# Conditions Weaker than Locally Euclidean

This is a very general question, but hopefully some people find it interesting. I'm working in the setting of compact metric spaces, so most of the basic topological properties will be satisfied.

When the space $X$ is also connected, then I know that being locally connected is sufficient to be locally path connected, and is also sufficient for the metric on $X$ to be equivalent to a convex metric. Thus for compact, connected, metric spaces being locally connected is very strong.

But there are also locally connected spaces like dendrites that look very far from manifolds. There are a lot of characterizations of the arc and the circle, and I know that there are a few characterizations in dimension two, since there are the Moore and Bing characterizations of the 2-sphere. But I am wondering if there are general local conditions that are strong enough to imply that a point is contained in a neighborhood that looks like some $\mathbb{R}^n$ or $\mathbb{R}^\omega$, the Hilbert Cube?

Basically, I am looking for some local condition that a point $x \in X$ can satisfy so that, assuming $X$ is a compact, connected, locally connected metric space, it follows that $X$ is locally Euclidean at $x$ - or has a neighborhood homeomorphic to the closed upper half-plane, i.e. looks like a boundary point of a manifold with boundary. I am wondering if any of the following three (plus something) are good enough, or if anyone knows some helpful counterexamples:

1) $X$ is locally contractible (not enough by itself, e.g. dendrite or simple triod)

2) $X$ is locally homogeneous (every point has a local basis of homogeneous open neighborhoods)

3) $X$ is strongly locally homogeneous (every $x$ has a local basis $U_i$ of open neighborhoods such that for every $y \in U_i$ there is a homotopy on $U$ sending $x$ to $y$ composed of homeomorphisms)

3a) $X$ is strongly locally homogeneous on every $n$-point set for every $n$ (i.e. the homotopy can be chosen to send any $n$ distinct points to any other $n$ distinct points).

We could also add homogeneity conditions to the boundaries, or conditions on the extensibility of the local homeomorphisms to boundaries.

Especially curious if there has been progress since the annulus theorem was extended to all dimensions by Moise/Kirby/Quinn, or since the theory of Cantor Manifolds was fully developed. In particular, are any of the above strong enough if the $U_i$ are assumed to be Cantor Manifolds? I think maybe for dimension 2 it might be relevant, at least, but probably the Cantor Manifold property would have to be replaced with some ($n-2$)-dimensional non-separating sets in higher dimensions.

To be honest, besides some classification theorems for surfaces, I really haven't come across much stuff on this local topological aspect. Maybe an algebraic topologist has an explanation for why it's so much harder in high dimensions? This is a paper I found that seems relevant, but the results aren't very satisfying except for surfaces.

https://www.jstor.org/stable/1969469

First of all, you should replace "homotopy" by "isotopy" in your question. Next, you should add "locally compact", to eliminate, say, infinite-dimensional Banach spaces. This is still not enough though. For instance, the Hilbert cube is compact, homogeneous, contractible, locally contractible, etc. However, it is infinite-dimensional (its covering dimension is infinite).

There is a substantial body of literature on "generalized manifolds", which are finite-dimensional homology manifolds satisfying some further conditions. You will find many references on this page maintained by Andrew Ranicky, including Ranicky's MR review of the paper

J.Bryant, S.Ferry, W.Mio, S.Weinberger, Topology of homology manifolds, Annals of Maths. 143, (1996) 435-467.

The authors construct some exotic finite-dimensional "manifold-like" spaces which are not topological manifolds. They conjecture that their spaces are homogeneous.

Edit. The examples are quite hard, unless your specialty is higher-dimensional topology, you will be unable to understand them. A baby version of these examples is the Pontryagin surface, which is a 2-dimensional ${\mathbb Q}$-homology manifold. It is obtained from $S^2$ by taking infinitely many connected sums with $RP^2$ "everywhere": Start with a triangulated $S^2$, then remove a small disk from each 2-simplex and identify its boundary by the antipodal map. Triangulate the resulting surface again and repeat. Continue inductively and then take a direct limit. The result is the Pontryagin surface $\Pi_2$. According to this, Pontryagin surface is homogeneous. It is not locally contractible, of course, so this is only a baby version of the actual example, which is based on a sequence of surgeries on a manifold of dimension $\ge 5$.

• Ok cool, I will check out that paper! Since I'm in the compact setting, all my spaces are locally compact and also separable, so the Hilbert Cube is the only infinite-dimensional vector space I might run into. Also, the Hilbert Cube is fine; I am wondering about local point-set conditions that imply $I^n$ or $I^\omega$. – John Samples Sep 25 '17 at 21:15
• @JohnSamples: Hilbert cube is compact, it is not a vector space. – Moishe Kohan Sep 25 '17 at 21:21
• You know what I mean. Btw, can you share any of those sweet counterexamples? I'm not really an algebraic topologist. – John Samples Sep 25 '17 at 21:24