The Topology of Fiber Bundles: Lecture Notes
by Ralph L. Cohen · Stanford University Department of Mathematics
Stanford graduate lecture notes, 171 pages, developing locally trivial fibrations through clutching functions and structure groups, then principal bundles, pullbacks, connections and curvature, classifying spaces, characteristic classes, and the Leray-Serre spectral sequence. Chapter 1 builds bundles explicitly from transition/clutching functions and structure groups, section 2.5 does connections and curvature, chapter 2 does universal bundles and classifying spaces. Book-length with full proofs, free.
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.
The Topology of Fibre Bundles (PMS-14)
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.