One of Milnor's best-known works is his proof in 1956 of the existence of
7-dimensional spheres with nonstandard differentiable structure, which marked the beginning of a new field – differential topology. He coined the term
exotic sphere, referring to any
n-sphere with nonstandard differential structure. Kervaire and Milnor initiated the systematic study of exotic spheres by
Kervaire–Milnor groups, showing in particular that the 7-sphere has 15 distinct
differentiable structures (28 if one considers orientation).
Egbert Brieskorn found simple algebraic equations for 28 complex hypersurfaces in complex 5-space such that their intersection with a small sphere of dimension 9 around a
singular point is diffeomorphic to these exotic spheres. Subsequently, Milnor worked on the
topology of isolated
singular points of complex hypersurfaces in general, developing the theory of the
Milnor fibration whose fiber has the
homotopy type of a bouquet of
μ spheres where
μ is known as the
Milnor number. Milnor's 1968 book on his theory,
Singular Points of Complex Hypersurfaces, inspired the growth of a huge and rich research area that continues to mature to this day. In 1961, Milnor disproved the
Hauptvermutung by illustrating two
simplicial complexes that are
homeomorphic but
combinatorially distinct, using the concept of
Reidemeister torsion. In 1984, Milnor introduced a definition of
attractor. The objects generalize standard attractors, include so-called unstable attractors, and are now known as Milnor attractors. Milnor's current interest is dynamics, especially holomorphic dynamics. His work in dynamics is summarized by Peter Makienko in his review of
Topological Methods in Modern Mathematics: It is evident now that low-dimensional dynamics, to a large extent initiated by Milnor's work, is a fundamental part of general dynamical systems theory. Milnor cast his eye on dynamical systems theory in the mid-1970s. By that time the Smale program in dynamics had been completed. Milnor's approach was to start over from the very beginning, looking at the simplest nontrivial families of maps. The first choice, one-dimensional dynamics, became the subject of his joint paper with
Thurston. Even the case of a unimodal map, that is, one with a single critical point, turns out to be extremely rich. This work may be compared with
Poincaré's work on
circle diffeomorphisms, which 100 years before had inaugurated the qualitative theory of dynamical systems. Milnor's work has opened several new directions in this field, and has given us many basic concepts, challenging problems and nice theorems. His other significant contributions include
microbundles, influencing the usage of
Hopf algebras, theory of
quadratic forms and the related area of
symmetric bilinear forms, higher
algebraic K-theory,
game theory, and three-dimensional
Lie groups. ==Awards and honors==