Relations among smooth Kervaire classes and smooth involutions on homotopy spheres

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...

Smooth biproximity spaces and P-smooth quasi-proximity spaces

The notion of smooth biproximity space  where $delta_1,delta_2$ are gradation proximities defined by Ghanim et al. [10]. In this paper, we show every smooth biproximity space $(X,delta_1,delta_2)$ induces a supra smooth proximity space $delta_{12}$ finer than $delta_1$ and $delta_2$. We study the relationship between $(X,delta_{12})$ and the $FP^*$-separation axioms which had been introduced by...

Finite Group Actions on Kervaire Manifolds

Let M K be the Kervaire manifold: a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product S × S of spheres. We show that a finite group of odd order acts freely on M K if and only if it acts freely on S × S. If MK is smoothable, then each smooth structure on MK admits a free smooth involution. If k 6= 2 − 1, then M K does not admit any free TOP in...

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...

Fixed Point Sets of Involutions on Spheres