Smooth Involutions

Abstract

One would like to classify smooth involutions of homotopy spheres in the way we did it in the previous chapter for p.l. involutions, But one cannot compute the set of normal invariants [Pⁿ, G/0], not even in terms of the (mostly unknown) homotopy groups of G/0. The most we can do is get an estimate as follows: let a _i = order of A _i = π _i (G/0), a ⁽²⁾ _i = order of A _i ⊗ℤ₂. Then an inductive application of Lemma 1, IV.1 for X = G/0 gives

$$ {\text{order}}[{P^{2k}},G/0]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \prod\limits_{i = 1}^{2k} {a_i^{(2)}} $$

$$ {\text{order}}[{P^{2k + 1}},G/0]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \prod\limits_{i = 1}^{2k} {a_i^{(2)}} \cdot {a_{2k + 1}} $$

and equality would hold if, and only if, G/0 satisfied condition (!) of IV.1. But an example given in V.5 in dimension 9 shows that G/0 does not satisfy condition (!). So this upper bound can be divided by 2 for n ≧ 10.