**0**

votes

**0**answers

35 views

### Submersion on open and dense subset

If $f \colon M \to N$ is a smooth map between smooth manifolds, is it possible to find an open and dense set $M_0$, such that $f(M_0)$ is a manifold and
$f \colon M_0 \to f(M_0)$ is a surjective ...

**1**

vote

**0**answers

42 views

### Does a homotopy sphere that bounds a highly connected manifold also bound a parallelizable manifold?

Suppose that the homotopy sphere $\Sigma^{n}$ can be realized as the boundary of a smooth $(n+1)$-dimensional cobordism that is $(n-1)/2$-connected for $n$ odd (respectively, $(n-2)/2$-connected for ...

**9**

votes

**2**answers

150 views

### Triangulation with simplices of same volume

Let $M$ be a Riemannian smooth compact manifold.
It is known that $M$ has a triangulation, for any dimension. But do we know if there exists a triangulation such that all simplices have same volume ?
...

**0**

votes

**0**answers

40 views

### Lower bounds on the measure of balls in attractor sets

I'm looking for a source for the following result.
Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset ...

**0**

votes

**0**answers

56 views

### Lower bound on the diameter of a ball contained in the stable manifold of a critical point

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Consider the negative of the gradient ...

**3**

votes

**0**answers

92 views

### Can we use the “size” of smooth structure set to predict the information geometry or other topological information?

The "size" can mean the number of elements or the diameter of the set of smooth structures. Y. Shikata defined a distance function on it and proved that it is a distance. He then used it to prove that ...

**0**

votes

**0**answers

27 views

### A problem from Sakai's book on derivations on C(K) and differential structure on K

In his book, Operator Algebras in Dynamical Systems, at page 59 Sakai poses the following question.
Problem: Let K be a compact space and suppose that C(K) has a non-zero closed *-derivation. Then ...

**2**

votes

**0**answers

252 views

### Measure of the Attractor of Critical Points of a Manifold

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Let $P$ be a property of some subset ...

**29**

votes

**1**answer

848 views

### Can a topological manifold have different tangent bundles?

We know that the tangent bundles of the sphere arising from different smooth structures are equivalent as vector bundles. Is it right in general? I want to know the relationship between the set of ...

**3**

votes

**1**answer

167 views

### Is the number of intervals you get by slicing the closed region under a nice graph continuous from below?

Let $I$ be a closed interval in $\mathbb{R}$, and let $f:I \to \mathbb{R}$ be a bounded function, smooth except at finitely many points (piece-wise smooth). There are also only finitely many critical ...

**0**

votes

**0**answers

41 views

### Connected Cobordisms

Perhaps someone who knows some differential topology will be able to give a simple answer to this:
Since the (unoriented) cobordism group $\Omega_{3} \cong 0$, every closed 3-manifold is cobordant to ...

**9**

votes

**1**answer

244 views

### Are there non-smoothable homotopy/homology spheres?

A homotopy sphere is a topological $n$-manifold $M$ which is homotopy equivalent to $S^n$.
A homology sphere is a topological $n$-manifold $M$ such that $H_i(M) \cong H_i(S^n)$ for all $i$.
Note, by ...

**23**

votes

**2**answers

677 views

### Why is it true that if two 4-manifolds are homeomorphic then their squares are diffeomorphic?

Near the top of the second page of this paper, it is claimed that if two 4-manifolds $X$ and $Y$ are homeomorphic, then their squares $X \times X$ and $Y \times Y$ are diffeomorphic. Why is this true? ...

**2**

votes

**0**answers

92 views

### How many linear independent vector fields can be constructed on a general manifold with $\chi(M)=0$?

We have known how many linear independent vector fields can be constructed on $S^n$:https://en.wikipedia.org/wiki/Vector_fields_on_spheres
So how many linear independent vector fields can be ...

**5**

votes

**0**answers

87 views

### $H(M)$ necessarily highly non-integrable, i.e. forms contact structure?

Let$$M^{2n - 1} = \{z \in \textbf{C}^n : \textbf{h}(\textbf{z}, \textbf{z}) \equiv \textbf{z} \cdot \overline{\textbf{z}} = \textbf{1}\}$$be the unit sphere in $\textbf{C}^n$. Consider the ...

**21**

votes

**1**answer

291 views

### Is the normal bundle of a torus trivial?

Question:
Let $T^k \subseteq \mathbb{R}^n$, $ n > k$, be a smoothly embedded $k$-torus. Is its normal bundle trivial?
What about the normal bundle of $S^k \subseteq \mathbb{R}^n$, $n > k$, the ...

**7**

votes

**0**answers

244 views

### Can any smooth triangulation of a smooth manifold be blurred?

For the purposes of this question, let's say that a blurring
of a smooth triangulation $T$ of a smooth manifold $X$
is a smooth homotopy $h\colon [0,1] \times X \to X$ such that ...

