**8**

votes

**1**answer

203 views

### Reference for push-pull formula in cohomology

I would like a precise reference for the following fact.
Assume that
$$
\begin{array}{ccc}
M\times_B N & \stackrel{f'}{\to} & N \newline
\quad\downarrow g' & & \quad\downarrow g \...

**5**

votes

**0**answers

119 views

### Short and elegant definition of the $C^1$ topology

A friend told me that the $\mathbf{C^1}$-topology on the set $C^\infty(M,N)$ of smooth functions between two smooth manifolds $M$ and $N$ can be defined as the coarsest topology making the map
$$
C^\...

**4**

votes

**1**answer

162 views

### Does every smooth manifold admit a metric with bounded geometry and uniform growth?

Let $M$ be a smooth manifold, $g_M$ a Riemannian metric, and consider for $x\in M$ the volume growth function, $gr_x$ that maps $r>0$ to the volume $vol_{g_M}(B(x,r))$. My interest is to see ...

**2**

votes

**0**answers

47 views

### Thom form of holomorphic bundle over Kaehler manifolds/orbifolds

Consider a holomorphic vector bundle $\pi:E\rightarrow X$ of complex rank $m$ over a Kaehler manifold $X$. Can we find a Thom form $\Theta$ of $E$ such that as a form on the complex manifold $E$, it ...

**4**

votes

**1**answer

185 views

### A proof of Theorem 9.2.12. in the Gompf-Stipsicz

I'm seeking for a proof of Theorem 9.2.12. in the Gompf-Stipsicz "4-Manifolds and Kirby Calculus" (for the statement, see the following image). But the textbook omits any proofs and only gives a ...

**1**

vote

**1**answer

232 views

### Homotopy type of an oriented, closed, simply connected manifold

It is well known that every closed, oriented, simply-connected four-manifold $M$ is homotopy equivalent to a CW-complex consisting on a 0-cell, a wedge of two spheres and a 4-cell.
I was wondering ...

**5**

votes

**2**answers

550 views

### Can a Morse function define a unique structure on a closed manifold?

I was thinking about the doubt that if $M$ and $N$ are closed manifold and if there exists two Morse function $f$ and $g$ respectively on $M$ and $N$ with the following property that they both have ...

**4**

votes

**0**answers

166 views

### Under what condition is a fiber bundle cobordant to the trivial bundle?

Let $E$ be the total space of a fiber bundle with base $B$ and fiber $F$, where $B$ and $F$ are smooth manifolds.
Under what condition is $E$ unoriented cobordant to $B\times F$?
And what happens ...

**3**

votes

**0**answers

163 views

### How far can one reconstruct the boundary of a manifold M given its interior $int M$? [duplicate]

Suppose I keep in my pocket a manifold with boundary $M$ , and I provide you access to $int M := M \setminus \partial M$ up to homeomorphism/diffeomorphism. What can you deduce about $\partial M$? can ...

**1**

vote

**1**answer

95 views

### Stability of the critical points set

Let $F:\mathbb{S}^{2}\times\lbrack0,1]\rightarrow\mathbb{R}$ be a smooth
($C^{\infty}$) function and $f_{t}(x)=F(x,t)$. Suppose that $f_{0}=f_{1\text{
}}$is the projection over $z$-axis, so point $P=(...

**5**

votes

**2**answers

261 views

### Generalizations of the handle trading techniques

As Theorem 8.1 in "Lectures on the h-cobordism theorem (written by J.Milnor)" show, we can choose a handle decomposition of cobordism (satisfying some connectivity and dimensional assumptions) with no ...

**15**

votes

**1**answer

339 views

### Are homology spheres stably trivial?

A homology sphere is a closed smooth $n$-dimensional manifold with the same homology groups as $S^n$. Igor Belegradek's answer to a previous question of mine shows that the smoothness hypothesis is ...

**15**

votes

**5**answers

665 views

### Smoothness of the closest point on a submanifold

Let $(M,g)$ be a smooth Riemannian manifold, and let $S \subseteq M$ be a compact submanifold.
Assume that for each $p \in M$, there exist a unique closest point on $S$, i.e a unique point $\tilde s(...

**7**

votes

**2**answers

466 views

### Spin^c structures on manifolds with almost complex structure

Let $M$ be a smooth even-dimensional manifold.
Is it true that for each almost-complex structure $J$ on $M$ there exists a canonical spin$^c$ structure $S_J$ associated to $J$ ?
(I've read this ...

**2**

votes

**0**answers

108 views

### If $X$ is a compact smooth Riemannian manifold, why don't we integrate on a fundamental domain in the universal cover? [closed]

Let $X$ be a compact connected Riemannian manifold. The metric gives a local volume form. The universal cover is orientable, and has a precompact subspace locally isometric (with the covering metric) ...

**0**

votes

**0**answers

48 views

### Small perturb a continuous map

I am reading the book: Convex Integration Theory by D. Spring and encounter a
question which I subtract as follows.
Let $p: X \rightarrow V$ be a smooth fibre bundle and $\mathcal{R} \subset X^{(1)}$...

