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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?

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

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