Let P be an arbitrary set of primes. The P-nilpotent completion of a group G is defined by the group homomorphism η: G → GP' where GP' = inv lim(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 of an 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 of circles is R-bad, where R is any subring of rationals.
@article{urn:eudml:doc:41311,
title = {P-nilpotent completion is not idempotent.},
journal = {Publicacions Matem\`atiques},
volume = {41},
year = {1997},
pages = {481-487},
mrnumber = {MR1485497},
zbl = {0897.20027},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41311}
}
Tan, Geok Choo. P-nilpotent completion is not idempotent.. Publicacions Matemàtiques, Tome 41 (1997) pp. 481-487. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41311/