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A 15-vertex triangulation of the quaternionic projective plane

Denis Gorodkov

Abstract

In 1992, Brehm and Kühnel constructed a 8-dimensional simplicial complex $M^8_{15}$ with 15 vertices as a candidate to be a minimal triangulation of the quaternionic projective plane. They managed to prove that it is a manifold "like a projective plane" in the sense of Eells and Kuiper. However, it was not known until now if this complex is PL homeomorphic (or at least homeomorphic) to $\mathbb{H}P^2$. This problem was reduced to the computation of the first rational Pontryagin class of this combinatorial manifold. Realizing an algorithm due to Gaifullin, we compute the first Pontryagin class of $M^8_{15}$. As a result, we obtain that it is indeed a minimal triangulation of $\mathbb{H}P^2$.

A 15-vertex triangulation of the quaternionic projective plane

Abstract

In 1992, Brehm and Kühnel constructed a 8-dimensional simplicial complex with 15 vertices as a candidate to be a minimal triangulation of the quaternionic projective plane. They managed to prove that it is a manifold "like a projective plane" in the sense of Eells and Kuiper. However, it was not known until now if this complex is PL homeomorphic (or at least homeomorphic) to . This problem was reduced to the computation of the first rational Pontryagin class of this combinatorial manifold. Realizing an algorithm due to Gaifullin, we compute the first Pontryagin class of . As a result, we obtain that it is indeed a minimal triangulation of .

Paper Structure

This paper contains 17 sections, 14 theorems, 27 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Manifolds "like a projective plane"
  4. Brehm--Kühnel complexes
  5. Gaifullin's algorithm of computing the first Pontryagin class
  6. Bistellar moves
  7. The graph $\Gamma_2$
  8. Decomposition of cycles in the graph $\Gamma_2$ into linear combinations of elementary cycles
  9. The case of odd $b$ .
  10. $b=1$.
  11. $b=3$.
  12. $b=5$.
  13. The case of even $b$ .
  14. $b=0.$
  15. $b=2.$
  16. ...and 2 more sections

Key Result

Theorem 1

The first rational Pontryagin class $p_1(M^8_{15})$ is equal to $2u$ where $u$ is the image of one of the two generators of the group $H^4(M^8_{15},\mathbb{Z})\cong \mathbb{Z}$ under the natural embedding $H^4(M^8_{15},\mathbb{Z})\subset H^4(M^8_{15},\mathbb{Q})$.

Figures (5)

  • Figure -8: The changed elementary cycle of type (2a)
  • Figure 0: Now suppose that the diagonal $u_1u_2$ is present in the sphere $L$. This case can be solved in the same way as in case (1) (see figure above). Elementary cycles $\gamma(L,\sigma_1,w_1w_2)$ and $\gamma(L,\sigma_2,w_1w_2)$ are defined. The chain $\beta_1 + \beta_2 + \gamma(L_1,\sigma_1,w_1w_2) - \gamma(L,\sigma_2,w_1w_2)$ can then be represented as a sum of moves with complexity less than $a$ and two moves that can be represented in the desired way according to the precious case. So, we described all the cases when the cycle $\gamma(L_1,\sigma_1,\sigma_2)$ is not defined.
  • Figure 1: Moves in dimension 2
  • Figure 2: Elementary cycles of the first type
  • Figure 3: Elementary cycles of the second type

Theorems & Definitions (24)

  • Theorem 1
  • Remark 2.1
  • Corollary 1
  • proof
  • Corollary 2
  • Proposition 1
  • Definition 3.1
  • Remark 3.1
  • Theorem 2: Eells, Kuiper EK
  • Definition 3.2
  • ...and 14 more