Topology of Fiber Bundles: Homotopy Theory of Bundles

Mahima Ranjan Adhikari
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Mahima Ranjan Adhikari: Institute for Mathematics, Bioinformatics, Information Technology and Computer Science (IMBIC)

Chapter Chapter 5 in Basic Topology 3, 2022, pp 303-419 from Springer

Abstract: Abstract This chapter continues the study topology of fiber bundles, which has created general interest as it involves interesting applications of topology to other areas such as algebraic topology, geometry, physics and gauge groups and addresses the homotopy theory of bundles. Covering spaces provide tools to study the fundamental groups. On the other hand, fiber bundles provide similar tools to discuss higher homotopy groups, which are natural generalizations of fundamental groups. The concept of fiber spaces is the most important generalization of the concept of covering spaces. The importance of fiber spaces was greatly realized during 1935–1950 to solve several problems involving homotopy and homology. The motivation of the study of fiber bundles and vector bundles was born through the distribution of signs of the derivatives of the plane curves at every point. Historically, the recognition of the theory of fiber bundles as a discipline of mathematics came through the work of H. Whitney (1907–1989), H. Hopf (1894–1971), E. Stiefel (1909–1978), J. Feldbau (1914–1945) and some others.

Date: 2022
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DOI: 10.1007/978-981-16-6550-9_5

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