**23**

votes

**1**answer

698 views

### When is there a submersion from a sphere into a sphere?

(First posted on math.SE, with no answers.)
That is:
For which positive integers $n, k \ge 1$ does there exist a submersion $S^{n+k} \to S^k$?
The discussion at this math.SE question has ...

**0**

votes

**0**answers

45 views

### cobordism and smoth-manifold [on hold]

Let M, N , N' and M' be smooth n-manifolds with nonempty boundaries , and suppose h:∂M→∂N , g:∂M'→∂N' are diffeomorphisms . Let M∪_h N be the adjunction space formed by identifying each xϵ∂M with ...

**2**

votes

**0**answers

40 views

### Is there any topological information encoded by the zero locus of a complex Hessian?

On a Kahler manifold one can always find a function $K$ so that (up to some constant) the Kahler form $\omega$ can be written as $\omega = \partial \bar{\partial}K$ (sometimes, under conditions that I ...

**8**

votes

**1**answer

203 views

### Non-orientable $6$-manifold with $H_4(M)=\mathbb{Z}/2$?

Does there exist a smooth, closed, non-orientable $6$-manifold $M$ such that $H_4(M;\mathbb{Z})=\mathbb{Z}/2$?

**0**

votes

**0**answers

28 views

### Preimage of singular points of smooth map between vector space and $SU(n)$

(Moved from Math SE as no answer was forthcomming: http://math.stackexchange.com/q/1294521/161684)
Given a smooth ($C^{\infty}$) map $\phi: V \rightarrow SU(n)$ (which is taken to be surjective) ...

**0**

votes

**1**answer

78 views

### Is the space of smooth maps $C^{\infty}(M,N)$ with the Whitney $C^{\infty}$ topology locally compact, if $M$ is compact

The title says it all:
Let $M$ be a compact manifold and $N$ a (possible non compact) manifold. Equip the space of smooth functions $C^{\infty}(M,N)$ with the Whitney $C^{\infty}$ topology. (The ...

**2**

votes

**1**answer

98 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 ...

**16**

votes

**1**answer

412 views

### Differential Topology over $\mathbb{Q}$

I have a specific question in mind, but it requires some explanation and context before it can be formally stated. To summarize it in a sentence, this is it:
Are every two rational manifolds of the ...

**14**

votes

**3**answers

526 views

### When can a class in $H^1(M;\mathbb{Z})$ be represented by a fiber bundle over $S^1$

For a topological space M, It is known from homotopy theory that the elements of the first cohomology $H^1(M;\mathbb{Z})$ are in 1-1 correspondence with homotopy classes of maps $[M,S^1]$
In my case ...

**-3**

votes

**1**answer

168 views

### Loop space of manifold [closed]

Question A: The free loop space of a manifold is also a manifold?
Question B: The free loop space of an algebraic variety is also a algebraic variety ?
Are these questions asked or answered anywhere ...

**0**

votes

**0**answers

120 views

### Gromov's defenition of Content of Ball

Let $B(p, R)$ denote the metric ball of radius $R$ centered at $p$ in a manifold.
Then Gromov defined the content of the ball by
$$Cont(B(p,R))=rank(H_*(B(p, R/5))\to H_*(B(p,R)))
$$
and he remark ...

**8**

votes

**1**answer

226 views

### is there a diffeomorphism with only finite orbits but of infinite order?

I asked this in stackexchange, but got no answer, so I am trying here.
Is it possible for a diffeomorphism $\phi$ (of a smooth manifold $M$) to have the following properties:
All its orbits are ...

**5**

votes

**3**answers

328 views

### When a homeomorphism is a diffeomorphism w.r.t to a suitable smooth structure?

Assume we have a homeomoprhism $\phi:M\rightarrow M$, where $M$ is a topological manifold which admits at least one smooth structure.
Is it always possible to construct a smooth structure on $M$ ...

**3**

votes

**0**answers

170 views

### Do J-holomorphic curves “very nearly” fail to be an immersion near the bubbling points?

Let $u_{t}: \mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be a family
of degree $2$ maps defined (for $t$ small and non zero) by
$$u_t([X,Y]) := [X^2, t Y^2, XY].$$
Note that as $t$ goes to zero, ...

**-1**

votes

**0**answers

22 views

### Is C^0 fine topology is finer that metric topology? [migrated]

Let C(E,F) be a set of continious maps between metric spaces E and F. Suppose we are given $C^0$ fine topology and a metric topology on C(E,F). We now that the fine topology is finer than compat-open ...

**6**

votes

**2**answers

282 views

### Charts needed for an atlas

I just read this question link and asked myself, if there is any easy way to decide how many charts you actually need to cover a given compact manifold in $\mathbb{R}^3$, maybe at least in this ...

**4**

votes

**0**answers

146 views

### manifold branched covering space for orbifolds

An orbifold structure on some topological space $X$ is a covering of $X$ with local quotient charts $V/G$, where $V$ is some connected manifold and $G$ effectively acts on $V$ via a finite group of ...

