Mathematical Journal of Okayama University 63巻 1号

2021-01 発行

Seita, Kohei
Department of Mathematics, Graduate School of Natural Science and Technology, Okayama University

Publication Date

2021-01

Abstract

Let G be a ﬁnite group and let V and W be real G-modules. We call V and W dim-equivalent if for each subgroup H of G, the H-ﬁxed point sets of V and W have the same dimension. We call V and W are Smith equivalent if there is a smooth G-action on a homotopy sphere Σ with exactly two G-ﬁxed points, say a and b, such that the tangential G-representations at a and b of Σ are respectively isomorphic to V and W . Moreover, We call V and W are d-Smith equivalent if they are dim-equivalent and Smith equivalent. The diﬀerences of d-Smith equivalent real G-modules make up a subset, called the d-Smith set, of the real representation ring RO(G). We call V and W P(G)-matched if they are isomorphic whenever the actions are restricted to subgroups with prime power order of G. Let N be a normal subgroup. For a subset F of G, we say that a real G-module is F-free if the H-ﬁxed point set of the G-module is trivial for all elements H of F. We study the d-Smith set by means of the submodule of RO(G) consisting of the diﬀerences of dim-equivalent, P(G)-matched, {N}-free real G-modules. In particular, we give a rank formula for the submodule in order to see how the d-Smith set is large.

Keywords

Real G-module

Smith equivalence

representation ring

Oliver group

Comments

Mathematics Subject Classiﬁcation. Primary 57S25, Secondary 20C15.