The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.

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21 views

### How to use differential system to predict error in eigenvectors

What is the method to use differential system to predict error in eigenvectors?
If there is time t eigenvector and time t+1 eigenvector
It usually fluctuating around 1/2 , sometimes like a E , L, H, ...

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**2**answers

311 views

### Measures and differential forms on manifolds

Let $M$ be a differentiable manifold. Let $\mu$ be a (probability) measure on $M$.
What are the conditions under which $\mu$ is given by a differential form on $M$? I imagine some sort of ...

**2**

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**0**answers

17 views

### Theorems similar to Tischler fibering theorem

Tischler theorem states that the existence of a nowhere vanishing closed $1$-form in a compact manifold $M$ implies that the manifold fibers over $S^1$. Do you know any other diffential topology ...

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**1**answer

91 views

### Possible orders of automorphisms for the Poincare homology sphere

Let $M^3$ denote the Poincare homology sphere. I am wondering what the possible orders of (smooth) automorphisms of $M$ are (I'm not sure if allowing arbitrary homeomorphisms changes things?). By ...

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98 views

### Geometric interpretation of torsion homology classes

Suppose I have a homology class $x \in H_1(M)$ which is torsion of order $k$ say. Suppose furthermore that $M$ has Dimension big enough, such that every element of $H_1$ and $H_2$ can be relalized as ...

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**1**answer

88 views

### Computing relative cohomology class of differential form

When dealing with a top degree differential form $\mu$ in a manifold $M$, a way of "computing" its cohomology class is integrating it through the whole manifold. For instance, if the integral $ \int_M ...

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72 views

### Trying to understand why this local coordinates parametrizes a manifold

First of all, I would like to say that I think this question fits better on Math Overflow than on Math Stack Exchange, in view of the proposal of the two sites. However, if my analysis of the ...

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**1**answer

197 views

### Every unorientable 4-manifold has a $Pin^c$, $Pin^{\tilde c+}$ or $Pin^{\tilde c-}$ Structure

The precise statement on J. W. Morgan's "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds (MN-44)" that 4-manifold $X$ admits a Spinc structure (Lemma 3.1.2) ...

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**1**answer

200 views

### A possible characterization of sphere or projective space

Is there a compact Riemanian manifold $M$ not diffeomorphic to sphere or real or complex or quaternion projective space which admit a diffeomorphism $f$ with the property that $$\forall x \in M, \...

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**0**answers

122 views

### Separating submanifolds in $\mathbb R^n$

Let $M\subset\mathbb{R}^n$ be a submanifold. If M is diffeomorphic to $\mathbb{R}^{n-1}$ and $\mathbb{R}^n\smallsetminus M$ has two components, then must the components be diffeomorphic to $\mathbb{R}^...

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47 views

### Is there a transitive Lie group action on the space of matrices with rank bigger than $k$?

$\newcommand{\GL}{\operatorname{GL}}$
Let $H_{>k}$ be the space of real $d \times d$ matrices of rank bigger than $k$, for some fixed $k$. $H_{>k}$ is an open connected submanifold of $ \mathbb{...

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113 views

### Non-spin 5-manifold and $2^2$-Bockstein homomorphism

The $2^2$-Bockstein is $\beta_4$ is associated to
$$0\to\mathbb{Z}/2\to\mathbb{Z}/{8}\to\mathbb{Z}/{4}\to 0,$$
(The $2^n$-Bockstein homomorphism
$$\beta_{2^n}:H^*(-,\mathbb{Z}/{2^n})\to H^{*+1}(-,\...

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votes

**3**answers

86 views

### On the properness of the graph of a convex function

Let $f : \Omega \subseteq \mathbb{R}^n \to \mathbb{R}$ be a smooth and convex function. Let us assume that $\Gamma_f = \mathrm{graph}(f) $ is a complete hypersurface of $\mathbb{R}^{n+1}$. Then I know ...

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**1**answer

123 views

### Stable normal framings of parallelizable manifolds

Suppose $M$ is a compact, connected, orientable manifold ($\dim M=m$) with trivial tangent bundle and let $j \colon M \to \mathbb R^n$ be an embedding. Suppose we choose a trivialization of $TM$. Then ...

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**1**answer

125 views

### Exponential map/ Lie derivative in variation for constant formula for ODE

In short: The question is how to go from the first equation on page 8, of this paper to the second equation.
Some background
I'm working in optimization and I am currently reading a paper
see page ...

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**1**answer

135 views

### Are isotopic and conjugate homeomorphisms, conjugate by an element in $\mathrm{Homeo}_0(M)$?

An answer to this question would also answer Isotopy of periodic homeomorphisms of a surface along periodic homeomorphisms
Let $M$ be a topological manifold and let $f,g$ be two orientation ...

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**1**answer

445 views

### What can we say about the Cartesian product of a manifold with its exotic copy?

Let $M$ be a smooth oriented manifold, and let $M^E$ be an exotic copy, i.e homeomorphic but not diffeomorphic to $M$.
Is it true that $M\times M$ is diffeomorphic to $M\times M^E$?
I am ...

