**4**

votes

**0**answers

35 views

### Lie subalgebras of $\chi^{\infty}(M)$ of codimension $n=dim M$

For a connected $n$ manifold $M$, the Lie algebra of all smooth vector fields is denoted by $\chi^{\infty}(M)$. For a pointe $p\in M$ we define $L_{p}=\{X\in \chi^{\infty}(M)\mid X(p)=0\} $....

**2**

votes

**1**answer

209 views

### Manifolds whose diffeomorphism group has the homotopy type of a manifold itself

I have a very stupid question.
Let $M$ be a closed smooth manifold. In particular cases the homotopy type of the diffeomorphism group $Diff(M)$ can be very pathological. For example, in the case of $M=...

**0**

votes

**1**answer

30 views

### Orientability and higher dimensional Moebius strip

A compact surface is non-orientable if and only if there is a Moebius strip on it. Is there a similar result in higher dimension?
More specifically, at least in the smooth setting, we can define a ...

**2**

votes

**0**answers

104 views

### Decay of the Fourier transform of a surface area measure

Let $\mu$ be a surface area measure of a manifold $M\hookrightarrow\mathbb{R}^{n+1}$. If $M$ is the unit sphere $S^n$, it's known that surprisingly the Fourier transform of $\mu$ decays: $$|\hat\mu(\...

**3**

votes

**0**answers

163 views

### How far can one reconstruct the boundary of a manifold M given its interior $int M$? [duplicate]

Suppose I keep in my pocket a manifold with boundary $M$ , and I provide you access to $int M := M \setminus \partial M$ up to homeomorphism/diffeomorphism. What can you deduce about $\partial M$? can ...

**19**

votes

**2**answers

2k views

### Questions on J. F. Nash's answer about his errors in the proof of embedding theorem

In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked
Is it true, as rumours have it, that
you started to work on the embedding ...

**3**

votes

**1**answer

129 views

### Framed bordism class of the circle has order $2$

As the question title says, how do I see that the framed bordism class of the circle has order two?

**12**

votes

**1**answer

295 views

### Measuring the failure of pushforward to commute with Steenrod squares

Let $f \colon X \rightarrow Y$ be a map of topological spaces. Let's say that they are (closed) manifolds (not necessarily orientable), though to be honest I'm really interested in the more general ...

**2**

votes

**0**answers

79 views

### Is a one-dimensional unstable manifold of an ODE a union of the associated equilibrium point and two full orbits? [closed]

Consider an ordinary differential equation (ODE) system \begin{align}
\frac{dx}{dt} = f(x)
\end{align} where $x \in \mathbb{R}^n$ ($n \geq 2$) and the vector field $f$ is defined on an open subset $X$ ...

**4**

votes

**0**answers

116 views

### Classifying countable sets of weighted dots on a real line

Each dot is located on the real line and assigned a weight that can be positive or negative. A dot is equivalent to two(or more) dots located at the same place whose weights sum is equal to that of ...

**9**

votes

**1**answer

281 views

### Elements of infinite order in the topological mapping class group

Let $M$ be a closed topological manifold, and let $\operatorname{MCG}(M):=\operatorname{Homeo}(M)/\operatorname{Homeo}_0(M)$ denote the topological mapping class group of $M$ ($\operatorname{Homeo}_0(...

**2**

votes

**0**answers

118 views

### does there exist a generalization of a manifold [closed]

Does there exist a generalization of a manifold whereby instead of being locally $\mathbb{R}^n$, it's locally another specified space?

**16**

votes

**2**answers

790 views

### What is an example of an orbifold which is not a topological manifold?

In Thurston's book The Geometry and Topology of Three-Manifolds it is proven that the underlying space of a two-dimensional orbifold is always a topological surface.
Are there any easy examples of ...

**10**

votes

**1**answer

469 views

### What is the Status of Borel conjecture today?

Let me recall the conjecture: $M$ and $N$ two aspherical closed $n$-manifolds with isomorphic fundamental groups, then $M$ and $N$ are homeomorphic.

**6**

votes

**1**answer

148 views

### Rational cohomology of the Rosenfeld projective planes

The bioctonionic plane $(\mathbb{C} \otimes \mathbb{O})\mathbb{P}^2$, the quarteroctonionic plane $(\mathbb{H} \otimes \mathbb{O})\mathbb{P}^2$ and the octooctonionic plane $(\mathbb{O} \otimes \...

**2**

votes

**1**answer

186 views

### Smallest homotopy equivalent space inside a manifold with boundary

It is well known that any compact manifold with boundary is homotopy equivalent to its interior. Is there a notion of some smallest space in the interior of the manifold that is homotopy equivalent to ...

**1**

vote

**1**answer

265 views

### On compact, orientable 3-manifolds with non-empty boundary

I recall my Professor having stated something along the lines of the following, but I am not quite certain about the precise statement she gave:
Let $M$ be a compact, orientable 3 manifold with non-...

