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### 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 ...

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32 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 ...

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137 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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151 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 ...

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votes

**1**answer

92 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 $Y$ ...

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**1**answer

36 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 ...

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votes

**1**answer

160 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 ...

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votes

**1**answer

344 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 ...

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**1**answer

115 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 ...

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**1**answer

326 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 ...

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votes

**1**answer

66 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 ...

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**1**answer

208 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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45 views

### Looking for Information on Local Degree of Maps on Homology Manifolds

By a homology $n$-manifold, we mean a topological space $X$ such that for all $x \in X$:
1: if $k \neq n$ then $H_k(X, X-x)=0$
2: $H_n(X,X-x) \cong \mathbb{Z}$.
Given homology $n$-manifolds $X$ and ...

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vote

**1**answer

166 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 ...

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votes

**1**answer

158 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 ...

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**2**answers

364 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 ...

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**1**answer

905 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 ...

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**1**answer

288 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 ...

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132 views

### Generalization of the Riemann curvature tensor

The Riemannian curvature tensor (also holding for manifolds with torsion) is for the vector fields $X,Y,Z$ formally given by: $R(X,Y)Z=(∇ X ∇ Y −∇ Y ∇ X −∇ [X,Y] )Z$ .
This tensor clearly exist for ...

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**1**answer

75 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 $M$ ...

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**2**answers

337 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?

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**1**answer

160 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 ...

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387 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 ...

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114 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 ...

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**1**answer

224 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 ...

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131 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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116 views

### A consequenc of a Lie group act on a Riemannian manifold by isometry

I am learning differential geometry for using this topic in my research. I am stuck to prove following Result which I got in a article.
Formulation:
Let $ f: [0, 1]\rightarrow \mathbb{R}^2$ be a ...

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63 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 ...

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19 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 ...

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127 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 ...

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**2**answers

295 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.
...

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**1**answer

326 views

### If all balls around two points are isometric… — manifold version

This question is a natural follow-up of this other question, asked earlier today by wspin.
Let's say that a metric space $(X,d)$ has two poles if:
there are two distinct points $x$, $y$ such that ...

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vote

**1**answer

115 views

### Zariski closure of the boundary of a closed convex subset of ${\mathbb R}^n$

Let $\eta$ be a closed convex subset of ${\mathbb R}^n$ of convex dimension $n$ (not necessarily compact) with nonempty boundary $\partial \eta$. Then $\eta$ is an $n$-dimensional topological ...

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### (Very) High dimensional manifolds

Usually one regards manifolds up to dimension 4 as a part of low dimensional topology. There are plenty of various results which work only in low dimensional topology; especially in dimension 4. ...

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144 views

### Can the graph Laplacian be well approximated by a Laplace-Beltrami operator?

It seems rather well known that given a Laplace-Beltrami operator $\mathcal{L}_{M}$ on a manifold $M$ we can approximate its spectrum by that of a graph Laplacian $L_{G}$ for some $G$ (where $G$ is ...

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361 views

### Inverse cohomological isomorphisms

Let $\ M'\ M''\ $ be simply-connected Hausdorff compact manifolds (possibly with boundary for another variant of the question). Let $\ f:M'\rightarrow M''\ $ be a continuous function which induces an ...

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votes

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578 views

### Canonical Metric on Grassmann Manifold

I was curious and quite clueless as to how we can equip the Grassmann Manifold with a canonical metric - I have yet to find anything upon this subject.

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**1**answer

330 views

### Manifolds such that every homeomorphism of a submanifold to itself extends to the full manifold

Let manifold $S$ (connected, without boundary) have next property: for every submanifold $D \subset S$ (connected, compact, without boundary), every homeomorphism $f:D \to D$ extends to a ...

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**0**answers

63 views

### Codimension $k$ homeomorphism extensions

Let $f:D \to D$ homeomorphism of $k$ codimension manifold (closed, compact, without boundary) to itself. ($D \subset \mathbb{R}^{n+k}$). For which $f$ does homeorophism $g: \mathbb{R}^{n+k} \to ...

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**1**answer

430 views

### On the fundamental group of closed 3-manifolds

I know that every finitely presented group can be realized as the fundamental group of a compact, connected, smooth manifold of dimension 4 (or higher). In dimension 2 there are strong restriction on ...

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**1**answer

88 views

### Orientability of Stiefel manifold V2(R4) [closed]

What is an easy proof of orientability of Stiefel manifold $V_2(\mathbb{R}^4)$ (pairs of orthonormal vectors from $\mathbb{R}^4$ - subset of $\mathbb{R}^8$)? All proofs I found deal with Lie groups ...

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191 views

### Action of the mapping class group on separating spheres in a connected sum of aspherical 3-manifolds

Let $M$ be a connected sum of $g$ closed aspherical 3-manifolds $M_1, \ldots, M_g$. [Update: I also assume that all the $M_i$-s are diffeomorphic, i.e. $M$ is a connected sum of copies of the same ...

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### Can the second order frame bundle of a 2-d manifold be embeded inside the ordinary frame bundle of a 3-d manifold?

I would like to know whether the second order frame bundle of a 2-d manifold can be embedded inside the ordinary frame bundle of a 3-d manifold.
Explanation
Suppose we have a 3D smooth manifold ...

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votes

**2**answers

716 views

### good covers of manifolds

It is well-known and easy to prove (see for instance this post) that every smooth manifold admits a "good cover", i.e. a locally finite cover by open balls such that all nonempty intersections of the ...

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votes

**1**answer

204 views

### homology of punctured manifolds

Let $M$ be an $n$-dimensional closed manifold. Choose $x \in M$. Using long exact sequence of pairs $(M,M - x)$, we have
$$H_k(M - x, \mathbb{Z}) \cong H_k(M, \mathbb{Z})$$
for $k<n-1$. For ...

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177 views

### Does there exist any subspace of R^n, homeomorhic to a manifold but not a C^0 submanifold of R^n?

At first I thought that if a subspace of $\mathbb{R}^n$ is homeomorphic to a manifold, then it is a $C^0$ submanifold of $\mathbb{R}^n$. But I found an asterisked exercise in the book Differential ...

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708 views

### How can gauge theory techniques be useful to study when topological manifolds can be triangulated?

I was reading a review article arXiv:1310.7644 and it was explained there that in the last few years it was proven that there are topological manifolds of dimension greater than four that cannot be ...

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67 views

### Tubular neighbourhood which is nowhere piecewise linear

I recently asked this question.
I think, if the following were true, then I would solve my problem.
Let $E\subset\{(x_1,\dots,x_n)\in\mathbb R^n\;|\;x_i\geq 0\, \&\, \sum_ix_i=1\}$ be a convex ...

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**1**answer

566 views

### Are there geometrically formal manifolds, which are not rationally elliptic?

Formality of a space is meant in the sense of Sullivan, i.e. a space $X$ is called formal, if it's commutative differential graded algebra of piecewise linear differential forms $(A_{PL}(X),d)$ is ...

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148 views

### A question about something like “shelling” in a PL manifold

If $P$ and $Q$ are compact codimension zero submanifolds of a PL manifold, say that they meet nicely if $P\cap Q$ is a codimension zero submanifold of both $\partial P$ and $\partial Q$. In ...