### Abstract

Serre and Swan showed that the category of vector bundles over a compact space X is equivalent to the category of finitely generated projective modules over the ring of continuous functions on X. In this paper, titled after the famous paper by Swan, this result is extended to an arbitrary topological space X. Also the well-known homotopy classification of the vector bundles over compact X up to isomorphism is extended to arbitrary X. It is shown that the Ko-functor and the Witt group of the ring of continuous functions on X coincide, and they are homotopy-type invariants of X.

Original language | English (US) |
---|---|

Pages (from-to) | 749-755 |

Number of pages | 7 |

Journal | Transactions of the American Mathematical Society |

Volume | 294 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1986 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

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*Transactions of the American Mathematical Society*, vol. 294, no. 2, pp. 749-755. https://doi.org/10.1090/S0002-9947-1986-0825734-3

**Vector bundles and projective modules.** / Vaserstein, Leonid N.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Vector bundles and projective modules

AU - Vaserstein, Leonid N.

PY - 1986/1/1

Y1 - 1986/1/1

N2 - Serre and Swan showed that the category of vector bundles over a compact space X is equivalent to the category of finitely generated projective modules over the ring of continuous functions on X. In this paper, titled after the famous paper by Swan, this result is extended to an arbitrary topological space X. Also the well-known homotopy classification of the vector bundles over compact X up to isomorphism is extended to arbitrary X. It is shown that the Ko-functor and the Witt group of the ring of continuous functions on X coincide, and they are homotopy-type invariants of X.

AB - Serre and Swan showed that the category of vector bundles over a compact space X is equivalent to the category of finitely generated projective modules over the ring of continuous functions on X. In this paper, titled after the famous paper by Swan, this result is extended to an arbitrary topological space X. Also the well-known homotopy classification of the vector bundles over compact X up to isomorphism is extended to arbitrary X. It is shown that the Ko-functor and the Witt group of the ring of continuous functions on X coincide, and they are homotopy-type invariants of X.

UR - http://www.scopus.com/inward/record.url?scp=84967774544&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84967774544&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-1986-0825734-3

DO - 10.1090/S0002-9947-1986-0825734-3

M3 - Article

VL - 294

SP - 749

EP - 755

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -