[permalink] [id link]
Haken sketched out a proof of an algorithm to check if two Haken manifolds were homeomorphic or not.
from
Wikipedia
Some Related Sentences
Haken and out
In particular, he was the first to investigate the notion of " discharging ", which turned out to be a fundamental ingredient of the eventual computer-aided proof by Kenneth Appel and Wolfgang Haken.
Haken and proof
To dispel remaining doubt about the Appel – Haken proof, a simpler proof using the same ideas and still relying on computers was published in 1997 by Robertson, Sanders, Seymour, and Thomas.
In 1986, Appel and Haken were asked by the editor of Mathematical Intelligencer to write an article addressing the rumors of flaws in their proof.
In 1996, Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas created a quadratic time algorithm, improving on a quartic algorithm based on Appel and Haken ’ s proof (; ).
At first, The New York Times refused as a matter of policy to report on the Appel – Haken proof, fearing that the proof would be shown false like the ones before it.
A computer-aided proof produced in 1976 by Kenneth Appel and Wolfgang Haken makes fundamental use of the notion of " discharging " developed by Heesch.
The Fields Medal was awarded to Thurston in 1982 partially for his proof of the geometrization conjecture for Haken manifolds.
Thurston's geometrization conjecture, formulated in the late 1970s, offered a framework that suggested geometry and topology were closely intertwined in low dimensions, and Thurston's proof of geometrization for Haken manifolds utilized a variety of tools from previously only weakly linked areas of mathematics.
The preface of the 1979 edition of Laws of Form repeats that claim, and further states that the generally accepted computational proof by Appel, Haken, and Koch has ' failed ' ( page xii ).
After the 1977 success of Appel and Haken, Heesch worked on refining and shortening their proof, even after his retirement.
This is the fibered part of Thurston's geometrization theorem for Haken manifolds, whose proof requires the Nielsen – Thurston classification for surface homeomorphisms as well as deep results in the theory of Kleinian groups.
Haken and algorithm
gave an algorithm to determine if a 3-manifold was Haken.
Since there is an algorithm to check if a 3-manifold is Haken ( cf.
Haken and if
Haken also showed that there was a finite procedure to find an incompressible surface if the 3-manifold had one.
Then one proves that if the theorem is true for pieces resulting from a cutting of a Haken manifold, that it is true for that Haken manifold.
Haken and manifolds
To complete the picture, Thurston proved a hyperbolization theorem for Haken manifolds.
Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture.
Other examples are given by the Seifert – Weber space, or " sufficiently complicated " Dehn surgeries on links, or most Haken manifolds.
A second route to Geometrization is the method of Bessières et al., which uses Thurston's hyperbolization theorem for Haken manifolds and Gromov's norm for 3-manifolds.
Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface.
Haken manifolds were introduced by.
proved that Haken manifolds have a hierarchy, where they can be split up into 3-balls along incompressible surfaces.
Normal surfaces are ubiquitous in the theory of Haken manifolds and their simple and rigid structure leads quite naturally to algorithms.
We will consider only the case of orientable Haken manifolds, as this simplifies the discussion ; a regular neighborhood of an orientable surface in an orientable 3-manifold is just a " thickened up " version of the surface, i. e. a trivial I-bundle.
The hierarchy makes proving certain kinds of theorems about Haken manifolds a matter of induction.
Jaco-Oertel ), the basic problem of recognition of 3-manifolds can be considered to be solved for Haken manifolds.
proved that closed Haken manifolds are topologically rigid: roughly, any homotopy equivalence of Haken manifolds is homotopic to a homeomorphism ( for the case of boundary, a condition on peripheral structure is needed ).
In addition, Waldhausen proved that the fundamental groups of Haken manifolds have solvable word problem ; this is also true for virtually Haken manifolds.
The hierarchy played a crucial role in William Thurston's hyperbolization theorem for Haken manifolds, part of his revolutionary geometrization program for 3-manifolds.
Haken has introduced several important ideas, including Haken manifolds, Kneser-Haken finiteness, and an expansion of the work of Kneser into a theory of normal surfaces.
Haken and were
In 1976, while other teams of mathematicians were racing to complete proofs, Kenneth Appel and Wolfgang Haken at the University of Illinois announced that they had proven the theorem.
These were the first such examples ; previously it had been believed that except for certain Seifert fiber spaces, all irreducible 3-manifolds were Haken.
In the following century, a vast amount of work and theories were developed to reduce the number of colors to four, until the four color theorem was finally proved in 1976 by Kenneth Appel and Wolfgang Haken.
Haken were used only in villages remaining under old Slavic law ( predominantly on the islands ), whereas Hufen were used for new villages placed under German law ( in Pomerania sometimes referred to as Schwerin Law ).
Between 1967 and 1971 Heesch came to the United States of America several times, where bigger and faster computers were available and where he worked with Haken and Y. Shimamoto.
Haken and .
Kenneth Appel and Wolfgang Haken finally proved this in 1976.
The four color theorem was proven in 1976 by Kenneth Appel and Wolfgang Haken.
Appel and Haken used a special-purpose computer program to confirm that each of these maps had this property.
Appel and Haken concluded that no smallest counterexamples existed because any must contain, yet not contain, one of these 1, 936 maps.
Using mathematical rules and procedures based on properties of reducible configurations, Appel and Haken found an unavoidable set of reducible configurations, thus proving that a minimal counterexample to the four-color conjecture could not exist.
These include the gesture-controlled Buchla Thunder, sonomes such as the C-Thru Music Axis, which rearrange the scale tones into an isometric layout, and Haken Audio's keyless, touch-sensitive Continuum playing surface.
Influential mathematicians such as Bing, Haken, Moise, and Papakyriakopoulos attacked the conjecture.
In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface.
A 3-manifold finitely covered by a Haken manifold is said to be virtually Haken.
The Virtually Haken conjecture asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken.
0.179 seconds.