Science Fair Project Encyclopedia
RH Bing
RH Bing (1914-1986) was an influential American mathematician. He worked mainly in the area of topology, where he made many important contributions. His influence can be seen through the number of mathematicians that can trace their academic lineage through him.
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Mathematical contributions
Bing's mathematician research was almost exclusively in 3-manifold theory and in particular, the geometric topology of 
. The term Bing-type topology was coined to describe 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 1951 he proved results regarding metrizability of topological spaces, including what would later be called the Bing-Nagata-Smirnov Theorem.
In 1952, Bing showed that the double of an Alexander horned ball 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 .
Bing was fascinated by the Poincaré conjecture and made several major attacks which ended unsuccessfully. His failure is a major factor in contributing to the reputation of the conjecture as a very difficult one.
Bing came "close" to proving the conjecture several times, for example, by showing 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. The conjecture can be seen as a more tractable version of the Poincaré conjecture. This was proven recently in 2004 as a culmination of work from several areas of mathematics. Ironically, 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.
Notable Examples
The House with Two Rooms
The House with Two Rooms is a contractible 2-complex that is not collapsible. Another such example, popularized by E.C. Zeeman, is the Duncehat.
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 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 is the quotient space obtained from a cellular decomposition of into points and polygonal arcs. The quotient space, B, is not a manifold, but 
is homeomorphic to 
.
Service and Educational contributions
Bing served as president of the MAA (1963-1964), president of the AMS (1977-78), and was department chair at University of Wisconsin, Madison (1958-1960), and at University of Texas, Austin (1975-1977).
Before entering graduate school to study mathematics, he graduated from Southwest Texas State Teacher's College 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 National Academy of Sciences (1965)
Chairman of Division of Mathematics of the National Research Council (1967-1969)
United States delegate to International Mathematical Union (1966, 1978)
Colloquium Lecturer of American Mathematical Society (1970)
Award for Distinguished Service to Mathematics from Mathematical Association of America (1974)
What does RH stand for?
His father was named Rupert Henry, but Bing's mother apparently thought that "Rupert Henry" was too British for Texas, and compromised by abbreviating it to "RH". Consequently, "RH" did not stand for a first and middle name, and Bing favored writing the "initials" as "RH" instead of "R. H." in order to emphasize this point.
Bing, according to a famous anecdote, would tell people he was named after his uncle. When asked what his uncle's name was, he would answer "RH Bing".
Another anecdote states that when Bing was made professor at Wisconsin, he was asked what name to put on his nameplate. He answered, "R only H only Bing". When he arrived and looked at his door, it said "Ronly Honly Bing".
Published works
The Geometric Topology of 3-Manifolds, American Mathematical Society, 1983, ISBN 0821810405
The Collected Papers of R.H. Bing, American Mathematical Society, ISBN 0821801171
"Topology", Encyclopedia Britannica, ? ed.
References
Singh, S., "R. H. Bing (1914--1986): a tribute", Special volume in honor of R. H. Bing (1914--1986), Topology and Its Applications. 24 (1986), no. 1-3, 5--8.
External Links
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