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Questions tagged [differential-topology]

Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

Differential topology is the field dealing with differentiable functions on differentiable manifolds. Somewhat simplified view of this field is that it describes the setting to which the notion of differentiable function can be generalized from the more familiar case of functions $\mathbb R^n\to\mathbb R^k$. Differential topology is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

7287 questions
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Why is there no natural metric on manifolds?

One of the things that always bothered me after learning introductory differential geometry (as a physics student) and then delving deeper into this field on my own is that, the usual construction of differential manifolds are such that $M$ is an…
Bence Racskó
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Product of spheres embeds in Euclidean space of 1 dimension higher

This problem was given to me by a friend: Prove that $\Pi_{i=1}^m \mathbb{S}^{n_i}$ can be smoothly embedded in a Euclidean space of dimension $1+\sum_{i=1}^m n_i$. The solution is apparently fairly simple, but I am having trouble getting a…
John Miller
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Why abstract manifolds?

If we can use Whitney embedding to smoothly embed every manifold into Euclidean space, then why do we bother studying abstract manifolds, instead of their embeddings in $\mathbb{R}^n$? A vague explanation I have heard is that from this abstract…
Beginner
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Simply connected manifolds are orientable

For a simply connected $n$-manifold $M\subseteq\Bbb{R}^k$, I want to show that $M$ is orientable. Take a point $p\in M$ and take an $n$-disc, $D^n$, around $p$ (we can take it as small as we please). Since $S^{n-1}$ is orientable and $M$ (and…
Xena
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Question about Milnor's talk at the Abel Prize

I don't quite follow the rough outline Milnor gives of the fact that the 7-sphere has different differentiable structures. The video is available here, and the slides he used can be found here. Here's what I got from the talk. Unless otherwise…
Olivier Bégassat
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Local diffeomorphism is diffeomorphism onto image provided one-to-one.

For the problem Guillemin & Pallock's Differential Topology 1.3.5, I am not confident with my proof. Prove that a local diffeomorphism $f: X \rightarrow Y$ is actually a diffeomorphism of $X$ onto an open subset of $Y$, provided that $f$ is…
WishingFish
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Inverse of regular value is a submanifold, Milnor's proof

In Milnor's famous book "Topology from the Differential Viewpoint" he proves the following on page 11: If $f: M\to N$ is a smooth map between manifolds of dimension $m\geq n$ and if $y\in N$ is a regular value, then the set $f^{-1}(y) \subset M$ is…
M.B.
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Is $\mathbb{S}^\infty$ exotic

During construction of universal bundles one considers (for example) the infinite real projective space $\mathbb{R}\mathbb{P}^\infty$, coming from the sphere $\mathbb{S}^\infty$. My question is, are there exotic $\mathbb{S}^\infty$'s ? Edit: Maybe…
Willem Noorduin
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Top homology of an oriented, compact, connected smooth manifold with boundary

Let $M$ be an oriented, compact, connected $n$-dimensional smooth manifold with boundary. Then, is it true that $n$-th singular homology of M, that is $H_n(M)$, is vanish? I can't make counterexamples for this statement, but I don't have the…
Shinichiro Nakamura
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What's so special about a homotopy $15$-sphere?

I just saw a table which counts diffeomorphism classes of homotopy $n$-spheres (that is, spaces homotopy equivalent to $n$-dimensional spheres). Such a table can be seen on the first page of this paper. Most of these numbers are less than $10$, and…
Joe
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The tangent space to the intersection is the intersection of the tangent spaces.

Let $X$ and $Z$ be transversal submanifolds of $Y$. Prove that if $y \in X \cap Z$, then $$T_y(X \cap Z) = T_y(X) \cap T_y(Z).$$ ("The tangent space to the intersection is the intersection of the tangent spaces.") Given $X$ and $Z$ transversal…
1LiterTears
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Can the composite of two smooth relations fail to be smooth?

Definition 0. Let $A$ and $B$ denote smooth manifolds. Then a smooth relation $A \rightarrow B$ is a smooth submanifold of $A \times B$. Definition 1. Given relations $P : A \rightarrow B$ and $Q : B \rightarrow C$, define a relation $Q \circ P : A…
goblin GONE
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Is there a condition for the closure of an open subset to be a manifold with boundary?

A necessary condition for a manifold with non-empty manifold boundary embedded in $\mathbb{R}^n$ to have its topological boundary coincide with its manifold boundary is that its manifold interior is an open subset of $\mathbb{R}^n$. Questions: Is…
Chill2Macht
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Construction of Exotic Spheres

Milnor was constructing exotic spheres (at least in dimension 7) by bundle theory. Having proven the existence of such an exotic beast, I wonder if something as this is possible: Let $\mathbb{S}^n$ be the n-sphere with a standard structure,…
Willem Noorduin
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Diffeomorphisms of manifolds with boundary

If two compact manifolds have diffeomorphic interiors and diffeomorphic boundaries, are they then diffeomorphic? Is it true for surfaces? Some context: there seems to exist an example by Barden and Mazur of a nontrivial cobordism between some…
Jesus RS
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