**3**

votes

**0**answers

80 views

### Is $C^\infty(M,\mathbb{R})$ an ind-(smooth manifold)?

Let $\mathrm{Man}$ be the category of smooth manifolds (2nd countable, Hausdorff, no boundary, not necessarily compact) and smooth maps, and let $M$ be an object thereof. Is the presheaf $S \mapsto ...

**1**

vote

**1**answer

147 views

### How to extend an equivariant map from a compact Lie group

Let $G$ be a compact Lie group and let $H$ be a closed subgroup of it. Let $g$ be a torsion element of $G$ and $C_G(g)$ the centralizer of it. Let $Y$ be a $C_G(g)-$space. I'm working on the space ...

**16**

votes

**4**answers

1k views

### How to get convinced that there are a lot of 3-manifolds?

My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that ...

**7**

votes

**1**answer

154 views

### first Chern class of complex vector bundles and first Pontrjagin class of quaternionic vector bundles [on hold]

Let $\xi$ be a (real) vector bundle of dimension $n$. Then the first Stiefel-Whitney class
$$
w_1(\xi)=0
$$
if and only if $\xi$ is orientable, i.e. the structure group of $\xi$ can be reduced to ...

**5**

votes

**1**answer

420 views

### Intuition behind salient numbers in number of h-cobordism classes of smooth homotopy n-spheres

The Wikipedia article on Exotic Sphere displays this sequence of numbers (see also OEIS A001676 and the Milnor link therein) for the order of the classses as
$$1, \;1, \;1,\; 1,\; 1, \;1, \;28,\; ...

**2**

votes

**0**answers

55 views

### What is known about topological equivalence of polynomial dynamical systems on two different domains in R^n?

The question is mainly about $\it flows$, not maps (i.e., continuous time, not discrete time).
Is it known if the study of polynomial dynamical systems on $\mathbb R^n$ can be reduced to the study ...

**7**

votes

**0**answers

54 views

### Embedding of parallelizable closed smooth manifold

It seems that any closed parallelizable smooth $n$-manifold can be immersed in the $(n+1)$-dimensional Euclidean space (compared to the general $(2n-1)$-dimensional case). Is there any analogous ...

**10**

votes

**1**answer

228 views

### Reference request: sheaves on the site of d-manifolds

I believe I know how to prove the following results. I also know to whom to cite fancy-shmancy results that have these as a very special case. My question is: what are the correct citations for ...

**9**

votes

**3**answers

714 views

### Is each closed convex set a manifold with corners?

Assume that $C$ is a convex set in $\mathbb{R}^{n}$ with non empty interior.
Then consider its closure, is it a smooth manifold with corners?
Edit:
1) The closure of $C$ should be a smooth manifold ...

**0**

votes

**0**answers

49 views

### Three-manifolds related by degree 1 map, whose products with the two-sphere are diffeomorphic

Suppose $M$ and $N$ are compact oriented smooth 3-manifolds such that there is an orientation-preserving diffeomorphism between the products $F:M \times S^2 \to N \times S^2$. Further, there are ...

**5**

votes

**0**answers

310 views

### Prerequisites for reading Gregory Perelman's work

What are the prerequisites for understanding the work of Perelman concerning the Poincaré conjecture?
I am referring to the last three papers here.

**1**

vote

**0**answers

60 views

### Surjectivity of maps between spheres [closed]

I am wondering how to prove that a non-zero degree map from $S^n \to S^n$ is surjective. For example, identifying $S^1 \subset \mathbb{C}$, we can take $f:S^1 \to S^1$ via $f(z) = z^k$ with $k\neq 0$. ...

**0**

votes

**0**answers

50 views

### Homogeneous regular manifolds

In order to solve the well-known Plateau-Problem on a general (non-compact) Riemannian 3-manifold, Morrey first introduced the condition of homogeneous regularity and defined it in the following way:
...

**-2**

votes

**0**answers

81 views

### Degree of map into Lie group representation

Suppose $M$ is a smooth manifold with unit volume and that $G$ is a compact Lie group of the same dimension. Given a smooth map $\phi: M\rightarrow G$, we can compute the degree of $\phi$ as:
...

**3**

votes

**1**answer

74 views

### Set of density matrices

A density matrix is a matrix $\rho \in \mathscr{D}:=\{A \in \mathbb{C}^{n \times n}; A^*=A; \operatorname{tr}(A)=1; A \ge 0\}.$
In Quantum Mechanics it is natural to look at a group action
$\Phi: ...

**9**

votes

**0**answers

161 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

136 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

313 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 ...

**7**

votes

**1**answer

223 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

193 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

76 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

202 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

147 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**

vote

**0**answers

49 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 ...

**3**

votes

**2**answers

173 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

244 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 ...

**5**

votes

**0**answers

155 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

85 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 ...

**2**

votes

**0**answers

89 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

89 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

197 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 ...

**3**

votes

**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 ...

**1**

vote

**1**answer

116 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

102 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

139 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

45 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

161 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

203 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

189 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

76 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

154 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: ...

**4**

votes

**0**answers

87 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

46 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 ...

**28**

votes

**2**answers

1k 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

70 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

49 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

118 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

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 ...

**17**

votes

**1**answer

504 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 ...