  We construct a geometric analog of the analytic surgery group of Higson and
Roe for the assembly mapping for free actions of a group with values in a
Banach algebra completion of the group algebra. We prove that the geometrically
defined group, in analogy with the analytic surgery group, fits into a six term
exact sequence with the assembly mapping and also discuss mappings with domain
the geometric group. In particular, given two finite dimensional unitary
representations of the same rank, we define a map in the spirit of $\eta$-type
invariants from the geometric group (with respect to assembly for the full
group $C^*$-algebra) to the real numbers.
