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See: Perspective

Bundles (of fibers, vectors, etc.) are the machinery needed for perspectives. They let us define something like a function that takes a value at each point of a manifold, but in a different copy of the relevant vector space. In this sense, they output their own point. And so that point is also relevant as a "zero", for example, for projective geometry.

Note that the three-cycle is expressed by long exact sequences of bundles. Understand this!

Readings

• Wikipedia: Fibration: Long exact sequence of homotopy groups
• Eckmann-Hilton duality
• Wikipedia: Whitehead bracket

Videos

• Videos: Gerginski: Dimensions
• Jacob Lure: Lie Algebras and Homotopy Theory
• Pierre Deligne: What do we mean by "equal"
• Daniel Chan: Vector Bundles

Ideas

• Note importance of base points in homotopy theory, fundamental group.
• What can I say about the intractible problem, homotopy groups of spheres?