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Title Topology of fiber bundles
Publication Date
Date Accessioned
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
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…