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 Formality and finite ambiguity
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Formality and finite ambiguity Verster, Jan Frans
Abstract
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 EckmannHilton, 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, Hspaces, complex and quaternionic projective spaces, Kahler manifolds and MoiSezon spaces.
Item Metadata
Title 
Formality and finite ambiguity

Creator  
Publisher 
University of British Columbia

Date Issued 
1982

Description 
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 EckmannHilton, 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, Hspaces, complex and quaternionic projective spaces, Kahler manifolds and MoiSezon spaces.

Genre  
Type  
Language 
eng

Date Available 
20100415

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 
10.14288/1.0080288

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
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.