Abstract
Let P be an arbitrary set ofprimes. The P-nilpotent completion ofa group G is defined by the group homomorphism η : G → GP where GP = invlim(G/ΓiG)P . Here Γ2G is the commutator subgroup [G, G] and ΓiG the subgroup [G, Γi−1G] when i > 2. In this paper, we prove that P-nilpotent completion ofan infinitely generated free group F does not induce an isomorphism on the first homology group with ZP coefficients. Hence, P-nilpotent completion is not idempotent. Another important consequence of the result in homotopy theory (as in [4]) is that any infinite wedge ofcircles is R-bad, where R is any subring ofrationals.
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