The block structure spaces of real projective spaces and orthogonal calculus of functors

Given a compact manifold $X$, the set of simple manifold structures on $X \times \Delta^k$ relative to the boundary can be viewed as the $k$-th homotopy group of a space $\widetilde{\mathcal{S}}^s (X)$. 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 $\leq 1$ 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 the block structure spaces of projective spaces of finite-dimensional real vector spaces and is closely related to some automorphisms spaces of real projective spaces.