**15**

votes

**0**answers

193 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

267 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

143 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

156 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

90 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

293 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

228 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

123 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

280 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

151 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

100 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

194 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

47 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

312 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.

**13**

votes

**1**answer

458 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

165 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. ...

**8**

votes

**2**answers

265 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

78 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

85 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

296 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

226 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 ...

**7**

votes

**2**answers

351 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

160 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

214 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 $x\in D^m$ ...

**0**

votes

**0**answers

66 views

### Foliation values of Exotic spheres

In the following question, we defined the foliation values of an smooth manifold;
Foliation values of a manifold
Let $S_{i}$'s, $i\in \{0,1,\ldots,27\}$, be the smooth structures of topological ...

**2**

votes

**0**answers

110 views

### Foliation values of a manifold

Let $M$ be a smooth n dimensional manifold. The foliation values of $M$, denoted by $F(M)$, is defined as
\begin{equation} F(M)=\{ 1\leq k\leq n\mid \text{there exist an smooth $k$ dimensional ...

**2**

votes

**1**answer

197 views

### Frobenius rank of a manifold

The rank of an smooth manifold M is defined by Milnor, as follows:
"The maximum number of independent commuting vector fields on M"
For example it is well known that the rank of $S^{3}$ is 1 (Lima, ...

**3**

votes

**0**answers

127 views

### A symplectic version of critical points

According to the interesting comment of Mohammad F Tehrani, I revise the question as follows:
Assume $n>2$. For what type of compact n dimensional manifolds $M$ we can say:
For every smooth ...

**6**

votes

**1**answer

180 views

### A lagrangian version of the Withney theorem

Let $M$ be a smooth n dimensional manifold. Is there an smooth embedding $f:M \to \mathbb{R}^{2n}$ which image is a Lagrangian submanifold of $\mathbb{R}^{2n}$?

**3**

votes

**1**answer

167 views

### Fundamental proof of the baby case of Hofer's theorem about displacement energy

In 1990, Hofer proved that the displacement energy of a standard ball in $C^{n}$ equals it's Gromov area.
Here is the baby case: Consider a smooth bounded function $f:R^{2}\rightarrow R$. Consider ...

**0**

votes

**0**answers

78 views

### Interpolating between two points on Stiefel manifold

I'm looking for a formula to interpolate between two given matrices from the Stiefel manifold (orthogonal n by k matrices).
I do not know the tangent direction, I only know the start and end points ...

**-1**

votes

**1**answer

143 views

### $S^n$ admit a real polarization $D\subset TS^n$?

When the $n$-sphere, $S^n$,admit a real polarization $D\subset TS^n$

**15**

votes

**1**answer

745 views

### Periodic Orbit property

A topological space $X$ satisfies "Periodic orbit property", briefly POP, if for every continuous map
$f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that ...

**-2**

votes

**1**answer

119 views

### A question on parallelizable manifolds [closed]

Let $M$ be a manifold with the property that $f^{*}(TM)$ is isomorphic to TM, for every diffeomorphism $f$ on $M$. Does this imply that $M$ is parallelizable?

**4**

votes

**1**answer

267 views

### Totally non parallelizable manifold

Does there exist a manifold M which all iterated tangent bundles are non parallelizable manifolds? That is$ M, TM , T^2(M), \ldots ,T^n(M)\ldots$ are non parallelizable manifold?
What is ...

**8**

votes

**1**answer

477 views

### Pontryagin numbers on manifolds with an $S^1$-action

Let $M$ is a smooth compact manifold with an $S^1$-action with isolated fixed points. Suppose the representation of $S^1$ at tangent spaces at all fixed points is known. Can one then find all ...

**4**

votes

**1**answer

375 views

### Quantization of symplectic vector space and choice of lagrangian subspaces

My question is related to Geometric Quantization.
I don't undrestand the philosophy of following assertion
If $(V,\omega)$ be a symplectic vector space then the quantizations of
$V$ ...

