# 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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### Given an embedded disk in $\mathbb{R}^n$, is there always another disk which intersects it nontrivially in a disk?

We call an open subset $D\subset X$ of a manifold $X$ an embedded disk, if there exists a homeomorphism $D\cong \mathbb{R}^n$.
The precise formulation of the question in the title is as follows:
Let $...

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### Is $\left|\frac{\det(A_{\mu\nu})}{\det(B_{\mu\nu})}\right|$ an invariant for two tensors $A_{\mu\nu}$ and $B_{\mu\nu}$ in a manifold?

I was doing some math around the determinant of 2nd-order covariant tensors. In a general $n$-dimensional manifold, I deduced that the determinant of a tensor $A_{\mu\nu}$ can be defined as
$$
\det(A_{...

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### Homeomorphism groups on manifolds and topological properties

Let $M$ be a compact $n$-dimensional manifold let $H(M)$ denote the homeomorphism group of $M$.
If $n=2$ then $H(M)$ enjoys nice properties such as being an ANR, is locally contractible, separable. ...

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### Quotient by freely acting group on Banach manifold

I have a Banach manifold $\mathcal{M}$ and I have a Lie group $G$, that is finite dimensional, such that $G$ acts freely on $\mathcal{M}$. I would like to know if $\mathcal{M} / G$ is a Banach ...

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### A detail in Brown's proof of the generalized Schoenflies theorem

Consider a homeomorphic embedding $h:S^{n-1}\times [0,1]\rightarrow S^n$ and denote
$$S^{n-1}_t=h(S^{n-1}\times \{t\}).$$
The generalized Schoenflies theorem states the closure of each connected ...

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### Detecting topology change of tubular neighbourhoods via smoothness of volume function

Let $M$ be an embedded closed manifold in $\mathbb R^n$, define $M_r=\{x\in\mathbb R^n:d(x,M)<r\}$.
Define $r\in\mathcal S_M$ iff $M_r\subset M_{r+\epsilon}$ is not a homotopy equivalence for all ...

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### Mappings of reducible 3 manifolds with boundary

In section 3 of his paper "Mappings of reducible 3 manifolds" McCullough, proves that every self-homeomorphism of a reducible 3 manifold can up to isotopy be written as a composition of ...

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### Reference request - Fibrations between spaces of embeddings

This is a cross-post of this question from MSE.
Given topological manifolds $M$ and $N$ of the same dimension, let $\operatorname{Emb}(M,N)$ denote the subspace of $\operatorname{Map}(M,N)$ ...

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### Is this limit a tangent vector? [closed]

Let $M$ be a smooth compact sub-manifold of $\mathbb R^d$. Let $p\in M$ and $x_n,y_n \in M$ be sequences such that $x_n,y_n\rightarrow p$. Does the following hold when passing to a convergent sub-...

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### Lower bounds for Betti numbers of a manifold given its boundary?

Let $B$ be some compact, path connected $n$-manifold without boundary such that its cobordism class is trivial, so that there exists some other $n+1$ manifold $M$ with $\partial M= B$. While there is ...

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### Is $\pi_m(M) = 0$ if $\pi_m(M-X) = 0$ for a low-dimensional subset $X$?

I am doing a problem where I am stuck at this point.
Let $M$ be a connected smooth manifold of dimension $n$ and let $X$ be any subset of $M$. Assume that there is a positive integer $m$ such that $n&...

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### When a support of an isotopy is disjoint from a subset

Let $M$ be a compact connected manifold, $X\subset M$ a closed subset, and $f:M \times [0;1] \to M$ an isotopy such that each $f_t:M \to M$ is fixed on some open neighborhood $N_t$ of $X$, but there ...

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### Determinant of SU(N) elements, and radius of associated manifold

I'm wondering if the fact $SU(2)$ group elements have $det = 1$ is connected with the radius of the unitary $S^{3}$ manifold associated.
The context is demonstration of dU being an Haar invariant ...

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### Question about stable manifold theorem and Frobenius integrability theorem

I have a question about Anosov diffeomorphism (Wikipedia: Anosov diffeomorphisms)
For hyperbolic fixed point $p$, $W^{s}(p)$ is a smooth manifold and its tangent space has the same dimension as the ...

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### Can vector fields in manifolds with corner and sharp edges still satisfy Poincare-Hopf theorem?

We know that the sum of singularity index of the vector fields on a sphere equal to Euler characteristics of sphere, satisfying the Poincare-Hopf theorem. But how about situations of the geometry with ...

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### Does every triangulable manifold have a vertex-transitive triangulation?

Does every triangulable manifold have a vertex-transitive triangulation?
When I talk about a vertex-transitive triangulation of a manifold, I mean in the sense of realizing a manifold homeomorphically ...

