Borsuk-Ulam theorems for elementary abelian 2-groups
DOI:
https://doi.org/10.12775/TMNA.2022.042Keywords
Borsuk-Ulam theorem, Bourgin-Yang theorem, equivariant mapping, Euler classAbstract
Let $G$ be a fct Lie group and let $U$ and $V$ be finite-dimensional real $G$-modules with $V^G=0$. A theorem of Marzantowicz, de Mattos and dos Santos estimates the covering dimension of the zero-set of a $G$-map from the unit sphere in $U$ to $V$ when $G$ is an elementary abelian $p$-group for some prime $p$ or a torus. In this note, the classical Borsuk-Ulam theorem will be used to give a refinement of their result estimating the dimension of that part of the zero-set on which an elementary abelian $p$-group $G$ acts freely or a torus $G$ acts with finite isotropy groups. The methods also provide an easy answer to a question raised in \cite{DM}.References
I. Axelrod-Freed and P. Soberón, Bisections of mass assignments using flags of affine spaces (2021), arXiv math.CO: 2109.13106.
P.V.M. Blagojević, A.S. Dimitrijević Blagojević and G.M. Ziegler, Polynomial partitioning for several sets of varieties, J. Fixed Point Theory Appl. 19 (2017), 1653–1660.
P.V.M. Blagojević and R. Karasev, Extensions of theorems of Rattray and Makeev, Topol. Methods Nonlinear Anal. 40 (2012), 189–213.
Z. Blaszczyk, W. Marzantowicz and M. Singh, Equivariant maps between representation spheres, Bull. Belg. Math. Soc. Simon Stevin 24 (2017), 621–630.
Z. Blaszczyk, W. Marzantowicz and M. Singh, General Bourgin-Yang theorems, Top. Appl. 249 (2018), 112–126.
Y.H. Chan, S. Chen, F. Frick and J.T. Hull, An optimal Borsuk–Ulam theorem for products of spheres and Stiefel manifolds, Topol. Methods Nonlinear Anal. 55 (2020), 553–564.
M.C. Crabb, Connective K-theory and the Borsuk–Ulam theorem, Algebraic Topology and Related Topics (M. Singh et al., eds.), Trends in Mathematics, Birkhäuser–Springer, 2019, pp. 51–66.
M.C. Crabb and J. Jaworowski, Aspects of the Borsuk–Ulam theorem, J. Fixed Point Theory Appl. 13 (2013), 459–488.
M.C. Crabb and M. Singh, Some remarks on the parametrized Borsuk–Ulam theorem, J. Fixed Point Theory Appl. 20 (2018), article 79.
Z. Dzedzej, A. Idzik and M. Izydorek, Borsuk–Ulam type theorems on product spaces II, Topol. Methods Nonlinear Anal. 14 (1999), 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), 73–85.
E.R. Fadell, S.Y. Husseini and P.H. Rabinowitz, Borsuk–Ulam theorems for arbitrary S 1 actions and applications, Trans. Amer. Math. Soc. 274 (1982), 345–360.
L. Guth, Polynomial partitioning for a set of varieties, Math. Proc. Camb. Philos. Soc. 159 (2015), 459–469.
J. Jaworowski, A continuous version of the Borsuk–Ulam theorem, Proc. Amer. Math. Soc. 82 (1981), 112–114.
W. Marzantowicz, D. de Mattos and E.L. dos Santos, Bourgin–Yang version of the Borsuk–Ulam theorem for Zpk -equivariant maps, Algebr. Geom. Topology 12 (2012), 2245–2258.
W. Marzantowicz, D. de Mattos and E.L. dos Santos, Bourgin–Yang versions of the Borsuk–Ulam theorem for p-toral groups, J. Fixed Point Theory Appl. 19 (2017), 1427–1437.
D. Miklaszewski, Borsuk–Ulam theorem for the loop space of a sphere, Top. Appl. 250 (2018), 74–79.
H.J. Munkholm, On the Borsuk–Ulam theorem for Zpa -actions on S 2n−1 and maps S 2n−1 → Rm , Osaka J. Math. 7 (1970), 451–456.
I. Nagaski, Determination of compact Lie groups with the Borsuk–Ulam property (2021), arXiv math.AT: 2107.13158.
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