Differential Topology

Smooth manifolds, transversality, Morse theory, and cobordism.

foundation tier

Differential Topology. Smooth manifolds, transversality, Morse theory, and cobordism.

Foundations and canonical references

The standard treatments of differential topology approach the subject from complementary angles. Guillemin, Differential Topology (1974) is the anchor reference for the subject and lays out the core definitions, theorems, and worked examples that practitioners return to. Milnor, Topology from the Differentiable Viewpoint (1965) gives a parallel, more proof-oriented exposition of the same material and is widely used as a graduate text.

Open methodological questions for differential topology include sharpening the bridges between foundational theory and computational practice, extending classical results to broader or more structured settings, and integrating the techniques surveyed above with adjacent mathematical disciplines. The references listed in this page are the entry points that current work builds on.

Prerequisites

Sources

  • textbook · primary · 1974
    Differential Topology
    guillemin-1974, pollack-1974
  • textbook · primary · 1965
    Topology from the Differentiable Viewpoint
    milnor-1965

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  1. 01

    Morse Theory

    Critical points, handle decompositions, and Morse homology.
  2. 02

    Cobordism Theory

    Thom's theorem, oriented and complex cobordism.
  3. 03

    Exotic Smooth Structures

    Donaldson and Seiberg–Witten invariants for 4-manifolds.

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