The Hauptvermutung of geometric topology

Geometric topology

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...

 is the conjecture

Conjecture

A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...

 that any two triangulations

Triangulation (topology)

In mathematics, topology generalizes the notion of triangulation in a natural way as follows:A triangulation of a topological space X is a simplicial complex K, homeomorphic to X, together with a homeomorphism h:K\to X....

 of a triangulable space have a common refinement, a single triangulation that is a subdivision of both of them. It was originally formulated in 1908, by Steinitz

Ernst Steinitz

Ernst Steinitz was a German mathematician.- Biography :Steinitz was born in Laurahütte , Silesia, Germany , the son of Sigismund Steinitz, a Jewish coal merchant, and his wife Auguste Cohen; he had two brothers. He studied at the University of Breslau and the University of Berlin, receiving his Ph.D...

 and Tietze

Heinrich Franz Friedrich Tietze

Heinrich Franz Friedrich Tietze was an Austrian mathematician, famous for the Tietze extension theorem. He also developed the Tietze transformations for group presentations, and was the first to pose the group isomorphism problem.He was born in Schleinz, Austria and died in Munich,...

.

This conjecture is now known to be false. The non-manifold version was disproved by John Milnor

John Milnor

John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University...

 in 1961 using Reidemeister torsion

Analytic torsion

In mathematics, Reidemeister torsion is a topological invariant of manifolds introduced by Kurt Reidemeister for 3-manifolds and generalized to higher dimensions by and ....

.

The manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

 version is true in dimension

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

s . The cases were proved by Tibor Radó

Tibor Radó

Tibor Radó was a Hungarian mathematician who moved to the USA after World War I. He was born in Budapest and between 1913 and 1915 attended the Polytechnic Institute. In World War I, he became a First Lieutenant in the Hungarian Army and was captured on the Russian Front...

 and Edwin E. Moise

Edwin E. Moise

Edwin Evariste Moise was an American mathematician and mathematics education reformer. After his retirement from mathematics he became a literary critic of 19th century English poetry and had several notes published in that field.-Early life and education:...

 in the 1920s and 1950s, respectively.

An obstruction to the manifold version was formulated by Andrew Casson

Andrew Casson

Andrew John Casson FRS is a mathematician, an expert on geometric topology.Casson is the Philip Schuyler Beebe Professor of Mathematics at Yale University in the United States where he served as department chair between 2004 and 2007. His Ph.D. advisor at the University of Liverpool was C. T. C...

 and Dennis Sullivan

Dennis Sullivan

Dennis Parnell Sullivan is an American mathematician. He is known for work in topology, both algebraic and geometric, and on dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center, and is a professor at Stony Brook University.-Work in topology:He...

 in 1967–9 (originally in the simply-connected

Simply connected space

In topology, a topological space is called simply connected if it is path-connected and every path between two points can be continuously transformed, staying within the space, into any other path while preserving the two endpoints in question .If a space is not simply connected, it is convenient...

 case), using the Rochlin invariant and the cohomology group H³(M;Z/2Z).

A homeomorphism

Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

  of m-dimensional piecewise linear manifold

Piecewise linear manifold

In mathematics, a piecewise linear manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions.An isomorphism of PL manifolds is called a PL...

s has an invariant such that for , ƒ is isotopic to a piecewise linear (PL) homeomorphism if and only if . In the simply-connected case and with , ƒ is homotopic to a PL homeomorphism if and only if
The obstruction to the manifold Hauptvermutung is now seen as a relative version of the triangulation obstruction of Rob Kirby

Robion Kirby

Robion Cromwell Kirby is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology...

 and Larry Siebenmann, obtained in 1970. The Kirby–Siebenmann obstruction is defined for any compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 m-dimensional topological manifold M
again using the Rochlin invariant. For , M has a PL structure (i.e. can be triangulated by a PL manifold) if and only if , and if this obstruction is 0 the PL structures are parametrized by H³(M;Z/2Z). In particular there is only a finite number of essentially distinct PL structures on M.

For compact simply-connected manifolds of dimension 4 Simon Donaldson

Simon Donaldson

Simon Kirwan Donaldson FRS , is an English mathematician known for his work on the topology of smooth four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London...

 found examples with an infinite number of inequivalent PL structures, and Michael Freedman

Michael Freedman

Michael Hartley Freedman is a mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born...

 found the E8 manifold

E8 manifold

In mathematics, the E8 manifold is the unique compact, simply connected topological 4-manifold with intersection form the E8 lattice.The E8 manifold was discovered by Michael Freedman in 1982...

which not only has no PL structure, but is not even homeomorphic to a simplicial complex. In dimensions greater than 4 the question of whether all compact manifolds are homeomorphic to simplicial complexes is an important open question.

External links

• http://www.maths.ed.ac.uk/~aar/haupt Additional material, including original sources
• Andrew Ranicki (ed.) The Hauptvermutung Book ISBN 0-7923-4174-0
• Andrew Ranicki High-dimensional manifolds then and now