We say that an involution (T, ∑ n ) of a homotopy sphere ∑ n desuspends if there is an embedded (smoothly or p.1., depending on the category we’re working on) homotopy sphere S n −1 ⊂ ∑ n which is invariant under T. We also say that (T, ∑ n ) is a suspension of (T|Sn −1, Sn −1 ) and that (T|S n −1, S n−1 ) is a desuspension of (T, ∑ n ). See also IV.3.2 and V.1.