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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If tangent vectors are a vector space of same dimension at every point, does one has a manifold? [closed]

Let $M$ be a non-empty subset of $\mathbb R^n$, $n \geq 2$. Recall that a vector $v$ is tangent to $M$ at the point $m \in M$ if it exists a differentiable curve $\gamma : I \to M$ such that $\gamma(0)...
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Good resources that talk about geodesically convex sets for riemannian manifolds?

Are there any good resources that talk about geodesically convex sets, and in particular convex hulls, for riemannian manifolds? I’ve been wanting to learn more about them, their geometry and ...
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3 votes
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Geometric intuition behind definition of $\delta$-necklike points of the Ricci flow

In "The Ricci Flow: An Introduction", the authors define a $\delta$-necklike point of the Ricci flow as a point $(x, t)$ where $$\|\text{Rm} - R (\theta \otimes \theta)\| \leq \delta \|\text{...
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15 votes
3 answers
919 views

Converse to Hopf degree theorem

Below, I mean smooth oriented closed connected manifolds and smooth maps (but am happy to hear about the topological category, or unoriented manifolds, etc instead). Say that $X^n$ has the Hopf ...
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composition of two tangent vector fields and connection [closed]

Let $X, Y$ be two tangent vector fields on a manifold. Consider their composition: $(X \circ Y)(f) = X(Y(f)) = X^i\frac{\partial Y(f)}{\partial u^i} = X^i\frac{\partial Y^j}{\partial u^i}\frac{\...
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3 votes
1 answer
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Given a Heegaard splitting $M = V\cup_F W$, then $V\setminus N(D_1)$ is ambient isotopic to $V\cup N(D_2)$ for a meridian pair $\{D_1,D_2\}$

I sincerely apologize if MathOverflow is not the appropriate place to ask this question. I also tried consulting M.SE but it seems that this question gained little to no interest . Consider a ...
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14 votes
2 answers
690 views

Must a space that is locally injective image of $\mathbb{R}^n$ be a manifold?

Suppose $X\subseteq\mathbb{R}^m$ s.t. for any $x\in X$ and any open $U\subseteq\mathbb{R}^m$ that contains $x$, there exists a smaller open set $V\subseteq U$ also containing $x$, so that $V\cap X$ is ...
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3 votes
1 answer
237 views

(Homotopy) colimit and manifold

Suppose that I have an arbitrary regular CW complex. By associating a topological space to each vertex of the CW complex, I can have a diagram of topological spaces, denoted by $D$, over the CW ...
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8 votes
1 answer
330 views

Universal cover with one end

Suppose that $M$ is a non-compact manifold of finite topological type with one end which is the universal cover of some closed manifold $N$. Is $M $ necessarily homeomorphic to the total space of some ...
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Persistence of planar trajectory converging to a node / focus

I consider a planar system $\dot u =F(u,p)$ where p is a scalar parameter. Suppose that the flow $\phi^t(u_0; 0)$ from $u_0$ converges to a stable node / focus $x^{eq}_0$ for the parameter value $p=0$....
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21 votes
1 answer
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Is $\mathbb{CP}^3$ minus two points the universal cover of a compact manifold?

After reading some recent questions on mathoverflow about universal coverings, I am curious about the following: Is it possible to construct a closed $6$-manifold $M$, with universal cover ...
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8 votes
1 answer
248 views

Universal covers of non-prime 3-manifolds

Let $M$ be a closed, connected, oriented 3-manifold. If $M$ is prime, then we know what the universal cover of $M$ looks like: it is either $S^3, \mathbb{R}^3$ or $S^2 \times \mathbb{R}$ depending on ...
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11 votes
1 answer
338 views

Is there a closed manifold whose universal cover is $\mathbb{R}^n\setminus\{x_1, \dots, x_k\}$ for some $k > 1$?

There are many closed manifolds with universal cover homotopy equivalent to $\mathbb{R}^n$, they are precisely the closed aspherical manifolds. There are also many closed smooth manifolds with ...
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2 votes
0 answers
54 views

Which quantity could hold its convergence under Gromov-Hausdorff convergence?

Recently I've been reading T.H.Colding's paper of Ricci curvature and volume convergence. A proof of the continuity of volume functions was given under the lower Ric bounded condition. Having searched ...
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6 votes
0 answers
156 views

Sobolev embedding theorems on manifolds

I had asked the following question on math.stackexchange but did not get any response: I'm looking for a reference which states the Sobolev embedding theorems on Riemann manifolds for fractional ...
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9 votes
1 answer
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When does the diagonal cohomology class of a non-compact oriented manifold vanish?

Let $M$ be a non-compact, connected and oriented topological $d$-manifold without boundary. My understanding is that there are two (equivalent) ways of defining the diagonal class $\delta_M \in H^{d}(...
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Can someone explain this proof on aspherical manifolds?

