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The Topology of Quasibundles

Published online by Cambridge University Press:  20 November 2018

H. Movahedi-Lankarani  and
R. Wells
H. Movahedi-Lankarani
Affiliation:
Department of Mathematics Penn State Altoona Campus Altoona, Pennsylvania 16601 U.S.A. e-mail: hml@math.psu.edu
R. Wells
Affiliation:
Department of Mathematics Penn State University University Park, Pennsylvania 16802 U.S.A. e-mail: wells@math.psu.edu
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Abstract

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Let ℳ(N, N) be the space of all N × N real matrices and let 𝒢(N) be the set of all linear subspaces of ℝN. The maps ker and coker from ℳ(N, N) onto 𝒢(N) induce two quotient topologies, the right and left respectively. A quasibundle over a space X is defined as a continuous map from X into 𝒢(N)\ it is a right quasibundle if 𝒢(N) = ℳ(N,N)/ ker and a left quasibundle if 𝒢(N) = ℳ(N,N)/ coker. The following is established. Theorem: Let ξ be a left quasibundle over a closed subset of some Euclidean space. Then the following statements are equivalent: (i) ξ has enough sections pointwise. (ii) Sections zero at infinity over closed subsets may be extended globally, (iii) A vector subbundle over a closed subset extends to a vector subbundle over a neighborhood, (iv) ξ is a fibrewise sum of local vector subbundles. (v) There exist finitely many global sections spanning ξ. (vi) ξ is an image quasibundle. (vii) ξ results from a Swan construction. These results are used to prove a version of the Hirsch-Smale immersion theorem for locally compact subsets of Euclidean space.

Keywords

Primary: 55R5555R6554C20quasibundlesimage bundleskernel bundlesfibreclassifying spacemonotopyimmersion
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 6 , 01 December 1995 , pp. 1290 - 1316
Copyright
Copyright © Canadian Mathematical Society 1995

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