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Session 31. Representation Theory, Transformation Groups, and Applications
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Note on tangential representations on spheres |
Toshio Sumi, Kyushu University, Japan
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Our targets are Smith sets for finite Oliver groups.
For any finite group \(G\), the Smith set \(\text{Sm}(G)\) consists
of the differences \([U]-[V]\in \text{RO}(G)\) such that for a smooth
homotopy \(G\)-sphere with just two fixed points, \(U\) and \(V\) are
the tangential representation spaces at the two fixed points.
In general, \(\text{Sm}(G)\) is not an additive subgroup of the real representation ring \(\text{RO}(G)\).
For \([U]-[V]\in\text{Sm}(G)\), \(U\) and \(V\) are isomorphic as \(G_{\{p\}}\)-modules for any Sylow \(p\)-subgroup \(G_{\{p\}}\),
for an odd prime \(p\). Morimoto has shown that the maximal additive subgroup of \(\text{Sm}(G)\) is a subset of \(\text{RO}(G)_{\mathcal{P}(G)}^{\mathcal{N}_2(G)}\).
Here \(\mathcal{P}(G)\) is the set of all subgroups of \(G\) of prime power order, \(\mathcal{N}_2(G)\) is the set of all normal subgroups of \(G\) with index \(1\) or \(2\), and
\[
\begin{array}{ll}
\text{RO}(G)_{\mathcal{P}(G)}^{\mathcal{N}_2(G)}=&
\displaystyle\bigcap_{P\in \mathcal{P}(G)} \text{ker}(\text{Res}^G_P\colon \text{RO}(G)\to \text{RO}(P)) \\
&\quad\cap \,
\displaystyle\bigcap_{N\in \mathcal{N}_2(G)} \text{ker}(\text{Fix}^N\colon \text{RO}(G)\to \text{RO}(G/N)).
\end{array}\]
So far, finite solvable Oliver groups possessing non-trivial Smith sets are not determined
completely [3]. However, we know the full answer to the question
for finite non-solvable groups [4].
In this talk, we give many examples of finite non-solvable groups whose Smith sets are non-trivial additive groups.
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References
- Masaharu Morimoto, Nontrivial P(G)-matched S-related pairs forfinite gap
Oliver groups, J. Math. Soc. Japan 62 (2010), no. 2, 623-647.
- Masaharu Morimoto and Yan Qi, The primary Smith sets of finite Oliver
groups, Group actions and homogeneous spaces, Fak. Mat. Fyziky Inform. Univ.
Komenskeho, Bratislava, 2010, pp. 61-73.
- Krzysztof Pawałowski and Toshio Sumi, The Laitinen Conjecture for finite solvable Oliver groups, Proc. Amer. Math. Soc. 137 (2009), no. 6, 2147-2156.
- Krzysztof Pawałowski and Toshio Sumi, The Laitinen Conjecture for finite nonsolvable groups,
Proc. Edinburgh Math. Soc. 56 (2013), issue 01, 303-336.
- Toshio Sumi, Richness of Smith equivalent modules for finite gap Oliver groups,
preprint.
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