SMITH PROBLEM FOR A FINITE OLIVER GROUP WITH NON-TRIVIAL CENTER

摘要

The Smith problem is that two tangential representations are isomorphic or not for a smooth action on a homotopy sphere with exactly two fixed points. Two real G-modules U and V are called Smith equivalent if there exists a smooth action of G on a sphere I such that SG = {x,y} for two points x and y at which TX(S) s U and Ty(T) = V as real G-modules. We will consider a subset Sm(G) of the real representation ring RO{G) of G consisting of the differences U- V of real G-modules U and V which are Smith equivalent. We also define a subset CSm(G) of RO(G) consisting of the differences U - V e Sm(G) of real G-modules U and V such that for the sphere E appearing in the notion of Smith equivalence of U and V satisfies that Sp is connected for every P e P(G). Moreover, we assume that 0 e CSm(G) as definition.