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**0**answers

70 views

### When are two codimension-one foliations of a manifold M “diffeomorphic”?

By the question, I mean: Given two different codimension-one foliations of a manifold M, $\mathcal{F}_i$ and $\mathcal{F}_j$, when does there exist an element of Diff(M) that maps each leaf of $\...

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**2**answers

298 views

### Torus action implying infinite fundamental group

Suppose that a $d$-dimensional torus $T$ acts smoothly and effectively on an $n$-dimensional closed manifold $M$. What conditions on $d$ and $n$ imply that $\pi_1(M)$ must be infinite?
Consider the ...

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**1**answer

271 views

### Thom space, homotopy group and cohomology group

In Thom's 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, and the Steenrod operations were all related. He used these ideas to prove in the 1954 paper Quelques propriétés ...

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**1**answer

269 views

### Deforming a section to a section without zeros

Let $M$ be an oriented manifold of dimension $n$. Suppose furthermore that $E$ is an oriented vector bundle of rank $n-1$ over $M$. Let $s$ be a section of $E$ transversal to the zero section in $E$. ...

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**0**answers

45 views

### Must bounded, closed, smooth curves with long straights have sharp bends?

Consider the family of bounded, closed, and continous curves $\Gamma$, i.e. for all $\gamma \in \Gamma$, we have $\gamma : [0, 1) \to [0, 1]^2$. Within this family, I am interested in curves that ...

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**0**answers

92 views

### Quotient by a non-free action of a Lie group and manifolds with corners

The quotient manifold theorem says that
If $G$ is a Lie group acting freely and properly on a smooth manifold $M$ then $M/G$ has a (unique) smooth structure such that the projection $\pi:M\to M/G$...

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**1**answer

221 views

### Non-triangulable 4-manifold as a boundary of some 5 manifold

We know that there are non-triangulable 4-manifolds, such as the E$_8$ manifold.
Can E$_8$ manifold be a boundary of some 5-manifold $M_5$? Can such a $M_5$ be triangulable or non-triangulable? What ...

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**2**answers

421 views

### Any 3-manifold can be realized as the boundary of a 4-manifold

We know
"Any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$." See this post: Elegant proof that any closed, oriented 3-manifold is the boundary of some ...

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**2**answers

422 views

### What is the weakest negative curvature condition ensuring a manifold is a $K(G,1)$?

The only statement I'm sure of is that any hyperbolic or Euclidean manifold is a $K(G,1)$ (i.e. its higher homotopy groups vanish), since its universal cover must be $\mathbb H^n$ or $\mathbb E^n$. ...

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**1**answer

361 views

### Is there a PL, or topological, bordism hypothesis?

The bordism hypothesis says that the $(\infty, n)$-category of smooth, framed $n$-bordisms, $(n-1)$-dimensional boundaries, and corners down to points, is freely generated symmetric monoidal with ...

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**1**answer

177 views

### Thom's first isotopy lemma

Thom's first isotopy lemma says that given $f:M\to P$ a smooth map between smooth manifolds and a closed Whitney stratified subset $S$ of $M$, such that
$f|_S:S\to P$ is proper and $f|_X:X\to P$ is a ...

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**0**answers

201 views

### Is a Difference of Fiber Bundles a Fiber Bundle?

I have a seemingly very basic question in differential topology, but I could not find the answer by a short google search.
Let $M,N$ be smooth manifolds, and let $f:M\to N$ be a smooth fiber-bundle,
...

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53 views

### Locally trivial fibration of hyperplane arrangement complement

I have a very specific question. How does one check the following map
${\mathbb C}^n-\cup \{z_i=\pm z_j\ for\ i\neq j\}\to {({\mathbb C}^*)}^{n-1}-\cup\{z_i= z_j\ for\ i\neq j\}$ defined by $(z_1,z_2,...

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**1**answer

112 views

### What is symplectic cut of a 4-ball?

Lerman's symplectic cut construction applied on 4-ball by collapsing its boundary 3-sphere along the $\mathbb{S}^1$ orbits of Hopf fibration gives a closed 4-dimensional symplectic manifold. ...

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**2**answers

501 views

### Topological obstructions to existence of immersion

Let $M$ be a smooth, non-compact manifold.
a) Can one always find a smooth, compact manifold $N$ with $\dim(N) = \dim(M)$ and a smooth embedding $i: M \to N$ ?
b) If not, are there some concrete ...

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**1**answer

130 views

### How to calculate the Chern class of the tensor product of a torsion free sheaf with a line bundle

I'am try to work with Chern class of the coherent sheaves, in this sense. If I have a vector bundle $E$ of rank $r$ and $L$ a line bundle we have the Chern class property
$$c_{r}(E\otimes L) = \sum_{...

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votes

**1**answer

222 views

### What are some surprising facts that happen after you remove a point to a space? [closed]

There are some facts that are really impressive after you remove a point to a space. Some typical examples are the existence of exotic spheres or the fact that
$S^4$ is not almost complex. Or some not ...

