The block structure spaces of real projective spaces and orthogonal calculus of functors II
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Tibor Macko
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 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 T1F (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 T1F(V) is when V = 0. Here we show that T1F(0) is the standard L-theory space associated with the group ℤ/2, except for a deviation in π0. 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 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∞.
© de Gruyter 2009
Articles in the same Issue
- A Hopf theorem for open surfaces in product spaces
- On R. Steinberg's theorem on algebras of coinvariants
- Natural pseudo-distances between closed curves
- Clifford semigroups of ideals in monoids and domains
- Lp norm estimates of eigenfunctions restricted to submanifolds
- Central values of generalized multiple sine functions
- Lp-independence of spectral bounds of non-local Feynman-Kac semigroups
- On pairwise mutually permutable products
- The block structure spaces of real projective spaces and orthogonal calculus of functors II
- Erratum to: “Intrinsic ultracontractivity for non-symmetric Lévy processes” [Forum Math. 21 (2009) 43–66]
Articles in the same Issue
- A Hopf theorem for open surfaces in product spaces
- On R. Steinberg's theorem on algebras of coinvariants
- Natural pseudo-distances between closed curves
- Clifford semigroups of ideals in monoids and domains
- Lp norm estimates of eigenfunctions restricted to submanifolds
- Central values of generalized multiple sine functions
- Lp-independence of spectral bounds of non-local Feynman-Kac semigroups
- On pairwise mutually permutable products
- The block structure spaces of real projective spaces and orthogonal calculus of functors II
- Erratum to: “Intrinsic ultracontractivity for non-symmetric Lévy processes” [Forum Math. 21 (2009) 43–66]
