# Questions tagged [differential-topology]

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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### Roadmap for a third-year undergraduate student to study and research the Hodge conjecture [closed]

I am a third-year undergraduate student majoring in mathematics, and I am fascinated by the Hodge conjecture. I am eager to delve deeper into this topic and eventually conduct research in the field. ...

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### Characterization of the homotopy type of the $C^\infty$ topology

One of the subtle aspects of geometric topology is the interaction of function space topologies and homotopy theory. There are many reasonable topologies to put on the space of smooth embeddings $\...

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### Does composition on the right by a volume-preserving diffeomorphism preserve homotopy class?

Let $M, N$ be smooth manifolds with $M$ orientable and compact. Let $\sigma$ be some volume form on $M$ and consider the set $\mathcal{M}$ of smooth maps from $M$ to $N$ in a fixed homotopy class. Now ...

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### Does $\overline{\mathbb{C}P^2 \setminus B^4}$ embed (smoothly/topologically) in $\mathbb R^5$?

Question: Does $\overline{\mathbb{CP}^2 \setminus B^4}$ (that is the closure of complex projective plane minus a 4-ball) embed (smoothly/topologically/piecewise linearly) in $\mathbb R^5$?
Background: ...

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### Homogenization of Morse-Bott functions

Let $M$ be a compact manifold of dimension $n$.
A smooth function $f:M \to \mathbb{R}$ is called Morse-Bott if the set critical points of $f$ is a disjoint union of compact submanifolds $C_1,\ldots,...

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### S¹ action on a manifold which generates "non-torsion" loop in diffeomorphism group

I am interested in $S^1$-actions on smooth, closed, and oriented manifolds $M$. I suppose that the action has a fixed point (I also suppose $M$ is connected). Let $\operatorname{Diff}(M)$ denote the ...

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### Geometric interpretation of transfer map on homology

Let $f\colon M\to N$ a smooth surjective map of compact oriented manifolds of the same dimension. Then there is a map $f_!\colon H_i(N)\to H_i(M)$ obtained from the induced map on cohomology combined ...

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### Schrödinger representation of the Heisenberg group

Let $\Pi_{\lambda}$ be the the Schrödinger representations of the Heisenberg group $H^n=\Bbb C^n\times\Bbb R$. For $\phi\in L^2(\Bbb R^n)$, we have
$$\Pi_{\lambda} (x,y,t)\phi(\xi)=e^{i\lambda t} e^{...

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### Finding a real-analytic diffeomorphism

Let $U_1\subset \mathbb R^3$ be a simply connected bounded open set with a smooth boundary and let $U_2$ be a neighborhood of $U_1$. Does there exist a real-analytic diffeomorphism $\psi: U_2 \to W_2$ ...

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### What is a quasi-isomorphism of complexes of vector bundles?

Consider a homomorphism $f$ between two complexes of vector bundles over a fixed smooth manifold $M$.
$$
\cdots \to V_{i - 1} \xrightarrow{\delta_{i-1}} V_i \xrightarrow{\delta_i} V_{i + 1} \to \cdots
...

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### Changing coordinate to smoothen a function

Let $U\subset \mathbb{R}^2$ be an open neighborhood of the origin $0$, and let $f:U\to \mathbb{R}$ be a continuous function which is smooth on $U\setminus\left\{0\right\}$. Let's say that $f$ is ...

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### Homotopy between sections

Let $f:X \to S$ be a proper $C^\infty$-morphism between real manifolds. Assume that each fiber of $f$ is connected of real dimension 2 (the fiber may not be smooth, but it is the union of smooth ...

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### Leaf holonomy of Reeb foliation on mobius strip

I am trying to understand the leaf holonomy of the Reeb foliation on the mobius strip, the first problem being visualization. I have been unable to find a visualization of this anywhere. I am ...

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### Doubles of 2-handlebodies

Let $X$ denote a $4$-manifold with boundary obtained by adding $k_1$ $1$-handles to $B^4$ and $k_2$ many $2$-handles to the resulting manifold i.e. $X$ is an arbitrary $4$-dimensional $2$-handlebody. ...

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### exterior differentiation of foliations

Let $M$ be a differentiable manifold.
Let $T^*M$ be the cotangent bundle of $M$.
Consider the exterior differentiation $d: A^p(M)\longrightarrow A^{p+1}(M)$, where $A^p(M)=\Gamma(\Lambda^...