**5**

votes

**2**answers

214 views

### Poisson structures on non-smooth manifolds with singularities

It's very known how we can describe a Poisson structure on a manifold $M$, where $M$ is a smooth manifold, but what about a Non-smooth manifold with singularities? In section $(2)$ of the paper The ...

**0**

votes

**1**answer

94 views

### A connection on fibred manifold always exists?

May be $\pi:Y \mapsto X$ a general fibred manifold. Is it true that in fibred manifold a connection always exists? This wiki article states this: https://en.wikipedia.org/wiki/Connection_(...

**2**

votes

**1**answer

118 views

### Is the complement of the ends of a manifold bounded?

Let $M$ be a connected manifold with precisely $k$ ends $\epsilon_1,...,\epsilon_k$. Choose a collection $(U_i)_{i=1}^k$ of pairwise disjoint open $\epsilon_i$-neighborhoods. Then I wonder how to ...

**4**

votes

**2**answers

346 views

### Gluing two 3 manifolds along their boundary

Let $X,Y$ be two compact, smooth, orientable 3 manifolds, each with an incompressible boundary component diffeomorphic to some genus $g $ surface $S_g$. Under an orientation-reversig diffeomorphism $...

**15**

votes

**1**answer

205 views

### Smooth manifolds as idempotent splitting completion

The nlab has a particularly interesting thing to say about the category of smooth manifolds: it is the idempotent-splitting completion of the category of open sets of Euclidean spaces and smooth maps.
...

**4**

votes

**1**answer

115 views

### Is the signature of inverse images of diffeomorphic submanifolds (along a homotopy equivalence) the same?

Suppose it is given an orientation preserving homotopy equivalence $h:N→M$ between closed oriented connected manifolds. Let $X,Y\subset M$ be diffeomorphic submanifolds, and assume $h$ to be ...

**3**

votes

**0**answers

48 views

### Cubulating non-compact hyperbolic manifolds

Let $X$ be a hyperbolic manifold of arbitrary dimension. When does $X$ admit a cell structure of a CAT(-1) cube complex? of a hyperbolic CAT(0) cube complex?
I suspect that the question is widely ...

**10**

votes

**0**answers

212 views

### Mapping class groups in high dimension

Let $M$ be a $1$-connected, closed, smooth manifold with $dim(M)>4$ and let us set $MCG(M)=\pi_0(Diff(M))$. Dennis Sullivan proved that $MCG(M)$ is commensurable to an arithmetic group.
I was ...

**8**

votes

**0**answers

171 views

### Geometric argument for “easy” part of Jordan-Brouwer separation theorem without local flatness

Let $M^n \subset \mathbb{R}^{n+1}$ be an $n$-dimensional compact connected topological submanifold. The Jordan-Brouwer separation theorem says that $\mathbb{R}^{n+1} \setminus M^n$ contains two ...

**3**

votes

**1**answer

115 views

### Existence of tubular neighborohoods of locally flat topological embeddings

Suppose $X$ is a topological manifold and $Y \subset X$ is a locally flat submanifold. We know that $Y$ doesn't necessarily have a tubular neighborhood. My definition of a tubular neighborhood of $Y$ ...

**1**

vote

**1**answer

58 views

### Tube formula for r-neighbourhood of a manifold

Let $P$ be a topologically embedded submanifold in a Riemannian manifold $M$. Then the tube $T(P, r)$ of radius $r \geq 0$ about $P$ is the set of all points $m \in M$ such that there exists a ...

**5**

votes

**1**answer

175 views

### Which combinations of normality, separability, and paracompactness do complex manifolds possess?

I am interested in what kinds of non-paracompact complex manifolds may exist and which topological properties they may have.
Is there a non-separable complex manifold? Can a non-separable complex ...

**5**

votes

**1**answer

381 views

### Homeomorphism of closed manifold

Suppose that we have two closed n-manifold $M$ and $N$ such that
the topological group of homeomorphisms $Homeo(M)$ is homotopy equivalent to $Homeo(N)$ (maybe as topological groups if needed), can ...

**4**

votes

**1**answer

131 views

### Exotic C^k manifolds

How much is known about exotic $C^k$-manifolds? For example,
Is it known whether there are $C^k$-differentiable manifolds that are homeomorphic, but not $C^k$ diffeomorphic?
More generally, is it ...

**11**

votes

**1**answer

398 views

### Homology spheres and fundamental group

I have a curiosity about homology spheres: I was wondering if they were uniquely characterized by their fundamental group. I.e. given two $n-$dimensional (integral) homology spheres with isomorphic ...

**3**

votes

**1**answer

71 views

### Can any $n$ dimensional (smooth, PL, topological) closed manifold be covered by $2^n$ pieces of $n$ dimensional real spaces?

For any $n$ dimensional closed manifold $M^n$, can we find an open covering $\{U_i\}_{i\in[2^n]}$ such that $M=\cup U_i$ and each $U_i\cong \mathbb R^n$? How about complex manifolds (replacing $\...

