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.
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Examples of manifolds that cannot be embedded in $\mathbb R^4$
Could someone give me an example of a (smooth) $n$-manifold $(n=2, 3)$ which cannot be embedded (or immersed) in $\mathbb R^4$?
Thanks in advance!
S. L.
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3answers
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Does having a codimension-1 embedding of a closed manifold $M^n \subset \mathbb{R}^{n+1}$ require $M$ to be orientable?
I'm trying to follow a proof about immersing/embedding $\mathbb{RP}^n$ into $\mathbb{R}^{n+1}$, which goes roughly as follows:
Write $\tau=T\mathbb{RP}^n$. The normal bundle $\nu$ has rank 1, so its ...
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463 views
Diffeomorphism group of the unit circle
I am given to understand that the group of diffeomorphisms of the unit circle, $\operatorname{Diff}(\mathbb{S}^1)$, has two connected components, $\operatorname{Diff}^+(\mathbb{S}^1)$ and ...
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106 views
No Smooth Onto Map from Circle to Torus
My professor was lecturing today and he made this statement which I was unable to verify.
(I worded it nicer)
There is no map which is both smooth and onto from $S^1$ to $S^1$$\times$ $S^1$.
When ...
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4answers
127 views
holomorphic exoticness
A topological manifold is an exotic copy of another smooth manifold if it is homeomorphic to it, but not diffeomorphic (and when you switch diffeomorphic by homotopic, you get a fake copy, following ...
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964 views
Intuitive explanation of covariant, contravariant and Lie derivatives
I would be glad if someone could explain in intuitive terms what these different derivatives are, and possibly give some practical, understandable examples of how they would produce different results.
...
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1answer
91 views
Are close maps homotopic?
Consider a smooth manifold $M = M^m$ and a smooth submanifold $N = N^n \subset M$. Suppose that two maps $f, g: M \to N$ are close to each other, in the sense that there exists $\epsilon > 0$ such ...
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196 views
Do all manifolds have a densely defined chart?
Let $M$ be a smooth connected manifold. Is it always possible to find a connected dense open subset $U$ of $M$ which is diffeomorphic to an open subset of R$^n$?
If we don't require $U$ to be ...
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101 views
Classification of all 28 Exotic 7-Spheres
In http://en.wikipedia.org/wiki/Exotic_sphere#Explicit_examples_of_exotic_spheres
Wikipedia says "As shown by Egbert Brieskorn (1966, 1966b) (see also (Hirzebruch & Mayer 1968)) the intersection ...
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1answer
126 views
Existence of geodesic on a compact Riemannian manifold
I have a question about the existence of geodesics on a compact Riemannian manifold $M$. Is there an elementary way to prove that in each nontrivial free homotopy class of loops, there is a closed ...
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126 views
is the fixed set of a smooth involution a submanifold?
Let $f:X\rightarrow X$ be a smooth map of a smooth manifold with $f^2=\operatorname{id}$.
Is the subset $\{x\in X\mid f(x)=x\}$ a smooth submanifold?
I tried to find an argument with the implicit ...
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324 views
Examples of Computations in Algebraic Topology
I have started reading "Differential Forms in Algebraic Topology" by Bott, Tu, recently. While I'm quite happy with the exposition of the theorems and explanation of theoretical results, I'm missing ...
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297 views
Proof that two spaces that are homotopic have the same de Rham cohomology
I know this is true, but how do I prove it? Specifically, I'm trying to calculate the de Rham cohomology of the 3-sphere by using the Mayer-Vietoris sequence and covering $S^3$ with two hemispherical ...
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1answer
130 views
Conceptual error in Kosinski's “Differential Manifolds”?
This seems like such an egregious error in such an otherwise solid book that I felt I should ask if anyone has noticed to be sure I'm not misunderstanding something basic.
In his section on connect ...
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How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?
I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded ...
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171 views
Nontrivial h-cobordism
I'm learning the h-cobordism theorem as I want to use it in a talk. I'd like to be able to give an example of an h-cobordism that isn't a cylinder, if possible by drawing a picture. What is the ...
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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 ...
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Compactification of Manifolds
It is known that for any locally compact Hausdorff space X, we can define a Hausdorff one-point compactification containing X.
In the case of the (differentiable) manifold $\mathbb R^n$ this one-point ...
