**9**

votes

**0**answers

147 views

### Do homology classes have “special” representatives?

Recall that, according to Hodge, de-Rham cohomology classes of "nice enough" manifolds have "special" representatives - namely, harmonic forms.
Now, how does one choose a "special" one among ...

**2**

votes

**1**answer

124 views

### Isometry Group of real Hilbert space?

Does the isometry group of a real separable infinite-dimensional Hilbert space have two connected components? Or, conversely, is the there even a Kuiper's theorem in the real case?
How does the ...

**5**

votes

**2**answers

297 views

### Riemannian metrics preserved by diffeomorphisms

Let $f \neq Id$ be a diffeomorphism (of a smooth manifold $M$) which admits some Riemannain metric on $M$ making it an isometry. How many different metrics are preserved by $f$?
Note that ...

**6**

votes

**1**answer

213 views

### Differential geometry without the Hausdorff condition or the second axiom of countability

I would like to know how the standard differential geometry of manifolds would change if we didn't assume the Hausdorff condition and/or the second axiom of countability. There are some simple things ...

**6**

votes

**1**answer

179 views

### How does one identify flow lines on a vector bundle with those on the base in Morse theory?

In Chapter 4.2 of Schwarz's book on Morse homology there is a brief discussion of Morse theory on the total space of a smooth vector bundle $E \to M$. In particular, one can take the Morse function ...

**1**

vote

**0**answers

73 views

### Integral curves on non compact manifolds [closed]

Define a vector field on $\mathbb{R}^d$ by $X = \frac{\partial}{\partial x_{d}}$. That is a vector field that always points upward along the $x_{d}$-axis. Consider starting at any point $p \in ...

**3**

votes

**1**answer

192 views

### Generalising the parametric transversality theorem to a foliation

The parametric transversality theorem states that, given a parameterised family of smooth maps of $C^{\infty}$ manifolds $\phi_s:M \rightarrow N$ and a submanifold $R < N$ then for almost all ...

**1**

vote

**1**answer

145 views

### Orientability of Surfaces and the Fundamental Group [closed]

Let $(M,g)$ be a compact riemannian 3-manifold and $\Sigma \subset M$ an embedded compact surface homeomorphic to the projective plane. Consider the application $i_\#:\pi_1(\Sigma)\to \pi_1(M)$ given ...

**-1**

votes

**0**answers

68 views

### diagonal stratum in symmetric product

Fix a Riemann surface $(\Sigma, j )$ and a partition $\pi: r=\Sigma a_in_i$ for integer $a_i \ge 1, n_i \ge 1$, there is a diagonal stratum $\chi_{\pi}$ indexed by $\pi$ comprising the image of the ...

**1**

vote

**0**answers

34 views

### A questions related to the Markus conjecture for special affine manifolds

An affine manifold $M$ is called special if there is a parallel volume form $\omega$ on $M$,
and a nowhere vanishing vector field $\mathcal{V}.$
Here we need to point out that any affine manifold of ...

**2**

votes

**2**answers

160 views

### Conditions on a Lorentzian manifold to ensure existence of global proper-time foliation?

I am wondering what conditions a Lorentzian manifold $(M,g)$ must satisfy to ensure the existence of a global proper-time foliation (i.e. a decomposition of $M$ into spacelike Cauchy hypersurfaces and ...

**6**

votes

**1**answer

225 views

### Four-dimensional vector bundles over $S^4$, intuition?

I know that $\pi_3(SO(4)) = \mathbb{Z} \oplus \mathbb{Z}$. We can choose an explicit identification as follows: given $(i, j) \in \mathbb{Z}$, we have a map $\phi: S^3 \to SO(4)$ which sends a unit ...

**6**

votes

**0**answers

147 views

### Intuition behind the following theorem of Reeb?

What is the intuition behind the following theorem of Reeb?
If a compact manifold admits a function with only two critical points which are non degenerate, it is homeomorphic to the sphere.

**4**

votes

**1**answer

79 views

### Submersion theorem for smooth tame Frechet manifolds

If $M$ and $N$ are Banach manifolds, $f:M\rightarrow N$ is a smooth map, and $q\in N$ is a regular value, so $f$ is a submersion on $f^{-1}(q)$, it is well known that the level set $f^{-1}(q)$ is a ...

**3**

votes

**0**answers

71 views

### What is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration? [migrated]

As the question suggests, what is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration? I not a topologist by trade and I find the concept kind of hard to understand... Thanks ...

**2**

votes

**0**answers

87 views

### cross-sections of a sphere bundle

Let $M$ be a $m$-manifold and $M_0$ a submanifold of $M$. Let $X$ be a pointed topological space. In the paper On the homology of configuration spaces, Bodigheimer-Cohen-Taylor, Topology 1989, ...

