R. H. Bing

Mathematical contributions

Bing's mathematical research focused almost exclusively on 3-manifold theory and the geometric topology of ${\displaystyle {\mathbb{R}}^3}$. He was a powerful problem solver who laid the foundations for several areas of topology, and his papers have continued to serve as sources for major theoretical developments. One notable example was Michael Freedman's use of Bing's shrinking criterion to prove the four-dimensional Poincaré conjecture in 1982.

Early results

Bing established his reputation in June 1945, just one month after completing his dissertation, by solving the Kline sphere characterization problem. This famous, longstanding conjecture stated that a metric continuum in which every simple closed curve separates but for which no pair of points separates the space is homeomorphic to the 2-sphere. When word spread that an unknown young mathematician had settled this old conjecture, some were skeptical; when a famous professor wired Moore asking whether any first-class mathematician had checked the proof, Moore replied, "Yes, Bing had."

In 1948, Bing proved that the pseudo-arc is homogeneous, contradicting a published but erroneous "proof" to the contrary and contradicting most mathematicians' intuition about the pseudo-arc.

Metrization

Around 1950, one of the great unsolved problems in general topology was the problem of giving a topological characterization of the metrizability of topological spaces. In 1951, Bing gave such a characterization in his paper "Metrization of Topological Spaces". Jun-iti Nagata and Yuri Smirnov proved similar, independent results at about the same time, so the result is now known as the Bing–Nagata–Smirnov metrization theorem. This paper has probably been cited more than any other of Bing's works, even though he later became more closely identified with geometric topology.

In this paper, Bing also introduced the important concept of collectionwise normality and proved that a Moore space is metrizable if and only if it is collectionwise normal. He constructed an example of a normal space that is not collectionwise normal, known as "Example G", which became influential in point-set topology.

Geometric topology

Bing's first major paper in geometric topology appeared in 1952 in the Annals of Mathematics. He showed that the double of a solid Alexander horned sphere was the 3-sphere, demonstrating the existence of wild involutions on the 3-sphere with fixed point set equal to a wildly embedded 2-sphere. This meant that the original Smith conjecture needed to be rephrased in a suitable category, and the result jump-started research into crumpled cubes. The proof involved a shrinking method 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.

In 1957 alone, three of Bing's papers appeared in the Annals of Mathematics, concerning decompositions of Euclidean 3-space and the approximation of surfaces by polyhedral surfaces. He later proved that polyhedral approximations can be constructed to lie "mostly" on one side of the surface being approximated.

Poincaré conjecture

Bing was fascinated by the Poincaré conjecture and made several major attempts to prove it, contributing to its reputation as an extremely difficult problem. In 1958, he proved a partial result: a simply connected, closed 3-manifold in which every loop is contained in a 3-ball is homeomorphic to the 3-sphere.

Bing initiated research into what became known as the Property P conjecture, including giving it its name, as a potentially more tractable approach to the Poincaré conjecture. Property P was proven in 2004 as a culmination of work from several areas of mathematics, with some irony, this proof was announced shortly after Grigori Perelman announced his proof of the Poincaré conjecture itself.

Side-approximation theorem

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. In 1959, Bing published an independent proof of the triangulation theorem, which had recently been proved by Edwin E. Moise using more complicated methods.

Notable examples

Along the way, Bing produced many counterexamples with nicknames: "The Bing Sling"—a simple closed curve that pierces no disk (1956); "Bing's Sticky Foot Topology"—a connected countable Hausdorff space (1953); and "Bing's Hooked Rug"—a wild 2-sphere in 3-space that contains no wild arc (1961).

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 be thickened and triangulated to be unshellable, despite topologically being a 3-ball. It appears in various contexts in topology, including 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 ${\displaystyle {\mathbb{R}}^3}$ into points and polygonal arcs. The quotient space ${\displaystyle B}$ is not a manifold, but ${\displaystyle B\times \mathbb{R}}$ is homeomorphic to ${\displaystyle {\mathbb{R}}^4}$.

Career

Bing's solution of the Kline sphere characterization problem brought him offers from Princeton University and the University of Wisconsin–Madison. He chose Wisconsin in 1947, preferring not to work in the shadow of established topologists at Princeton. He remained at Wisconsin for 26 years, except for leaves at the University of Virginia (1949–50), the Institute for Advanced Study (1957–58, 1962–63, 1967), and the University of Texas (1971–72). At Wisconsin he was appointed Rudolph E. Langer Research Professor in 1968 and served as department chairman from 1958 to 1960.

Bing returned permanently to the University of Texas at Austin in 1973 with the goal of building the mathematics department into a top-ten state university program. He was chairman of the department from 1975 to 1977 and retired in 1985 as the Mildred Caldwell and Blaine Perkins Kerr Centennial Professor in Mathematics.

During his years at Wisconsin, Bing directed an effective training program for future topologists; his first-year graduate topology classes sometimes numbered more than 40 students. He directed the Ph.D. dissertations of 35 students.

Published works

• Bing, R. H. (1983), The geometric topology of 3-manifolds, American Mathematical Society Colloquium Publications, vol. 40, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1040-8, MR 0728227
• Bing, R. H. (1988), Collected papers. Vol. 1, 2, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0117-8, MR 0950859