I am interested in learning more about general vector bundle theory. More specifically, vector bundles of class $C^k$ for $k\in\mathbb{N}$ or $C^\infty$ or real-analytic whose fibers can be given the structure of finite-dimensional real vector space, or even complex-analytic ones whose fibers can be given the structure of finite-dimensional complex vector space. Even class $C^1$ through $C^\infty$ vector bundles whose fibers can be given the structure of a real Banach space. I've only been able to find maybe two or three textbooks exclusive to fibre bundle and/or vector bundle theory. Two of them are Norman Steenrod's The Topology of Fibre Bundles as well as Allen Hatcher's Vector Bundles and K-Theory. To be frank, most of the time I usually find snippets or a chapter dedicated to these topics in corresponding differential geometry textbooks. I enjoy reading about it from entire textbooks dedicated to the development of the tools made by geometers/topologists which are intended to be employed by geometers/topologists and other mathematicians, and they tend to spell out more details for the reader than you would usually find in differential geometry textbooks (this is not to imply that differential geometry textbooks do not explain the details).
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1$\begingroup$ Did you have a look at Husemoller's book? I think it is one of the standard texts on fibre bundle theory and has a chapter devoted to vector bundles. $\endgroup$pyridoxal_trigeminus– pyridoxal_trigeminus2025-11-30 18:45:44 +00:00Commented 10 hours ago
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$\begingroup$ I will take a look at this. Thank you for sharing a standardized textbook! It seems to have a variety of topics :) $\endgroup$Man-I-Fold– Man-I-Fold2025-11-30 20:03:26 +00:00Commented 9 hours ago
1 Answer
It's true that vector bundle theory is usually fragmented across DG texts. If I were in your shoe I would look into the following books for the foundational theory of real and complex vector bundles over finite dimensional manifolds.
- Milnor, J. and Stasheff, J., Characteristic Classes:- Chapters 1–5 are arguably the clearest existing exposition of vector bundle structure (definitions, Whitney sums, Grassmannians, and the classification theorem). It is concise and rigorously topological.
- Husemöller, D., Fibre Bundles:- I think a lot of people agree that this is the standard modern alternative to Steenrod. It covers general fibre bundles, principal bundles, and vector bundles in depth before moving to K-theory.
And as for bundles with complex fiber structures and holomorphic transition functions (as opposed to just smooth complex bundles) my go to text book is Kobayashi, S., Differential Geometry of Complex Vector Bundles. It bridges the gap between algebraic and differential geometry, covering Hermitian structures, Chern connections, and stability.
As for Banach Bundles, it is my expirence that most geometry texts assume finite dimensions, but Lang, S., Fundamentals of Differential Geometry book systematically treats the infinite dimensional case as the default. the text book defines manifolds and vector bundles over Banach spaces from Chapter 1. This is the primary reference for class $C^p$ bundles where fibers are Banach spaces; finite dimensionality is treated merely as a special case.
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$\begingroup$ Thank you. This is a wonderful response. I, too, got the Banach bundle portion from S. Lang's textbook. S. Lang recommended us N. Steenrod's textbook. Although, Abraham, Marsden, and Ritiu also have a treatment for vector bundles in this scenario. To encourage even more answers, I am going to withhold on accepting it for a few more days. Hopefully other members have other textbooks to share :) $\endgroup$Man-I-Fold– Man-I-Fold2025-11-30 20:01:45 +00:00Commented 9 hours ago