Session 31. Representation Theory, Transformation Groups, and Applications

Study of representation spaces from the viewpoint of algebraic/differential topology

Masaharu Morimoto, Okayama University, Japan
The theory of representation spaces of finite groups was developed not only from the viewpoint of algebra but also from the algebraic/differential topology perspective.

Let \(G\) be a finite group. The classification problem of real \(G\)-representation spaces has been considered so far up to \(G\)-diffeomorphism, \(G\)-homeomorphism, \(G\)-homotopy equivalence, or the Smith equivalence for smooth actions of \(G\) on spheres [1,2].

The Burnside ring \(\Omega (G)\) of \(G\) was interpreted in terms of proper continuous \(G\)-maps defined on real \(G\)-representation spaces ([1]) and the Dress induction theory was explicitly/implicitly applied to the classification problem above.

In this talk, we recall some classical results of the field, and we also discuss new results on the Smith equivalence of real \(G\)-representation spaces [3,4].

References
  1. T. tom Dieck; Transformation Groups, Walter de Gruyter, Berlin-New York, 1987.
  2. T. Pertie and J. D. Randall; Transformation Groups on Manifolds, Marcel Dekker, Inc., New York-Basel, 1984.
  3. M. Morimoto; Tangential representations of one-fixed-point actions on spheres and Smith equivalence, to appear in J. Math. Soc. Japan.
  4. M. Morimoto; A necessary condition for the Smith equivalence of G-modules and its suficiency, to appear in Math. Slovaca.
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