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289 views
Uncountable disjoint union of $\mathbb{R}$
I'm doing 1.2 in Lee's Introduction to smooth manifolds: Prove that the disjoint union of uncountably many copies of $\mathbb{R}$ is not second countable.
So first, let $I$ be the set over which we ...
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2answers
143 views
normal bundle of level set
Let $M$ be a Riemannian manifold and $S \subset M$ a regular level set of a smooth function $f:M\rightarrow \mathbb{R}^k$. How can I show that the normal bundle of $S$ is trivial?
If $k=1$ then ...
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1answer
191 views
Is an immersed submanifold second-countable?
By general manifold I mean Hausdorff differential manifold not necessarily second-countable. By standard manifold I mean Hausdorff, second-countable differential manifold.
So my question is, we have ...
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117 views
Structure of a $ C^{\infty} $-manifold
I was studying differentiable manifolds (an introduction) and found the following example, but I am confused.
Example
The function
\begin{align}
f: &\mathbb{R}^{3} \to \mathbb{R}, \\
f: ...
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237 views
Manifold with 3 nondegenerate critical points
Suppose $M$ is a n-dimensional (compact) manifold and $f$ is a differentiable function with exactly three (non-degenerate) critical points. Then one can show, using Morse theory, that $M$ is ...
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1answer
111 views
Visualising a specific orbifold
Let $1 < k \in \mathbb N$ and $M = \{(z_1, z_2) \in \mathbb C^2 : k|z_1|^2 + |z_2|^2 = 1\}$. Let $S^1$ act on $M$ via $e^{i\theta}(z_1,z_2) = (e^{ik\theta} z_1, e^{i\theta} z_2)$. Then I am told ...
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173 views
Confusion on Cech cohomology
From Harvard math qualification exam, 1990.
Let $X$ be a smooth manifold with an open cover $N<\infty$ sets $\{B_{n}\}^{N}_{1}$ which are contractible. Assume that $$\pi_{0}(B_{n}\cap B_{m})\le ...
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126 views
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 $\mathbb{S}^n$ ...
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1answer
326 views
Compact submanifolds of $\mathbb{R}^n$ without boundary
I'm having a little trouble seeing how to do Exercise 7.5 in Lee Smooth Manifolds:
Let $M$ be a smooth compact manifold. Show there is no submersion $F:M\rightarrow\mathbb{R}^k$ for any $k>0$. ...
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245 views
How calculate the De Rham cohomology group of $3$-torus: $T^3$?
How do I calculate the De Rham cohomology group of the $3$-torus $T^3$? Here $T^3=S^1 \times S^1 \times S^1 $.
Using the Mayer-Vietoris sequence, I can show that $\dim H_3(T^3)=\dim H_0(T^3)=1$. But ...
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2answers
194 views
$D^m\cup_h D^m$, joining $D^m \amalg D^m$ along the boundary $\partial D^m$
Given an orientation-preserving diffeomorphism $h: \partial D^m \to \partial D^m$, we can glue two copies of the closed unit disk $D^m$ along the boundary by identifying $x \sim h(x)$ to form the ...
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284 views
Diffeomorphisms and Stokes' theorem
Problem:
Let $\omega\in\Omega^r(M^n)$ suppose that $\int_\sum \omega = 0$ for every oriented smooth manifold $\sum \subseteq M^n$ that is diffeomorphic to $S^r$. Show that $d\omega = 0$.
Proof:
...
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Let $A$, $B$ be subsets of $S^n$, n≥2. Show that if $A$ and $B$ are closed, disjoint, and neither separates $S^n$, then…
Let $A$,$B$ be subset of $S^n$, $n\geq 2$. Show that if $A$ and $B$ are closed, disjoint, and neither separates $S^n$, then $A\cup B$ does not separate $S^n$.
I've thought to do it by contradiction ...
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1answer
57 views
Example of a diffeomorphism of class $C^{k}$ which is not $C^{k+1}$
Can anyone give me an example of a map $f:\mathbb{R}\to\mathbb{R}$, which is a diffeomorphism of class $C^{k}$ but it is not a diffeomorphism of class $C^{k+1}$?
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Lemma 6.3 of Milnor's Lectures on the h-cobordism theorem
Milnor's statement is: "Let $M^r$ and $N^s$ be sub-manifolds of $V^{r+s}$ which are all smooth, compact, oriented and without boundary. If $p$ is a point of $M^r$ contained in an $r$-cell $U$, ...
