  Given a compact manifold X, the set of simple manifold structures on X x
\Delta^k relative to the boundary can be viewed as the k-th homotopy group of a
space \S^s (X). This space is called the block structure space of X.
  We study the block structure spaces of real projective spaces. Generalizing
Wall's join construction we show that there is a functor from the category of
finite dimensional real vector spaces with inner product to the category of
pointed spaces which sends the vector space V to 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 show that it is polynomial of degree <= 1
in the sense of orthogonal calculus.
  This result suggests an attractive description of the block structure space
of the infinite dimensional real projective space via the Taylor tower of
orthogonal calculus. This space is defined as a colimit of block structure
spaces of projective spaces of finite-dimensional real vector spaces and is
closely related to some automorphisms spaces of real projective spaces.
