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Flexihedra

Take the infinite ladder surface, a 2-manifold with 2 ends and infinitely generated $H_1$. One description, an embedding in $\mathbb{R}^3$, is the boundary of an $\epsilon$-neighborhood of the ladder …

I will assume you mean that $U$ is a "deleted" neighborhood, i.e. that it does not contain $S$, since that is the more general assumption. Then there are two equivalence classes: one represented by th …

Is Eberlein's theorem relevant to your question? If $M$ is a compact Riemannian manifold of nonpositive section curvature, Eberlein's theorem MR0674166 characterizes when $M$ is a Riemannian symmetric …

One way to distinguish points of a hyperbolic surface $S$ is to show that the local geometry of the Voronoi graph of a point $p$ is changed when $p$ is perturbed. The Voronoi graph is a connected 1-co …

You can avoid the Kunneth formula, "all" you need is the existence of a single nontrivial homology group of $M$ in positive dimensions, namely $H_{\text{dim} \\, M}(M;\mathbb{Z}/2\mathbb{Z}) = \mathbb …

Let me address your third question, about the Thurston geometries. What I have to say is covered in standard references such as Peter Scott's article "The geometries of 3-manifolds". Thurston's resul …

I don't know about the specific sums you suggest, but here are some well established alternatives. Try Kaimanovich's paper "The Poisson boundary of hyperbolic groups", which is about boundaries arisin …

There are negatively curved, complete cylinders whose ends are asymptotic to parallel planes; I've seen lots of pictures of these but don't have one available. Or even more simply, a one-sheeted hype …

Let $C \subset S^1 \subset \mathbb{R}^2$ be a Cantor set. Let $H_C$ be its convex hull, the smallest closed subset of $\mathbb{R}^2$ containing $C$. Then $H_C$ is not a manifold with corners. Its boun …

I doubt there can be a simple answer to the first question. Even the analogous question regarding $S^1$ has no simple answer. According to work of Denjoy, given a homeomorphism $f : S^1 \to S^1$ with …

I like to think that compact 3-manifolds will not be known until we know "the list" of compact, oriented, hyperbolic 3-manifolds, in the way that we know "the list" of compact, oriented, hyperbolic su …

Orbifold Euler characteristic satisfies the same formula with respect to orbifold coverings that ordinary Euler characteristic satisfies for ordinary coverings: the Euler characteristic of the cover e …

There is a ``quasi-isometric rigidity'' context in which a finite kernel arises naturally. I am not sure how general this is so let me state it in the narrower context of constant sectional curvature …

In the topological category the proof that connected sum is well-defined depends on the Annulus Theorem, first proved by Kirby; the necessity of the Annulus Theorem is seen from Bruno Martelli's answe …