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I give a quick summary of my proposal for simulating an improvement on quenched QCD with dynamical fermions which interact with the gluon configuration only via the topological index of the latter. It amounts to include only the topological part of the functional determinant into the measure, thereby absorbing a correction factor into the observable. I discuss the prospects of this concept from a study in the massive N f-flavour Schwinger model, where the correction factor is indeed found to be of order O(1). Simulations of lattice QCD have, to date, used the quenched and the partially quenched approximations , where internal fermion loops are omitted or heavily suppressed by giving the dynamical quarks a mass heavier than that of the propagating current quarks. Such an approximation turns out to be necessary, since presently available update algorithms with dynamical fermions tend to slow down rather dramatically if the fermion mass is taken light. From a physical point of view this critical slowing down does not come as a surprise , since it reflects the very nonlocal nature of the fermion determinant. The problem with the quenched and, to a milder extent, the partially quenched approximations, however, is that they introduce unphysical degrees of freedom which persist in low energy observables and render the whole theory sick, if the chiral limit is taken [1,2]. An alternative path towards full QCD is " To-pologically unquenched QCD " [3]. The aim is to simulate full QCD, but in an unconventional way: The functional determinant is split into a " topo-logical factor " , which is included into the measure and a " correction factor " which is absorbed in the observable. The " topological factor " is defined as the determinant of a suitably chosen " standard configuration " which shares with the actual configuration nothing but the overall topological charge. This means that the importance sampling pays attention only to the first (" topologi-cal ") factor in the decomposition det(D / + m) = det(D / std + m) det(D / + m) det(D / std + m). (1) In the language of full QCD this amounts to a statistical guess of the determinant of a configuration on the basis of its topological index. The last factor (i.e. the ratio of the true determinant of the configuration to the guess on the basis of which it has gotten accepted in the " …