Nilpotent spaces: some inequalities on nilpotency degrees

Abstract

Our purpose in this work is to compare the nilpotency degree of the action of π 1 ( X ) {\pi _1}(X) on π n ( X ) ( 2 ≤ n ≤ 7 ) {\pi _n}(X)\quad (2 \leq n \leq 7) with the one of the action of π 1 ( X ) {\pi _1}(X) on H n ( X ~ ) Hn(\tilde X) . We work in the category of the nilpotent spaces (here X ~ \tilde X means the universal cover of X X ). The main point in the proof of the main theorems, which yields such inequalities, is the reiterated use of the Serre spectral sequence.