Some curious involutions of spheres

Abstract

SOME CURIOUS INVOLUTIONS OF SPHERES BY MORRIS W. HIRSCH AND JOHN MILNOR Communicated by Deane Montgomery, December 31, 1963 Consider an involution T of the sphere S n without fixed points. Is the quotient manifold S n /T necessarily isomorphic to projective n- space? This question makes sense in three different categories. One can work either with topological manifolds and maps, with piecewise linear manifolds and maps, or with differentiable manifolds and maps. For n^3 the statement is known to be true (Livesay [6]). In these cases it does not matter which category one works with. On the other hand, for n = 7> in the differentiable case, the statement is known to be false (Milnor [lO]). This note will show that, in the piecewise linear case, the statement is false for all n*z5. Furthermore, for w = 5, 6, we will construct a differentiable involution T: S n —>S n so that the quotient manifold is not even piecewise linearly homeomorphic to projective space. Our proofs depend on a recent theorem of J. Cerf. Let us start with the exotic 7-sphere M as described by Milnor [7]. This differentiable manifold MI is defined as the total space of a certain 3-sphere bundle over the 4-sphere. It is known to be homeo- morphic, but not diffeomorphic, to the standard 7-sphere. Taking the antipodal map on each fibre we obtain a differentiable involution T: Ml—>Ml without fixed points. (The quotient manifold M\/T can be considered as the total space of a corresponding projec- tive 3-space bundle over S 4 .) The following lemma was pointed out to us, in part, by P. Conner and D. Montgomery. LEMMA 1. There exists a differentiably imbedded 6-sphere, S$C.Ml, which is invariant under the action of T, and a differentiably imbedded 3JIC»So which is also invariant. Thus in this way one constructs a differentiable involution of the standard sphere in dimensions 5, 6. The proof will depend on the explicit description of Ml (or more generally of Ml) which was given in [7]. Take two copies of R A XS Z and identify the subsets (R*— (0)) XS Z under the diffeomorphism (u, v) -> («', v') = (u/\\u\ u*vu?/\\u\\), using quaternion multiplication, where h-\-j= 1, h—j = k. The involu- tion T changes the sign of v and v'. Let S% be the set of all points of Ml such that ( 0ft(v') = dl(uv)=O 1 where dt(uv) *=$t(vu) denotes the real