Open AccessArtin relation for smooth representations
Author(s) -
Karl Heinz Dovermann,
Ted Petrie
Publication year - 1980
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.77.10.5620
Subject(s) - mathematics , dimension (graph theory) , combinatorics , prime (order theory) , group (periodic table) , homotopy , order (exchange) , function (biology) , finite group , pure mathematics , discrete mathematics , chemistry , organic chemistry , finance , evolutionary biology , economics , biology
LetG be a finite group. IfG acts smoothly on a closed homotopy sphereS , we callS a smooth representation ofG . The main result is: There is a functionhG such that for every smooth representationS ofG , dimensionSG =hG {dimensionSH ǀH proper subgroup ofG } if and only ifG has prime power order andG is not cyclic. In other words, only for a noncyclicp -groupG is dimensionSG a universal function of the dimensions of the fixed setsSH asH ranges over proper subgroups ofG . This result is compared with an old theorem of Artin's dealing with dimensions of fixed sets of orthogonal representations ofG .