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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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\} $....
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1answer
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=...
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1answer
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
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0answers
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
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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 ...
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2answers
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
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1answer
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
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1answer
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
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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
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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
1answer
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(...
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0answers
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?
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2answers
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
1answer
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.
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votes
1answer
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
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1answer
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
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1answer
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
2answers
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
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1answer
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
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1answer
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
2answers
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
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1answer
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
1answer
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
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0answers
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
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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 ...
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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
1answer
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
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1answer
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
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1answer
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
1answer
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
1answer
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
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1answer
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
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1answer
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
1answer
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 ...
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1answer
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
1answer
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
2answers
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
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1answer
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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 ...
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1answer
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
1answer
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
2answers
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
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1answer
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
1answer
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
0answers
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
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1answer
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)...
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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 \...
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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 ...
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0answers
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
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0answers
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
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2answers
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. ...