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A star shaped open set in $\mathbb{R}^n$ is diffeomorphic to $\mathbb{R}^n$
After googling this fact (which is used in Bott and Tu to show the existence of good covers of manifolds) I've gotten the impression this is somewhat difficult to prove. But I also came across this ...
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311 views
Smooth structure on the topological space
Consider a topological space $X$. Lee in Introduction to Smooth Manifolds wrote that it is impossible to introduce a smooth structure on the topological manifold based only on topology (i.e. ...
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261 views
Smooth closed real plane curve intersecting itself at infinitely many points
Can a smooth closed real plane curve intersect itself at infinitely many points? It seems intuitively obvious that the answer should be no, yet I have no idea how to prove this or construct a ...
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2answers
104 views
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 ...
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2answers
579 views
Showing that a level set is not a submanifold
Is there a criterion to show that a level set of some map is not an (embedded) submanifold? In particular, an exercise in Lee's smooth manifolds book asks to show that the sets defined by $x^3 - y^2 = ...
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1answer
231 views
When does the topological boundary of an embedded manifold equal its manifold boundary?
Suppose I embed a manifold-with-boundary $M$ in some $\mathbb{R}^n$. Are there conditions (necessary, sufficient, or both) that can help determine when the topological boundary of $M$ is equal to the ...
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1answer
55 views
Is manifold mapping degree equal to algebraic degree for polynomials?
If $M$ and $N$ are oriented $n$-manifolds and $f: M \to N$ then the degree of $f$ is given by
$$
\deg f = \sum_{p \in f^{-1}(q)} sign_p f
$$
where $q$ is a regular value and the sign is $+1$ if $f$ is ...
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146 views
Is a vector bundle orientable if and only if its dual bundle is orientable?
I was reading up on my dual spaces today and I made the following hypothesis:
A vector bundle $\xi$ is orientable if and only if $\xi^*$ is orientable.
This seems rather intuitive, and although ...
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113 views
Obstructions to lifting a map for the Hopf fibration
This is a bit of an elementary question, but
suppose $\pi: \mathbb{S}^3\to \mathbb{S}^2$ is the Hopf fibration, are there reasonably computable obstructions to when a map $f:M\to \mathbb{S}^2$ can be ...
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4answers
169 views
How do you imagine the shape of a manifold $S^2 \times S^1$?
In 3-dimensional manifold theory, I have encountered the manifold $S^2 \times S^1$ many times.
(The following story can be applied not only this manifold but also for any 3-dimensional manifold.)
But ...
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1answer
167 views
Proof that vector area is a boundary integral?
Let $M \subset \mathbb{R}^2$ be a closed topological disk and let $f: M \rightarrow \mathbb{R}^3$ be a smooth embedding; let $N$ be the corresponding unit normal field on $M$. The vector area is ...
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1answer
227 views
embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$
Consider the classic map
$$F:\mathbb{RP}^2\rightarrow \mathbb{R}^4$$
defined by $$F[x,y,z]=(x^2-y^2,xy,xz,yz)$$.
This defines a smooth embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$. It is clearly ...
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3answers
319 views
Proper maps and families of compact complex manifolds
Kodaira defines a complex analytic family of compact complex manifolds as the data $(E,B,\pi)$, where $E$ and $B$ are complex manifolds, and $\pi$ is a surjective holomorphic submersion such that the ...
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271 views
Which continuous functions are polynomials?
Suppose $f \in C(\mathbb{R}^n)$, the space of continuous $\mathbb{R}$-valued functions on $\mathbb{R}^n$. Are there conditions on $f$ that guarantee it is the pullback of a polynomial under some ...
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94 views
Morse homology of $P^2$
I have seen and worked through the explicit computation of the Morse homology of the sphere and the torus (with signs and all). But trying it for $\mathbb{R}\mathbb{P}^2$ has lead me to dead ends. Is ...
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66 views
Identification of integration on smooth chains with ordinary integration
Let $M$ be a smooth oriented $n$-dimensional manifold and denote by $A \in H_n(M;\mathbb{Z})$ the fundamental class of $M$ (a generator of singular homology consistent with the orientation of $M$). ...
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330 views
On the Use of the Topology on Tangent Bundles
On learning about how to define smooth vector fields on a manifold $M$, I learned that one should first define a tangent bundle , $T(M)$, as $\cup T_p(M)$ together with a topology(smooth structure). ...
