# 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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### Holomorphic foliation and Morse-Bott function

Let $f$ be a holomorphic Morse-Bott function on a complex manifold $X$, with critical points being a compact complex manifold $Y$. Here Morse-Bott means the Hessian of $f$ is non-degenerate in the ...

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### Is differential topology a dying field?

I recently had a post doc in differential topology advise against me going into the field, since it seems to be dying in his words. Is this true? I do see very little activity on differential topology ...

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**1**answer

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### Reduction of the structure group of $\mathbb{R}^n$-fiber bundles to a special subgroup of $\mathrm{Homeo}(\mathbb{R}^n)$

Let $G$ be the group of all self-homeomorphisms $f$ of $\mathbb{R}^n$ which satisfy $$f(x+m)=f(x)+m,\quad \forall m\in \mathbb{Z}^n.$$
In other words, $G$ is the group of all equivariant self-...

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**1**answer

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### Is $π:\mathcal{C}^∞(M,N)→\mathcal{C}^∞(S,N)$, $π(f)=f|_S$ a quotient map in the $\mathcal{C}^1$ topology?

This question was previously posted on MSE.
Let $M, N$ be smooth connected manifolds (without boundary), where $M$ is a compact manifold, so we can put a topology in the space $\mathcal C^\infty(M, N)...

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### Realizing cohomology classes by submanifolds

In "Quelques propriétés globales des variétés différentiables", Thom gives conditions for a class in singular homology of a compact manifold to be realized by a smooth oriented submanifold (see e.g. ...

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### How much is the theory of piecewise-smooth manifolds different from the PL and smooth ones?

There are many differences between the smooth manifolds category and piecewise linear manifolds category when it comes to classification, embedding of these manifolds inside each other or otherwise. ...

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**1**answer

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### Example of two exotic closed 4-manifolds s.t. SW(X)=0

I am interested in seeing examples of two closed 4-manifolds $X_1,X_2$ such that $SW(X_i)=0$ and they are homeomorphic but not diffeomorphic.
So far in the literature I've only found examples which ...

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### Is Cohen immersion conjecture (theorem) known for vector bundles?

R. Cohen proved the immersion conjecture in a 1985 Annals paper:
Cohen, Ralph L., The immersion conjecture for differentiable manifolds, Ann. Math. (2) 122, 237-328 (1985). ZBL0592.57022.
Any smooth ...

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### Rado-Kneser-Choquet Theorem, surjectivity [closed]

This question is there on MSE, but the answer (Comment mode) is not satisfactory. I tried to prove the surjectivity, but unable to complete my argument.
It is clear that $f$ is a closed map, since $\...

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**1**answer

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### Every symplectic submanifold is J-holomorphic

I am trying to show that every symplectic submanifold $N$ of a 2n- dimensional symplectic manifold $(M,\omega)$ is J-holomorphic for some compatible almost complex structure $J$.
The way I am ...

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### Minimum rank of inverse complex vector bundles

When considering vector bundles (real or complex) over a compact manifold, i know about the existence of inverse bundles. That is, if $\xi$ is a vector bundle over $M$, then there is a bundle $\nu$ ...

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**1**answer

144 views

### A special non vanishing vector field on odd dimensional compact manifolds

Edit: According to the comment of Michael Albanese we revise the question.
Assume that $n$ is an odd integer and $M\subset \mathbb{R}^{2n}$ is a compact orientable $n$ dimensional submanifold.
Does ...

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**0**answers

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### Reference request; Hirsch's structure functors

In Hirsch's Differential Topology book is decribed the notion of a structure functor (page 52 of the first edition). Subsequently appears a Globalization Theorem involving such functors. The ...

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**1**answer

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### Do submersions induce open maps between spaces of differentiable maps?

Let $X$, $Y$ and $Z$ be smooth manifolds.
Any differentiable map $f \colon Y \rightarrow Z$ induces a continuous map $f_{\ast} \colon C^{\infty}(X, Y) \rightarrow C^{\infty}(X, Z)$ via composition $g \...

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### Is polar decomposition of a smooth map Sobolev?

Motivation:
Let $\mathbb{D}^2$ be the closed unit disk. I am studying the "elastic energy" functional $E(f)=\int_{\mathbb{D}^2} \text{dist}^2(df,\text{SO}_2)$, where $f \in C^{\infty}(\mathbb{D}^2,\...