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

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dc.creatorMacko, Tibor
dc.creatorWeiss, Michael
dc.date2007-03-11
dc.date.accessioned2026-07-07T07:51:25Z
dc.date.available2026-07-07T07:51:25Z
dc.descriptionFor 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 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) > 5 which has the L-theory of the group Z/2 as one layer, and a form of unreduced homology of RP (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^\infty.
dc.description34 pages
dc.identifierhttps://arxiv.org/abs/math/0703303
dc.identifierhttp://arxiv.org/abs/math/0703303
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/125505
dc.subjectAlgebraic Topology
dc.subjectGeometric Topology
dc.subjectPrimary 57N99, 55P99, Secondary: 57R67
dc.titleThe block structure spaces of real projective spaces and orthogonal calculus of functors II
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