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

Abstract

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 launchedin (Ma), weinvestigateF asafunctoronvector spaceswith innerproduct,relyingonfunctor calculus ideas. It was shown in (Ma) that F agrees with its first Taylor approximation T1F (which is a polynomial functor of degree 1) on vector spacesV with dim(V) ≥ 6. To convert this theorem into a functorial homotopy-theoretic description of F(V), one needs to know in addition what T1F(V )i s whenV = 0.HereweshowthatT1F(0) isthestandardL-theoryspaceassociatedwith thegroupZ/2, except for a deviation in π0. The main corollary is a functorial two-stage decomposition of F(V )f or dim(V) ≥ 6whichhas theL-theoryofthe groupZ/2asonelayer,andaform ofunreducedhomology ofRP(V) with coefficientsin the L-theory ofthe trivial groupas theother layer.Exceptfor dimension shifts, theseare also the layersin the traditional Sullivan-Wall-Quinn-Ranicki decompositionof F(V). But the dimension shifts are serious and the SWQR decomposition of F(V) is not functorial in V. Becauseof the functoriality, our analysis of F(V) remains meaningful and valid when V = R ∞ .