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H* and some rather nice spaces

University of British Columbia

The Problem The integral cohomology algebra functor, H*, was introduced to algebraic topology in hopes of deciding when spaces are homotopy-equivalent. With this in mind, let T(A) ≃{ X|H* (X) ≃ A} , the collection of all simply-connected finite complexes X , for which the cohomology algebra H* (X) is isomorphic to A . We ask: when are there only a finite number of homotopy equivalence classes in T(A) ? The Result Let A satisfy the condition: [See Thesis for Equation] Then there are only a finite number of homotopy-equivalence classes in T(A) . The Methods For a given A we construct a "model space" M and show that for any X ε T(A) there exists a continuous map X [sup θ/sub →]M "within N" . The concept of a map within N is less restrictive than that of a homotopy-equivalence, but more restrictive than the concept of a rational equivalence. We then show that in the category T[sup N]/M , whose objects are [sup θ/sub →] , maps within N , having range M , there are only a finite number of equivalence classes. This is proved with the use of Postnikov Towers and algebraic arguments similar to Serre's mod C theory. The result then follows. Applications The result applies to spaces having rational cohomology isomorphic to the rational cohomology of topological groups, H-spaces, Stiefel manifolds, the complex, and quaternlonic projective spaces and of some other homogeneous spaces.

Text

2011-03-14

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http://hdl.handle.net/2429/32425

Mathematics

University of British Columbia

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