在這份論文中，我們研究投影空間上的Stiefel-Whitney 特徵類，並發現一些計算上的技巧。將之應用到上面，我們可以呈現出投影空間上正切叢的平凡性的必要條件。

In this thesis, we study the Stiefel-Whitney classes of projective space, and develope some calculation technique. As an application, we present the necessary condition for the triviality of the tangent bundle of the projective space.

Contents
1 Introduction                                1
2 Main Theorem                                3
3 Vector Bundle                               7
4 Cohomology                                 11
5 Stiefel-Whitney class                      17
  5.1 Definition and Proposition . . . . . . 17
  5.2 Existence of Stiefel-Whitney class . . 19
6 Appendix                                   22
7 Reference                                  29

目錄
1 前言                                   30
2 主要證明                               32
3 向量叢                                 35
4 上同調                                 38
5 Stiefel-Whitney 特徵類                 44
  5.1 定義跟定理 . . . . . . . . . . . . 44
  5.2 Stiefel-Whitney 特徵類的存在性 . . 46
6 附錄                                   49
7 參考資料                               56

Reference
[1] James R.Munkres, Elements of Algebraic Topology, Addison-Wesley Publishing Company, Taipei.
[2] JohnW. Milnor and James D. Stasheff, Characteristic Classes, Princetion University Press and University of Tokyo Press Pricetion, New Jersey, 1974.
[3] William M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Second Edition, Academic Press, INC.
[4] Bott R. and Milnor J., On the parallelizability of spheres, Bull. Amer. Math. Soc. 64 (1958), 87–89.
[5] Gamkrelidze, Algebra I, Basic Notions of Algebra, Springer-Verlag.
[6] James R. Munkres, Topology, Second Edition, Prentice Hall, INC.

參考資料
[1] James R.Munkres, Elements of Algebraic Topology, Addison-Wesley Publishing Company, Taipei.
[2] JohnW. Milnor and James D. Stasheff, Characteristic Classes, Princetion University Press and University of Tokyo Press Pricetion, New Jersey, 1974.
[3] William M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Second Edition, Academic Press, INC.
[4] Bott R. and Milnor J., On the parallelizability of spheres, Bull. Amer. Math. Soc. 64 (1958), 87–89.
[5] Gamkrelidze, Algebra I, Basic Notions of Algebra, Springer-Verlag.
[6] James R. Munkres, Topology, Second Edition, Prentice Hall, INC.