**12**

votes

**2**answers

607 views

### Does every compact manifold exhibit an almost global chart

Let $M$ be a compact connected manifold.
Is there a chart $\Psi:U \to \mathbb{R}^n$ such that the closure of $U$ is $M$?
This is true for $S^n, T^n, K$, all compact surfaces, etc.
If it is not true in ...

**3**

votes

**0**answers

116 views

### Is a continuous map between smoothable manifolds of the same dimension always smoothable?

(My question is inspired by this math.SE question, whose
negative answer I showed by a dimension-increasing map.)
Is it the case that for all smoothable manifolds $M_0$ and $M_1$ with the same ...

**0**

votes

**1**answer

71 views

### A question on parallelizability

Is there a manifold $M$ such that for every $x\in M$, $M-\{x\}$ is not parallelizable but there is a finite set $S\subset M$, with $\# S>1$, such that $M-S$ is parallelizable?

**2**

votes

**0**answers

180 views

### The Cauchy Problem in General Relativity: Existence of a Hausdorff Development

This is related to a problem that I posed about a year ago. I was given several references by a number of experts who were kind enough to entertain my rather arcane question. Those references were ...

**4**

votes

**1**answer

223 views

### The embeddings $S^{6} \to S^{7}$ and $S^{7}\to S^{8}$

Is the standard smooth structure of $S^{7}$ the only structure for which each of the following canonical embedding is an smooth embedding?:
1)$S^{6}\to S^{7}$
2)$S^{7}\to S^{8}$

**10**

votes

**1**answer

307 views

### heat kernel on n-sphere

I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way ...

**1**

vote

**2**answers

224 views

### A question on involutions on the Lie algebra of vector fields

Edite According to the essential comment of Ian Agol I revise the question as follows
For a smooth manifold $M$, is there a non identity involution $\theta$ on the lie algebra $\chi^{\infty}(M)$ ...

**4**

votes

**2**answers

423 views

### A question about Dehn surgery and Brieskorn homology 3-spheres

I have been learning about Brieskorn homology 3-spheres $\Sigma(a_1,...,a_n)$ and Seifert manifolds. My reference is the first few pages of Saveliev's "Invariants of Homology 3-spheres."
If I ...

**5**

votes

**2**answers

243 views

### What 3-manifolds can be obtained by gluing $ S^1 \times P $ and two copies of $S^1 \times D^2$

Let P denote the pair of pants e.g. a sphere minus three small discs $D_1$,$D_2$,$D_3$ about marked points $x_1,x_2,x_3$. I then consider $P \times S^1$. We have boundary components $T_1$,$T_2$,$T_3$. ...

**0**

votes

**0**answers

109 views

### Question about a particular estimate in Riemannian geometry

I have been studying the book Some Nonlinear Problems In Riemannian Geometry - Thierry Aubin. On page $46$ he begins the proof of the Sobolev imbedding theorem to manifolds. The proof is divided in ...

**23**

votes

**2**answers

783 views

### Ellipses on spheres (and other surfaces)

Define an ellipse $E$ on a sphere as the locus of points whose sum of
shortest geodesic distances to two foci $p_1$ and $p_2$ is a constant $d$.
There are conditions on $\{ p_1, p_2, d \}$ for this ...

**30**

votes

**9**answers

6k views

### Why differential forms are important?

Importance of differential forms is obvious to any geometer and some analysts dealing with manifolds, partly because so many results in modern geometry and related areas cannot even be formulated ...

**2**

votes

**0**answers

304 views

### Closed 4-manifolds with uncountably many differentiable structures

I know that $\mathbb{R}^4$ admits uncountably many differentiable structures and I was wandering what happen if we consider closed 4-manifolds. Are there any closed 4-manifolds with uncountably many ...

**12**

votes

**1**answer

471 views

### Homotopy spheres with vanishing and non-vanishing $\alpha$-invariant

I'm unsure whether this question is appropriate for mathoverflow, so feel free to criticize.
All manifolds are closed, smooth and have dimensions $n\ge 5$.
The Atiyah-Shapiro-Bott-Orientation gives ...

**9**

votes

**0**answers

122 views

### When does a cobordism factorize the sphere?

Let $W: \emptyset \Rightarrow M$ be a smooth cobordism from $\emptyset$ to a smooth closed $n$-manifold $M$. Are there reasonably simple conditions on $W$ which guarantee the existence of another ...

**2**

votes

**0**answers

95 views

### Smooth function over a manifold into an algebra

Let $M$ be a manifold and $A$ a $*$-algebra. Does is hold that
$$C^{\infty}(M,A) \cong C^{\infty}(M) \otimes A$$
where the RHS means that you take smooth functions which map into $A$. If this holds, ...

**1**

vote

**0**answers

154 views

### Foliation of the tangent bundle of $n$-sphere

Is there a smooth $n$-dimensional foliation of $TS^{n}$,( here $n\neq1,3,7$) such that the zero section be a leaf of this foliation?

