UPDATE: This is solved by Denis Gorodkov in https://arxiv.org/abs/1603.05541
Computation of the total -class would also solve some manifold recognition problems. Here is an example of such a problem and my proposed approach. Brehm and Kühnel in  exhibit a few different combinatorial 15-vertex triangulations of an 8-manifold “like” the quaternionic projective plane . One of these examples is especially symmetric, and likely PL homeomorphic to .
Question. Is there a PL homeomorphism between the 15-vertex complex of Brehm–Kühnel and ?
Despite more recent work (e.g., ) 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 -class by implementing the procedure in [???]. It is perhaps surprising that this can be done effectively. The relevant steps are to
find a simplicial map of nonzero degree,
consider the preimage of some point , and
compute the signature of by computing the cup product pairing on .
Of these, finding the map has proven to be more involved than I would have hoped; the stabilized copy of needs to be subdivided to get a map to a sphere, and this subdivision quickly increases the number of simplexes that need to be stored, in spite of how small the 15-vertex triangulation is at first. More work needs to be done.