# 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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### Existence of a cone neighbourhood in an open disc with a point in the border

I started reading C. P. Rourke and B. J. Sanderson’s “Introduction to Piecewise-Linear Topology” which is recommended by D. Rolfsen in “Knots and Links”, they provide the following definition: 1.1 A ...
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### Finite subcover of an open cover with common boundary point

Let $S$ be a set of open sets. Suppose all open sets in $S$ have a common boundary point. That is, $\exists$ a point $p$ such that $\forall s\in S$, $p$ is a boundary point of $s$. It is known that $S$...
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### Can the ambient isotopy connecting two smooth embeddings close to each other be taken close to the identity in the Whitney topology?

Let $M,N$ be smooth manifolds, of which $M$ is compact. It is a well known result that for any embedding $f:M\to N$ there is some neighborhood $\mathcal{W}$ of $f$ in the Whitney topology such that ...
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### Prove that every manifold is paracompact

Following Lee's book on smooth manifolds. I'm trying to understand the proof of the theorem (Every manifold is paracompact) and there are some topological claims that i don't understand. I've marked ...
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### Existence of connected neighborhood in product via projection

Definition: Let $X$ be a space, a neighborhood of a subset $A$ of $X$ is a set $N$ containing an open set (in $X$) that contains $A$. That is, a neighborhood need not be open. Assume $Y'$ is a ...
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### Local to global theorem for quasi isometry

I am looking for a local - to - global principle in quasi-isometry. Suppose $Z$ and $X$ are proper, geodesic, hyperbolic metric spaces such there is a local quasi-isometry $f: Z \to X$. That it for ...
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### Is there a complete set of topological invariants for smooth 3-manifolds?

Does there exist a complete set of topological invariants for smooth 3-manifolds? (If not for all smooth 3-manifolds, maybe for a certain class?) By equipping the manifold with a metric, can some ...
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Fix $d\in\mathbb{N}$. Consider the following sets as topological spaces with the subspace topology from $\mathbb{R}^{d+1}$. $$S^d = \{ (x_0,\ldots,x_d)\in\mathbb{R}^{d+1}\mid \sum x_i^2 = 1\}$$ $$D^{... • 133 1 vote 1 answer 36 views ### Inclusion of continuous functions with compact open topology into product topology is continuous Let X,Y be topological spaces and C(X,Y) the set of continuous functions from X to Y equipped with the compact open topology. It has a subbase consisting of sets$$V(K,U):=\{f\in C(X,Y)\ |\ f(...
I am studying spin structures on $SO(n)$-bundles using some lecture notes. Right after defining the twist of a spin structure $(P,\psi,\rho)$ on $Q\xrightarrow{} X$ by a double cover \$\pi:R\...