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where <f = (F, p, B, F) is the induced fibre space. Eilenberg and Moore [6] have constructed a spectral sequence {Er, dr} with (i) ET => //*(£; k), (ii) E2 = ForH.iBoM(H*(B; k), H*(E0; k)), where A: is a field. In Part I we shall give a short summary of how one constructs this spectral sequence. Various elementary properties are developed. In Part II we develop some simple devices to compute ForA(A, B) when A is a polynomial algebra. These results while basically not new are spread throughout the literature. This material owes much to Borel and the presentation here is based on ideas of J. Moore and P. Baum. These algebraic considerations lead us to a collapse theorem for the spectral sequence in several situations of geometric interest. We close with some applications. This is a portion of the author's doctoral dissertation completed under the direction of Professor W. S. Massey, whom we wish to thank for much useful guidance. We also wish to thank J. P. May, P. F. Baum and E. O'Neil for useful discussions and suggestions.

Homological algebra and the Eilenberg-Moore spectral sequence