# Nilpotency of topological groups

A group $$G$$ is said to be nilpotent if $$G$$ has a central series of finite length, that is, a series of normal subgroups $$\{1\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G$$ where $$G_{i+1}/G_i$$ is a subgroup of $$Z(G/G_i)$$.

A (based) topological space $$X$$ is said to be nilpotent if the fundamental group $$\pi_1 X$$ is a nilpotent group, and $$\pi_1 X$$ acts nilpotently on the higher homotopy groups, $$\pi_i X$$ for $$i \geq 2$$, i.e., there is a central series $$\{1\} \triangleleft G_0^i \triangleleft G_1^i \triangleleft \cdots G_n^i = \pi_i X$$ such that the induced action of $$\pi_1X$$ on the quotient $$G_{k+1}^i/G_k^i$$ is trivial for all $$k$$.

Here are the two questions.

Q1. Is there an example of a topological group $$G$$ which is nilpotent as a group, but not nilpotent as a topological space?

Q2. Is there an example of a topological group $$G$$ which is nilpotent as a space, but not nilpotent as a group?

• Aren't all topological groups nilpotent as spaces? (Because $G = \Omega BG$?) – Najib Idrissi Jan 14 at 16:46
• In any case, Q2 is trivial: take a simply connected semi-simple group, e.g. $SU(n)$. – abx Jan 14 at 16:49