**11**

votes

**1**answer

276 views

### A symmetric embedding of manifolds

Assume that $M$ is a manifold.
Is there an embedding of $M$ in some $\mathbb{R}^{n}$ such that the image of $M$ in $\mathbb{R}^{n}$ is invariant under each reflection $(x_{1},x_{2},\ldots ...

**0**

votes

**0**answers

152 views

### About the homotopy type of diffeomorphism groups

In this paper by Antonelli, Burghelea and Kahn (Topology, 1972), a homomorphism $L :\pi_i(\operatorname{Diff}(S^n, D_+^{n})) \rightarrow \Gamma^{n+i+1}$ was used as a tool to detect non-triviality of ...

**13**

votes

**2**answers

241 views

### Classification of $O(2)$-bundles in terms of characteristic classes

I had asked this question in stackexchange but there seems to be no consensus in the answer
It is well-known that $SO(2)$-principal bundles over a manifold $M$ are topologically characterized by ...

**2**

votes

**2**answers

134 views

### Finding a specific Global Smooth Function

Any help with this problem would be appreciated. Thanks
Suppose $(M^3,g)$ is a smooth compact Riemannian manifold with smooth boundary and $\gamma$ is a simple smooth orientable curve in $M$. Does ...

**0**

votes

**1**answer

97 views

### Variation of normals along loops

I've constructed a quotient space $M/\sim$ in $\mathbb{R}^d$ that must be a $2$-manifold. If $M/ \sim$ is a sphere then I know that its normal spaces must vary at least 90 degrees. That is $M / \sim$ ...

**3**

votes

**1**answer

168 views

### The space of homotopy classes of maps of products of spheres

Proposition 17.6.1 of "Differential form in Algebraic Topology" by Bott and Tu proves the following beautiful result:
$[S^{q}, X]\simeq \frac{\pi_{q}(X,x)}{\pi_{1}(X,x)}$
where $S^{q}$ is the ...

**1**

vote

**0**answers

80 views

### Tightening a loop

Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose ...

**2**

votes

**0**answers

60 views

### Singularities of Families of Differential Equations

Suppose you are given a family of differential equations with rational coefficients: $\partial_x^ny+a_1(x,x_1,\dots,x_m)\partial_x^{n-1}y+\dots+a_n(x,x_1,\dots, x_m)y=0$. At each point $X=(x_1,\dots, ...

**1**

vote

**1**answer

50 views

### Existence of a continuous map of a disk with a given boundary image on a surface to its complement in $\mathbb{R}^3$

I am considering the following problem:
given an embedded closed surface in $\mathbb{R}^3$ (unknotted) and a non-trivial simple closed curve on it, does there exist a continuous map of a disk to the ...

**1**

vote

**0**answers

84 views

### “Nice” limits of sequences of smooth embeddings

Consider smooth embeddings of a manifold $M$ into some $\mathbb{R}^n$. If a sequence $f_k : M \to \mathbb{R}^n$ of such embeddings converges to some continuous function $f : M \to \mathbb{R}^n$, then ...

**9**

votes

**2**answers

206 views

### non-triviality of the underlying real vector bundle of the complexification of a real vector bundle

Let $M$ be a given manifold and $\xi$ be a given $k$-dimensional vector bundle over $M$. How to determine whether the underlying real vector bundle of $\xi\otimes\mathbb{C}$, i.e. the Whitney sum ...

**0**

votes

**0**answers

49 views

### map of constant rank

Let $f_1, \dots, f_m \colon M^n \to \mathbb{R}$ be smooth functions on a manifold $M$, such that $F := (f_1, \dots , f_m) \colon M \to \mathbb{R}^m$ has constant rank $k <m,n $ on $M$.
I'm trying ...

**3**

votes

**2**answers

107 views

### Extension of a group action beyond the boundary

Let $M$ be a compact manifold with boundary and suppose a compact group $G$ acts on it. Can one always extend the action beyond the boundary? More precisely, does there always exist a $G$-manifold ...

**7**

votes

**1**answer

120 views

### Can any path in the diffeomorphism group of a smooth compact manifold be approximated by a smooth path?

Given a smooth compact $n$-dimensional manifold $M^{n}$, let $\operatorname{Diff}(M)$ denote the group of smooth diffeomorphisms $M \rightarrow M$ equipped with the Whitney $C^{\infty}$-topology. Let ...

**2**

votes

**1**answer

135 views

### Can one smooth open star shaped domains from the inside by star shaped domains?

Let $O\subset\mathbb{R}^n$ be a open set which is star shaped with respect to the origin. How does one prove that there exists an increasing sequence of star shaped (w.r.t the origin) domains $O_i$ ...

**1**

vote

**0**answers

44 views

### Convex Hull and Least Area Discs in Riemannian 3-Manifolds

Let $M$ be a complete Riemannian 3-manifold and $\gamma \subset M$ a simple closed curve that bounds a least-area disc $D$ - a disc that minimizes the area among all discs bounded by $\gamma$.
Let ...

