  The notion of a Bing cell is introduced, and it is used to define invariants,
link groups, of 4-manifolds. Bing cells combine some features of both surfaces
and 4-dimensional handlebodies, and the link group \lambda(M) measures certain
aspects of the handle structure of a 4-manifold M. This group is a quotient of
the fundamental group, and examples of manifolds are given with \pi_1(M) not
equal to \lambda(M). The main construction of the paper is a generalization of
the Milnor group, which is used to formulate an obstruction to embeddability of
Bing cells into 4-space. Applications to the A-B slice problem and to the
structure of topological arbiters are discussed.