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### Modality of a point under a Lie group action

Let $X$ be a smooth manifold and $G$ a Lie group acting on it. V. I. Arnold defines the modality of a point $x\in X$ as follows [1] (see also [2]):
We say that a point $x$ has modality $m$ (under the ...

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### Existence (or non existence) of principal bundle charts compatible with an $f$-reduction

I asked this question on math stack exchange here, but I wonder if it would be better received here.
Let $\pi:P\rightarrow M$ and $\pi':P'\rightarrow M$ be principal $G$ and $H$ bundles respectively, ...

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### Basic question on the de Rham theorem

There is a modern nice proof of the de Rham theorem based on sheaf theory.
The de Rham theorem says that for a smooth manifold $M$ there is a canonical isomorphism
$$H^i_{dR}(M,\mathbb{R})\simeq H^i_{...

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### Is $\pi_m(M) = 0$ if $\pi_m(M-X) = 0$ for a low-dimensional subset $X$?

I am doing a problem where I am stuck at this point.
Let $M$ be a connected smooth manifold of dimension $n$ and let $X$ be any subset of $M$. Assume that there is a positive integer $m$ such that $n&...

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### Classification of surface bundles over surfaces

Can anyone recommend one place or a few places that describe what is known about the classification of (real) surface bundles over (real) surfaces?
Now, if the fibre F and the base B are both ...

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### Obstructions to maximal number of independent constants of motion in a given symplectic manifold

Given a compact symplectic manifold $(X, \omega)$, are there any invariants (topological or easily computable geometric/analytic ones) which give an estimate of the maximal number of independent ...

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### Foliation of $X$ by once punctured planes without any singularities

Let $n=3.$
Take $X=(0,1)^n.$ Fix points $p,q$ s.t. $\text{dist}_n(p,q)=\sqrt{n}.$ Construct a smooth regular foliation of $X$ with $(n-1)-$dim. leaves which are topologically $(0,\sqrt{n})\times S^{n-...

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### Is the minimal volume a topological invariant?

On Wikipedia, it is said that the minimal volume
$$\operatorname{MinVol}(M):=\inf\{\operatorname{vol}(M,g) :g\text{ a complete Riemannian metric with }|K_{g}|\leq 1\}$$
is a topological invariant, ...

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### Compact closed aspherical manifolds with vanishing second homology in all the covering spaces

I wonder if there exists a compact closed smooth aspherical manifold $M$ of dimension at least $4$, so that for any covering space $\tilde{M}$ over $M,$ we always have $H_2(\tilde{M},\mathbb{Z})=0$ ...

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### Relative Thom transversality and the D-topology

Suppose we are given a smooth manifold $M$ and, for the sake of simplicity, some compact submanifold $L\subseteq M$ of the same dimension, as well as $f\in C^{\infty}(M,N)$ and some submanifold $V\...

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### When is compactness of fiber components an open condition?

Consider a smooth map $f:M\rightarrow N$ between smooth manifolds.
Ehresmann's theorem states that if $f$ is a proper submersion, then it is locally trivializable; in particular, this implies that ...

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### In what topology does Gromov's lemma hold on noncompact symplectic manifolds?

In symplectic geometry, it is commonly said that ``the set of almost complex
structures tamed to a symplectic form is contractible'' even on noncompact symplectic manifolds. In my understanding
one ...

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### Explicit parameterizations of complicated unlinks?

I have a somewhat empirical question which I hope is still welcome here. I would like to know how to write down explicit parameterizations of "complicated unlinks", say with 2 or 10 ...

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### Open book decompositions in dimension 4

The question about the existence of open book decompositions for a closed oriented $n$-dimensional manifold seems to be answered in all dimensions except dimension $4$, where as far as I can tell the ...

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### Homological restrictions on certain $4$-manifolds

I am not very familiar with the non-compact $4$-manifold theory. So I apologize if the following question is very silly.
Let $X$ be a non-compact, orientable $4$ manifold that is homotopic to an ...

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### When can vector fields span the tangent space at each point?

If the tangent bundle of a smooth manifold is a smoothly trivial smooth fiber bundle, is it a trivial smooth vector bundle?
Since this question got no answer in MathExchange, I am migrating it here.
...

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### Does every triangulable manifold have a vertex-transitive triangulation?

Does every triangulable manifold have a vertex-transitive triangulation?
When I talk about a vertex-transitive triangulation of a manifold, I mean in the sense of realizing a manifold homeomorphically ...

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### A question regarding how Thom-Boardman strata sit in their closures

I just started learning about these things, so there is a chance I might have misunderstood some things. My apologies if that is the case.
Some context. Suppose that we are given a differentiable map $...

