Orientation reversing involutions on Brieskorn spheres

Relations among smooth Kervaire classes and smooth involutions on homotopy spheres

“…We have already shown(2) in Lemma 2.7. We have also shown(3) in case n=l{I)-2 in Lemma 2.7. So we assume that n^3.…”

“…If ι=2'~2 then we have t-2 s~2 -2 d and d must be equal to s-3 since if d<s-3, we have ί^2 s~3 +2 s " 4 contradicting to the allowability. When d = s-3, we have ί=2 s " 3 and A is nonzero. Thus in this case, Σ" contains w if and only if t-2 s~% .…”

“…We shall consider now two special cases. The first one is the involution (Wί n+1 , T 3 ) studied in [3]. According to Corollary C above, this involution is not Msovariantly homotopy equivalent to a linear involution L on the standard sphere S 4n+1 with Fix(L)=S 2n although it is equivariantly homotopy equivalent to the linear involution (S 4n+1 , L) and more than that it is even equivariantly normally cobordant to the linear involution (S iΐl+1 , L).…”

Surgery obstructions on closed manifolds and the inertia subgroup

“…Recall that W 4i−3 (d) denotes the Brieskorn variety given by intersecting the solution set of the equation Proof. These involutions were studied by Kitada [19], who gave necessary and sufficient conditions for (…”

“…Proof. These involutions were studied by Kitada [19], who gave necessary and sufficient conditions for (W 4i−3 (d), T d ) to be Z/2-equivariantly diffeomorphic to (W 4i−3 (d ′ ), T d ′ ). We need only the easy part of his argument, namely that (W 4i−3 (d), T d ) is Z/2-equivariantly normally cobordant to S 4i−3 (2i − 1) by a normal cobordism which is the identity on a neighbourhood of the fixed set.…”