Fibre bundles are ubiquitous in mathematical physics, and a great introduction to fibre bundles is provided by Husemoller's, Fibre Bundles. For a book on understanding the topology of fibre bundles, such as the homotopy theory, especially on a conceptual level, Steenrod's Topology of Fibre Bundles while dated will ensure a thorough understanding.
Steenrod Topology Of Fibre Bundles Djvu Files
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