Is 2a¹H-cyclopenta[cd]indene aromatic?
It's a modified form of [10] annulene (https://en.wikipedia.org/wiki/Cyclodecapentaene) where internal bridges are used to hold the outer, conjugated ring in a planar geometry and thus with the $4n+2$ rule that outer ring is aromatic.

You may be more familiar with the methylene bridged version of [10] annulene that works the same way, but with a simpler bridging structure.  See the above mentioned Wikipedia link.

Addendum:

As mentioned in comments, it is overly simplistic to use the $4n+2$ rule.  IUPAC justifies it only for simple annulene systems.  When you have internal bonds in the ring like the compound here, you no longer have the symmetry and degeneracy structure on which the rule is based.

When you have saturated interior atoms surrounded by an outer conjugated ring, the $\pi$ orbitals are only slightly altered and that is why we are tempted to still apply the $4n+2$ rule.  But beware: if we unsaturate any internal atoms, in this case by removing the hydrogen from the central carbon, the $\pi$ orbitals are altered more considerably and the $4n+2$ rule is well past its use-by date.

Maybe in the end we can use the rule as a guide, but ultimately we need to back our conclusions up with hard evidence.  In this case the conclusion of aromaticity is supported by https://doi.org/10.1039/C39800000691, in which the authors studied substituted versions of the compound involved here.  They found proton n m r shifts consistent with an aromatic outer ring.