Full Record

Author | Zhang, Hainan |

Title | Topology of fiber bundles |

URL | http://hdl.handle.net/2097/18185 |

Publication Date | 2014 |

Date Accessioned | 2014-08-06 16:40:22 |

Degree | MS |

Discipline/Department | Department of Mathematics |

Degree Level | masters |

University/Publisher | Kansas State University |

Abstract | This report introduces the fiber bundles. It includes the definitions of fiber bundles such as vector bundles and principal bundles, with some interesting examples. Reduction of the structure groups, and covering homotopy theorem and some specific computation using obstruction classes, Cech cohomology, Stiefel-Whitney classes, and first Chern classes are included. |

Subjects/Keywords | Introduction to Fiber Bundles; obstruction theory from characteristic class; 1st SW class and 1st Chern class; Mathematics (0405) |

Contributors | David Auckly |

Language | en |

Country of Publication | us |

Record ID | handle:2097/18185 |

Repository | ksu |

Date Retrieved | 2018-01-02 |

Date Indexed | 2018-01-03 |

Issued Date | 2014-08-06 00:00:00 |

Note | [degree] Master of Science; [level] Masters; [department] Department of Mathematics; [advisor] David Auckly; |

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…which implies that it is a fiber bundle, hence, a fibration since
the base space S2 is a paracompact CW-complex.
24
Chapter 6
*Obstruction* *Theory*
Suppose we want to construct a section from a CW-complex X into a bundle E with fiber
F . We do this by…

…induction: given a section σ : X (k) → E on the kth-skeleton X (k) and a
(k+1)-cell i : Dk+1 → X, we want to extend σ over the i. The *obstruction* to extend over a
(k+1)-cell is an element of πk (F ), the k-th…

…fiber space F . Now we can define
k+1
the *obstruction* class Oσ (i) = [P2 ◦ φ(b, σ ◦ i)] ∈ πk (F ), where Oσ ∈ CCW
(X, πk (F )). These
obstructions fit together to give a cellular cochain O on X…

…with coefficients in this πk . In
fact, this cochain is a cocycle, so it defines an *obstruction* class O(E) in H k+1 (X, πk (E)).
Then there exists a cross-section over the (k+1)-skeleton if and only if a certain…

…well defined
*obstruction* class is zero. If the cochain is 0, then there exists a map µ : Dk+1 → F . Then
the section extending to (k+1)-skeleton σ
e : X k+1 → E is defined to be P2 (φ−1 (v, µ(v))),
where v ∈ Dk+1…

…map. Note that the fiber Z2 is a
group so that its 0th-homotopy group π0 (Z2 ) = Z2 . In this case, the *obstruction* cocycle is
1
the 1st Stiefel-Whitney class w1σ (E) ∈ CCW
(X, Z2 ). Hence, a section defined on 0-cells is…

…x29;(D− ) = w1τ (E)(D+ ) = 0, so w1τ (E)(S 1 ) = 0. It actually shows that
28
the *obstruction* class is independent with the choice of the functions. Therefore, the zero
*obstruction* class implies that…

…*obstruction* class is independent with the choice of the functions,
thus, w1τ (E 0 )(D− ) = 0 and w1τ (E 0 )(D+ ) = 1 which can also be proved by the graph. So,
w1τ (E 0 )(S 1 ) = 1 + 0 = 1. Hence, we…