R. H. Bing
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R. H. Bing (October 20, 1914 – April 28, 1986) was an American
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
who worked mainly in the areas 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 ...
and
continuum theory In the mathematical field of point-set topology, a continuum (plural: "continua") is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space. Continuum theory is the branch of topology devoted to the st ...
. His father was named Rupert Henry, but Bing's mother thought that "Rupert Henry" was too British for Texas. She compromised by abbreviating it to R. H. Consequently, R. H. does not stand for a first or middle name.


Mathematical contributions

Bing's mathematical research was almost exclusively in 3-
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 ...
theory and in particular, the
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 ...
of \mathbb R^3. The term Bing-type topology was coined to describe the style of methods used by Bing. Bing established his reputation early on in 1946, soon after completing his Ph.D. dissertation, by solving the Kline sphere characterization problem. In 1948 he proved that the
pseudo-arc In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. The pseudo-arc is an arc-like homogeneous continuum, and played a central role in the classification of homogeneous planar continua. R. H. Bi ...
is
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
, contradicting a published but erroneous 'proof' to the contrary. In 1951 he proved results regarding the metrizability of topological spaces, including what would later be called the Bing–Nagata–Smirnov metrization theorem. In 1952, Bing showed that the double of a solid
Alexander horned sphere The Alexander horned sphere is a pathological object in topology discovered by . Construction The Alexander horned sphere is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting ...
was the
3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...
. This showed the existence of an
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
on the 3-sphere with fixed point set equal to a wildly embedded
2-sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
, which meant that the original
Smith conjecture In mathematics, the Smith conjecture states that if ''f'' is a diffeomorphism of the 3-sphere of Order (group theory), finite order, then the fixed point set of ''f'' cannot be a nontrivial knot (mathematics), knot. showed that a non-trivial or ...
needed to be phrased in a suitable category. This result also jump-started research into
crumpled cubes In geometric topology, a branch of mathematics, a crumpled cube is any space in R3 homeomorphic to a 2-sphere together with its interior. Lininger showed in 1965 that the union Union commonly refers to: * Trade union, an organization of worke ...
. The proof involved a method later developed by Bing and others into set of techniques called
Bing shrinking In geometric topology, a branch of mathematics, the Bing shrinking criterion, introduced by , is a method for showing that a quotient of a space is homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, o ...
. Proofs of the
generalized Schoenflies conjecture In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often referred to as the Jordan–Schoenflies theorem. O ...
and the
double suspension theorem In geometric topology, the double suspension theorem of James W. Cannon () and Robert D. Edwards states that the double suspension ''S''2''X'' of a homology sphere ''X'' is a topological sphere.James W. Cannon, "Σ2 H3 = S5 / G", ''Rocky Mounta ...
relied on Bing-type shrinking. Bing was fascinated by the
Poincaré conjecture In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dim ...
and made several major attacks which ended unsuccessfully, contributing to the reputation of the conjecture as a very difficult one. He did show that a simply connected, closed 3-manifold with the property that every loop was contained in a
3-ball Three-ball (or "3-ball", colloquially) is a folk game of pool played with any three standard pool and . The game is frequently gambled upon. The goal is to () the three object balls in as few shots as possible.
is
homeomorphic 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 ...
to the 3-sphere. Bing was responsible for initiating research into the
Property P conjecture In mathematics, the Property P conjecture is a statement about 3-manifolds obtained by Dehn surgery on a knot (mathematics), knot in the 3-sphere. A knot in the 3-sphere is said to have Property P if every 3-manifold obtained by performing (non-tri ...
, as well as its name, as a potentially more tractable version of the Poincaré conjecture. It was proven in 2004 as a culmination of work from several areas of mathematics. With some irony, this proof was announced some time after
Grigori Perelman Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
announced his proof of the Poincaré conjecture. The side-approximation theorem was considered by Bing to be one of his key discoveries. It has many applications, including a simplified proof of
Moise's theorem In geometric topology, a branch of mathematics, Moise's theorem, proved by Edwin E. Moise in , states that any topological 3-manifold has an essentially unique piecewise-linear structure and smooth structure. The analogue of Moise's theorem in ...
, which states that every 3-manifold can be triangulated in an essentially unique way.


