Mod 2 unstable equivariant spherical characteristic classes and Poincaré duality bordism generators
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.
scientific article; zbMATH DE number 3935976