Balanced presentations of the trivial group and four-dimensional geometry.

*(English)*Zbl 1362.58004For a compact manifold \(M\), if \(I(M)\) is the space of Riemannian structures on \(M\), i.e. of isometry classes of Riemannian metrics on \(M\), endowed with the Gromov-Hausdorff metric \(d_{GH}\), then \(I_\varepsilon(M)\subset I(M)\) is its subset formed by all Riemannian structures such that \(\text{vol}(\mu)=1\) and \(\text{inj}(\mu)\geq\varepsilon\), where \(\text{inj}(\mu)\) denotes the injectivity radius of \(\mu\). In the papers [Commun. Pure Appl. Math. 48, No. 4, 381–428 (1995; Zbl 0845.57023); Geom. Funct. Anal. 6, No. 4, 703–725 (1996; Zbl 0872.53030); Commun. Math. Phys. 181, No. 2, 303–330 (1996; Zbl 0863.57015)], A. Nabutovsky studied properties of isometry classes of Riemannian metrics on a compact manifold \(M\) whose dimension \(n\geq 5\). For example, the author proved that for all sufficiently small \(\varepsilon>0\), \(I_\varepsilon(M)\) is the union of two non-empty subsets \(A\) and \(B\) such that \(d_{\mathrm{GH}}(A,B)>\epsilon/9\). It is also shown that for arbitrarily small \(\varepsilon\), there exist two Riemannian structures \(\mu_1\) and \(\mu_2\) in \(I_\epsilon(M)\) such that any path in the space of Riemannian structures of \(\mathrm{Vol}(M)=1\) from \(\mu_1\) to \(\mu_2\) must pass through Riemannian structures whose injectivity radii are uncontrollably smaller than those of \(\mu_1,\mu_2\).

In this paper, the authors extend Nabutovsky’s results to the case of the dimension four. They show that for any closed four-dimensional Riemannian manifold \(M\), \(I_\varepsilon(M)\) is disconnected, and can be represented as the union of two non-empty subsets \(A_1\), \(A_2\) such that \(d_{GH}(\mu_1,\mu_2)>\frac{\varepsilon}{10}\) for any \(\mu_1\in A_1\), \(\mu_2\in A_2\). Furthermore, there exist \(\mu, \nu\in I_\epsilon(M)\) such that if a sequence \(\mu_1=\mu\), \(\mu_2\),\(\dots\), \(\mu_N=\nu\) of isometry classes of Riemannian metrics on \(M\) of volume one and \(d_{\mathrm{GH}}(\mu_i,\mu_{i+1})\leq\frac{\varepsilon}{10}\) for each \(i\), then \(\inf_i\text{inj}(\mu_i)\leq\frac1{\exp_m(\frac1\varepsilon)}\), where \(\text{inj}(\mu)\) denotes the injectivity radius of \(\mu\) and \(\exp_m(N)=\exp(\exp(\cdots \exp(N)))\) (\(m\) times). Next, they show that if \(M\) is a closed four-dimensional manifold such that either its Euler characteristic is not equal to zero, or its simplicial volume is not equal to zero, then the diameter regarded as a functional on \(AL_1(M)\) has infinitely many local minima, and the set of values of the diameter at its local minima on \(AL_1(M)\) is unbounded. Also, the authors prove that for each four-dimensional closed PL-manifold \(M\) and each positive integer \(m\), there exist arbitrarily large values of \(N\) and two triangulations \(T_1\), \(T_2\) with \(\leq N\) simplices such that \(d_{\text{Bist}}(T_1,T_2)>\exp_m(N)\), in other words, \(T_1\) and \(T_2\) cannot be connected by any sequence of less than \(\exp_m(N)\) bistellar transformations.

In this paper, the authors extend Nabutovsky’s results to the case of the dimension four. They show that for any closed four-dimensional Riemannian manifold \(M\), \(I_\varepsilon(M)\) is disconnected, and can be represented as the union of two non-empty subsets \(A_1\), \(A_2\) such that \(d_{GH}(\mu_1,\mu_2)>\frac{\varepsilon}{10}\) for any \(\mu_1\in A_1\), \(\mu_2\in A_2\). Furthermore, there exist \(\mu, \nu\in I_\epsilon(M)\) such that if a sequence \(\mu_1=\mu\), \(\mu_2\),\(\dots\), \(\mu_N=\nu\) of isometry classes of Riemannian metrics on \(M\) of volume one and \(d_{\mathrm{GH}}(\mu_i,\mu_{i+1})\leq\frac{\varepsilon}{10}\) for each \(i\), then \(\inf_i\text{inj}(\mu_i)\leq\frac1{\exp_m(\frac1\varepsilon)}\), where \(\text{inj}(\mu)\) denotes the injectivity radius of \(\mu\) and \(\exp_m(N)=\exp(\exp(\cdots \exp(N)))\) (\(m\) times). Next, they show that if \(M\) is a closed four-dimensional manifold such that either its Euler characteristic is not equal to zero, or its simplicial volume is not equal to zero, then the diameter regarded as a functional on \(AL_1(M)\) has infinitely many local minima, and the set of values of the diameter at its local minima on \(AL_1(M)\) is unbounded. Also, the authors prove that for each four-dimensional closed PL-manifold \(M\) and each positive integer \(m\), there exist arbitrarily large values of \(N\) and two triangulations \(T_1\), \(T_2\) with \(\leq N\) simplices such that \(d_{\text{Bist}}(T_1,T_2)>\exp_m(N)\), in other words, \(T_1\) and \(T_2\) cannot be connected by any sequence of less than \(\exp_m(N)\) bistellar transformations.

Reviewer: Andrew Bucki (Edmond)

##### MSC:

58D17 | Manifolds of metrics (especially Riemannian) |

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

57Q15 | Triangulating manifolds |

20F05 | Generators, relations, and presentations of groups |

57Q45 | Knots and links in high dimensions (PL-topology) (MSC2010) |

##### Keywords:

“thick” knots; disconnectedness of sublevel sets of Riemannian functions; spaces of triangulations; Baumslag-Gersten group; Andrews-Curtis conjecture; injectivity radius##### References:

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