Notable examples


The house with two rooms

The ''
house with two rooms House with two rooms or Bing's house is a particular contractible, 2-dimensional simplicial complex that is not collapsible. The name was given by R. H. Bing.Bing, R. H., ''Some Aspects of the Topology of 3-Manifolds Related to the Poincaré C ...
'' is a contractible 2-complex that is not collapsible. Another such example, popularized by E.C. Zeeman, is the '' dunce hat''. The house with two rooms can also be thickened and then triangulated to be unshellable, despite the thickened house topologically being a 3-ball. The house with two rooms shows up in various ways in topology. For example, it is used in the proof that every compact 3-manifold has a standard spine.


Dogbone space

The ''
dogbone space In geometric topology, the dogbone space, constructed by , is a quotient space of three-dimensional Euclidean space \R^3 such that all inverse images of points are points or tame arcs, yet it is not homeomorphic to \R^3. The name "dogbone spac ...
'' is the quotient space obtained from a
cellular decomposition In geometric topology, a cellular decomposition ''G'' of a manifold ''M'' is a decomposition of ''M'' as the disjoint union of cells (spaces homeomorphic to ''n''-balls ''Bn''). The quotient space ''M''/''G'' has points that correspond to the ce ...
of \mathbb R^3 into points and polygonal arcs. The quotient space, B, is not a manifold, but B \times \mathbb R is homeomorphic to \mathbb R^4.


Service and educational contributions

Bing was a visiting scholar at the
Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...
in 1957–58 and again in 1962–63. Bing served as president of the MAA (1963–1964), president of the
AMS AMS or Ams may refer to: Organizations Companies * Alenia Marconi Systems * American Management Systems * AMS (Advanced Music Systems) * ams AG, semiconductor manufacturer * AMS Pictures * Auxiliary Medical Services Educational institutions * A ...
(1977–78), and was department chair at
University of Wisconsin, Madison A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, t ...
(1958–1960), and at
University of Texas at Austin The University of Texas at Austin (UT Austin, UT, or Texas) is a public research university in Austin, Texas. It was founded in 1883 and is the oldest institution in the University of Texas System. With 40,916 undergraduate students, 11,075 ...
(1975–1977). Before entering graduate school to study mathematics, Bing graduated from Southwest Texas State Teacher's College (known today as
Texas State University Texas State University is a public research university in San Marcos, Texas. Since its establishment in 1899, the university has grown to the second largest university in the Greater Austin metropolitan area and the fifth largest university ...
), and was a high-school teacher for several years. His interest in education would persist for the rest of his life.


Awards and honors

*Member of the
National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nati ...
(1965) *
Lester R. Ford Award Lester is an ancient Anglo-Saxon surname and given name. Notable people and characters with the name include: People Given name * Lester Bangs (1948–1982), American music critic * Lester W. Bentley (1908–1972), American artist from Wisc ...
from the MAA (1965) *Chairman of Division of Mathematics of the National Research Council (1967–1969) *United States delegate to the
International Mathematical Union The International Mathematical Union (IMU) is an international non-governmental organization devoted to international cooperation in the field of mathematics across the world. It is a member of the International Science Council (ISC) and supports ...
(1966, 1978) *Colloquium Lecturer of the
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
(1970) *Award for Distinguished Service to Mathematics from the
Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...
(1974) *Fellow of the
American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and ...
(1980)


What does R. H. stand for?

As mentioned in the introduction, Bing's father was named Rupert Henry, but Bing's mother thought that "Rupert Henry" was too British for Texas. Thus she compromised by abbreviating it to R. H. It is told that once Bing was applying for a visa and was requested not to use initials. He explained that his name was really "R-only H-only Bing", and ended up receiving a visa made out to "Ronly Honly Bing". Krantz 2002: page 34


Published works

* *


References


Sources

* * *


External links

* *
MAA presidents: R. H. BingNational Academy of Sciences Biographical MemoirR.H. Bing
at
Texas State University Texas State University is a public research university in San Marcos, Texas. Since its establishment in 1899, the university has grown to the second largest university in the Greater Austin metropolitan area and the fifth largest university ...
{{DEFAULTSORT:Bing, RH 1914 births 1986 deaths 20th-century American mathematicians Fellows of the American Academy of Arts and Sciences Members of the United States National Academy of Sciences Presidents of the American Mathematical Society Presidents of the Mathematical Association of America Topologists Institute for Advanced Study visiting scholars University of Texas at Austin alumni University of Texas at Austin faculty People from Oakwood, Texas Mathematicians from Texas