Let G = ℤpk be a cyclic group of prime power order and let V and W be orthogonal representations of G with V G = WG = {0}. Let S(V ) be the sphere of V and suppose f : S(V ) → W is a G–equivariant mapping. We give an estimate for the dimension of the set f−1{0} in terms of V and W. This extends the Bourgin–Yang version of the Borsuk–Ulam theorem to this class of groups. Using this estimate, we also estimate the size of the G–coincidences set of a continuous map from S(V ) into a real vector space W′.
Keywords: equivariant maps, covering dimension, orthogonal representation, equivariant $K$–theory
Marzantowicz, Wacław 1 ; de Mattos, Denise 2 ; dos Santos, Edivaldo 3
@article{10_2140_agt_2012_12_2245,
author = {Marzantowicz, Wac{\l}aw and de Mattos, Denise and dos Santos, Edivaldo},
title = {Bourgin{\textendash}Yang version of the {Borsuk{\textendash}Ulam} theorem for {\ensuremath{\mathbb{Z}}pk{\textendash}equivariant} maps},
journal = {Algebraic and Geometric Topology},
pages = {2245--2258},
year = {2012},
volume = {12},
number = {4},
doi = {10.2140/agt.2012.12.2245},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2012.12.2245/}
}
TY - JOUR AU - Marzantowicz, Wacław AU - de Mattos, Denise AU - dos Santos, Edivaldo TI - Bourgin–Yang version of the Borsuk–Ulam theorem for ℤpk–equivariant maps JO - Algebraic and Geometric Topology PY - 2012 SP - 2245 EP - 2258 VL - 12 IS - 4 UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2012.12.2245/ DO - 10.2140/agt.2012.12.2245 ID - 10_2140_agt_2012_12_2245 ER -
%0 Journal Article %A Marzantowicz, Wacław %A de Mattos, Denise %A dos Santos, Edivaldo %T Bourgin–Yang version of the Borsuk–Ulam theorem for ℤpk–equivariant maps %J Algebraic and Geometric Topology %D 2012 %P 2245-2258 %V 12 %N 4 %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2012.12.2245/ %R 10.2140/agt.2012.12.2245 %F 10_2140_agt_2012_12_2245
Marzantowicz, Wacław; de Mattos, Denise; dos Santos, Edivaldo. Bourgin–Yang version of the Borsuk–Ulam theorem for ℤpk–equivariant maps. Algebraic and Geometric Topology, Tome 12 (2012) no. 4, pp. 2245-2258. doi: 10.2140/agt.2012.12.2245
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