  Based on the analogies between mapping class groups and absolute Galois
groups, we introduce an arithmetic pro-$\ell$ analogue of Orr invariants for a
Galois element associated with Galois action on \'etale fundamental groups of
punctured projective lines. At the same time, we also introduce pro-$\ell$ Orr
space as an arithmetic analogue of Orr space whose third homotopy group is a
target group of Orr invariant. We then determine its rank as
$\mathbb{Z}_{\ell}$-module following Igusa-Orr's computation. Moreover, we
investigate its relation with Ellenberg's obstruction to $\pi_1$-sections
associated with lower central series filtration in the context of
Grothendieck's section conjecture.
