L-class computations

UPDATE: This is solved by Denis Gorodkov in https://arxiv.org/abs/1603.05541

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 [1] 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., [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 L -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 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}) .

Of these, finding the map f has proven to be more involved than I would have hoped; the stabilized copy of X^8 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, 15 -vertex triangulations of an 8 -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.