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# 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. ...
1 vote
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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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### 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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### 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. ...
287 views

### 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 ...
1 vote
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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 ...
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 ...