R. H. Bing

RH Bing (October 20, 1914 – April 28, 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.

Mathematical contributions

Bing's mathematical research was almost exclusively in 3-manifold theory and in particular, the geometric topology of $\mathbb {R} ^{3}$. 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 1948 Bing proved that the pseudo-arc is homogeneous, contradicting a published but erroneous 'proof' to the contrary.

In 1951 he proved results regarding 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. Bing's proof involved a shrinking method which was later developed by Bing and others into a powerful 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. 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 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 is the quotient space obtained from a cellular decomposition 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 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 at 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 the National Academy of Sciences (1965)
• Chairman of Division of Mathematics of the National Research Council (1967-1969)
• United States delegate to the International Mathematical Union (1966, 1978)
• Colloquium Lecturer of the American Mathematical Society (1970)
• Award for Distinguished Service to Mathematics from the 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 Colloquium Publications, vol 40. American Mathematical Society, Providence, RI, 1983. x+238 pp. ISBN 0-8218-1040-5

• The collected papers of R.H. Bing, Vol. 1, 2. Edited and with a preface by Sukhjit Singh, Steve Armentrout and Robert J. Daverman. American Mathematical Society, Providence, RI, 1988. Vol. 1: xxii+886 pp.; Vol. 2: pp. i--xxii and 895--1654. ISBN 0-8218-0117-1

• 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

• R. H. Bing at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F., "R. H. Bing", MacTutor History of Mathematics Archive, University of St Andrews
• Starbird's memoir on Bing
• Memorial Resolution - Univ. of Texas, Austin
• R. H. Bing Papers, 1934-1986 (archive)