On the Hilbert-Smith conjecture and related problems

Dušan Repovš

Institute for Mathematics, Physics and Mechanics Jadranska 19, P.O.Box 2964 University of Ljubljana Ljubljana, Slovenia 1001 dusan.repovs@uni-lj.si

We shall present a survey of results concerning one of the most important conjectures of topology of the previous century, the so-called Hilbert-Smith Conjecture, which asks whether only Lie groups can act effectively on topological manifolds. Equivalently, the conjecture asserts that the group of padic integers A_{p} cannot act effectively on any topological manifold. We shall present some results (including Malešič, Martin, Repovš, Skopenkov, Ščepin) and also some attempts at the conjecture in recent years (including Černavski, Edwards). We shall also list some open problems and open conjectures from this and related areas of geometric topology (including the smooth version of the classical Bing Borsuk Conjecture).