Homotopy linear involutions on spheres in the unstable range (Q1085848)

This paper considers the classification of smooth involutions that are topologically equivalent to linear involutions on spheres. If the dimension of the fixed point set is less than half that of the ambient manifold, then established techniques from surgery theory yield a reasonable classification. The author shows that the general case can be handled by a combination of these methods and a homotopy theoretic desuspension obstruction (which vanishes if the dimension of the fixed point set is small). This observation and some additional machinery are applied to obtain results on realizing exotic spheres as fixed point sets and to study some related geometric questions.