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**0**answers

122 views

### Is the image of the map $A \to \bigwedge^{k}A $ a weakly embedded submanifold?

$\newcommand{\End}{\operatorname{End}}$
$\newcommand{\GL}{\operatorname{GL}}$
Let $V$ be a $d$-dimensional real vector space. ($d \ge 4$). Fix an odd $2 \le k \le d-2$. Define
$H_{>k}=\{ A \in \End(...

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**0**answers

57 views

### Smooth structures on connected sums of high dimensional manifolds

Are there examples of manifolds $M$ of dimension $n\ge 5$ which are nontrivial connected sums, and such that the classification of smooth structures on $M$ is known?
For instance, a connected sum of ...

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**1**answer

166 views

### Is the map $A \to \bigwedge^{k}A $ from matrices above rank $k$ proper?

$\newcommand{\End}{\operatorname{End}}$
Let $V$ be a $d$-dimensional real vector space. ($d \ge 3$). Fix an odd $2 \le k \le d-1$. Define
$H_{>k}=\{ A \in \End(V) \mid \operatorname{rank}(A) > ...

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votes

**1**answer

216 views

### Chern classes of generators of $K(S^{2n})$

Calculate the Chern classes $$ c_n \in H^{2n}(S^{2n})$$ for the generator of the group $$ K(S^{2n})$$ where $S^{2n} $ - sphere of dimension $ 2n $, $ K(S^{2n})$ - group from K-theory.
I found the ...

**11**

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**1**answer

346 views

### Sheaf-theoretically characterize a Riemannian structure?

A smooth structure on a topological manifold can be characterized as a sheaf of local rings, see for example the discussion here.
Q: Is there a way to characterize a Riemannian structure on a smooth ...

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votes

**2**answers

241 views

### Are there vector fields which are gradients with respect to one metric but not another? [closed]

Is it possible for a vector field on a smooth manifold $M$ to be a gradient with respect to a Riemannian metric $g$, but not a gradient with respect to a different Riemannian metric $h$?
For ...

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**1**answer

112 views

### Isotopy extension theorem: how non-unique is ambient isotopy

Let $M$ and $N$ be smooth manifolds. Consider an isotopy of $M$ inside $N$. This means that we have a level preserving embedding $J\colon M\times [0,1] \to N \times [0,1]$. Put $J(x,t)=(\phi_t(x),t)$. ...

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**0**answers

91 views

### Why according to Takens' theorem (1981) we need 2m+1 observation functions (or time series) to reconstruct the attractor?

In the paper [1], page 369 we have Theorem 1 which says:
Let $M$ be a compact manifold of dimension $m$. For pairs $(\phi,y),\;\phi: M\to M $ a smooth diffeomorphism and $y: M \to \mathbb{R}$ is a ...

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**0**answers

108 views

### Classification of fibrations for classifying spaces $B^2\mathbb{Z}_2$ and $BSO(2)$ or $BO(2)$

Thanks to a suggestion by @Igor Belegradek, I am interested also in a simpler problem of this earlier question 301523, by knowing what can we say about the classification of fibrations for classifying ...

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**0**answers

37 views

### Is a Sobolev map with smooth minors smooth on the whole domain?

This question is related (but not identical) to this question.
Let $d>2$ be an integer and let $2 \le k \le d-1$ be a fixed integer. Suppose that at least one of $k,d$ is not even. Let $\Omega$ be ...

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**1**answer

210 views

### Is the sheaf associated to a differential structure of a specific type?

On a set $X$, let us define a set $\mathcal{D}$ of functions from $X$ to $\mathbb{R}$. Consider first the initial topology $\tau_\mathcal{D}$ on $X$ with respect to $\mathcal{D}$, i.e. the coarsest ...

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**0**answers

64 views

### Bounding the dimension of the euclidean space in which any $n$-manifold embeds “$k$-uniquely” in it

(The question will be interesting for topological/Pl as well but in order to not be too vague I will restrict the meaning of manifold to smooth manifold without boundary).
I'm interested in the ...

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**2**answers

252 views

### A map of spaces implementing the Pontryagin Thom collapse map? (collapse maps in families)

Let $M$ be an $n$ dimensional smooth manifold and let $j: M \to \mathbb{R}^{m}$ be an embedding. Associated to this embedding we can form the "collapse map" which is a pointed map from a sphere to the ...

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**0**answers

144 views

### Differential topology on arbitrary fields

What do the differential topology theories on arbitrary fields have in common?
Different differential topology theories
There is "ordinary" differential topology on real manifolds, with its rich ...

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votes

**2**answers

2k views

### Can one hear the (topological) shape of a drum?

Let $(M,g)$ be a (say closed) Riemannian manifold. One can try to understand the geometry/topology of $(M,g)$ by studying the eigenvalues of the Laplacian (this I guess has two versions: when ...

**5**

votes

**1**answer

134 views

### $S^{2}$-bundles over complex projective varieties

Is there an example of a smooth complex projective variety and an $S^{2}$-bundle over it which is not diffeomorphic to a complex projective variety?