Vol. 108, No. 2, 1983

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Characteristic classes for spherical fibrations with fibre-preserving free group actions

Benjamin Michael Mann and Edward Yarnell Miller

Vol. 108 (1983), No. 2, 327–377

Let G(H) be the monoid of H equivariant self maps of S(nV ), the unit sphere of n copies of a finite dimension orthogonal representation V of a finite group H, stabilized over n in an appropriate way. Let SG(H) be the submonid of G(H) consisting of all degree 1 maps. If H1 is a subgroup of H there is a natural forgetful map SG(H) SG(H1) and if Z is the center of H there is a natural action map BZ × SG(H) SG(H) induced by the natural action of Z on H. The main results of this paper are the calculations of the Hopf algebra structures of H(SG(Z∕pn),Z∕p) and H(BSG(Z∕pn),Z∕p) for all n and all primes p, the calculations in homology of forgetful maps induced by the natural inclusions Z∕pn1 Z∕pn and, for H = Z2, the calculation of the action map H(RP,Z2)H(BSG(Z2),Z2) H(BSG(Z2),Z2).

Mathematical Subject Classification 2000
Primary: 55P47
Secondary: 57S99
Received: 10 February 1981
Revised: 1 March 1982
Published: 1 October 1983
Benjamin Michael Mann
Edward Yarnell Miller