The Topology of Fibre Bundles (PMS-14)
by Norman Steenrod · Princeton University Press
The 1951 monograph that first organized fibre bundle theory systematically, constructing bundles from coordinate transformations and structure groups, then applying homotopy and cohomology theory to bundle classification, obstruction theory, and characteristic classes. Reissued by Princeton, 224 pages.
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More resources on Fiber Bundles
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
nLab: Fiber Bundle
Category-theoretic reference page defining bundles as maps E to B plus local triviality over site covers, then specializing to G-bundles, principal, vector and associated bundles. Shows how the classical definition generalizes to higher and noncommutative settings.
Geometrical Anatomy of Theoretical Physics
Learn fiber-bundles with Frederic Schuller's "Geometrical Anatomy of Theoretical Physics" course! Explore the math behind modern physics.
Hatcher's Algebraic Topology (free PDF)
Allen Hatcher's standard graduate textbook, freely downloadable from Cornell, covering fundamental groups, homology, cohomology and homotopy theory, with CW complexes introduced in Chapter 0 and used throughout. Readers learn to build spaces from cells and compute their algebraic invariants.
Fibre Bundles (Graduate Texts in Mathematics 20)
Where Steenrod is the classical source, Husemöller is the modern working reference: the book that treats associated bundles and gauge groups at length, which the topic note calls out directly. Graduate Texts in Mathematics volume covering bundle basics, homotopy classification, principal and associated bundles, K-theory, and, added in the third edition, the gauge group of a bundle and differential forms representing characteristic classes of complex vector bundles.
Vector Bundles and K-Theory
Hatcher's free in-progress book, roughly 120 pages posted, covering vector bundles, topological K-theory and Bott periodicity, Adams' theorem on the Hopf invariant, division algebras, sphere parallelizability, and Stiefel-Whitney, Chern, Euler and Pontryagin classes. Directly serves the associated-vector-bundles half of the topic note, from an author whose exposition is the field's default recommendation.