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Mark Steinberger


Research interests

Currently, my main interest is in De Rham's Problem, a.k.a. the Nonlinear Similarity Problem.

De Rham's Problem was originally stated for matrices. In this form, it has a particularly clean, elementary statement, which we give in the following link.

De Rham's problem may also be stated in terms of representations of finite groups. (The matrix statement corresponds to the case of cyclic groups.) We first take a moment to discuss representations.

Write Gln(R) for the group (under matrix multiplication) of n by n invertible matrices over the real numbers R. An n-dimensional real representation, rho, of a group G is a group homomorphism

rho: G --> Gln(R).

Thus, for g in G, the matrix rho(g) defines an operator on Euclidean n-space Rn. By the representation space of rho, we mean Rn, with G acting on it via rho as a group of operators.

Two representations, rho and theta, are linearly equivalent (linearly similar) if there is an invertible matrix A such that

A rho(g) A-1 = theta(g)

for all g in G. Representation theory shows that two representations of a finite group are linearly equivalent if and only if for each g in G, the matrices rho(g) and theta(g) have the same trace, so linear equivalence is easy to detect.

Let rho and theta be real representations of dimension n. We say that rho and theta are topologically similar if there is a homeomorphism (i.e., a continuous, 1-1 and onto function whose inverse is continuous)

f: Rn --> Rn

which commutes with the actions of G in the sense that

f(rho(g)x) = theta(g)f(x)

for all g in G and x in Rn. We call such an f a G-homeomorphism from the representation space of rho to the representation space of theta. Note that since f is invertible, this says that, regarding rho(g) and theta(g) as operators, we have

f rho(g) f -1 = theta(g)

for all g in G. (Thus topological similarity means that rho and theta are similar as homomorphisms from G to the group of homeomorphisms from Rn to itself.)

Suppose rho and theta are topologically similar via a G-homeomorphism

f: Rn --> Rn.

If rho and theta are not linearly similar, we say f is a nonlinear similarity.

De Rham's Problem: When does there exist a nonlinear similarity between a given pair of linearly inequivalent representations?

In 1935, de Rham conjectured this couldn't happen. His conjecture was false, and the first counterexamples were found by Cappell and Shaneson in 1979.

Hsiang and Pardon, and, independently, Madsen and Rothenberg, showed in 1980 that nonlinear similarity may only occur for groups whose order is properly divisible by 4.

My own contributions to this problem (including joint work) include the following:

  • A pair of linearly inequivalent representations is nonlinearly similar if and only if their unit spheres are h-cobordant in the category of locally linear G-manifolds [1]. The classification of such h-cobordisms, also given in [1], has been applied in [1], as well as the papers below, to obtain applications regarding the nonlinear similarity problem.

  • Nonlinear similarity begins in dimension 6 [2], i.e., 6 is the minimal dimension in which nonlinear similarity may occur.

  • Nonlinear similarities of cyclic 2-groups are classified in [3].
The six-dimensional examples have an interesting property. By [1], (and by construction) their unit spheres are G-h-cobordant in the category of locally linear actions. But if these G-h-cobordisms had G-smoothings of any kind, then an analysis of the tangent representations and the exponential map would produce five dimensional nonlinear similarities. Since such similarities cannot exist, the study of nonsmoothable G-h-cobordisms of [1] is essential to an understanding of the Nonlinear Similarity Problem.

The above results are based on theoretical results regarding locally linear actions of finite groups on manifolds, as well as on calculations in algebraic K-theory and surgery theory. See also a more technical discussion of surgery theory.

I have always enjoyed calculational mathematics. Earlier in my career, I did some calculations of homology operations for spectra with enriched ring structure [4].

References

[1] M. Steinberger, The equivariant topological s-cobordism theorem, Invent. Math. 91 (1988), 61-104.

[2] S.E. Cappell, J.L. Shaneson, M. Steinberger and J.E. West, Nonlinear similarity begins in dimension six, Amer. J. Math. 111 (1989), 717-752.

[3] S.E. Cappell, J.L. Shaneson, M. Steinberger, S. Weinberger and J.E. West, The classification of nonlinear similarities over Z2r, Bull AMS. 22 (1990), 51-57.

[4] R. Bruner, J.P. May, J. McClure, and M. Steinberger, H_infinity Ring Spectra and their Applications, Lecture Notes in Mathematics vol. 1176, Springer-Verlag, Berlin, 1986.



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