
Mathematics 2007
The block structure spaces of real projective spaces and orthogonal calculus of functors IIAbstract: For a finite dimensional real vector space V with inner product, let F(V) be the block structure space, in the sense of surgery theory, of the projective space of V. Continuing a program launched in part I, we investigate F as a functor on vector spaces with inner product, relying on functor calculus ideas. It was shown in part I that F agrees with its first Taylor approximation T_1 F (which is a polynomial functor of degree 1) on vector spaces V with dim(V) > 5. To convert this theorem into a functorial homotopytheoretic description of F(V), one needs to know in addition what T_1 F(V) is when V=0. Here we show that T_1 F(0) is the standard Ltheory space associated with the group Z/2, except for a deviation in \pi_0. The main corollary is a functorial twostage decomposition of F(V) for dim(V) > 5 which has the Ltheory of the group Z/2 as one layer, and a form of unreduced homology of RP (V) with coefficients in the Ltheory of the trivial group as the other layer. (Except for dimension shifts, these are also the layers in the traditional SullivanWallQuinnRanicki decomposition of F(V). But the dimension shifts are serious and the SWQR decomposition of F(V) is not functorial in V.) Because of the functoriality, our analysis of F(V) remains meaningful and valid when V=R^\infty.
