(𝑛-1)-axial 𝑆𝑂(𝑛) and 𝑆𝑈(𝑛) actions on homotopy spheres

Abstract

Let G ( n ) = O ( n ) G(n) = O(n) or U ( n ) U(n) and S G ( n ) = S O ( n ) SG(n) = SO(n) or S U ( n ) SU(n) . For each integer m ⩾ 1 m \geqslant 1 a family { S γ , σ : γ ∈ H , σ ∈ K } \{ {S_{\gamma ,\sigma }}:\gamma \in H,\sigma \in K\} of ( n − 1 ) (n - 1) -axial S G ( n ) SG(n) homotopy spheres S γ , σ {S_{\gamma ,\sigma }} is constructed. Each S γ , σ {S_{\gamma ,\sigma }} has fixed point set of dimension ( m − 1 ) ⩾ 0 (m - 1) \geqslant 0 and orbit space of dimension r = 1 2 n ( n − 1 ) + ( m − 1 ) r = \tfrac {1} {2}n(n - 1) + (m - 1) (resp. r = ( n − 1 ) 2 + m − 1 r = {(n - 1)^2} + m - 1 ) if S G ( n ) = S O ( n ) SG(n) = SO(n) (resp. S U ( n ) SU(n) ). H H is π r − 1 ( S G ( n ) / G ( n − 1 ) ) {\pi _{r - 1}}(SG(n)/G(n - 1)) . K K is trivial if S G ( n ) = S O ( n ) SG(n) = SO(n) and is a homotopy theoretically defined subgroup of sections of an S 2 {S^2} bundle depending only on m m and n n if S G ( n ) = S U ( n ) SG(n) = SU(n) . Assume that m m and n n satisfy the mild restriction § 5 \S 5 , (1). It is shown that the above family is universal for ( n − 1 ) (n - 1) -axial S G ( n ) SG(n) homotopy spheres and provides a classification analogous to the classification of fibre bundles: for each ( n − 1 ) (n - 1) -axial S G ( n ) SG(n) homotopy sphere Σ \Sigma there is a S γ , σ {S_{\gamma ,\sigma }} and a unique equivariant stratified map Σ → S γ , σ \Sigma \to {S_{\gamma ,\sigma }} . Σ \Sigma is equivariantly diffeomorphic to the pullback of S γ , σ {S_{\gamma ,\sigma }} via the map B ( Σ ) → B ( S γ , σ ) B(\Sigma ) \to B({S_{\gamma ,\sigma }}) of orbit spaces. If S G ( n ) = S O ( n ) SG(n) = SO(n) then γ \gamma is unique (and σ = 1 \sigma = 1 ). If S G ( n ) = S U ( n ) SG(n) = SU(n) then γ \gamma is unique modulo the image of \[ π r − 1 S ( U ( n − 2 ) × U ( 2 ) ) / U ( k − 1 ) × U ( 1 ) in H . {\pi _{r - 1}}S(U(n - 2) \times U(2))/U(k - 1) \times U(1)\quad {\text {in}}\;H. \] An example is given showing that the differentiable structure of the underlying smooth manifold of S γ , σ {S_{\gamma ,\sigma }} may be exotic.