# L-class computations

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 [1] 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., [2]) 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.

[1] U. Brehm, W. Kühnel, -vertex triangulations of an -manifold, Math. Ann. 294 (1992) 167–193. https://doi.org/10.1007/BF01934320.

[2] F. Chapoton, L. Manivel, Triangulations and Severi varieties, Exp. Math. 22 (2013) 60–73. https://doi.org/10.1080/10586458.2013.743854.