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 H* and some rather nice spaces
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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 homotopyequivalent. With this in mind, let T(A) ≃{ XH* (X) ≃ A} , the collection of all simplyconnected 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 homotopyequivalence 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 homotopyequivalence, 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, Hspaces, Stiefel manifolds, the complex, and quaternlonic projective spaces and of some other homogeneous spaces.
Item Metadata
Title  H* and some rather nice spaces 
Creator  Body, Richard A. 
Publisher  University of British Columbia 
Date Issued  1972 
Description 
The Problem The integral cohomology algebra functor, H*,
was introduced to algebraic topology in hopes of deciding when spaces
are homotopyequivalent. With this in mind, let T(A) ≃{ XH* (X) ≃ A} ,
the collection of all simplyconnected 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 homotopyequivalence 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 homotopyequivalence, 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, Hspaces, Stiefel manifolds, the complex, and quaternlonic projective spaces and of some other homogeneous spaces.

Genre  Thesis/Dissertation 
Type  Text 
Language  eng 
Date Available  20110314 
Provider  Vancouver : University of British Columbia Library 
Rights  For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. 
DOI  10.14288/1.0080448 
URI  
Degree  Doctor of Philosophy  PhD 
Program  Mathematics 
Affiliation  Science, Faculty of; Mathematics, Department of 
Degree Grantor  University of British Columbia 
Campus  UBCV 
Scholarly Level  Graduate 
Aggregated Source Repository  DSpace 
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For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.