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 functor from the category of real finite-dimensional vector spaces with inner product to pointed spaces which to a vector space V assigns the block structure space of the projective space of V . We study this functor from the point of view of orthogonal calculus of functors, we prove that the 6-fold delooping of the first derivative spectrum of this functor is an Ω-spectrum. The proof uses mainly codimension 1 surgery theory. This result suggests an attractive description of the block structure space of the infinite-dimensional real projective space which is a colimit of block structure spaces of projective spaces of finitedimensional real vector spaces. The description is via the Taylor tower of orthogonal calculus.