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### Sufficient condition for the union of two submanifolds to be a submanifold

I have two smoothly embedded orientable surfaces $S_1,S_2\subset S^3 \times [0,1]$ with boundary such that
$(i)$ $S_1\cap S_2$ is a smoothly embedded surface without boundary and
$(ii)$ $\overline{...

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### Applications of Strong Whitney Embedding

I am looking for applications of the strong Whitney's embedding theorem that have an advantage over weak theorems. That is, applications where it's important that the dimension of the Euclidian space ...

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### "Neck cutting" and why gauge theory doesn't work on homotopy 4-spheres

I attended a talk recently and the speaker said, essentially, that gauge theory invariants are expected to never be able to detect exotic 4-spheres because they always vanish, for a reason related to ...

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### Can every homeomorphism of the 4-ball fixing the boundary be approximated by diffeomorphisms smoothly isotopic to the identity?

Let $B^{n}$ denote the euclidean closed ball of dimension $n$. Alexander's trick shows that every homeomorphism of $B^{n}$ fixing $\partial B^n$ is $C^{0}$-isotopic to the identity via a boundary ...

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### Applications of diffeological spaces to ordinary differential geometry

Recently I've been learning more about differential geometry, and I came upon the notion of a diffeological space, which encompasses a number of already known extensions of smooth manifolds or related ...

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### Example of a non $\pi_1$-injective, degree one, self-map of a three-manifold

All manifolds will be assumed to be closed, oriented, and connected.
Let $f\colon M\to M$ be a map of degree $\pm 1$. It is not hard to show that $\pi_1(f)$ is surjective.
What is an example of a non ...

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### Is every Riemannian metric conformally equivalent to one that is geodesically complete?

The question in the title seems a natural one to ask, and I suspect that it is already considered, even completely solved, somewhere in the literature. Although I prefer some explanation of the idea(s)...

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### Approximating PL homeomorphism by diffeomorphisms in Euclidean space

The question is whether a piecewise function can be approximated by diffeomorphic functions in the following two situations.
I'm not really familiar with these piecewise stuffs. So it may be stupid ...

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### Are these two concepts of a differential form on the loop space equivalent?

Notation:
Let $X$ denote a smooth manifold (without boundary) and define $LX = C^{\infty}(S^1, X)$ to be the loop space on $X$.
In the context of loop space homology and the supersymmetric path ...

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### Minimal dimension for immersions to be dense in the continuous function space

Let $f:[0,1]^n \rightarrow \mathbb{R}^m$ be an arbitrary continuous function.
My question is, under what conditions for $m$, there exists an immersion $g:(-\epsilon,1+\epsilon)^n \rightarrow \mathbb{R}...

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### Morse theory for isotopies of codimension 1 submanifolds and generalized bigon moves

I seem to have managed to convince myself of the validity of a certain result. If it does indeed hold (modulo non-catastrophic adjustments) I could really use a reference. Otherwise, I would also be ...

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### Computation on characteristic classes

I am organizing a reading seminar on characteristic classes. The audience in the seminar is interested in symplectic and contact manifolds. I work in categorification and would like to compute some ...

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### Understanding dimension of gradient flow trees for product on Morse complex

I am trying to square my intuition with the facts and hope this question is not too vague.
I am reading this paper which describes the $A_\infty$-category of Morse functions on a manifold $M$ of ...

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### Consequences of Nash-Tognoli Theorem

The Nash-Tognoli theorem states that every closed and smooth manifold is diffeomorphic to a real algebraic variety. This appears to me as a very strong and surprising fact.
However, I am not aware of ...

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### What are known properties of the boundary curves of J-holomorphic curve with boundary

Suppose $\Sigma$ be a punctured Riemann surface with punctured boundary, and $(M, J)$ be a $2n$-manifold with almost complex structure $J$. Let $L$ be a totally real submanifolds of $M$ in smooth ...

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### Types of differential structures on higher dimensional spheres

This problem comes from the smooth Poincaré conjecture:
Is a homotopy equivalent manifold to sphere is differential
homeomorphic to standard sphere?
Since the general Poincaré conjecture has been ...

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### Vanishing directional derivatives on $S^2$

Let $u$ be a smooth function defined on the unit sphere $S^2$. Does there exist a plane $P$ passing through the origin such that $P\cap S^2$ contains at least three points $x_1,x_2,x_3$ with $\nabla u(...