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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 [Macko, Trans. Amer. Math. Soc. 359: 349–383, 2007], we investigate  F  as a functor on vector spaces with inner product, relying on functor calculus ideas. It was shown in [Macko, Trans. Amer. Math. Soc. 359: 349–383, 2007] 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 ) ≥ 6. To convert this theorem into a functorial homotopy-theoretic 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  L -theory space associated with the group ℤ/2, except for a deviation in π 0 . The main corollary is a functorial two-stage decomposition of  F ( V ) for dim( V ) ≥ 6 which has the  L -theory of the group ℤ/2 as one layer, and a form of unreduced homology of ℝ P ( V ) with coefficients in the  L -theory of the trivial group as the other layer. Except for dimension shifts, these are also the layers in the traditional Sullivan-Wall-Quinn-Ranicki 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 ∞ .

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