# Questions tagged [manifolds]

A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.

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### Non-density of continuous functions to interior in set of all continuous functions

Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed with the ...
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### Classification of involutions on $G_{2}$-homogeneous spaces

Are you aware of a systematic classification of involutions on $G_{2}$-homogeneous spaces?
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### Every _______ $d$-manifold has an $S$-structure

I am looking for some analogous nontrivial but known statements and references about statements of the form: Every _______ $d$-manifold has an $S$-structure. Here _______ is a placeholder for ...
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### When is a manifold boundary a deformation retract of its open neighborhood?

For this question a manifold is a locally upper-Euclidean Hausdorff space; paracompactness or second-countability is not assumed, and boundary may be present. Let $M$ be a manifold. What are ...
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### Find a circle that intersects the image of $[0,1]$ in a manifold $\mathcal{M}$ at only 1 point

Let $\gamma : [0,1] \rightarrow \mathcal{M}$ be a continuous map so that $[0,1]$ is homeomorphic to $\gamma([0,1])$, where $\mathcal{M}$ is a manifold (Hausdorff, second countable, and locally ...
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### What is a sufficient set of links in a simplicial complex to represent any PL manifold?

The link of a vertex in a $n$-dimensional simplicial complex is the $(n-1)$-dimensional simplicial complex formed by the $(n-1)$-simplices that together with the vertex span a $n$-simplex. A ...
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### Physical interpretation of the Manifold Hypothesis

Motivation: Most dimensionality reduction algorithms assume that the input data are sampled from a manifold $\mathcal{M}$ whose intrinsic dimension $d$ is much smaller than the ambient dimension $D$. ...
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### Is open subspace of manifold with collared boundary a manifold with collared boundary?

An $n$-manifold (for this question) is a locally upper-$n$-Euclidean Hausdorff space. Hence, a manifold possibly has a boundary and possibly is not paracompact. An $n$-precell (for this question) is a ...
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### Which properties, a warped product manifold $M$, can benefit from in having a complex subspace in its tangent space?

I have a Riemannian warped product manifold $M=B \times_f F$ where $M$ is not compatible with an almost-complex structure $J$, but (for example) $B$ is compatible with an almost-complex structure $J$. ...
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### Collared boundary of a non-metrizable manifold

For this question a manifold-with-boundary is a topological space which is Hausdorff and locally upper-Euclidean. Every metrizable manifold-with-boundary has a collared boundary, as shown in "Locally ...
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### Transitive embedding of the projective plane $\Bbb R P^2$ into the $4$-sphere
Is there an embedding (i.e. injective continuous map) $$\phi:\Bbb R \Bbb P^2\hookrightarrow S^4\subseteq\Bbb R^5$$ of the projective plane $\Bbb R\Bbb P^2$ into the $4$-sphere, that is ...
Is there a closed connected $n$-dimensional topological manifold $M$ ($n\geq 2$) such that $\pi_i(M)\neq 0$ for all $i>0$ and $H_i(M, \mathbb{Z})=0$ for $i\neq 0$, $n$? The manifold $S^1\times S^2$ ...