L-class computations

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 $H^\star(f^{-1}(x);\mathbb{Q})$.

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.