I am trying to understand this proof that the fundamental group of an aspherical manifold is torsion free. The proof is lemma 4.1 from Aspherical manifolds at the Manifold Atlas Project. The proof is: ...
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2 votes
0 answers
104 views

Proper isotopy of proper embeddings of manifolds

We know from Theorem 2.2 in(http://www.map.mpim-bonn.mpg.de/Embeddings_of_manifolds_with_boundary:_classification#6.2) that Any two smooth embeddings of closed oriented $n$-manifold(n>1) in $\...
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2 answers
525 views

Mathematics Roadmap [closed]

I immediately apologize for my English, Google translator is my assistant. I couldn't find the information in my own language. My question is addressed to people who understand mathematics. I hope for ...
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7 votes
1 answer
381 views

Is there a version of the Poincaré–Hopf theorem for manifold with corners?

As we know, the square $S=[0,1]\times[0,1]$ is not a manifold with boundary. Instead, it's a manifold with corners. For a tangent vector field on a compact manifold with boundary, we have the Poincaré–...
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4 votes
1 answer
297 views

On eigenfunctions of the Laplace Beltrami operator [closed]

How can we generate the eigenspace for the Laplace Beltrami operator on SU(2)?
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3 votes
1 answer
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Is every (not necessarily PL-) triangulation of a manifold pure, non-branching and strongly-connected?

A triangulation of a topological manifold $\mathcal{M}$ possibly with boundary is an abstract simplicial complex $\Delta$ together with a homeomorphism $\varphi:\vert\Delta\vert\to\mathcal{M}$, where $...
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3 votes
0 answers
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Analogue of Kolmogorov/Arnold superposition for general manifolds?

Previously asked and bountied at MSE with slightly different language: Given a topological space $\mathcal{X}$, let $$\mathsf{Cl_C}(\mathcal{X})=\bigcup_{n\in\mathbb{N}}C(\mathcal{X}^n,\mathcal{X})$$ ...
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7 votes
1 answer
221 views

Decomposition of manifolds with toroidal boundary

Let $\mathcal{M}$ be a compact, connected, oriented 3-manifolds with non-empty connected boundary $\partial\mathcal{M}$. Then, following this article, it is stated that $\mathcal{M}$ can be written as ...
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13 votes
1 answer
501 views

Classification of 3-dimensional manifolds with boundary

It is well-known that every closed, connected and orientable 3-manifold $\mathcal{M}$ can uniquely be decomposed as $$\mathcal{M}=P_{1}\#\dots\# P_{n}$$ where $P_{i}$ are prime manifolds, i.e. ...
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4 votes
0 answers
279 views

Obstruction of smooth structure

The first 24 lectures of Jacob Lurie on Geometric Topology [1] gave a concise introduction to the comparison of smooth manifolds and piecewise-linear manifold. In the first five lectures, it is shown ...
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1 vote
0 answers
108 views

Equivariant embeddings

According to the article I got from this question, closed Riemann manifolds together with their isometry groups (which is always a compact Lie group) can be smoothly and equivariently embedded into ...
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3 votes
1 answer
185 views

Relation between cohomological dimensions of manifolds

$\DeclareMathOperator\Ch{Ch}$Let $M$ be a connected manifold of finite type. We denote $\Ch_{\mathbb{Q}}(M),$ $\Ch_{\mathbb{Z}}(M)$ and $\Ch_{\mathbb{\pm}\mathbb{Z}}(M)$ by cohomological dimensions of ...
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3 votes
0 answers
181 views

CW-complexes that cannot be homotopically compressed

Definition. A CW-complex $A$ can be elementary compressed to a CW-complex $B$ if there is a deformation retraction $A \to B$ or a quotient map by a contractible subcopmlex $A \to B$ (meaning according ...
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2 votes
0 answers
57 views

Domain of definition of a certain mapping

Suppose we have a compact smooth riemannian manifold $(M,g)$ and a $\mathcal{C}^2$ diffeomorphism $f$ of this manifold. I am studying the mapping $$\tilde{f}(x)= \exp_{f(x)}^{-1} \circ f \circ \exp_x \...
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5 votes
1 answer
270 views

Set of proper homotopy classes of arcs in a manifold

Let $M^n$ be an $n$-manifold with nonempty boundary and let $\partial_0 M$ be a specific connected component of $\partial M$. I am interested in the set of continuous maps $f : [0,1] \to M$ such that $...
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2 votes
0 answers
99 views

Approximating maps by embedding

A continuous map $f$ between two metric spaces is said to be a $r$-map if preimage of each point under $f$ has diameter atmost $r$. Suppose $D^n=\{x\in \mathbb{R}^n\mid ||x||\leq 1\}\subset \mathbb{R}^...
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5 votes
0 answers
356 views

Does there exist a manifold with finitely generated homology groups that is not homotopy equivalent to a compact manifold with boundary?

Does there exist a manifold with finitely generated homology groups that is not homotopy equivalent to a compact manifold with boundary? I am also interested in several variations of this question. ...
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2 votes
1 answer
144 views

A Riemann surface is automatically paracompact

[A question I remember from many years ago.] Definition A Riemann surface is a connected complex manifold $X$ of complex dimension one. This means that $X$ is a connected Hausdorff space that is ...
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3 votes
0 answers
72 views

non-smooth manifold with circle action (with fixed points)

I am interested to know if there a non-smooth manifold (i.e. a closed topological manifold admitting no smooth structure) $M$, having a continuous action $M \times S^1 \rightarrow M$, and the number ...
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3 votes
1 answer
149 views

Open manifolds which have stable $\pi_1$ at infinity but are not inward tame

Let $M$ be a $1$-ended open manifold. An important result of Siebenmann states that (in dimension $\geq6$) if $M$ is $(i)$ inward tame, i.e. for every closed neighborhood of infinity there exists a ...
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11 votes
2 answers
626 views

Can we embed a closed manifold into a homotopy equivalent CW complex?

Suppose $X$ is a CW complex and $M$ is a closed manifold and suppose further that there exists a homotopy equivalence $X \simeq M$. Does there exists an embedding of $M$ into $X$ (i.e. an injective (...
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4 votes
0 answers
134 views

Classifying singularities of the Ricci flow

Context: A solution $(M^n, g(t))$ of the Ricci flow is said to encounter a Type III Singularity if $g(t)$ is defined for all $t \geq 0$ and: $$ \sup _{\mathcal{M}^{n} \times[0, \infty)} \|\...
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12 votes
1 answer
436 views

Are there examples of Einstein manifolds with unbounded curvature?

Are there any known examples of Einstein manifolds $(M, g)$ such that $$\sup_{x \in M} \|\text{Rm}(x) \| = \infty$$ I'm looking for these examples because they might provide a counter-example to a ...
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12 votes
1 answer
383 views

Roadmap for L-Theory

Background: I spent sometime reading about algebraic K-theory and started reading research papers on the subject with relative facility at least I do understand constructions, statements of the ...
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7 votes
0 answers
144 views

Homotopy equivalent cartesian product of closed manifold

I'm little bit lost with the following question: I have four connected closed orientable manifolds $M,N,S,S'$ such that $S$ and $S'$ are homotopy equivalent and $M\times S$ is homotopy equivalent to $...
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22 votes
0 answers
585 views

Do most manifolds have symmetries? or not?

Let us say that a (closed, connected) manifold has a symmetry if it admits a non-trivial action by a finite group. Note that I am not asking the action to be free. So for example rotating the 2-sphere ...
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6 votes
3 answers
460 views

Contractible set in a manifold

Let $M$ be an $n$-dimensional topological closed manifold. Suppose $K$ is a compact subset of $M$ which is contractible in the sense that there exists a continuous map $F:K \times [0,1] \to M$ with $F(...
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9 votes
2 answers
382 views

Is $\operatorname{Spin}(8)$ a direct product of $\operatorname{Spin}(7)$ and $S^7$?

Is $\textrm{Spin}(8)$ a direct product of $\textrm{Spin}(7)$ and $S^7$? I met this statement in the literature, but without a reference. If it is true, where is it explicitly written?
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0 votes
0 answers
64 views

Hypothesis on $W$ to achieve $d(x,F(W))<\sum\epsilon_n$

Writing a paper, I'm trying to formulate the following technical result: Let $X$ be a manifold, $W\subset\subset X$. Let $f_k:U_k\to X$ be continuous, where $U_k\subset X$ is an open neighborhood of $...
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10 votes
2 answers
513 views

Do colimits of manifolds coincide with underlying colimits as topological spaces?

Categories of manifolds (possibly with extra structure) tend not to have all colimits. Other questions have addressed when colimits of manifolds exist. I'd like to know what we can say in general ...
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8 votes
1 answer
552 views

When is a triangulation of sphere two-colorable?

Let $T$ be a triangulation of sphere. We say that $T$ is $k$-colorable if the triangles of $T$ can be assigned with $k$ colors such that any two triangles with a common edge have different colors. I ...
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2 votes
0 answers
105 views

Birth of chaos due to nonautonomous perturbation

Let $\sigma, b>0$. I want to study the dynamics of the map $$ T \colon \mathbb{N} \times \mathbb{S}^1 \times \mathbb{R} \to \mathbb{S}^1 \times \mathbb{R}$$ such that $$T_{\sigma,b}(n,\theta,y) = (\...
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3 votes
2 answers
130 views

Embedded submanifold in a cylinder

Let $M^n$ be an $n$-dimensional topological closed manifold. Suppose there exists an embedding $i:M \to M \times [0,1]$ such that $i(M)$ is contained in the interior of $M \times [0,1]$ and separates $...
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17 votes
2 answers
1k views

Suspension of a topological space

Let $X$ be a topological space such that its suspension is a topological manifold. Can we prove that $X$ itself is a topological manifold?
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