Введение
Опровергнутое предположение геометрической топологии. Hauptvermutung геометрической топологии — это теперь опровергнутое предположение, утверждавшее, что любые две триангуляции триангулируемого пространства имеют подразделения, которые комбинаторно эквивалентны, то есть подразделенные триангуляции построены по одной и той же комбинаторной схеме. Оно было впервые сформулировано как предположение в 1908 году Эрнстом Штайницем и Генрихом Францем Фридрихом Титце, но сейчас известно, что оно неверно.
The Hauptvermutung of geometric topology is a now refuted conjecture asking whether any two triangulations of a triangulable space have subdivisions that are combinatorially equivalent, i. e. the subdivided triangulations are built up in the same combinatorial pattern. It was originally formulated as a conjecture in 1908 by Ernst Steinitz and Heinrich Franz Friedrich Tietze, but it is now known to be false.
История
Неразмноженная версия была опровергнута Джоном Милнором в 1961 году с использованием торсии Ридемейстера. Многомерная версия верна в размерностях . Случаи и были доказаны Тибором Радо и Эдвином Э. Моисом в 1920-х и 1950-х годах соответственно. Обструкция для многообразной версии была сформулирована Эндрю Кассоном и Деннисом Салливаном в 1967–69 годах (первоначально для простосвязного случая) с использованием инварианта Рохлина и группы когомологий .
In dimension , a homeomorphism of m dimensional piecewise linear manifolds has an invariant such that is isotopic to a piecewise linear (PL) homeomorphism if and only if In the simply connected case and with , is homotopic to a PL homeomorphism if and only if
This quantity is now seen as a relative version of the triangulation obstruction of Robion Kirby and Laurent C. Siebenmann, obtained in 1970. The Kirby–Siebenmann obstruction is defined for any compact m dimensional topological manifold M
again using the Rochlin invariant. For , the manifold M has a PL structure (i. e., it can be triangulated by a PL manifold) if and only if , and if this obstruction is 0, the PL structures are parametrized by In particular there are only a finite number of essentially distinct PL structures on M.
For compact simply connected manifolds of dimension 4, Simon Donaldson found examples with an infinite number of inequivalent PL structures, and Michael Freedman found the E8 manifold which not only has no PL structure, but (by work of Casson) is not even homeomorphic to a simplicial complex. In 2013, Ciprian Manolescu proved that there exist compact topological manifolds of dimension 5 (and hence of any dimension greater than 5) that are not homeomorphic to a simplicial complex. Thus Casson's example illustrates a more general phenomenon that is not merely limited to dimension 4.
В размерности , гомеоморфизм м-мерного кусочно-линейного многообразия имеет инвариант , такой что является изотопным к кусочно-линейному (PL) гомеоморфизму тогда и только тогда, если . В простосвязном случае и при , является гомотопным к PL-гомеоморфизму тогда и только тогда, если .
In dimension , a homeomorphism of m dimensional piecewise linear manifolds has an invariant such that is isotopic to a piecewise linear (PL) homeomorphism if and only if In the simply connected case and with , is homotopic to a PL homeomorphism if and only if
This quantity is now seen as a relative version of the triangulation obstruction of Robion Kirby and Laurent C. Siebenmann, obtained in 1970. The Kirby–Siebenmann obstruction is defined for any compact m dimensional topological manifold M
again using the Rochlin invariant. For , the manifold M has a PL structure (i. e., it can be triangulated by a PL manifold) if and only if , and if this obstruction is 0, the PL structures are parametrized by In particular there are only a finite number of essentially distinct PL structures on M.
For compact simply connected manifolds of dimension 4, Simon Donaldson found examples with an infinite number of inequivalent PL structures, and Michael Freedman found the E8 manifold which not only has no PL structure, but (by work of Casson) is not even homeomorphic to a simplicial complex. In 2013, Ciprian Manolescu proved that there exist compact topological manifolds of dimension 5 (and hence of any dimension greater than 5) that are not homeomorphic to a simplicial complex. Thus Casson's example illustrates a more general phenomenon that is not merely limited to dimension 4.
Эта величина теперь рассматривается как относительная версия триангуляционной обструкции Робиона Кирби и Лорана Сибенмана, полученной в 1970 году. Обструкция Кирби–Сибенмана определена для любого компактного м-мерного топологического многообразия M, вновь используя инвариант Рохлина. Для , многообразие M имеет PL-структуру (т.е. его можно триангулировать PL-многообразием) тогда и только тогда, если , и если эта обструкция равна 0, PL-структуры параметризуются . В частности, существует лишь конечное число по существу различных PL-структур на M.
In dimension , a homeomorphism of m dimensional piecewise linear manifolds has an invariant such that is isotopic to a piecewise linear (PL) homeomorphism if and only if In the simply connected case and with , is homotopic to a PL homeomorphism if and only if
This quantity is now seen as a relative version of the triangulation obstruction of Robion Kirby and Laurent C. Siebenmann, obtained in 1970. The Kirby–Siebenmann obstruction is defined for any compact m dimensional topological manifold M
again using the Rochlin invariant. For , the manifold M has a PL structure (i. e., it can be triangulated by a PL manifold) if and only if , and if this obstruction is 0, the PL structures are parametrized by In particular there are only a finite number of essentially distinct PL structures on M.
For compact simply connected manifolds of dimension 4, Simon Donaldson found examples with an infinite number of inequivalent PL structures, and Michael Freedman found the E8 manifold which not only has no PL structure, but (by work of Casson) is not even homeomorphic to a simplicial complex. In 2013, Ciprian Manolescu proved that there exist compact topological manifolds of dimension 5 (and hence of any dimension greater than 5) that are not homeomorphic to a simplicial complex. Thus Casson's example illustrates a more general phenomenon that is not merely limited to dimension 4.
Для компактных простосвязных многообразий размерности 4 Саймон Дональдсон нашёл примеры с бесконечным числом неэквивалентных PL-структур, а Майкл Фридман нашёл многообразие E8, которое не только не имеет PL-структуры, но (по работам Кассона) даже не гомеоморфно симплициальному комплексу. В 2013 году Сиприан Манолеску доказал, что существуют компактные топологические многообразия размерности 5 (и, следовательно, любой размерности, большей 5), которые не гомеоморфны симплициальному комплексу. Таким образом, пример Кассона иллюстрирует более общее явление, которое не ограничивается только размерностью 4.
In dimension , a homeomorphism of m dimensional piecewise linear manifolds has an invariant such that is isotopic to a piecewise linear (PL) homeomorphism if and only if In the simply connected case and with , is homotopic to a PL homeomorphism if and only if
This quantity is now seen as a relative version of the triangulation obstruction of Robion Kirby and Laurent C. Siebenmann, obtained in 1970. The Kirby–Siebenmann obstruction is defined for any compact m dimensional topological manifold M
again using the Rochlin invariant. For , the manifold M has a PL structure (i. e., it can be triangulated by a PL manifold) if and only if , and if this obstruction is 0, the PL structures are parametrized by In particular there are only a finite number of essentially distinct PL structures on M.
For compact simply connected manifolds of dimension 4, Simon Donaldson found examples with an infinite number of inequivalent PL structures, and Michael Freedman found the E8 manifold which not only has no PL structure, but (by work of Casson) is not even homeomorphic to a simplicial complex. In 2013, Ciprian Manolescu proved that there exist compact topological manifolds of dimension 5 (and hence of any dimension greater than 5) that are not homeomorphic to a simplicial complex. Thus Casson's example illustrates a more general phenomenon that is not merely limited to dimension 4.