# Selected publications in manifold theory

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

The term manifold includes

## 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

## 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$$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