**-1**

votes

**4**answers

397 views

### Studying topology: which first, algebraic or differential? [closed]

I have recently studying the basics of topology (ideas in point set, connectedness compactness) and I want to continue my studies but i'm interested in both differential and algebraic topology. which ...

**1**

vote

**0**answers

139 views

### How does a level set look like when the minimum point of a function degenerate?

I apologize in advance if this question is well-known. I would like to know an explicit example of a compact, connected manifold $M$ and a smooth function $f: M \to\mathbb{R}$ which satisfy the ...

**0**

votes

**0**answers

50 views

### Product structure on manifolds via lifting classifying maps

Let's say you want to study $d$-dimensional manifolds $M$ which decompose functorially into $M\cong N\times P$ for a fixed $P$. Can this structure be expressed by a lift of the stable normal bundle?
...

**10**

votes

**1**answer

274 views

### Morse number of the Poincaré homology sphere

What is the Morse number of the Poincaré homology sphere? What about the stable Morse number?

**6**

votes

**1**answer

115 views

### Closed geodesics in free smooth loop space?

I know very little about these subjects, so I apologise if this is a naive line of inquiry:
Let $M$ be a smooth $n$-dimensional Riemannian manifold. I understand that it is possible to construct an ...

**0**

votes

**0**answers

60 views

### Cohomology operators inducing local basis of $1-$forms

Suppose that $\partial$ is a non-trivial ($\partial \neq 0$) cohomology operator on an $m-$dimensional manifold $M$ (that is: $\partial:\Omega(M)\to\Omega(M)$ is a degree $1$ derivation such that ...

**0**

votes

**1**answer

256 views

### Shrinking $\mathcal{C}_c^{\infty}(M)$ to obtain a first countable space

This is a follow-up to this question.
Let $M$ be manifold (Hausdorff, countable base, finite dimensional, if it simplifies anything embedded in $\mathbb{R}^n$).
I'm interested in the topological ...

**4**

votes

**2**answers

373 views

### How many metrics of constant curvature exist on a Riemannian surface?

I have been trying to determine the number of metrics of constant curvature on a surface of genus $n$, say $\Sigma$. For low values, the answer is clear, the moduli space is a point for the sphere, ...

**7**

votes

**3**answers

310 views

### Classification of natural invariants of Riemannian structures

Before I formulate my question, let me remind P. B. Gilkey's characterization of Pontryagin forms,following the paper "On the heat equation and the index theorem" by Atiyah, Bott, Patodi.
By ...

**4**

votes

**1**answer

114 views

### Smoothing operator raising the smoothness exactly by one

Is there a continuous map $S: C^k(M)\to C^{k+1}(M)$ with the following properties?
(1) if $S(f)$ is $C^{k+2}$, then $f$ is $C^{k+1}$,
(2) if $f$ is $C^\infty$, then so is $S(f)$,
(3) $f$ and $S(f)$ ...

**2**

votes

**1**answer

191 views

### Ehresmann fibration theorem for manifolds with boundary

All manifolds in consideration may have nonempty boundary and may be disconnected.
Let me fix a definition first. A map between smooth manifolds $M\rightarrow N$ is a fiber bundle, iff it's locally ...

**6**

votes

**2**answers

351 views

### Relating different topologies on $C^{\infty}_c(M)$

This is somehow connected to this question.
I can think of at least four topologies to put on $C_c(M)$:
Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the weak Whitney ...

**5**

votes

**1**answer

230 views

### When does the Borel construction have the homotopy type of a CW-complex?

Suppose that $G$ is a Lie group acting smoothly on a manifold $M,$ does the Borel $M \times_G EG$ construction have the homotopy type of a CW-complex? If not, under what conditions would this be true? ...

**1**

vote

**1**answer

96 views

### Are normal deformations of an embedding open in the $C^{\infty}$-space of embeddings of a compact smooth manifold`

Let $M$ be a compact smooth manifold (closed, for simplicity), $n\in\mathbb{N}$ and equip the space of embeddings $Emb(M,\mathbb{R^n})$ wih the Whitney-$C^{\infty}$-topology. (The weak and the strong ...

**2**

votes

**0**answers

77 views

### Analogue of the Euler class of a circle bundle and the global angular form

This is a general question that asks whether there is geometric significance to differential forms that restrict to G-invariant volume elements on the fibers of a principal G-bundle.
For an SO(2) ...

**7**

votes

**2**answers

638 views

### Stokes theorem with corners

I've found the following version of Stokes' theorem in G. Stolzenberg's lecture notes 19:
Notation:
for $1 \le n \le m$
$\Lambda(m, n) = \{ \lambda: \{1,...,n\} \rightarrow \{1,...,m\} \ | \ ...

**2**

votes

**1**answer

132 views

### References on the Free Loop Space

I intend to approach the paper of Wolfgang Ziller: "The Free Loop Space of Globally Symmetric Spaces", but I need the proper background on the foundations of the study of Free Loop Spaces. I obtained ...

