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arXiv · 1504.05927

Balanced presentations of the trivial group and four-dimensional geometry

Boris Lishak·Alexander Nabutovsky
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Abstract

We prove that 1) There exist infinitely many non-trivial codimension one "thick" knots in $\mathbb{R}^5$; 2) For each closed four-dimensional smooth manifold $M$ and for each sufficiently small positive $ε$ the set of isometry classes of Riemannian metrics with volume equal to $1$ and injectivity radius greater than $ε$ is disconnected; 3) For each closed four-dimensional $PL$-manifold $M$ and any $m$ there exist arbitrarily large values of $N$ such that some two triangulations of $M$ with $<N$ simplices cannot be connected by any sequence of $<M_m(N)$ bistellar transformations, where $M_m(N)=\exp(\exp(\ldots \exp (N)))$ ($m$ times).

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Boris Lishak, Alexander Nabutovsky. 2015-04-22. Balanced presentations of the trivial group and four-dimensional geometry. https://doi.org/10.1142/s1793525317500042

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