Questions tagged [manifolds]

For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

In mathematics, a manifold of dimension $n$ is a topological space that near each point resembles $n$-dimensional Euclidean space. More precisely, each point of an $n$-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension $n$. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot.

7815 questions
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Sanity check about Wikipedia definition of differentiable manifold as a locally ringed space

Most textbooks introduce differentiable manifolds via atlases and charts. This has the advantage of being concrete, but the disadvantage that the local coordinates are usually completely irrelevant- the choice of atlas and chart is arbitrary, and…
36
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2 answers

Why maximal atlas

This has been on the back of my mind for one whole semester now. Its possible that in my stupidity I am missing out on something simple. But, here goes: Let $M$ be a topological manifold. Now, even though $C^\infty$-compatibility of charts is not…
Dactyl
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Do all continuous real-valued functions determine the topology?

Let $X$ be a topological space. If I know all the continuous functions from $X$ to $\mathbb R$, will the topology on $X$ be determined? I know the $\mathbb R$ here is somewhat artificial. So if this is wrong, will it be right if $X$ is a topological…
34
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CW complexes and manifolds

What is the strongest known theorem (if any) that classify which manifolds can be built as CW complexes? Thank you guys. This is just a question I thought of during class, obviously not homework...
LASV
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32
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Why do we need Hausdorff-ness in definition of topological manifold?

Suppose $M^n$ is a topological manifold, then $M^n$ locally looks like $\mathbb{R}^n$. $M^n$ is locally Hausdorff, since $\mathbb{R}^n$ is Hausdorff and Hausdorff-ness is a topological invariant. All I want to understand is: 1) Does locally…
Sepideh Bakhoda
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Can every manifold be turned into a Lie group?

I am studying Lie theory and just thought of this random question out of curiosity. Can any manifold be turned into a Lie group? More precisely, given a manifold $G$, can we always construct (or prove the existence of) some smooth map $m:G\times…
WillG
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PDEs on manifold: what changes from Euclidean case?

I know some PDE theory for nice open domains in $\mathbb{R}^n.$ I want to know what the changes are when I switch to other domains like manifolds. For example, do things like Poincare's inequality for $H^1_0$ hold, or like the norms $|\Delta \cdot…
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Are there p-adic manifolds?

Is there anything resembling a manifold on the field of p-adic or complex p-adic fields? If so is there a connection to algebraic geometry as rich as in the reals?
user106581
29
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The boundary of an $n$-manifold is an $n-1$-manifold

The following problem is from the book "Introduction to topological manifolds". Suppose $M$ is an $n$-dimensional manifold with boundary. Show that the boundary of $M$ is an $(n-1)$-dimensional manifold (without boundary) when endowed with the…
Saal Hardali
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When is a fibration a fiber bundle?

In this question I am using Wiki's definitions for fibration and fiber bundle. I want to be general in asking my question, but I am mostly interested in smooth compact manifolds and smooth fibrations and bundle projection between them. Under some…
29
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Moebius band not homeomorphic to Cylinder.

I have been trying to think of a rather basic way of proving this, but it seems a bit elusive. In the case with boundary (looking at them as quotients in $ [0,1] \times [0,1] $), they can be distinguished from the connectedness of the boundary…
Tom
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concrete examples of tangent bundles of smooth manifolds for standard spaces

I'm having trouble visualizing what the topology/atlas of a tangent bundle $TM$ looks like, for a smooth manifold $M$. I know that $$\dim(TM)=2\dim(M).$$ Do the tangent bundles of the following spaces have any "known form", i.e. can be constructed…
Leo
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Motivation behind the definition of a Manifold.

A manifold $M$ of dimension n is a topological space with the following properties: a) $M$ is Hausdorff b)$M$ is locally Euclidean of dimension n c) $M$ has a countable basis of open sets. Why is the first property necessary? I do not have…
Omkar
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Complexified tangent space

Let $M$ be a complex manifold of dimension $n$ and $p\in M$. So $M$ can be viewed as a real manifold of dimension $2n$ and we can consider the usual real tangent space at $p$, $T_{\mathbb{R},p}(M)$, that is the space of $\mathbb{R}$ linear…
gradstudent
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Global sections of a tensor product of vector bundles on a smooth manifold

This question is similar to Conditions such that taking global sections of line bundles commutes with tensor product? and Tensor product of invertible sheaves except that I am concerned with smooth manifolds. If $M$ is a smooth manifold and $V$…
Keenan Kidwell
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