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H* and some rather nice spaces Body, Richard A.
Abstract
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.
Item Metadata
Title |
H* and some rather nice spaces
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
1972
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Description |
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.
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Genre | |
Type | |
Language |
eng
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Date Available |
2011-03-14
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Provider |
Vancouver : University of British Columbia Library
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Rights |
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
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DOI |
10.14288/1.0080448
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URI | |
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Affiliation | |
Degree Grantor |
University of British Columbia
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Campus | |
Scholarly Level |
Graduate
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Aggregated Source Repository |
DSpace
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Item Citations and Data
Rights
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.