**1**

vote

**0**answers

49 views

### Flows and Lie brackets, $\beta$ not a priori smooth at $t = 0$ [closed]

Let $X$ and $Y$ be smooth vector fields on $M$ generating flows $\phi_t$ and $\psi_t$ respectively. For $p \in M$ define$$\beta(t) := \psi_{-\sqrt{t}} \phi_{-\sqrt{t}} \psi_{\sqrt{t}} ...

**4**

votes

**1**answer

152 views

### Linear extension operators for smooth functions: from compact sets to compact sets

I'm considering a situation where I have the linear restriction map of Fréchet spaces
$$
C^\infty(C_1) \to C^\infty(C_2)
$$
where $C_2 \hookrightarrow C_1$ are a pair of compact, connected subsets ...

**10**

votes

**1**answer

233 views

### Closed $3$-manifold, $2$-dimensional subbundle of this manifold, is this form exact or not?

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. I know the following.
There is a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector ...

**9**

votes

**1**answer

181 views

### Dimension in Whitney's theorem

There are some interesting examples of (classes of) manifolds for which I suspect that the Whitney theorem can be strenghtened. For example it is known that every (smooth) $n$-dimensional manifold can ...

**0**

votes

**1**answer

171 views

### Proper actions and diffeomorphism groups

Since the diffeomorphism group is not locally compact; is it true that there is no proper action of an infinite-dimensional diffeomorphism group on a finite-dimensional smooth manifold?
Edit: The ...

**20**

votes

**1**answer

265 views

### Primary obstruction to the existence of a cross-section of $V_{n - q}(\omega)$ is a cohomology class in $H^{2q+2}(B, \pi_{2q+1} V_{n - q}(F))$?

Let $V_{n - q}(\mathbb{C}^n)$ denote the complex Stiefel manifold consisting of all complex $(n - q)$-frames in $\mathbb{C}^n$, where $0 \le q < n$. This manifold is $2q$-connected, and$$\pi_{2q + ...

**3**

votes

**0**answers

57 views

### Linking circles inside an immersed surface

(Migrated from Math Stack Exchange)
A smooth embedding $f : D \to \mathbb{R}^3$ can be isotoped to a canonical inclusion $D \hookrightarrow \mathbb{R}^3$. (This is part of a proof that only the ...

**26**

votes

**1**answer

411 views

### Is there an explicit description of a cobordism between $\mathbb{CP}^n$ and $\mathbb{RP}^n\times\mathbb{RP}^n$?

With a little bit of work, one can show that $\mathbb{CP}^n$ and $\mathbb{RP}^n\times\mathbb{RP}^n$ have the same Stiefel-Whitney numbers, so by a theorem of Thom, they are (unorientedly) cobordant.
...

**16**

votes

**1**answer

300 views

### Is the restriction map for embeddings of manifolds with boundary a fibration?

Let $M$ and $W$ be smooth manifolds (possibly with boundary) and $V\subseteq W$ a submanifold. We have a map between embedding spaces
$$Emb(W,M)\rightarrow Emb(V,M)$$ given by restriction.
Richard ...

**7**

votes

**1**answer

191 views

### Chern-Simons forms, characteristic numbers, and boundary terms?

For any principal $G$-bundle $P \to M$ with principal connection $\omega$, given a $G$-invariant polynomial $p: \mathfrak{g} \to \mathbb{R}$ we can construct a form $p(F_\omega)$ on $P$ which descends ...

**2**

votes

**1**answer

160 views

### Holonomic splitting

I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not ...

**4**

votes

**1**answer

190 views

### Local product structure of determinantal variety

The variety $X_n$ of singular $n\times n$ real matrices is stratified by smooth strata $X_{n,k}$ where $k$ is the rank. Choose a rank $k$ matrix $A\in X_{n,k}$. Is there a local diffeomorphism sending ...

**3**

votes

**0**answers

100 views

### Intersection patterns of loops on surfaces

Let $a,b$ be to simple closed loops on a surface $S$ with homologically trivial intersection (more generally I'm also interested in the case when $b$ is 1-codimensional). Denote their intersection on ...

**3**

votes

**1**answer

240 views

### Differential characters, Chern-Simons forms, and differential cohomology

I've read through the classic Chern-Simons paper where they introduce the Chern-Simons forms. These are differential forms whose exterior derivative gives you the characteristic forms for any given ...

**31**

votes

**3**answers

2k views

### What is a foliation and why should I care?

The title says everything but while it is a little bit provocative let me elaborate a bit about my question. First time when I met the foliation it was just an isolated example in the differential ...

**9**

votes

**0**answers

296 views

### Milnor-Stasheff Characteristic Classes Problem 7B, Borel 1953

There is the following Proposition 11.1 from Borel's 1953 paper La cohomologie mod 2 de certains espaces homogènes (see here).
Proposition 11.1 The classes $w^i$ and $\overline{w}^j$ are related ...

**11**

votes

**2**answers

531 views

### Intuition/idea behind a proof of the splitting principle?

The splitting principle is as follows.
Given a vector bundle $E \to X$ with $X$ compact Hausdorff, there is a compact Hausdorff space $F(E)$ and a map $p: F(E) \to X$ such that the induced map ...