Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

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

The block structure spaces of real projective spaces and orthogonal calculus of functors II 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 ॠ 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 ∞ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

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

Macko, Tibor; Weiss, Michael
Forum Mathematicum , Volume 21 (6) – Nov 1, 2009
Download PDF
46 pages

Loading next page...
 
/lp/de-gruyter/the-block-structure-spaces-of-real-projective-spaces-and-orthogonal-gxUnrTvzZF

References (42)

Publisher
de Gruyter
Copyright
© de Gruyter 2009
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/FORUM.2009.055
Publisher site
See Article on Publisher Site

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 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 ॠ 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 ∞ .

Journal

Forum Mathematicum – de Gruyter

Published: Nov 1, 2009

There are no references for this article.