**13**

votes

**2**answers

432 views

### Homotopy groups of spaces of embeddings

Let $\mathrm{Emb}(M, N)$ be the space of smooth embeddings of a closed manifold $M$ into a manifold $N$ equipped with smooth compact-open topology.
Question 1. Are there conditions ensuring that ...

**4**

votes

**0**answers

214 views

### “Naïve”cobordism?

The naive view of (say, complex) cobordism of a space $X$ is that it should be the group generated by continuous families of closed manifolds parametrized by $X$. Acting even more naively, one may try ...

**3**

votes

**1**answer

228 views

### Is the space of smooth functions with compact support a DF space?

Is there a good criterion, when a (nuclear) LF space is DF (DFN)? What are references to find out about that? If not, is there a known way, to construct a projective resolution of LF spaces? Prosmans ...

**2**

votes

**0**answers

86 views

### Strictly Convex Smoothing of a function defined on an affine manifold

A function defined on an interval is strongly convex if it has positive definite second derivative. A function on an affine manifold is strongly convex
if its restriction to each line is. An affine ...

**3**

votes

**2**answers

330 views

### The group of diffeomorphisms with compact support

Let $M$ be a topological/differentiable manifold. Is there any topology on the group of homeomorphisms/diffeomorphisms with compact support, turning it into a (locally-)compact topological group?
(My ...

**1**

vote

**1**answer

387 views

### A multilinear question and its smooth version

Let $E$ and $F$ be two finite dimensional vector spaces. For every $k\in \mathbb{N}$, $E^{k}$ has a natural vector space structure and is isomorphic to $E\otimes \mathbb{R}^{k}$, in a natural way.
...

**1**

vote

**1**answer

212 views

### A special type of transitivity

Let $M$ be a smooth orientable manifold with volume form $\Omega$. Fix two pints $x,y \in M$. Put $A$=all volume preserving diffeomorphism of M which maps $x$ to $y$.
Define $B$=All linear volume ...

**3**

votes

**0**answers

166 views

### smoothing a current

Let $M$ be a smooth oriented manifold of dimension $n$ and $T$ a current of dimension $k$ on $M$. Let $\phi:P\times M \to M$ be a proper smooth family of diffeomorphisms of $M$ (i.e. $P$ is a smooth ...

**6**

votes

**1**answer

204 views

### Dimension of a set detected by a homology class

A colleague asked me a topology question which comes down to this: Suppose that $M$ is a smooth $n$-manifold, and $C\subset M$ is a closed set such that $H_{n-p}(M-C)\to H_{n-p}(M)$ is not surjective. ...

**22**

votes

**1**answer

282 views

### Finite limits in the category of smooth manifolds?

The category of smooth manifolds does not have all finite limits. However, it does have some limits: it has finite products, it has splittings of idempotents, and it has certain other limits if we ...

**3**

votes

**1**answer

320 views

### Compactly supported cohomology of homotopy equivalent manifolds

Are there examples of homotopy equivalent smooth, orientable manifolds $M$ and $N$ of the same dimension with non-isomorphic compactly supported cohomology rings?

**2**

votes

**1**answer

152 views

### Anti_symplectic 2-forms

A two form $\alpha$ on a n- manifold $M$ is called anti symplectic if for every $x\in M$, $\{ v\in T_{x} M \mid i_{v} \alpha=0 \}$ is a $n-2$ dimensional subspace of $T_{x}M$. So we obtain a $n-2$ ...

**2**

votes

**2**answers

200 views

### analytic vector bundles

Let $E$ be a real analytic vector bundle on an analytic manifold $M$. Assume that $E$, as a smooth vector bundle, is a trivial bundle.
Is $E$ a trivial analytic vector bundle?
I need to the ...

**2**

votes

**0**answers

119 views

### Analytic version of the Cartan lemma

Assume that $\beta$ is a real analytic 2-form on an analytic manifold $M$ and $\alpha$ is an analytic non vanishing 1-form on $M$. Assume that $\beta \wedge \alpha=0$. Is there an analytic 1-form ...

**11**

votes

**1**answer

424 views

### Are the mapping class groups of manifolds finitely presentable?

The mapping class group of a manifold is the group $\pi_0 Diff(M)$ of components of the diffeomorphism group. There are several variations: oriented manifolds and orientation preserving ...

**1**

vote

**2**answers

238 views

### A cohomology associated to a 1- form

In this question all objects are real analytic.(manifolds, differential forms..)
Assume that $M$ is a compact manifold and $\alpha \in \Omega^{1}(M)$ is a one form.
We define a map ...

**0**

votes

**1**answer

127 views

### Comparison of two infinite dimensional Lie Algebras

Is there an example of a real analytic (compact) manifold $M$ such that the following two lie algebras are isomorphic Lie algebras:
$\chi^{\infty}(M)$, the Lie algebra of all smooth vector ...