**2**

votes

**2**answers

295 views

### Is it true that all sphere bundles are some double of disk bundle?

Let's consider a smooth sphere bundle over a smooth manifold with structure group is equal to the diffeomorphism group of sphere. Then, can we say that this is a double of some disk bundle? Thank you ...

**13**

votes

**3**answers

544 views

### Nice things that can be proved easily with characteristic classes

A bit of context for this question: as a project for my master's degree my supervisor asked me to understand the construction of Milnor's exotic spheres. After learning the heavy material (I knew very ...

**1**

vote

**4**answers

181 views

### On a parallelizable manifold, is there always a frame satisfying $[X_i,X_j]=0$?

[This question was asked on MSE, but got no answers, I thought it could be more appropriate here]
Let $M$ be a parallelizable manifold.
Is there always a global frame $(X_i)$ such that $[X_i,X_j]=0$...

**1**

vote

**1**answer

129 views

### Open non-parallelizable 4-manifolds

Let $M$ be a connected orientable open 4-manifold (noncompact, without boundary).
Is it possible for $M$ to be non-parallelizable ?
If yes, what example of such $M$ is there ?
[EDIT : The answer ...

**10**

votes

**2**answers

675 views

### Homotopy of space of immersions, Smale-Hirsch theorem

If $M$ and $N$ are manifolds with $\dim M< \dim N$, we denote by $Imm\left(M,N\right)$ the space of immersions of $M$ in $N$.
Let $M$ and $M'$ be a simply connected manifolds of dimensions $m>...

**6**

votes

**2**answers

241 views

### Flat connections on 3-manifold with boundary

Suppose $Y$ is a 3-manifold and the boundary $\Sigma:=\partial Y$ is non-empty. Let $G$ be a Lie group with trivial center. Let $\overline {\mathcal A}_{flat}(\Sigma)$ and $\overline {\mathcal A}_{...

**0**

votes

**0**answers

100 views

### de Rham type cohomology for covariant derivative?

We know that in general for a covariant derivative $D$ on a vector bundle $\xi$ over $M$ we don't have $D \circ D = 0$. This prevents us from having the following cochain complex
\begin{equation}
\...

**1**

vote

**0**answers

48 views

### Hyper-Complex Connected Sums of Grassmannians?

As we all know, the Grassmannians are Kaehler manifolds. Is there anyway to take connected sums of Grassmannians and produce a examples of hypercomplex manifolds, hyper-Kaehler or Calabi--Yau exmples ...

**7**

votes

**1**answer

117 views

### Properties of connection Laplacian on vector fields

Let $(M,g)$ be a simply-connected compact surface with boundary $\partial M$ and metric $g$. Let $N$ denote the outward unit normal on $\partial M$, $\nabla$ the Levi-Civita connection and $\Delta_g$ ...

**0**

votes

**0**answers

42 views

### When is a critical value of a map contained in the interior of the image?

Let $M^n$ be a compact manifold, and $F\colon M \to \mathbb{R}^n$ a smooth map. The inverse function theorem implies that every regular value of $F$ lies in the interior of $F(M)$, hence every point ...

**1**

vote

**1**answer

219 views

### Invariance of Gauss-Bonnet theorem with respect to connection?

I am stuck with a basic understanding of the generalized (and even the ordinary version of) Gauss-Bonnet theorem. For a compact 2-dimensional Riemannian manifold $M$ with boundary $\partial M$, let $K$...

**3**

votes

**1**answer

141 views

### Is there a smooth polar decomposition for non-invertible matrices?

Every $n \times n$ real matrix $A$ has a polar decomposition $A=OP$, where $O \in O_n, P$ is symmetric positive semi-definite.
$P$ is uniquely determined by $A$, by $P(A)=\sqrt{A^TA}$,
and when $A$ ...

**5**

votes

**0**answers

95 views

### Smoothing a continuous section in 1-jet bundle

Here is a question I encountered when reading the book "Convex Integration Theory by D.Spring". My question lies in the second paragraph to the proof of theorem 4.2($C^{0}$-dense $h$-principle).
I ...

**0**

votes

**0**answers

67 views

### Submersion on open and dense subset

If $f \colon M \to N$ is a smooth map between smooth manifolds, is it possible to find an open and dense set $M_0$, such that $f(M_0)$ is a manifold and
$f \colon M_0 \to f(M_0)$ is a surjective ...

**1**

vote

**0**answers

50 views

### Does a homotopy sphere that bounds a highly connected manifold also bound a parallelizable manifold?

Suppose that the homotopy sphere $\Sigma^{n}$ can be realized as the boundary of a smooth $(n+1)$-dimensional cobordism that is $(n-1)/2$-connected for $n$ odd (respectively, $(n-2)/2$-connected for $...

**10**

votes

**2**answers

185 views

### Triangulation with simplices of same volume

Let $M$ be a Riemannian smooth compact manifold.
It is known that $M$ has a triangulation, for any dimension. But do we know if there exists a triangulation such that all simplices have same volume ?
...

**0**

votes

**0**answers

41 views

### Lower bounds on the measure of balls in attractor sets

I'm looking for a source for the following result.
Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \...

