Computation of the total $L$-class would also solve some manifold recognition problems. Here is an example of such a problem and my proposed approach. Brehm and Kühnel in [MR1180457] exhibit a few different combinatorial 15-vertex triangulations of an 8-manifold “like” the quaternionic projective plane $\mathbb{H}P^8$. One of these examples $X^8$ is especially symmetric, and likely PL homeomorphic to $\mathbb{H}P^8$.

Question. Is there a PL homeomorphism between the 15-vertex complex of Brehm–Kühnel and $\mathbb{H}P^8$?

Despite more recent work (e.g., [MR3038783]) which has placed these examples in a nice context, this question remains open. I propose answering this question with a direct computation of the rational $L$-class by implementing the procedure in [MR440554]. It is perhaps surprising that this can be done effectively. The relevant steps are to- find a simplicial map $f : X^8 \times S^n \to S^{n+8}$ of nonzero degree,
- consider the preimage $f^{-1}(x)$ of some point $x \in S^{n+8}$, and
- compute the signature of $f^{-1}(x)$ by computing the cup product pairing on $H^\star(f^{-1}(x);\mathbb{Q})$.