Mod 2 unstable equivariant spherical characteristic classes and Poincaré duality bordism generators
type: Item
instance of: scholarly article
title: Mod 2 unstable equivariant spherical characteristic classes and Poincaré duality bordism generators
zbMATH Open document ID: 0584.55016
author: Benjamin M. Mann
author: E. Y. Miller
published in: Illinois Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: This paper determines the mod 2 characteristic classes for unstable \({\mathbb{Z}}_ 2\) and \(S^ 1\) equivariant spherical fibrations and analyses the natural inclusion of the relevant classifying spaces \(BSG({\mathbb{Z}}_ 2,n)\) and \(BSG(S^ 1,2n)\) into their stable analogs \(BSG({\mathbb{Z}}_ 2)\) and \(BSG(S^ 1).\) Several authors (J. Becker, M. Crabb, and L. Woodward) have shown that the natural inclusion of \(SG({\mathbb{Z}}_ 2,n)\) in \(BSG({\mathbb{Z}}_ 2)\) admits a section. The main results of this paper give an explicit determination of the obstructions to the existence of such a section for the unstable inclusion \(SG({\mathbb{Z}}_ 2,n)\to SG(n)\). (That such obstructions exist was pointed out by this reviewer in a 1979 preprint.) The authors' results in this direction can be succinctly summarized by the following result which is their corollary 6.5. The map \(f(2n+1)_*: H_*(SG({\mathbb{Z}}_ 2,2n+1);{\mathbb{Z}}_ 2)\to H_*(SG(2n+1);{\mathbb{Z}}_ 2)\) is onto \[ H_*(SG(2n+1);{\mathbb{Z}}_ 2)/\{Q^{n+i}(-1)*Q^{n+i}(-1)*[5]\}_{i\geq 0}. \] The proofs depend on a detailed analysis of spectral sequences associated to the filtrations of \(G({\mathbb{Z}}_ 2,n)\) and \(G(S^ 1,2n)\) due to R. Schultz. This provides the calculations of the homology of the unstable spaces up to filtration. The product structure is determined with the aid of the construction of representatives for some specific elements in the homology. In a final section the authors give a result on the representation of polynomial generators of the Poincaré bordism ring by spherical projective bundles.
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 55R91
Mathematics Subject Classification ID: 55R25
Mathematics Subject Classification ID: 57P10
Mathematics Subject Classification ID: 55S12
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 3935976
zbMATH Keywords: mod 2 characteristic classes
zbMATH Keywords: equivariant spherical fibrations
zbMATH Keywords: classifying spaces
zbMATH Keywords: Poincaré bordism ring
zbMATH Keywords: spherical projective bundles
reviewed by: Jonathan Hodgson
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3935976