**0**

votes

**0**answers

58 views

### Lower bound on the diameter of a ball contained in the stable manifold of a critical point

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Consider the negative of the gradient ...

**3**

votes

**0**answers

103 views

### Can we use the “size” of smooth structure set to predict the information geometry or other topological information?

The "size" can mean the number of elements or the diameter of the set of smooth structures. Y. Shikata defined a distance function on it and proved that it is a distance. He then used it to prove that ...

**0**

votes

**0**answers

27 views

### A problem from Sakai's book on derivations on C(K) and differential structure on K

In his book, Operator Algebras in Dynamical Systems, at page 59 Sakai poses the following question.
Problem: Let K be a compact space and suppose that C(K) has a non-zero closed *-derivation. Then ...

**2**

votes

**0**answers

257 views

### Measure of the Attractor of Critical Points of a Manifold

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Let $P$ be a property of some subset ...

**30**

votes

**1**answer

995 views

### Can a topological manifold have different tangent bundles?

We know that the tangent bundles of the sphere arising from different smooth structures are equivalent as vector bundles. Is it right in general? I want to know the relationship between the set of ...

**4**

votes

**1**answer

175 views

### Is the number of intervals you get by slicing the closed region under a nice graph continuous from below?

Let $I$ be a closed interval in $\mathbb{R}$, and let $f:I \to \mathbb{R}$ be a bounded function, smooth except at finitely many points (piece-wise smooth). There are also only finitely many critical ...

**0**

votes

**0**answers

45 views

### Connected Cobordisms

Perhaps someone who knows some differential topology will be able to give a simple answer to this:
Since the (unoriented) cobordism group $\Omega_{3} \cong 0$, every closed 3-manifold is cobordant to ...

**11**

votes

**1**answer

293 views

### Are there non-smoothable homotopy/homology spheres?

A homotopy sphere is a topological $n$-manifold $M$ which is homotopy equivalent to $S^n$.
A homology sphere is a topological $n$-manifold $M$ such that $H_i(M) \cong H_i(S^n)$ for all $i$.
Note, by ...

**23**

votes

**2**answers

701 views

### Why is it true that if two 4-manifolds are homeomorphic then their squares are diffeomorphic?

Near the top of the second page of this paper, it is claimed that if two 4-manifolds $X$ and $Y$ are homeomorphic, then their squares $X \times X$ and $Y \times Y$ are diffeomorphic. Why is this true? ...

**2**

votes

**0**answers

99 views

### How many linear independent vector fields can be constructed on a general manifold with $\chi(M)=0$?

We have known how many linear independent vector fields can be constructed on $S^n$:https://en.wikipedia.org/wiki/Vector_fields_on_spheres
So how many linear independent vector fields can be ...

**7**

votes

**0**answers

92 views

### $H(M)$ necessarily highly non-integrable, i.e. forms contact structure?

Let$$M^{2n - 1} = \{z \in \textbf{C}^n : \textbf{h}(\textbf{z}, \textbf{z}) \equiv \textbf{z} \cdot \overline{\textbf{z}} = \textbf{1}\}$$be the unit sphere in $\textbf{C}^n$. Consider the real-...

**21**

votes

**1**answer

330 views

### Is the normal bundle of a torus trivial?

Question:
Let $T^k \subseteq \mathbb{R}^n$, $ n > k$, be a smoothly embedded $k$-torus. Is its normal bundle trivial?
What about the normal bundle of $S^k \subseteq \mathbb{R}^n$, $n > k$, the $...

**7**

votes

**0**answers

255 views

### Can any smooth triangulation of a smooth manifold be blurred?

For the purposes of this question, let's say that a blurring
of a smooth triangulation $T$ of a smooth manifold $X$
is a smooth homotopy $h\colon [0,1] \times X \to X$ such that $h_0=\operatorname{id}...

**10**

votes

**1**answer

287 views

### A symmetric embedding of manifolds

Assume that $M$ is a manifold.
Is there an embedding of $M$ in some $\mathbb{R}^{n}$ such that the image of $M$ in $\mathbb{R}^{n}$ is invariant under each reflection $(x_{1},x_{2},\ldots x_{i},\...

**0**

votes

**0**answers

157 views

### About the homotopy type of diffeomorphism groups

In this paper by Antonelli, Burghelea and Kahn (Topology, 1972), a homomorphism $L :\pi_i(\operatorname{Diff}(S^n, D_+^{n})) \rightarrow \Gamma^{n+i+1}$ was used as a tool to detect non-triviality of ...

**13**

votes

**2**answers

338 views

### Classification of $O(2)$-bundles in terms of characteristic classes

I had asked this question in stackexchange but there seems to be no consensus in the answer
It is well-known that $SO(2)$-principal bundles over a manifold $M$ are topologically characterized by ...

**2**

votes

**2**answers

136 views

### Finding a specific Global Smooth Function

Any help with this problem would be appreciated. Thanks
Suppose $(M^3,g)$ is a smooth compact Riemannian manifold with smooth boundary and $\gamma$ is a simple smooth orientable curve in $M$. Does ...