Free piecewise linear involutions on spheres

Free Piecewise Linear Involutions on Spheres

If T is a piecewise linear fixed-point free involution on S, the orbit space Q = S/T is a PL-manifold homotopy equivalent to P»(JR) ~ P n [2J; the affirmative solution to the Poincaré conjecture implies that conversely for n&Z, 4 the double covering manifold of any such Q can be identified with S. Write In for the set of (oriented if n is even) PL-homeomorphism classes of manifolds Q homotopy e...

Free Involutions on Homotopy (4&+3)-spheres

In [ l ] Browder and Livesay defined an invariant a(Tt 2 ) £ 8 Z of a free differentiate involution T of a homotopy (4&+3)-sphere 2 , k>0. I t is the precise obstruction to finding an invariant (4&+2)sphere of the involution. In [5] and [ô] Medrano showed how to construct free involutions with arbitrary Browder-Livesay invariant on some homotopy (4&+3) -spheres and hence that there exist infini...

The Join Construction for Free Involutions on Spheres

Given a manifold X , the set of manifold structures on X × ∆ relative to the boundary can be viewed as the k-th homotopy group of a space S̃s(X). This space is called the block structure space of X . Free involutions on spheres are in one-to-one correspondence with manifold structures on real projective spaces. We generalize Wall’s join construction for free involutions on spheres to define a fu...

Fixed Point Sets of Involutions on Spheres

Some Curious Involutions of Spheres

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 t...