**3**

votes

**1**answer

225 views

### Perturb a given smooth function to a Morse function relative to fixed level sets, which are already fine

This question was not answered on math.stackexchange.
Let $M$ be a manifold (not necessarily compact) , for the sake of clearness embedded in $\mathbb{R^n}$ and $f\colon M\rightarrow \mathbb{R}$ a ...

**1**

vote

**1**answer

180 views

### Homology of manifold with action of group

Sorry for my ignorance in advance, this should be a very naive question and I would be happy for a reference.
Let $G$ be an arbitrary group (not necessary finite) acting on two (connected) manifolds $...

**2**

votes

**1**answer

188 views

### Digital topology, animal problem, 2-sphere and torus

I have the following question relating digital topology, surfaces, particularly $S^2$ and torus.
Can a body $B$ constructed with cubes (without cavities or tunnels) and with frontier homeomorphic to ...

**9**

votes

**2**answers

431 views

### Topological $n$-manifolds have the homotopy type of $n$-dimensional CW-complexes

I search for a chain of clean references, which lead the fact of topological manifolds of dimension $n$ having the homotopy type of a CW of dimension $n$.
Milnor's On spaces having the homotopy type ...

**32**

votes

**1**answer

1k views

### Are there only countably many compact topological manifolds?

Up to homeomorphism, there are 2 one-dimensional topological manifolds and countably many 2- and 3-dimensional compact manifolds, respectively, since each manifold in these dimensions can be ...

**-1**

votes

**1**answer

316 views

### History of Poincare conjecture in higher dimension [closed]

As far as I know, when Poincare formulated his well known conjecture, the original statement was the follwoing: if a closed manifold has the same homology groups as the sphere it is homoeomorphic to ...

**3**

votes

**1**answer

82 views

### Whether the manifold part of an Alexandrov space is connected?

The title is my question.
Alexandrov space here means finite dimensional Alexandrov space with curvature bounded below ,denoted by CBB.
Let $\gamma$ be a simple curve in a $n$ dimensional CBB $M$ ...

**8**

votes

**2**answers

380 views

### Number of critical points of smooth functions on $S^1$

Let $u$ be a smooth function on the unit circle $S^1$ such that $\int_{S^1}ux_j=0$, for $j=1,2$. Is the number of critical points of $u$ strictly bigger than 2?

**6**

votes

**1**answer

265 views

### Dual cell structures on manifolds

Suppose that $M$ is a compact manifold without boundary (smooth if you like), and suppose further that $M$ is equipped with a regular CW-complex structure. Denote the face poset of this CW-complex by $...

**18**

votes

**1**answer

901 views

### Is there a manifold with fundamental group $\mathbb{Q}$?

It is known that the fundamental group of a locally path connected, path connected compact metric space is finitely presented or uncountable. Furthermore the fundamental group of every manifold is ...

**4**

votes

**0**answers

119 views

### Sampling from a Manifold

Suppose we were to obtain a uniform sample, $S=\{x_1,...,x_m\}$, of points on a closed Riemannian $n$-manifold $M$. Let $\Gamma(S)$ be the set of all geodesics between the points in $S$ and we are ...

**4**

votes

**1**answer

302 views

### The relation between Hausdorff dimension of an $n$-manifold and $n$

It is known that for a topological space with different metrics, the Hausdorff dimensions may not be equal in general.
For the case of manifolds, suppose $M$ is a $n$-manifold with a metric(distance)...

**2**

votes

**0**answers

166 views

### Caffarelli-Silvestre extension definition of fractional Laplace-Beltrami on hypersurface

Let $\Gamma \subset \mathbb{R}^{n+1}$ be a $n$-dimensional (closed) $C^k$-hypersurface. Consider the problem
$$\nabla_\Gamma \cdot (y^{1-2s}\nabla_\Gamma u)(x,y) =0 \quad \text{for $(x,y) \in \Gamma \...

**1**

vote

**0**answers

71 views

### Invariant subsets of a local action

I have also asked this in MSE, but it seems to me that my question wasn't very well received there and I think someone in here will be able to answer it more quickly, hence this post.
I don't ...

**0**

votes

**0**answers

31 views

### Approximate rank of the set formed by all delayed replicas of a bandlimited signals between 0 and T

Given a complex-valued signal with a certain delay $s(t-\tau)$ for which we sample $N$ instants
$$
\mathbf{s(\tau)}=\left[s(0-\tau),\ldots,s\left(\frac{N-1}{f_s}-\tau\right)\right]^T
$$
at Nyquist ...

**2**

votes

**0**answers

136 views

### Are there analogs of smooth partitions of unity and good open covers for PL-manifolds?

Smooth partitions of unity and differentiable good open covers are important technical tools in the realm of smooth manifolds.
Are there analogs of these tools for piecewise linear manifolds?
A PL ...

**10**

votes

**2**answers

343 views

### Homeo-Fixed point property

Edit: According to comment of Michał Kukieła I revised the question
A topological space $X$ satisfies "Homeo-fixed point" property if every homeomorphism $f$ on $X$ possess a fixed point.
...