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### (When) can you embed a closed map with finite discrete fibers into a (branched) cover?

Assume all spaces are topological manifolds. A branched cover is a continuous open map with discrete fibers. A finite branched cover is one with finite fibers.
Questions. Given closed map $X\to S$ ...

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### If there exists a function on a Riemannian manifold such that its Hessian matrix is the identity matrix?

In Euclidean space $\mathbb{R}^n$, $n\geq 2$, the Hessian matrix of the function $\frac{|x|^2}{2}$ is the identity matrix. While on a smooth manifold $(M^n, g)$, do there exists a function on $(M^n, g)...

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### Why does the tangent classifier classify the tangent (micro)bundle?

Let $\mathcal{M}\mathrm{fld}_n$ denote the $\infty$-category of topological manifolds (without boundary) and embeddings; more precisely, it is the homotopy coherent nerve of the simplicial category ...

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### Lens space bounding a topological, simply-connected 4-manifold with $b_2=1$

The following is written in section 1.6 (p.7) of this paper: https://arxiv.org/pdf/1010.6257.pdf.
($\cdots$) Which lens spaces bound a smooth, simply-connected 4-manifold $W$ with $b_2(W)=1$? ($\cdots$...

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### Non-compact three-manifolds with the same proper homotopy type are homeomorphic?

I am looking for some literature with some (counter) examples of the following fact (though I don't know if the fact is true or not):
Let $M, M'$ be two non-compact connected $3$-manifolds with the ...

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### Question about canonical divergence on a dually flat manifold

I am reading "Methods of Information geometry by Shun-Ichi-Amari" (chapter 3 sec 3.4) and I am stuck here, can someone explain or give any resource about how we got equation $(3.53)$?

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### The fundamental group of the complement of badly embedded open $n$-ball in $\Bbb R^n$

Let $\mathcal D^n$ be an open subset of $\Bbb R^n$ such that $\mathcal
D^n$ is homeomorphic to $\{x\in \Bbb R^n:|x|<1\}$. Suppose $\Bbb R^n\setminus \mathcal D^n$ is path-connected. How bad can
$\...

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### Obstruction to finding a framing for quotient manifolds

The question is rather open-ended but I hope it is concrete enough.
If $M$ be a closed parallelizable smooth manifold with a smooth properly discontinuous co-compact action of a Lie group $G,$ what ...

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### Is there a flat manifold with trivial first homology?

Is there a closed flat manifold whose fundamental group has trivial abelianization?
The famous Hantzsche–Wendt flat manifold has fundamental group with finite abelianization.

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### Stable torus that is not a torus [duplicate]

Let $M$ be a closed manifold such that $M\times \mathbb{S}^1$ is a torus.
Is it true that $M$ is homeomorphic to a torus?

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### Finite domination and Poincaré duality spaces

Here are some definitions:
A space is homotopy finite if it is homotopy equivalent to a finite CW complex.
A space finitely dominated if it is a retract of a homotopy finite space.
A space $X$ is a ...

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### Do rings of smooth functions differ from rings of continuous functions?

Let $M$, $N$ be connected nondiscrete compact smooth manifolds. Can the ring of continuous functions on $M$ be isomorphic to the ring of smooth functions on $N$?

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### Can a smooth manifold be realised as the image of a smooth function?

Consider, $M$, a smooth $m$ dimensional submanifold of $\mathbf R^n$. Does there exist a smooth map $X: \mathbf{R}^m\to\mathbf R^n$ such that $M=X(\mathbf R^m)$?
$X$ may have points at which the ...

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### On Euler angles decomposition of $\mathrm{SU}(N)$

$\DeclareMathOperator\SU{SU}$I am looking for a (generalized) Euler angles decomposition for $\SU(N)\ (N>1)$ in the following fashion:
$$
\SU(N)\ni m = a\, u \, b
$$
where $a,b$ are independent ...

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### If $M$ is contractible manifold and $X\subset \partial M$, does the cone over $X$ embed in $M$?

Let $M$ be a compact contractible manifold, $X\subset\partial M$ and $C_X$ the cone over $X$.
Question: Is it true that $C_X$ embeds in $M$ with its boundary $\partial C_X$ mapped to $X\subset \...

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### Does the (Poincare) dual complex represent the same topology?

To start with, consider some abstract $3$-dimensional simplicial complex $\Delta$ representing a manifold without boundary, for simplicity. Then, there is this well-known construction of the "(...

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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 ...

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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 $\gamma(I)\subset M$?

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 \...

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### 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
\...

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### 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 ...

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### 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 $M$...

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### 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 ...

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### 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 ...

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### 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 ...