke, J.E. Goodman and C. Toth, eds.), Chapman & Hall/CRC, 2017.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism