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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Manifolds as Cauchy completed objects

The category of smooth manifolds (SmoothMfld) can be thought of the Cauchy completion of the category $U$ of open subsets of Euclidean spaces (with smooth maps) [1]. This fact is shocking to me as it ...
Student's user avatar
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18 votes
1 answer
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Smooth curve in $\mathbb{R}^3$ not contained in real analytic surface?

Is there a $C^\infty$-smooth embedding $\gamma : I \to \mathbb{R}^3$ so that there is no real analytic $2$-dimensional submanifold $M \subset \mathbb{R}^3$ with $\gamma(I)\subset M$?
Otis Chodosh's user avatar
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11 votes
1 answer
492 views

Are different categories of manifolds non-equivalent (as abstract categories)?

Consider, for instance, the categories of $C^k$-manifolds, where $k=0,1,2,...,\infty,\omega$. ($C^\omega$ means real analytic.) Are these categories pairwise non-equivalent? Of course, the obviuos ...
igorf's user avatar
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3 votes
1 answer
244 views

Space filling curves

The classic Hahn-Mazurkiewicz theorem has the following consequence: Let $X$ be a compact, connected topological manifold. Then there is a continuous surjective map $f: [0,1] \rightarrow X$. It is ...
Rahul Sarkar's user avatar
4 votes
1 answer
219 views

What is the Freudenthal compactification of a wildly punctured n-sphere?

Let $C$ be a compact and totally-disconnected subspace of the $n$-sphere $\mathbb{S}^n$, where $n\geq 2$. Question: Must the Freudenthal compactification of $\mathbb{S}^n \setminus C$ be homeomorphic ...
Agelos's user avatar
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8 votes
1 answer
859 views

On the Euler characteristic of a Poincaré duality space

Background. Suppose that $M$ is an oriented, connected, closed manifold of dimension $d$ with fundamental class $\mu \in H_d(M;\Bbb Z)$. Let $\Delta : M \to M \times M$ be the diagonal map. Then the ...
John Klein's user avatar
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7 votes
0 answers
288 views

Can every finitely presented group be realized as a fundamental group of a compact four-dimensional smooth submanifold $\mathbb{R}^4$?

Every finitely presented group is realized as a fundamental group of a two-dimensional complex (a simple exercise on Van Kampen's theorem). I was told that a two-dimensional complex can be well ...
Arshak Aivazian's user avatar
0 votes
2 answers
117 views

Request for two articles on conformal transformations

I am looking for two articles for my research purpose. The first one is entitled with "Invariant metrics for groups of conformal transformations" (1993, preprint) by K. R. Gutschera and the ...
Ibrahim's user avatar
4 votes
0 answers
218 views

To what extent is the Nash embedding not unique?

Consider a smooth Nash embedding, $f$, of a Riemannian manifold $Σ$ into Euclidean space $\mathbb R^n$. To what extent is this embedding not unique? It is clear that the set of all such embeddings ...
dennis's user avatar
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13 votes
0 answers
221 views

Examples of manifolds with first nontrivial SW-class in degree 16 or bigger

As a module over the Steenrod algebra, $H^{\ast}(BO;\mathbb F_2) = \mathbb F_2[w_1, w_2, w_3, \dots]$ is generated by $w_{2^t}, t \geq 0$. Thus, the first nontrivial SW-class of any vector bundle $\xi ...
Jens Reinhold's user avatar
7 votes
0 answers
279 views

When does the tangent microbundle of a closed orientable topological $4k$-manifold have a trivial rank 2 subbundle?

$\DeclareMathOperator{\Top}{Top} \DeclareMathOperator{\co}{H}$Let $M$ be a closed orientable connected topological manifold of dimension $4k$ with $k > 1$. It is known (David Frank, On the index of ...
Cihan's user avatar
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1 vote
0 answers
140 views

Bijective continuous map from subset of $\mathbb{R}^n$ to a manifold of dimension $n$

I'd like to know if the following assertion is true or not (if true I'd like an example): There exists a positive integer $n$, and a manifold $M$ of dimension $n$ such that there is no subset $X \...
Rahul Sarkar's user avatar
2 votes
0 answers
186 views

Transition maps between coordinate charts on the Grassmann manifold

Let $\mathbf{Gr}_{n,k}$ be the manifold of $k$-dimensional subspaces of $\mathbb{R}^n$, and let $\mathbf{col}$ be the map that takes a matrix in $\mathbb{R}^{n\times k}$ to its columnspace. The map \...
RedRobin's user avatar
0 votes
0 answers
140 views

Prove that Takens' embedding is a smooth one-to-one map with a smooth inverse

Let $f: \mathcal{M} \rightarrow \mathcal{M}$ be a smooth diffeomorphism and $\phi: \mathcal{M} \rightarrow \mathbb{R}$ be a smooth function, where $\mathcal{M}$ is a $d$-dimensional manifold (which we ...
Mark's user avatar
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2 votes
0 answers
91 views

If a Compact $n$-Manifold Immerses in $\mathbb{R}^{n+1}$ is there a Locally Flat Immersion?

