UBC Theses and Dissertations
Formality and finite ambiguity Verster, Jan Frans
The integral cohomology algebra functor, H*( ;Z), was developed as an aid in distinguishing homotopy types. We consider the problem of when there are only a finite number of homotopy equivalence classes in the collection of simply connected, finite CW complexes, for which the cohomology algebra is isomorphic to a given algebra. We prove that, if we restrict ourselves to formal homotopy types, the set of such homotopy types is always finite. This is shown by using the concept of distance between homotopy types. We build model spaces so that the distance from a CW complex, whose cohomology is isomorphic to the given algebra, to one of the model spaces can be bounded. General results about distance then imply that the set of homotopy types is finite. Formality is a property of rational homotopy type and we use information obtained from calculations with the minimal models of Sullivan as a guide in the construction of spaces and maps. As a partial converse, we show that, for every nonformal space, X, there is a homology section, X', of X such that there are an infinite number of different homotopy types with cohomology algebras isomorphic to H*(X';Z) and with the same rational homotopy type as X'. The dual problem, in the sense of Eckmann-Hilton, is also shown to have similar answers. For the dual problem one would replace "cohomology algebra" with "Samuelson algebra", "formal" with "coformal", and "homology section" with "Postnikov section". The result applies to many naturally occuring spaces, such as topological groups, H-spaces, complex and quaternionic projective spaces, Kahler manifolds and MoiSezon spaces.
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