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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2
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0answers
90 views

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

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
0answers
98 views

Are induced Riemannian metrics weakly continuous?

Let $\Omega \subseteq \mathbb{R}^d$ a nice bounded domain. Let $F_n:\Omega \to \mathbb{R}^D$ be a sequence of smooth embeddings* in $W^{1,p}(\Omega,\mathbb{R}^D)$. Assume $F_n \rightharpoonup F$ in ...
0
votes
0answers
46 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 ...
2
votes
2answers
240 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 ...
12
votes
3answers
438 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
4answers
151 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 ...
1
vote
1answer
113 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 ...
7
votes
2answers
332 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 $M'$ manifolds of dimensions $m>0$. It is true that if ...
6
votes
2answers
216 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 ...
0
votes
0answers
85 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
0answers
47 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
1answer
99 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
0answers
38 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
1answer
186 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 ...
3
votes
1answer
125 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
0answers
94 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
0answers
64 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
0answers
48 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 ...
9
votes
2answers
175 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
0answers
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
0answers
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
0answers
99 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
0answers
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
0answers
254 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
1answer
985 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 ...
3
votes
1answer
172 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
0answers
42 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 ...
9
votes
1answer
258 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
2answers
692 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
0answers
94 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 ...
5
votes
0answers
88 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 ...
21
votes
1answer
303 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
0answers
249 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 ...
10
votes
1answer
279 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 ...
0
votes
0answers
152 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
2answers
269 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
2answers
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 ...
0
votes
1answer
100 views

Variation of normals along loops

I've constructed a quotient space $M/\sim$ in $\mathbb{R}^d$ that must be a $2$-manifold. If $M/ \sim$ is a sphere then I know that its normal spaces must vary at least 90 degrees. That is $M / \sim$ ...
3
votes
1answer
170 views

The space of homotopy classes of maps of products of spheres

Proposition 17.6.1 of "Differential form in Algebraic Topology" by Bott and Tu proves the following beautiful result: $[S^{q}, X]\simeq \frac{\pi_{q}(X,x)}{\pi_{1}(X,x)}$ where $S^{q}$ is the ...
1
vote
0answers
82 views

Tightening a loop

Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose ...
2
votes
0answers
61 views

Singularities of Families of Differential Equations

Suppose you are given a family of differential equations with rational coefficients: $\partial_x^ny+a_1(x,x_1,\dots,x_m)\partial_x^{n-1}y+\dots+a_n(x,x_1,\dots, x_m)y=0$. At each point $X=(x_1,\dots, ...
1
vote
1answer
51 views

Existence of a continuous map of a disk with a given boundary image on a surface to its complement in $\mathbb{R}^3$

I am considering the following problem: given an embedded closed surface in $\mathbb{R}^3$ (unknotted) and a non-trivial simple closed curve on it, does there exist a continuous map of a disk to the ...
1
vote
0answers
84 views

“Nice” limits of sequences of smooth embeddings

Consider smooth embeddings of a manifold $M$ into some $\mathbb{R}^n$. If a sequence $f_k : M \to \mathbb{R}^n$ of such embeddings converges to some continuous function $f : M \to \mathbb{R}^n$, then ...
9
votes
2answers
206 views

non-triviality of the underlying real vector bundle of the complexification of a real vector bundle

Let $M$ be a given manifold and $\xi$ be a given $k$-dimensional vector bundle over $M$. How to determine whether the underlying real vector bundle of $\xi\otimes\mathbb{C}$, i.e. the Whitney sum ...
0
votes
0answers
49 views

map of constant rank

Let $f_1, \dots, f_m \colon M^n \to \mathbb{R}$ be smooth functions on a manifold $M$, such that $F := (f_1, \dots , f_m) \colon M \to \mathbb{R}^m$ has constant rank $k <m,n $ on $M$. I'm trying ...
3
votes
2answers
112 views

Extension of a group action beyond the boundary

Let $M$ be a compact manifold with boundary and suppose a compact group $G$ acts on it. Can one always extend the action beyond the boundary? More precisely, does there always exist a $G$-manifold ...
7
votes
1answer
122 views

Can any path in the diffeomorphism group of a smooth compact manifold be approximated by a smooth path?

Given a smooth compact $n$-dimensional manifold $M^{n}$, let $\operatorname{Diff}(M)$ denote the group of smooth diffeomorphisms $M \rightarrow M$ equipped with the Whitney $C^{\infty}$-topology. Let ...
2
votes
1answer
139 views

Can one smooth open star shaped domains from the inside by star shaped domains?

Let $O\subset\mathbb{R}^n$ be a open set which is star shaped with respect to the origin. How does one prove that there exists an increasing sequence of star shaped (w.r.t the origin) domains $O_i$ ...
1
vote
0answers
44 views

Convex Hull and Least Area Discs in Riemannian 3-Manifolds

Let $M$ be a complete Riemannian 3-manifold and $\gamma \subset M$ a simple closed curve that bounds a least-area disc $D$ - a disc that minimizes the area among all discs bounded by $\gamma$. Let ...
1
vote
0answers
49 views

Flows and Lie brackets, $\beta$ not a priori smooth at $t = 0$ [closed]

Let $X$ and $Y$ be smooth vector fields on $M$ generating flows $\phi_t$ and $\psi_t$ respectively. For $p \in M$ define$$\beta(t) := \psi_{-\sqrt{t}} \phi_{-\sqrt{t}} \psi_{\sqrt{t}} ...