Suppose that $M$ is a compact, topological $n$-manifold and there is a topological immersion (i.e. local embedding) of $M$ into $\mathbb{R}^{n+1}$. Is there necessarily a locally flat immersion of $M$...
John Samples's user avatar
3 votes
0 answers
219 views

The geometry of the group of automorphisms of a manifold

Given a manifold $M$, the group $Aut(M)$ is made of diffeomorphisms $M\to M$. Since the complete vector fields on $M$ form an infinite dimensional Lie algebra, and each generates a 1 dimensional Lie ...
Carles Gelada's user avatar
2 votes
0 answers
212 views

What is the interpretation of Jacobi Identity on sympletic manifold?

Context (pg-321): We have a manifold with an anti symmetric metric tensor/sympletic form $S$ with components in a basis $S_{ab}$ satisfying the property that $$dS=0$$ Where $d$ is the exterior ...
tryst with freedom's user avatar
6 votes
1 answer
178 views

Uniqueness of the set of decomposing spheres in prime decomposition of a 3-manifold

At the end of Section 1.1 of 3-manifold groups it is written that "the decomposing spheres are not unique up to isotopy, but two different sets of decomposing spheres are related by ‘slide ...
LaFede's user avatar
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6 votes
0 answers
123 views

A particular case of the general converse to the preimage (submanifold) theorem

I was thinking whether it would be possible to develop a converse to the preimage theorem in differential topology and I found the following post: When is a submanifold of $\mathbf R^n$ given by ...
geooranalysis's user avatar
8 votes
0 answers
193 views

A modified version of the converse to the Sard's Theorem

When I learned Sard's Theorem in differential topology by myself, I was thinking whether it would be possible to prove a converse version of the theorem. That is to say, can we somehow show that each (...
pureorapplied's user avatar
4 votes
1 answer
237 views

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)...
Héhéhé's user avatar
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4 votes
0 answers
60 views

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 ...
Spencer Kraisler's user avatar
3 votes
0 answers
113 views

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{...
Matheus Andrade's user avatar
16 votes
3 answers
1k 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 ...
Otis Chodosh's user avatar
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3 votes
1 answer
74 views

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 ...
Zest's user avatar
  • 173
14 votes
2 answers
818 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 ...
183orbco3's user avatar
  • 271
2 votes
1 answer
284 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 ...
chriswest's user avatar
8 votes
1 answer
378 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 ...
Nick L's user avatar
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0 votes
0 answers
43 views

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 $p=0$....
herve's user avatar
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21 votes
1 answer
724 views

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 ...
Nick L's user avatar
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8 votes
1 answer
499 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 ...
Minkowski's user avatar
  • 571
11 votes
1 answer
538 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 ...
Michael Albanese's user avatar
2 votes
0 answers
65 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 ...
Fan Gauss's user avatar
6 votes
1 answer
290 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 ...
Guest's user avatar
  • 131
9 votes
1 answer
312 views

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}(...
Cihan's user avatar
  • 1,586
0 votes
0 answers
239 views

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: ...
user3308874's user avatar
3 votes
0 answers
187 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 $\...
Arnold's user avatar
  • 31
0 votes
2 answers
8k 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 ...
Student's user avatar
  • 25
9 votes
1 answer
706 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–...
Ya He's user avatar
  • 93
7 votes
1 answer
885 views

On eigenfunctions of the Laplace Beltrami operator [closed]

How can we generate the eigenspace for the Laplace Beltrami operator on SU(2)?
Lateef's user avatar
  • 91
3 votes
1 answer
198 views

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 $...
B.Hueber's user avatar
  • 987
3 votes
0 answers
138 views

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 $$\mathsf{Cl_C}(\mathcal{X})=\bigcup_{n\in\mathbb{N}}C(\mathcal{X}^n,\mathcal{X})$$ ...
Noah Schweber's user avatar
7 votes
1 answer
317 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 ...
G. Blaickner's user avatar
  • 1,137
13 votes
1 answer
808 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. ...
G. Blaickner's user avatar
  • 1,137
4 votes
0 answers
351 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 ...
Student's user avatar
  • 5,008
3 votes
1 answer
231 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 ...
King Khan's user avatar
  • 173
3 votes
0 answers
205 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 ...
Arshak Aivazian's user avatar
2 votes
0 answers
68 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 \...
Giuseppe Tenaglia's user avatar
5 votes
1 answer
294 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 $...
user101010's user avatar
  • 5,319
2 votes
0 answers
126 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}^...
user429294's user avatar

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