Deformation retracts and the Hochschild homology of polynomial rings

We assume given a ringA with unit, and a subcomplex of the reduced bar complex ofA. We assume that this subcomplex is a deformation retract of the whole complex and thus has homology equal to the Hochschild homology ofA, but it will typically be smaller and easier to calculate with. We use these to construct (accordingly small) deformation retracts for the reduced bar complexes ofA[t] andA[t,t ⁻¹]. WhenA is a Banach algebra, we also do this construction forC ^∞(S¹;A). Partially supported by N.S.F. Grant No. DMS 92-03398.