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Balanced presentations of the trivial group and four-dimensional geometry. (English) Zbl 1362.58004
For 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.

##### 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)
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