# Questions tagged [geometric-topology]

The corpus of tools and results that arose from studying manifold theory using non-algebraic techniques, that is, as opposed to (algebraic-topology). The focus of the field tends to be on special objects/manifolds/complexes and the topological characterisation and classification thereof. A key example is the literature surrounding the Poincaré Conjecture.

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### Question of deck transformation on double cover $\tilde{M}$ of non-orientable manifold $M$.

Suppose $M$ is a non-orientable connected manifold, $\tilde{M}$ is its orientable double cover. $\varphi$ is deck transformation of $\tilde{M}$ with $\varphi \not= id$. $\varphi$ is deck ...
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### Why does a CW-complex consist of *open* cells?

Wikipedia states A CW complex is a Hausdorff space X together with a partition of X into open cells (of perhaps varying dimension) that satisfies two additional properties: For each n-...
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### Complement of tubular neighborhood

Let $M$ be a closed, connected, orientable and embedded surface inside the unit 3-sphere $\mathbb{S}^3$ and consider a small tubular neighborhood $U$ of $M$: U = \{ x \in \mathbb{S}^3 : d(x, M) \...
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### Show that all triangulations of a compact surface are equivalent

I'm having trouble with solving the following question: Let $T_1, T_2$ be two finite triangulations of a compact surface. Show that if $E_{T_1}\cap E_{T_2}$ is a finite set of points, where $E_{T_i}$ ...
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### Manifold with a good exhaustion [duplicate]

Let $M$ a smooth manifold such that $M=\bigcup_{i=1}^{\infty} U_i$ for $U_i \subset U_{i+1}$, where $U_i$ is an open set of $M$ which is diffeomorphic to $\mathbb R^n$. Can we prove that $M$ is ...
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### On the connected sum of a surface with a torus

I am studying the classification of Surfaces, and run into the notion of connected sum. We define it in terms of triangulations. I want to show the following. Let $S$ be a triangulated surface. I want ...
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### Hyperbolic 3-manifolds of finite volume as link complements

This is an improved version of this question (sorry if I wasn't so clear there and sorry if this is well-known and I didn't find the reference). Let $N$ be a hyperbolic 3-manifold of finite volume ...
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### A compact set in $\Bbb R^n$ with smooth boundary is a manifold?

Can a compact set $\Omega \subset \Bbb R^n$ with smooth boundary be considered as a smooth manifold with boundary?(Smooth boundary probably means $\partial\Omega$ is a $n-1$-smooth manifold?) I think ...
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### Vogel's Algorithm - Why can we read braid words from nested coherent Seifert surfaces?

I was reading this paper from R. Goldstein-Rose: http://math.uchicago.edu/~may/REU2017/REUPapers/GoldsteinRose.pdf In Figure 12 it was mentioned that if a Seifert surface is coherent and nested, then ...
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### Understanding a comment by Thurston

In page 359 (right after Theorem 2.3) of the following paper Thurston, William P., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc., New Ser. 6, 357-379 (...
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### Is every hyperbolic 3-manifold of finite volume a link complement in some closed 3-manifold?

The question says all I need to know, but I will try to clarify it a little more. Let $M$ be a compact 3-manifold with nonempty torus boundary such that ${\rm int}(M)$ admits a complete hyperbolic ...
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