Geometric topology, infinite-dimensional topology
My research has focused on the topology and symmetries of manifolds of finite and infinite dimensions, and on the related topics of polyhedra, absolute neighborhood retracts, function spaces and spaces of sets.
An example of the interplay between these theories is that manifolds modeled on the Hilbert cube appear naturally in several ways as limits of stabilization processes for finite-dimensional objects, and, unlike standard function space stabilization, retain more of their important properties, e.g. simple homotopy type. Study of the Hilbert cube manifolds has produced several of the initial breakthroughs in introducing control into the homeomorphism theory of finite-dimensional manifolds. This in turn, has been useful in analyzing the failure of the classical matrix algebra to describe equivariant homeomorphisms and homotopy types of manifolds with locally linearizable transformation groups, which in turn has led to new results on the topological classification of linear representations of finite groups. I have been involved in these studies.
Involutions of l2 and s with unique fixed points, Trams. Amer. Math. Soc. 373 (2020), 7327- 7346.
- Absolute retract involutions of Hilbert cubes: fixed point sets of infinite codimension. New York Journal of Mathematics 24 (2018), 261-279.
- Involutions of Hilbert cubes that are hyperspaces of Peano continua, Topology and its Applications, 240 (2018), 238-248.
- Compact group actions that raise dimension to infinity (with A. N. Dranishnikov), Topology and its Applications 80 (1997), 101–114.
- Fibrations and Bundles with Hilbert Cube Manifold Fibers (with H. Torunczyk), Memoirs of the AMS 406, 1989, iv + 75 pp.
- Nonlinear similarity begins in dimension 6 (with S. Cappell, J. Shaneson and M. Steinberger), Amer. J. Math. 111 (1989), 717–752.
- Equivariant h-cobordisms and finiteness obstructions (with M. Steinberger), Bull. AMS (NS) 12 (1985), 217–220.
- Mapping Hilbert cube manifolds to ANR's, Ann. Math. 106 (1977), 1–18.