Gromov-Hausdorff limits of closed surfaces (Q6585791)

It follows from a theorem by \textit{S. C. Ferry} and \textit{B. L. Okun} [Proc. Am. Math. Soc. 123, No. 6, 1865--1872 (1995; Zbl 0828.53038)] that for any fixed closed smooth manifold \(M\) of dimension at least three, every length space that is a simply connected compact absolute-neighborhood-retract can be realized as the Gromov-Hausdorff limit of a sequence of length spaces homeomorphic to \(M\). On the contrary, this is no longer the case if the dimension of \(M\) is two or less. This paper investigates what topological spaces can be obtained as such a limit.\N\NA \textit{cactoid} is a compact space \(X\) obtained by joining 2-dimensional spheres in a tree-like fashion (i.e., such that all maximal cyclically connected subspaces of \(X\) are spheres). If other surfaces are also allowed, we then obtain a \textit{generalized cactoid}.\NA theorem of Whyburn shows that a Gromov-Hausdorff limit of 2-dimensional spheres must be a cactoid. A naive generalization of this result fails: if \(M\) is a closed surface of higher connectivity, it is easy to see that a Gromov-Hausdorff limit of \(M\) need not be a cactoid nor a generalized cactoid. In fact, pinching a non-trivial simple closed curve shows that a limit can also contain maximal cyclically connected subsets obtained by identifying two points of some other surface with smaller connectivity. In this paper, the author shows that this is more or less all that can happen.\N\NSpecifically, let \(X\) be a compact length space and \(c\in\mathbb N\) any number. The main theorem of this paper shows that the following are equivalent:\N\N\begin{itemize}\N\item[1.] \(X\) can be obtained as the Gromov-Hausdorff limit of closed length surfaces whose connectivity number is equal to \(c\).\N\item[2.] \(X\) can be obtained by \(k\) successive 2-point identifications of some generalized cactoid \(X'\) such that the sum of the connectivity numbers of the maximal cyclic subsets of \(X'\) is at most \(c - 2k\).\N\end{itemize}\N\NThe proof that 1 implies 2 really shows that the behavior outlined above is the only possible issue. Namely, it shows that if the sequence \(X_n\) converging to \(X\) does not contain non-trivial closed curves of length decreasing to zero, then \(X\) is a generalized cactoid. It then remains to show that if such a sequence of curves does exist, then the limit can be described as a 2-point identification of a simpler object and proceed by induction.\NThe converse implication also relies on a complexity-reduction argument.\N\NThis paper relies on the work of Whyburn and a number of other topological results, which are referenced as needed. It contains various beautifully drawn pictures that help illustrate the main definitions.