**6**

votes

**1**answer

260 views

### how to obtain a generalized Morse function out of a fiber bundle?

I already asked this question in MSE but did not get any answer/comment yet.
Let $M\to E\to B$ be a smooth fiber bundle. In "Parametrized Morse Theory and Its Applications,(Proceedings of the ICM, ...

**8**

votes

**0**answers

273 views

### Space of embeddings of an $n$-ball into an $n$-manifold

Let $M$ be a smooth $n$-manifold without boundary, and let $B$ be the open unit ball in $\mathbb{R}^n$. I am trying to understand the space $\text{Emb}(B,M)$ of smooth embeddings of $B$ into $M$. ...

**3**

votes

**0**answers

72 views

### Analytic stuctures on $\mathbb R^n$ and the nilpotent ideal of supermanifolds

I have two questions which are somewhat related:
(a) It is a well known result (of Freedman?) in differential topology that $\mathbb R^4$ has exotic smooth structures. Apparently, it is known that ...

**10**

votes

**3**answers

734 views

### Is the class of n-dimensional manifolds essentially small?

Question: Consider the proper class of all $n$-dimensional smooth manifolds. If we take the equivalence classes where two manifolds are identified if there exists a diffeomorphism between them, is ...

**2**

votes

**1**answer

513 views

### Is it possible to define a structure of differentiable manifold on the Hilbert cube $[0,1]^\mathbb{N}$?

Is it possible to define a structure of differentiable (smooth) manifold on the Hilbert cube $[0,1]^\mathbb{N}$ ?
Has it been done in the literature?
In textbooks, only the Banach case is treated, ...

**11**

votes

**1**answer

227 views

### Hyperbolic Manifolds which fiber over the circle

If $N^2$ is a closed, orientable surface of genus at least $2$, and if $\phi$ is an (orientation-preserving) pseudo-Anosov mapping on $N$, then one can form the closed orientable 3-manifold $M^3$ by ...

**1**

vote

**1**answer

169 views

### isotopy classes of embeddings of the torus

Let's consider $S^1$-bundle $E$ over a 2-manifold $M$. How many isotopy classes of embeddings of the torus $\mathbb{T}^2$ in $E$?
For each free homotopy classes $\gamma$ of mappings of the circle ...

**1**

vote

**0**answers

87 views

### Proving that two given functionally structured spaces are isomorphic

The relevant definitions are listed below. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups; and Section 2, Chapter II of Bredon's Topology and ...

**9**

votes

**1**answer

379 views

### Do partitions of unity exist if we impose additional conditions on the derivatives?

Let $ ~~\cup_{k=-1}^{\infty} U_k = \mathbb{R} $ be an open covering of
$\mathbb{R}$. It is a well known fact that partitions of unity subbordinate to
the cover exists, i.e. there exists smooth
...

**7**

votes

**2**answers

221 views

### Pseudofree $T^2$ actions on spheres

Is it possible to construct a smooth action of $S^1\times S^1$ on $S^{2n+1}$ ($n\ge 2$) such that no point on $S^{2n+1}$ has an infinite stabilizer?
Note that if such an action exists, it can not be ...

**6**

votes

**2**answers

212 views

### Fixed component of an $S^1$ action on $S^n$

Suppose $S^1$ is acting smoothly on $S^n$ and $M$ is a connected component of the set of fixed points of the action. What can be said about $M$?
Is it true that $\pi_1(M)=0$? (sorry this first bit ...

**8**

votes

**3**answers

393 views

### Configuration spaces of the torus

I would like a reference that calculates the rational homology of the unordered configuration spaces of the torus.

**2**

votes

**3**answers

251 views

### Fatou sets and topological entropy

Let us consider a diffeomorphism of a compact real manifold (complex manifold defined over the reals), and let us say that the diffeomorphism is birational. Hence, it extends to a birational map from ...