**-3**

votes

**1**answer

305 views

### Lie algebraic Grassmannian

Assume that $L$ is a Lie algebra structure on $\mathbb{R}^{n}$, and $1<k<n$ is given.
We define $Gr(k,n)_{L}$, the space of all $k$ dimensional Lie subalgebra of $(\mathbb{R}^{n}, L)$.
For ...

**3**

votes

**1**answer

164 views

### A Manifold for which $\chi^{\infty}(M)$ is rich

Is there a manifold $M$ for which $\chi^{\infty} (M)$, the lie algebra of smooth vector fields on $M$ contains all finite dimensional Lie algebras(Up to isomorphism)?
A weaker question:
Is there a ...

**1**

vote

**0**answers

107 views

### Riemann normal coordiantes and change of metric

Le $(M,g)$ be Riemannian manifold. Fix point $p\in M$. We can define the map
$$\exp: U \subset T_p M \rightarrow M$$
$$\exp(X) = \gamma_{p,X}(1)$$
where $t\mapsto \gamma_{p,X}(t)$ is geodesic such ...

**4**

votes

**1**answer

220 views

### Exercises around Diffeological Spaces or a Diffeologic Atlas Theory

Is there a Book or a bunch of exercises to get used to diffeological spaces from a practical point of view? It seems to me that papers on this topic are mostly concerned with their very good ...

**0**

votes

**0**answers

54 views

### Question on center-stable manifold

Assume that you have a gradient system smooth enough and a fixed point $x_{0}$. Is it true that if $x_{0} \in \omega(x)$ then $\gamma^{+}(x)$ must intersect the local center-stable manifold of ...

**3**

votes

**3**answers

336 views

### undergraduate handle decomposition. Reference

As the title says, I'm searching for a nice textbook for introducing the theory of handle decomposition of manifolds to undergraduate students.

**15**

votes

**1**answer

592 views

### Derivation on real analytic manifolds

Let $M$ be a real analytic manifold. By $C^{\omega}(M)$ we mean the algebra of all analytic functions from $M$ to $\mathbb{R}$. Assume that $D$ is a derivation on $C^{\omega}(M)$ .
Is there a ...

**3**

votes

**2**answers

261 views

### Markov Partitions for toral automorphisms

I know that my question is more practical than theoretical. But, I do know where to look for the theoretical sources.
I want to find a program in the case that it exists (does it?), or to program it. ...

**10**

votes

**2**answers

430 views

### The boundary of a domain whose interior is diffeomorphic to the ball

We know that there are compact manifolds with diffeomorphic interiors but their boundaries are not homeomorphic (see the question Manifolds with homeomorphic interiors).
My question is about a very ...

**1**

vote

**1**answer

83 views

### Integration from vector bundles

Let $(E,M,p)$ be a smooth n dimensional vector bundle. Then $(TE,TM,Dp)$ is a 2n dimensional vector bundle. We restrict this bundle to $M\subset TM$. We denote this restricted bundle by $F$, as a ...

**1**

vote

**1**answer

101 views

### Are these two bundles, stably equivalent?

Let $(E,M,p)$ be a n dimensinal smooth vector bundle where $M$ is a k dimensional manifold. We assign to $M$, two different vector bundles $F_{1}$ and $F_{2}$ over $M$ as follows:
1)$TE$ is a ...

**6**

votes

**1**answer

308 views

### Can the monoidal structure on manifolds be strictified?

I'm asking this question purely out of curiosity.
Let $\{M_\alpha\}_{\alpha\in A}$ be a collection of closed smooth manifolds, with exactly one in every diffeomorphism class of closed smooth ...

**3**

votes

**1**answer

287 views

### Exponential mapping versus flow

In Hamilton's article on the Nash-Moser Theorem, he gives the map that maps a vector field $X$ to its flow $e^{tX}$ in $\mathrm{Diff}(M)$ as an example where the implicit function theorem in Frechet ...

**9**

votes

**2**answers

647 views

### Reference Request: Compact manifolds with boundary have the homotopy type of a CW-complex

Let $M$ be a compact manifold (possibly non-smooth) manifold with boundary $\partial M$.
Is the inclusion $\partial M\hookrightarrow M$ homotopy equivalent to the inclusion of a subcomplex into a ...

**-1**

votes

**1**answer

176 views

### A closed manifold with a subset with the same ring cohomology

Is there an example of a closed manifold $M$ with a proper subset $A\subset M$ such the inclusion $i:A \to M$ gives a ring isomorphism $i^{*}$ between $\mathbb{Z}$-cohomologies?
In this question ...

**3**

votes

**1**answer

259 views

### Is any smooth homeomorphism isotopic to a smooth embedding?

Let $f:D^m\to \mathbb{R}^m$ be a smooth map ($D^m$ is the unit ball).
We call $f$ embedding if it is a homeomorphism on the image and the
derivative $D_xf$ is nonsingular at each point $x\in D^m$ ...