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Geometric topology studies manifolds, their embeddings and the ways they can be assembled, classified or transformed. Typical questions ask whether two manifolds are homeomorphic or diffeomorphic, whether two embeddings are related by an ambient isotopy, and how gluing simpler pieces changes the resulting space. The category of structure matters: a homeomorphism preserves topology, while a diffeomorphism also preserves smooth structure. Algebraic invariants and geometric constructions work together in these questions.^[ref]

For example, a closed connected oriented surface is classified by its genus of a surface $g$, with Euler characteristic $\chi =2-2g$. A sphere, torus and genus-two surface have Euler characteristics $2,0,-2$. A polygon construction for the genus-two surface has one vertex, four edges and one face, giving $1-4+1=-2$. Specifying closed, connected and oriented is essential; surfaces with boundary or noncompact surfaces require more data.^[ref]

In 1956 John Milnor constructed smooth seven-dimensional manifolds homeomorphic to the standard seven-sphere but not diffeomorphic to it. His construction used three-sphere bundles over the four-sphere and an invariant distinguishing smooth structures.^[ref] This shows why classifying the underlying topological space and classifying its smooth structures are distinct problems.

James D. Stasheff works on topology and its algebraic structures. With John Milnor, he wrote Characteristic Classes by John Milnor and James D. Stasheff, developing invariants that detect how vector bundles twist over a space.^[ref]

The total Pontryagin class illustrates the importance of coefficient conventions: its product formula is stated modulo elements of order two in their theorem. Keeping that qualification prevents an integral statement from being inferred from a result valid only after removing a torsion ambiguity.^[ref]

John Milnor works across topology, geometry and dynamical systems. His expositions connect local calculations with global structure: Morse Theory by John Milnor develops the topology of a manifold through critical points, while his complex-dynamics lectures study iteration of holomorphic maps.^[ref][ref]

Near a nondegenerate critical point, the Morse lemma reduces a smooth function exactly to a signed quadratic form in suitable coordinates. Its index controls the topological change in sublevel sets. With James D. Stasheff, Milnor also wrote Characteristic Classes by John Milnor and James D. Stasheff, developing invariants of vector bundles.

By John Milnor.

Verified locator: Annals of Mathematics 64(2), September 1956, 399–405. Introduction, section 3 and Theorem 3, printed p.403; S^3 bundles over S^4 with exotic smooth structures. Primary scan also at homepages.math.uic.edu/~kauffman/Milnor.pdf ; theorem statement verified in indexed original scan.