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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

$ (n-1)$-axial $ {\rm SO}(n)$ and $ {\rm SU}(n)$ actions on homotopy spheres


Author: R. D. Ball
Journal: Trans. Amer. Math. Soc. 292 (1985), 51-79
MSC: Primary 57S15
DOI: https://doi.org/10.1090/S0002-9947-1985-0805953-1
MathSciNet review: 805953
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Abstract: Let $ G(n) = O(n)$ or $ U(n)$ and $ SG(n) = SO(n)$ or $ SU(n)$. For each integer $ m \geqslant 1$ a family $ \{ {S_{\gamma ,\sigma }}:\gamma \in H,\sigma \in K\} $ of $ (n - 1)$-axial $ SG(n)$ homotopy spheres $ {S_{\gamma ,\sigma }}$ is constructed. Each $ {S_{\gamma ,\sigma }}$ has fixed point set of dimension $ (m - 1) \geqslant 0$ and orbit space of dimension $ r = \tfrac{1} {2}n(n - 1) + (m - 1)$ (resp. $ r = {(n - 1)^2} + m - 1$) if $ SG(n) = SO(n)$ (resp. $ SU(n)$). $ H$ is $ {\pi _{r - 1}}(SG(n)/G(n - 1))$. $ K$ is trivial if $ SG(n) = SO(n)$ and is a homotopy theoretically defined subgroup of sections of an $ {S^2}$ bundle depending only on $ m$ and $ n$ if $ SG(n) = SU(n)$. Assume that $ m$ and $ n$ satisfy the mild restriction $ \S5$, (1). It is shown that the above family is universal for $ (n - 1)$-axial $ SG(n)$ homotopy spheres and provides a classification analogous to the classification of fibre bundles: for each $ (n - 1)$-axial $ SG(n)$ homotopy sphere $ \Sigma $ there is a $ {S_{\gamma ,\sigma }}$ and a unique equivariant stratified map $ \Sigma \to {S_{\gamma ,\sigma }}$. $ \Sigma $ is equivariantly diffeomorphic to the pullback of $ {S_{\gamma ,\sigma }}$ via the map $ B(\Sigma ) \to B({S_{\gamma ,\sigma }})$ of orbit spaces. If $ SG(n) = SO(n)$ then $ \gamma $ is unique (and $ \sigma = 1$). If $ SG(n) = SU(n)$ then $ \gamma $ is unique modulo the image of

$\displaystyle {\pi _{r - 1}}S(U(n - 2) \times U(2))/U(k - 1) \times U(1)\quad {\text{in}}\;H.$

An example is given showing that the differentiable structure of the underlying smooth manifold of $ {S_{\gamma ,\sigma }}$ may be exotic.

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DOI: https://doi.org/10.1090/S0002-9947-1985-0805953-1
Article copyright: © Copyright 1985 American Mathematical Society