The ''Hauptvermutung'' of
geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originated i ...
is a now refuted conjecture asking whether any two
triangulations of a
triangulable space
In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic to a simplicial complex are called triangula ...
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
Ernst Steinitz (13 June 1871 – 29 September 1928) was a German mathematician.
Biography
Steinitz was born in Laurahütte (Siemianowice Śląskie), Silesia, Germany (now in Poland), the son of Sigismund Steinitz, a Jewish coal merchant, and ...
and
Heinrich Franz Friedrich Tietze
Heinrich Franz Friedrich Tietze (August 31, 1880 – February 17, 1964) was an Austrian mathematician, famous for the Tietze extension theorem on functions from topological spaces to the real numbers. He also developed the Tietze transformat ...
, but it is now known to be false.
History
The non-
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
version was disproved by
John Milnor
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...
in 1961 using
Reidemeister torsion
In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister–Franz torsion) is a topological invariant of manifolds introduced by Kurt Reidemeister for 3-manifolds and generalized to higher dimensions by and .
Analytic torsion (or Ray– ...
.
The manifold version is true in
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
s
. The cases
and
were
proved by
Tibor Radó
Tibor Radó (June 2, 1895 – December 29, 1965) was a Hungarian mathematician who moved to the United States after World War I.
Biography
Radó was born in Budapest and between 1913 and 1915 attended the Polytechnic Institute, studying civ ...
and
Edwin E. Moise
Edwin Evariste Moise (; December 22, 1918 – December 18, 1998)
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 severa ...
in the 1920s and 1950s, respectively.
An obstruction to the manifold version was formulated by
Andrew Casson
Andrew John Casson FRS (born 1943) is a mathematician, studying geometric topology. Casson is the Philip Schuyler Beebe Professor of Mathematics at Yale University.
Education and Career
Casson was educated at Latymer Upper School and Trinity Co ...
and
Dennis Sullivan
Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Ce ...
in 1967–69 (originally in the
simply-connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space ...
case), using the
Rochlin invariant and the
cohomology group
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
.
In dimension
, a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
of ''m''-dimensional
piecewise linear manifold
In mathematics, a piecewise linear (PL) 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 ...
s has an
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
such that
is
isotopic to a piecewise linear (PL) homeomorphism
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
. In the simply-connected case and with
,
is
homotopic
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
to a PL homeomorphism if and only if