Some Curious Involutions of Spheres
نویسنده
چکیده
Consider an involution T of the sphere S without fixed points. Is the quotient manifold S/T necessarily isomorphic to projective nspace? 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—>S 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 homeomorphic, 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 projective 3-space bundle over S.) The following lemma was pointed out to us, in part, by P. Conner and D. Montgomery.
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