Geometric complexity of 4-manifolds

We find a lower bound on how geometrically complicated 4-manifolds might be. Results of this type are, probably, most difficult for the sphere. We find Riemannian metrics (or triangulations) of the sphere which make it look very different from the round $S^4$. More precisely, for each closed four-dimensional smooth manifold $M$ and for each sufficiently small positive $\epsilon$ the set of isometry classes of Riemannian metrics with volume equal to $1$ and injectivity radius greater than $\epsilon$ is disconnected. 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 $<\exp_m(N)$ bistellar transformations, where $\exp_m(N)=\exp(\exp(\ldots\exp(N)))$ ($m$ times).

We construct families of trivial $2$-knots $K_i$ in $\mathbb{R}^4$ such that the maximal complexity of 2-knots in any isotopy connecting $K_i$ with the standard unknot grows faster than a tower of exponentials of any fixed height of the complexity of $K_i$. Here we can either construct $K_i$ as smooth embeddings and measure their complexity as the ropelength (a.k.a the crumpledness) or constrcut $PL$-knots $K_i$, consider isotopies through $PL$-knots, and measure the complexity of a $PL$-knot as the minimal number of flat 2-simplices in its triangulation.