Purdue Professor Clarence Wilkerson and his co-author,
Professor W. G. Dwyer of Notre Dame University, have in the
last three years solved fundamental problems first raised in the
1930's by pioneers of algebraic topology.
Purdue Professor Clarence Wilkerson and his co-author,
Professor W. G. Dwyer of Notre Dame University, have in the
last three years solved fundamental problems first raised in the
1930's by pioneers of algebraic topology.
Algebraic topology seeks to study the structure of geometric objects by abstracting from the geometry more tractable algebraic information. It is a well-known observation that, from a topological point of view, a donut and a coffee cup have the same "shape" (they each have one hole). It turns out to be very useful, both in mathematics and the study of nature, to extend this idea as widely as possible, and to consider the "shapes" of things that may not seem to be geometric objects at all, such as the set of all possible solutions to an equation in mathematical physics. Since objects like this cannot be visualized, but can be described in computational terms, it is important to have a way of analyzing the "shape" of an object by doing algebraic computations. This is what algebraic topology does: given a suitable description of an object (and many different kinds of descriptions are acceptable to this machine), it calculates "Betti numbers," "cup products," and "Steenrod operations," among other things, which give information about the shape. For example, for an ordinary surface the Betti numbers will predict the number of holes in the surface.
Lie groups are geometric objects modeled on the symmetry properties of geometric figures. Since ancient times (consider, for example, the Platonic regular solids), the importance of the collection of symmetries that preserve the basic properties of a figure have been recognized. For example, for an ordinary square with vertices A,B,C,D, there are symmetries that rotate by 90, 180, and 270 degrees, respectively, plus a flip that changes the order of the vertices to A,D,C,B. These symmetries are each reversible, and any two can be done in succession to yield a third symmetry. In the case of the square, there are a total of 8 distinct symmetries. If one replaces the square by a circular disk, rotation by an arbitrary angle is a symmetry, as well as a flip type symmetry. The properties of invertibility and composition were abstracted by 19th century mathematicians as the concept of a group. Early in the 20th century, S. Lie and others generalized from calculus ideas of differentiation and integration to certain infinite groups, now called Lie groups. Using these analytical tools, E. Cartan and others were able to derive from the differential geometrical properties of Lie groups much simpler algebraic invariants such as Weyl groups, root systems, Dynkin diagrams, and Lie algebras. This algebraic data was then used to completely classify the finite dimensional Lie groups (in the compact case).
By the 1950's it was established that the Lie groups were extremely interesting geometric objects with a pervasive influence on such diverse fields as quantum mechanics, number theory, and algebraic geometry. Lie groups crop up when symmetries are studied--for example, in elementary particle physics where certain Lie groups describe the symmetries of physical laws. Borel and others were able to compute in the 1950's and 1960's much of the algebraic topology of Lie groups by the use of analytically derived invariants.
In the 1930's the pioneer topologist, H. Hopf, had the brilliant insight that many of the important properties of Lie groups can be predicted by knowing only their Betti numbers and other topological invariants. Thus many questions about Lie groups can be answered without knowing all of the details of their precise differential structure if one knows the simpler kind of information provided by algebraic topology. Mathematics is driven by the desire to understand structure, and this sort of insight as to the appropriate level at which the structure arises is fundamental. Hopf introduced what are now called H-spaces as the algebraic topological analogs of Lie groups.
In the 1960's, Steenrod and Stasheff refined Hopf's original definitions and offered the view of H-spaces with classifying spaces as the closest analog one could provide in the more relaxed algebraic topological sense to the rigid geometrical idea of a Lie group. These were expected to provide the appropriate level of abstraction sought by Hopf. Still left open, however, were the questions of whether these H-spaces with classifying spaces are an adequate mirror of the real world of Lie groups, and exactly what H-spaces actually occur.
It is these long outstanding and fundamental questions that Wilkerson and Dwyer have answered, i.e., they have shown that these spaces do serve the function for which they were originally introduced, and they have completely classified these H-spaces. In other words, they have found a complete list of all the (generalized) geometric objects that look the same as a Lie group to the eyes of algebraic topology (i.e., have the same Betti numbers, cup products, etc.). The basic classification is similar to that of compact Lie groups, but in fact, one obtains a different theory for each prime number p. For example, at the prime 2, the building blocks are just the known Lie groups, plus one additional example discovered by Wilkerson and Dwyer. This latter example was the first new H-space at the prime 2 since Hopf's original work and had been mistakenly rejected by other researchers as impossible to construct.
The classification and tools developed by Dwyer and Wilkerson lead to the solution of a wide range of other questions on H-spaces. This work has produced considerable excitement in the algebraic topology community, not only because it completes a line of investigation going back almost 60 years, but also because of its unexpected elegance, and because it is comparatively rare to achieve a complete classification of a class of geometric objects or spaces based on algebraic evidence alone. These classifying spaces are one of the few examples known for which the cohomology algebras give such complete information.