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UBC Theses and Dissertations

The Eilenberg-Moore spectral sequence Yagi, Toshiyuki

Abstract

For any two differential modules M and N over a graded differential k-algebra Λ (k a commutative ring), there Is a spectral sequence Er, called the Eilenberg-Moore spectral sequence, having the following properties: Er converges to Tor Λ (M,N) and E2=TorH(Λ) (H(M),H(N)). This algebraic set-up gives rise to a "geometric" spectral sequence in algebraic topology. Starting with a commutative diagram of topological spaces [diagram omitted] where B Is simply connected, one gets a spectral sequence Er converging to the cohomology H*(X xBY) of the space X xBY, and for which E₂=TorH*(B) (H*(X),H*(Y)). In this thesis we outline a generalization of the above geometric spectral sequence obtained, by first extending the category of topological spaces and then, extending the cohomology theory H* to this larger category. The convergence of the extended spectral sequence does not depend, on any topological conditions of the spaces involved. It follows algebraically from the way the exact couple (from which the spectral sequence Is derived) Is set up and from the Suspension Axiom of the extended cohomology theory.

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