English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Brehm and K\"uhnel (1992) constructed three 15-vertex combinatorial8-manifolds `like the quaternionic projective plane' with symmetry groups$\mathrm{A}_5$, $\mathrm{A}_4$, and $\mathrm{S}_3$, respectively. Gorodkov(2016) proved that these three manifolds are in fact PL homeomorphic to$\mathbb{HP}^2$. Note that 15 is the minimal number of vertices of acombinatorial manifold that is not PL homeomorphic to $S^8$. In the presentpaper we construct a lot of new 15-vertex triangulations of $\mathbb{HP}^2$. Asurprising fact is that such examples are found for very different symmetrygroups, including those not in any way related to the group $\mathrm{A}_5$.Namely, we find 19 triangulations with symmetry group $\mathrm{C}_7$, onetriangulation with symmetry group $\mathrm{C}_6\times\mathrm{C}_2$, 14triangulations with symmetry group $\mathrm{C}_6$, 26 triangulations withsymmetry group $\mathrm{C}_5$, one new triangulation with symmetry group$\mathrm{A}_4$, and 11 new triangulations with symmetry group $\mathrm{S}_3$.Further, we obtain the following classification result. We prove that, up toisomorphism, there are exactly 75 traingulations of $\mathbb{HP}^2$ with 15vertices and symmetry group of order at least 4: the three Brehm-K\"uhneltriangulations and the 72 new triangulations listed above. On the other hand,we show that there are plenty of triangulations with symmetry groups$\mathrm{C}_3$ and $\mathrm{C}_2$, as well as the trivial symmetry group.

New examples and partial classification of 15-vertex triangulations of the quaternionic projective plane