Approximate and discrete Euclidean vector bundles
We introduce $\varepsilon $ -approximate versions of the notion of a Euclidean vector bundle for $\varepsilon \geq 0$ , which recover the classical notion of a Euclidean vector bundle when $\varepsilon = 0$ . In particular, we study Čech cochains with coefficients in the orthogonal...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2023-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509423000166/type/journal_article |
| _version_ | 1797867002800373760 |
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| author | Luis Scoccola Jose A. Perea |
| author_facet | Luis Scoccola Jose A. Perea |
| author_sort | Luis Scoccola |
| collection | DOAJ |
| description | We introduce
$\varepsilon $
-approximate versions of the notion of a Euclidean vector bundle for
$\varepsilon \geq 0$
, which recover the classical notion of a Euclidean vector bundle when
$\varepsilon = 0$
. In particular, we study Čech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that
$\varepsilon $
-approximate vector bundles can be used to represent classical vector bundles when
$\varepsilon> 0$
is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data and also allows for some tolerance to noise when working with vector bundles in an applied setting. As an example, we prove a reconstruction theorem for vector bundles from finite samples. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples. |
| first_indexed | 2024-04-09T23:33:24Z |
| format | Article |
| id | doaj.art-ca9103a08eb44841a174936845f01f66 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-09T23:33:24Z |
| publishDate | 2023-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-ca9103a08eb44841a174936845f01f662023-03-21T02:05:50ZengCambridge University PressForum of Mathematics, Sigma2050-50942023-01-011110.1017/fms.2023.16Approximate and discrete Euclidean vector bundlesLuis Scoccola0https://orcid.org/0000-0002-4862-722XJose A. Perea1https://orcid.org/0000-0002-6440-5096Department of Mathematics, Northeastern University, 43 Leon St, Boston, MA 02115, USA; E-mail:Department of Mathematics and Khoury College of Computer Sciences, Northeastern University, 43 Leon St, Boston, MA 02115, USA; E-mail:We introduce $\varepsilon $ -approximate versions of the notion of a Euclidean vector bundle for $\varepsilon \geq 0$ , which recover the classical notion of a Euclidean vector bundle when $\varepsilon = 0$ . In particular, we study Čech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that $\varepsilon $ -approximate vector bundles can be used to represent classical vector bundles when $\varepsilon> 0$ is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data and also allows for some tolerance to noise when working with vector bundles in an applied setting. As an example, we prove a reconstruction theorem for vector bundles from finite samples. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples.https://www.cambridge.org/core/product/identifier/S2050509423000166/type/journal_article55R9955N3168W0555U99 |
| spellingShingle | Luis Scoccola Jose A. Perea Approximate and discrete Euclidean vector bundles Forum of Mathematics, Sigma 55R99 55N31 68W05 55U99 |
| title | Approximate and discrete Euclidean vector bundles |
| title_full | Approximate and discrete Euclidean vector bundles |
| title_fullStr | Approximate and discrete Euclidean vector bundles |
| title_full_unstemmed | Approximate and discrete Euclidean vector bundles |
| title_short | Approximate and discrete Euclidean vector bundles |
| title_sort | approximate and discrete euclidean vector bundles |
| topic | 55R99 55N31 68W05 55U99 |
| url | https://www.cambridge.org/core/product/identifier/S2050509423000166/type/journal_article |
| work_keys_str_mv | AT luisscoccola approximateanddiscreteeuclideanvectorbundles AT joseaperea approximateanddiscreteeuclideanvectorbundles |