RH Bing
Encyclopedia
R. H. Bing was an American
mathematician
who worked mainly in the areas of geometric topology
and continuum theory. His first two names were just single letters that do not stand for anything: see below.
theory and in particular, the geometric topology
of
. 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
is homogeneous
, 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
was the 3-sphere
. This showed the existence of an involution on the 3-sphere with fixed point set equal to a wildly embedded 2-sphere, which meant that the original Smith conjecture
needed to be phrased in a suitable category. This result also jump-started research into crumpled cubes. The proof involved a method later developed by Bing and others into set of techniques called Bing shrinking
. Proofs of the generalized Schoenflies conjecture and the double suspension theorem
relied on Bing-type shrinking.
Bing was fascinated by the Poincaré conjecture
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 is homeomorphic to the 3-sphere. Bing was responsible for initiating research into the Property P conjecture
, 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
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
, which states that every 3-manifold can be triangulated in an essentially unique way.
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.
is the quotient space
obtained from a cellular decomposition
of
into points and polygonal arcs. The quotient space, 
, is not a manifold, but 
is homeomorphic to 
.
(1963–1964), president of the AMS
(1977–78), and was department chair at University of Wisconsin, Madison (1958–1960), and at University of Texas at Austin
(1975–1977).
Before entering graduate school to study mathematics, Bing graduated from Southwest Texas State Teacher's College (known today as Texas State University-San Marcos), and was a high-school teacher for several years. His interest in education would persist for the rest of his life.
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".
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
who worked mainly in the areas 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 :...
and continuum theory. His first two names were just single letters that do not stand for anything: see below.
Mathematical contributions
Bing's mathematical research was almost exclusively in 3-manifoldManifold
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....
theory and in particular, the 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 :...
of
. 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
Kline sphere characterization
In mathematics, a Kline sphere characterization, named after John Robert Kline, is a topological characterization of a two-dimensional sphere in terms of what sort of subset separates it. Its proof was one of the first notable accomplishments of R.H. Bing....
problem. In 1948 he proved that the pseudo-arc
Pseudo-arc
In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. Pseudo-arc is an arc-like homogeneous continuum. R.H...
is homogeneous
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
, 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
Alexander horned sphere
The Alexander horned sphere is a wild embedding of a sphere into space, discovered by . It is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting with a standard torus:...
was the 3-sphere
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
. This showed the existence of an involution on the 3-sphere with fixed point set equal to a wildly embedded 2-sphere, which meant that the original Smith conjecture
Smith conjecture
In mathematics, the Smith conjecture states that if f is a diffeomorphism of the 3-sphere, of finite order then the fixed point set of f cannot be a nontrivial knot....
needed to be phrased in a suitable category. This result also jump-started research into crumpled cubes. The proof involved a method later developed by Bing and others into set of techniques called Bing shrinking
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 to the space....
. Proofs of the generalized Schoenflies conjecture and the double suspension theorem
Double suspension theorem
In geometric topology, the double suspension theorem of and R. D. Edwards states that the double suspension S2X of a homology sphere X is a topological sphere....
relied on Bing-type shrinking.
Bing was fascinated by the Poincaré conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
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 is homeomorphic to the 3-sphere. Bing was responsible for initiating research into the Property P conjecture
Property P conjecture
In mathematics, the Property P conjecture is a statement about 3-manifolds obtained by Dehn surgery on a knot in the 3-sphere. A knot in the 3-sphere is said to have Property P if every 3-manifold obtained by performing Dehn surgery on the knot is non-simply-connected...
, 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 Perelman
Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...
announced his proof of the Poincaré conjecture.
The side-approximation theorem
Side-approximation theorem
In geometric topology, the side-approximation theorem was proved by . It implies that a 2-sphere in R3 can be approximated by polyhedral 2-spheres....
was considered by Bing to be one of his key discoveries. It has many applications, including a simplified proof of Moise's theorem
Moise's theorem
In geometric topology, a branch of mathematics, Moise's theorem, proved by , states that any topological 3-manifold has an essentially unique piecewise-linear structure and smooth structure....
, which states that every 3-manifold can be triangulated in an essentially unique way.
The house with two rooms
The house with two roomsHouse with two rooms
House with two rooms or Bing's house is a particular contractible 2-complex that is not collapsible.The name was given by R. H. Bing.-External links:*...
is a contractible 2-complex that is not collapsible
Collapse (topology)
In topology, a branch of mathematics, collapse is a concept due to J. H. C. Whitehead.- Definition :Let K be a simplicial complex, and suppose that s is a simplex in K. We say that s has a free face t if t is a face of s and t has no other cofaces. We call a free pair...
. Another such example, popularized by E.C. Zeeman, is the dunce hat
Dunce hat (topology)
For the item of clothing designed to be humiliating, now rarely used, see dunce cap.In topology, the dunce hat is a compact topological space formed by taking a solid triangle and gluing all three sides together, with the orientation of one side reversed...
.
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 spaceDogbone space
In geometric topology, the dogbone space, constructed by , is a quotient space of three-dimensional Euclidean space R3 such that all inverse images of points are points or tame arcs, yet it is not homeomorphic to R3. The name "dogbone space" refers to a fanciful resemblance between some of the...
is the quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
obtained from a cellular decomposition
Cellular decomposition
In geometric topology, a cellular decomposition G of a manifold M is a decomposition of M as the disjoint union of cells.The quotient space M/G has points that correspond to the cells of the decomposition. There is a natural map from M to M/G, which is given the quotient topology. A fundamental...
of
into points and polygonal arcs. The quotient space, , is not a manifold, but is homeomorphic to .Service and educational contributions
Bing served as president of the MAAMathematical Association of America
The Mathematical Association of America 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 and applied mathematicians; computer scientists;...
(1963–1964), president of the AMS
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
(1977–78), and was department chair at University of Wisconsin, Madison (1958–1960), and at University of Texas at Austin
University of Texas at Austin
The University of Texas at Austin is a state research university located in Austin, Texas, USA, and is the flagship institution of the The University of Texas System. Founded in 1883, its campus is located approximately from the Texas State Capitol in Austin...
(1975–1977).
Before entering graduate school to study mathematics, Bing graduated from Southwest Texas State Teacher's College (known today as Texas State University-San Marcos), 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 SciencesUnited States National Academy of SciencesThe National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...
(1965) - Chairman of Division of Mathematics of the National Research Council (1967–1969)
- United States delegate to the International Mathematical UnionInternational Mathematical UnionThe International Mathematical Union is an international non-governmental organisation devoted to international cooperation in the field of mathematics across the world. It is a member of the International Council for Science and supports the International Congress of Mathematicians...
(1966, 1978) - Colloquium Lecturer of the American Mathematical SocietyAmerican Mathematical SocietyThe American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
(1970) - Award for Distinguished Service to Mathematics from the Mathematical Association of AmericaMathematical Association of AmericaThe Mathematical Association of America 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 and applied mathematicians; computer scientists;...
(1974) - Fellow of the American Academy of Arts and SciencesAmerican Academy of Arts and SciencesThe American Academy of Arts and Sciences is an independent policy research center that conducts multidisciplinary studies of complex and emerging problems. The Academy’s elected members are leaders in the academic disciplines, the arts, business, and public affairs.James Bowdoin, John Adams, and...
(1980)
What does R. H. stand for?
His father was named Rupert Henry, but Bing's mother thought that "Rupert Henry" was too British for Texas, and compromised by abbreviating it to R. H. Consequently, R. H. did not stand for a first and middle name.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".