**1**

vote

**0**answers

82 views

### How close (Homology-wise) can we approximate a topological manifold with a PL or smooth one?

Sorry if this question is to naive or badly phrased. I am curious about the following problem, given a manifold $M$, how "close" can we find a smooth or PL manifold, $N$, with a map $f:N\to M$. The ...

**2**

votes

**1**answer

193 views

### Construction of appropriate Morse functions

I am interested in the properties of connectedness of level sets of Morse functions. Let $M$ a compact smooth $n$-manifold, and $1\leq k<n$. Is it possible to construct $k$ Morse functions ...

**2**

votes

**1**answer

110 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 ...

**1**

vote

**1**answer

114 views

### Reference request: $\mathcal{C}^\infty_c(M)$ is a topological vector space with the Whitney topology

Let $M$ be a smooth manifold and let $\mathcal{C}^\infty(M)$ be the space of all smooth real valued functions with the (strong) Whitney topology. This space is a topological vector space iff $M$ is ...

**1**

vote

**1**answer

101 views

### Decomposition of a closed surface

I know that I can decompose an hyperbolic closed surface of genus $g>1$ into $2(g−1)$ pants bounded by $3$ geodesics. It seems reasonable to think the same can be done for a closed surface of genus ...

**0**

votes

**1**answer

138 views

### Marcel Berger's “Sur les groupes d'holonomie homogènes de variétés à connexion affine et des variétés riemanniennes.”

I would appreciate any reference that contains either a translation or proof of the main theorem in this paper. Thank you in advanced.

**1**

vote

**0**answers

43 views

### Parametric Sard-Smale theorem - when is the generic set open?

I am learning Morse/Floer Theory and in my work on my master's thesis I want to apply the parametric Sard-Smale theorem. I.e. I consider a Banach bundle $\mathcal{L} \to \mathcal{M}\times \mathcal{G}$ ...

**3**

votes

**1**answer

157 views

### Reference request: Intrinsic definition of the strong Whitney topology on $\mathcal{C}^{\infty}(M,\mathbb{R})$ without using charts or jets

Let $M$ and $N$ be smooth manifolds. There is a description of the strong Whitney topology on $\mathcal{C}^{\infty}(M,N)$ in terms of partial derivative in charts (using locally finite sets of charts ...

**2**

votes

**1**answer

194 views

### existence of totally geodesic hypersurfaces

Assume we are on a smooth, complete Riemannian manifold $(M,g), dim(M) \geq 3$. What are the specific geometric/topological constraints for such a manifold to admit complete, totally geodesic ...

**4**

votes

**2**answers

186 views

### Exponential rule for Whitney-$\mathcal{C}^{\infty}$-topology

Let $M,N,X$ be smooth manifolds. Equip the space of smooth functions between two manifolds with the (strong) Whitney- $\mathcal{C}^\infty$-topology.
The evaluation map $$ev\colon ...

**2**

votes

**0**answers

74 views

### Gradient vector fields defined with respect to two different metrics and Morse theory

Given a differentiable manifold $M$, we can equip $M$ with a Riemannian metric $g$ or $g'$ to generate a pair of Riemannian manifolds $(M,g)$ and $(M,g')$, respectively. The gradient vector fields ...

**0**

votes

**2**answers

150 views

### Special connection of vector bundle over real manifold

Let $E \rightarrow M$ be a vector bundle over a smooth manifold $M$ and let $g$ be a bundle metric. Does there exists a conection (maybe unique) $\nabla$ which is compatible with $g$. By this I mean: ...

**3**

votes

**0**answers

75 views

### The Free Loop Space of a Manifold $M$ when $M$ is not compact

In Klingenberg's Lectures on Closed Geodesics, before constructing the differentiable structure of the free loop space of a compact manifold $M$, he states that:
A large part of the construction ...

**1**

vote

**0**answers

45 views

### Perturbation of a Fredholm sections which preserves compactness of 0-set

I am learning Morse-Bott-Floer theory and found the following cool paper
http://de.arxiv.org/abs/1310.5080
by P. Albers and D Hein. In order to prove a cup-length estimate on the number of critical ...

**25**

votes

**1**answer

996 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 ...

**2**

votes

**0**answers

69 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 ...

**7**

votes

**1**answer

223 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

46 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

112 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

182 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 ...

**17**

votes

**1**answer

488 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 ...

**15**

votes

**3**answers

572 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

183 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

134 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

242 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 ...

**7**

votes

**3**answers

394 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

181 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, ...

**6**

votes

**2**answers

296 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

172 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

524 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

170 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

53 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

307 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

134 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 ...