**8**

votes

**2**answers

229 views

### A kind of uniqueness for the double of a manifold

Given two smooth, connected manifolds, M, N, with the same boundary. If their doubles, D(M) and D(N) are diffeomorphic, does it follow that M and N are diffeomorphic ? The condition on the boundary is ...

**5**

votes

**0**answers

143 views

### proving the injectivity half of de Rham's theorem by construction in degrees other than $1$ and $n$

(This is a revision of a question I asked on MSE.)
Let $M$ be a smooth manifold of dimension $n$, and let $\omega$ be a differential form of degree $p$ on $M$. Then we have (I'm pretty sure) the ...

**2**

votes

**1**answer

102 views

### Why does the Gluck twist on a spun knot give the standard $S^4$?

Given a knotted arc $A \subset D^3$ (whose endpoints are, say, at $(\pm 1,0,0)$), the spun knot on this arc is $$\partial\left((D^3, A) \times D^2\right), = (\partial(D^3,A) \times D^2) \cup ((D^3,A) ...

**2**

votes

**0**answers

105 views

### Estimates of eigenvalues of elliptic operators on compact manifolds

The classical Weyl law says that if $\Delta$ is the Laplace operator on functions on a compact Riemannian manifold $(M^n,g)$, $n>2$, then its $k$th eigenvalue satisfies the asymptotic formula
...

**1**

vote

**0**answers

110 views

### Classification of $SU(n)$-principal bundles over a four-dimensional base

It is well-known that a principal $SU(2)$-bundle $P$ over a four-dimensional manifold $M$ is topologically classified by its second Chern-class $c_{2}(P)\in H^{4}(M,\mathbb{Z})$, as explained for ...

**1**

vote

**1**answer

112 views

### Boundary components of a subsurface

Consider the following situation. Suppose we have a closed oriented Riemannian surface $ \Sigma $ and a connected open subset $ \Omega \subseteq \Sigma $ with a boundary, consisting of finitely many ...

**5**

votes

**0**answers

201 views

### Is there a citeable reference for star-shaped open subsets of R^n being diffeomorphic to R^n?

A folk theorem says that star-shaped open subsets of R^n are diffeomorphic to R^n.
Is there a citeable reference for this result?
For the sake of being definite, let's say that
“citeable” means ...

**-2**

votes

**1**answer

176 views

### Degree of a rational function [closed]

I would like to have a simple proof for the following result:
Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined). ...

**2**

votes

**1**answer

115 views

### Is there a formula for the intersection of projectivized lines inside a projectivized vector bundle?

Let $E\rightarrow D$ be a complex rank two vector bundle over a compact complex
one dimensional manifold $D$. Let $L_1, L_2 \subset E$ be rank one subbundles of E
(i.e. line bundles). Let
$$ n_1:= ...

**0**

votes

**1**answer

221 views

### On Gromov's Theorem on Symplectic Homotopy

I want to understand the proof of the following theorem due to Gromov which I'll state in the context of Euclidean spaces. While I tried to read the proof from Macduff-Salamon, it turned out that my ...

**1**

vote

**0**answers

31 views

### Existence of triangulation of Lipschitz domains

Consider a bounded Lipschitz domain $\Omega \subset \mathbb R^n$.
Q1: Can its closure $\overline\Omega$ be triangulated?
Q2: If yes, can the triangulation be chosen as finite? Furthermore, how ...

**1**

vote

**1**answer

192 views

### Linearisation of Einstein operator

Let $(M,g)$ be a $(m+1)$-dimensional Riemannian manifold with Levi-Civita connection $\nabla$.
The Ricci curvature can be viewed as a differential operator ...

**2**

votes

**1**answer

150 views

### Lorentzian metrics on the torus up to continuos deformations

Any two Riemannian metrics can easily be deformed into each other, only obtaining positive definite metrics in between.
However, for metrics of other signatures this might not be possible.
Which ...

**5**

votes

**2**answers

174 views

### Is the strong Whitney topology connected?

$\def\bbR{\mathbb R}\def\ssp{\kern.4mm}$Put more precisely, let $C$ be the set of all continuous functions $f:\bbR\to\bbR$ when
$\bbR$ has its standard order topology. Let $\mathscr T$ be the set of ...

**6**

votes

**1**answer

208 views

### Is there an analog of compactified moduli spaces(/stacks) for smooth manifolds?

This question has been inspired by an answer to the question Reference request: Topology on the space of smooth compact submanifolds; I've asked it in a comment to that answer but then decided to make ...

**4**

votes

**0**answers

133 views

### When does a manifold 'deformation retract' into a small neighborhood of some k-dimensional subpolyhedron?

Under which conditions does a m-manifold $M^m$ admit a deformation retraction to a
small neighborhood of some k-dimensional subpolyhedron?
Or, under which conditions is the identity map $id_M$ of a ...