# 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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### No analytic surjection $f:M \to N$ when $\dim(M) >\dim(N)$

Inspired by comment discussions in this MO post smooth version of splitting principle we ask: Are there two compact real analytic manifolds $M,N$ of dimension $m>n$ such that there is not any ...
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### A question regarding weak Whitney embedding theorem

The weak Whitney embedding theorem states that any continuous map $f: D^n \to \mathbb{R}^{2n+1}$ (Let us focus on $D^n$ for this question) can be approximated (in $C^0$-norm) by embeddings. A counter ...
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### Is there a differential form which corresponds to an eigenvalue of the homomorphism in cohomology?

Let $M$ be a closed manifold and $f:M\to M$ be a diffeomorphism. Suppose the homomorphism $f^*:H^k(M;\mathbb R)\to H^k(M;\mathbb R)$ has an eigenvalue $\lambda\in\mathbb{R}$. Note that $\lambda$ is ...
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### Must the number of smooth structures be countable or continuum?

Let $M$ be a manifold. Must the number of non-diffeomorphic smooth structures on $M$ be either countable or continuum? Could it be something in between when the continuum hypothesis fails? Edit: By ...
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### Extending diffeomorphisms between surfaces

Suppose we have two smooth compact oriented surfaces $M_1$ and $M_2$ with boundary,both of them have genus $g$, and there are orientation preserving diffeomorphisms $\psi_1, \psi_2, \cdots, \psi_n$, ...
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### Is the wildness of 4-manifolds related to the diversity of their fundamental groups?

$n = 4$ is the smallest dimension such that the fundamental group of a closed $n$-manifold can be any finitely-presentable group (leading e.g. to various undecidability results stemming from the ...
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### Existence of a spin map from a standard sphere to any closed Riemaninan manifold with nonnegative curvature operator

Let $S^m$ be a standard sphere of dimension $m=n+4k$, and let $M$ be any closed Riemaninan manifold of dimension $n$ with nonnegative curvature operator. My question: Is there always a smooth spin map ...
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1 vote
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### Extending the natural thom form of a vector bundle from the boundary of a manifold

(Edited after taking into account Tom Goodwillie's answer.) Let $E \rightarrow X$ be an orientable vector bundle. In this MO answer it is explained how to obtain a representative of the Thom class (...
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### Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations

Given a smooth nested set of "partial" foliations $\mathcal F_{\alpha}=\big\lbrace e^{\frac{\alpha}{\log x}}: \alpha \in (1/k,k), x\in(0,1),k\in [1,\infty) \big\rbrace$ of $X^2=(0,1)^2$ with ...
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### Does a coarser topology lead to a non-Hausdorff topological manifold? [closed]

Take a topological manifold $M$. Suppose one considers a strictly coarser topology than the manifold topology. Can such topology result in a non-Hausdorff topological manifold? NOTE: PLEASE avoid the ...
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### Criteria for extending vector field on sphere to ball

Below is a theorem that is equivalent to Brouwer fixed-point theorem, which I found quite interesting. The proof is in this PDF file. Let $v: \mathbb S^{n-1} \to \mathbb R^n$ be a continuous map, ...
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### How to chart tubes around manifolds with boundary/corners?

Let $M \subset \mathbb{R}^d$ be a manifold with boundary/corners. For example, a piece of curve with endpoints or a $2d$ unit square in $\{ z = 0 \}$. I am interested in introducing local coordinates ...
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### Does the group of compactly supported diffeomorphisms have the homotopy type of a CW complex?

It is known that the group of diffeomorphisms of a compact manifold with the natural $C^{\infty}$ topology has the homotopy type of a countable CW complex. See for instance this thread: Is the space ...
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### Classification of bundles with fixed total space

I am aware of classification theorems for principal bundles, vector bundles, and covering spaces $\pi:E\to B$ over a fixed base space $B$. Principal and vector bundles over $B$ are classified by ...
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### Representing some odd multiples of integral homology classes by embedded submanifolds

Consider an $m$-dimensional compact closed orientable smooth manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]$ on $M$, with $1 \le n \le m-1$. Then does there exist an odd integer ...
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Does there exist a closed connected nonsmoothable 4-manifold $M$ such that: $\kappa(M)=0$ (Kirby-Siebenmann invariant vanishes, hence, there is no "classical" obstruction to smoothability) ...