Selected publications in manifold theory

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This is a chronological list of significant publications in the theory of manifolds: it is the personal choice of Andrew Ranicki, and is thus very biased!

The list is not intended to be encyclopedic: please contact Andrew Ranicki if you have suggestions for additions/deletions.

The dates listed are the publication dates, not necessarily when the work was actually created.

The term manifold includes

See also:


1 Up to 1890

Year Authors Publication and remarks
1868 Riemann Über die Hypothesen, welche der Geometrie zugrunde liegen

2 1891 - 1930

Year Authors Publication and remarks
1899 Poincaré Complément à l'Analysis situs.
1907 Dehn Heegaard Analysis situs
1908 Tietze Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten
1915 Alexander A proof of the invariance of certain constants of analysis situs

3 1930 - 1960

Year Authors Publication and remarks
1926 Kneser Die Topologie der Manigfaltigkeiten
1926 Lefschetz Intersections and transformations of complexes and manifolds
1931 Hopf Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche
1931 Morse The critical points of a function of n variables
1933 Borsuk Zur kombinatorischen Eigenschaft der Retrakte
1940 Cairns Homeomorphisms between topological manifolds and analytic manifolds
1944 Whitney The self-intersections of a smooth n-manifold in 2n-space
1947 Chern On the characteristic classes of Riemannian manifolds
1950 Whitehead Simple homotopy types
1952 Rokhlin New results in the theory of four-dimensional manifolds
1953 Hirzebruch On Steenrod's reduced powers, the index of inertia and the Todd genus
1954 Thom Quelques propriétés globales des variétés différentiables
1956 Milnor On manifolds homeomorphic to the 7-sphere

Showed the existence of several smooth structures on a fixed topological manifold: in this case the 7-sphere.

1959 Bott The stable homotopy of the classical groups
1959 Pontryagin Smooth manifolds and their applications in homotopy theory
1960 Adams On the non-existence of elements of Hopf invariant one

4 1961 - 1970

Year Authors Publication and remarks
1960 Kervaire A manifold which does not admit any differentiable structure

The first construction of a PL manifold without a smooth structure.

1961 Atiyah Bordism and cobordism
1961 Smale Generalized Poincaré's conjecture in dimensions greater than four

Proved that every smooth homotopy sphere of dimension 6 or more is homeomorphic to the standard sphere

1962 Browder Homotopy type of differentiable manifolds
1963 Kervaire Milnor Groups of homotopy spheres I

Applied surgery theory to the diffeomorphism classification of smooth structures on the n-sphere

1965 Novikov Rational Pontrjagin classes. Homeomorphism and homotopy type of closed manifolds I

The topological invariance of the rational Pontrjagin classes.

1966 Brieskorn Beispiele zur Differentialtopologie von Singularitäten
1968 Atiyah Singer The index of elliptic operators I
1970 Sullivan Geometric topology: localization, periodicity and Galois symmetry
1970 Wall Surgery on compact manifolds

5 1971 - 1980

Year Authors Publication and remarks
1974 Chapman Topological invariance of Whitehead torsion
1977 Kirby Siebenmann Foundational essays on topological manifolds, smoothings, and triangulations
1978 Waldhausen Algebraic K-theory of topological spaces I
1980 Edwards The topology of manifolds and cell-like maps

6 1981 - 1990

Year Authors Publication and remarks
1982 Freedman The topology of four-dimensional manifolds

Proves the Poincaré conjecture in dimension 4

1983 Donaldson An application of gauge theory to four-dimensional topology
1988 Quinn Topological transversality holds in all dimensions

7 1991 - 2000

Year Authors Publication and remarks
1993 Farrell Jones Topological rigidity for compact non-positively curved manifolds
1996 Casson Generalisations and applications of block bundles

8 2001 - present

Year Authors Publication and remarks
2002 Perelman The entropy formula for the Ricci flow and its geometric applications
2009 Hill Hopkins Ravenel On the non-existence of elements of Kervaire invariant one

9 Discussion

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