On cell-like resolutions of 2-polyhedra up to special ones
by
Salikhov Konstantin
Moscow State University
Coauthors: Dusan Repovs

We show that two important concepts of geometric topology - cell-like resolutions and special polyherdra - are closely related. A cell-like (resp. collapsible) resolutoin of a polyhedron P is a pair (Q, f) of a polyhedron Q and a PL proper surjective map f:Q --> P with cell-like (resp. collapsible) point-inverses. Our main result is:

Let P be a finite connected polyhedron of dimension at most two, distinct from a point. Then P admits a cell-like (or collapsible) resolution up to a special (in the sense of Matveev) polyhedron Q iff either P is a 2-shpere or (P - P' is a disjoint union of open 2-disks, and P is ``dimensionally homogeneous'')

Here P' is the subgraph of P, consisting of all points having no neighbourhood, homeomorphic to a closed 2-disk. Among the corollaries is a reduction of the Whitehead Conjecture on asphericalness.

Date received: June 15, 1998