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

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 $\S 5$, (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 \[ {\pi _{r - 1}}S(U(n - 2) \times U(2))/U(k - 1) \times U(1)\quad {\text {in}}\;H.\] An example is given showing that the differentiable structure of the underlying smooth manifold of ${S_{\gamma ,\sigma }}$ may be exotic.