For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.
50
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5answers
1k views
Defining a manifold without reference to the reals
The standard definition I've seen for a manifold is basically that it's something that's locally the same as $\mathbb{R}^n$, without the metric structure normally associated with $\mathbb{R}^n$. ...
36
votes
7answers
2k views
What's the point of studying topological (as opposed to smooth, PL, or PDiff) manifolds?
Part of the reason I think algebraic topology has acquired something of a fearsome reputation is that the terrible properties of the topological category (e.g. the existence of space-filling curves) ...
25
votes
2answers
608 views
PDEs on manifold: what changes from Euclidean case?
I know some PDE theory for nice open domains in $\mathbb{R}^n.$ I want to know what the changes are when I switch to other domains like manifolds.
For example, do things like Poincare's inequality ...
23
votes
3answers
1k views
Why are smooth manifolds defined to be paracompact?
The way I understand things, roughly speaking, the importance of smooth manifolds is that they form the category of topological spaces on which we can do calculus. The definition of smooth manifolds ...
19
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3answers
482 views
Sanity check about Wikipedia definition of differentiable manifold as a locally ringed space
Most textbooks introduce differentiable manifolds via atlases and charts. This has the advantage of being concrete, but the disadvantage that the local coordinates are usually completely irrelevant- ...
19
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1answer
183 views
If $S\times\mathbb{R}$ is homeomorphic to $T\times\mathbb{R}$, and $S$ and $T$ are compact, can we conclude that $S$ and $T$ are homeomorphic?
If $S \times \mathbb{R}$ is homeomorphic to $T \times \mathbb{R}$ and $S$ and $T$ are compact, connected manifolds (according to an earlier question if one of them is compact the other one needs to be ...
18
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3answers
353 views
What's the connection between derivatives and boundaries?
The (second) fundamental theorem of calculus says that
$$\int_a^b f'(x) dx = f(b) - f(a)$$
which can also be stated, if one knows enough about what's coming next, as:
The integral of the ...
16
votes
1answer
356 views
Decomposition of a manifold
As a kind of aside to this question, where one of the answers assumed that if $S^n=X \times Y$ then we can assume that $X$ and $Y$ are manifolds.
If we have a manifold $M$, such that $M$ is ...
16
votes
0answers
230 views
What is the algebraic structure of functions with fixed points?
So I just noticed that the set of functions with a fixed point
$$f(x_0)=x_0,$$
are closed under composition
$$(f*g)(x):=g(f(x)),$$
and with $e(x)=x$, the inverible functions even seem to form a ...
14
votes
1answer
151 views
Are locally homotopic functions homotopic?
Suppose we have two (smooth) functions $f,g:X\to Y$, where $X,Y$ are smooth (second-countable, Hausdorff) manifolds which are locally homotopic (that is, any point in $X$ has a neighbourhood $U$ such ...
13
votes
3answers
800 views
Is every embedded submanifold globally a level set?
It's a well-known theorem (Corollary 8.10 in Lee Smooth) that given a smooth map of manifolds $\phi:M\rightarrow N$ and a regular point $p\in N$ of $\phi$, the level set $\phi^{-1}(p)\subset M$ is a ...
13
votes
2answers
263 views
No hypersurface with odd Euler characteristic
Here is a classic problem which I encountered and could not solve:
Prove that a simply connected closed smooth manifold has no closed sub-manifold of co-dimension $1$ with odd Euler ...
13
votes
1answer
245 views
Manifold of Density Matrices
Let $\mathrm{M}_{d\times d}\left(\mathbb{C}\right)$ denote the set of all $d\times d$-matrices with complex entries.
My goal is to show that the set
$\mathcal{M}:= \left\{ \rho\in \mathrm{M}_{d\times ...
12
votes
1answer
139 views
Is this surface diffeomorphic to a 2-sphere?
Let $f:\mathbb{R}^3\to \mathbb{R}$ be defined by $f(x,y,z)=x^4+y^6+z^8$. Let $M=f^{−1}(1)$.
Is $M$ is diffeomorphic to a sphere $S^2$?
I tried to solve this problem, but I realized that I ...
12
votes
1answer
356 views
Showing $[0,1] \times [0,1]$ is a manifold with boundary
I'm familiarizing myself with manifolds. I tried to show $[0,1]\times[0,1]$ is a manifold with a boundary. Can you please tell me if my proof is correct:
The definition for manifold with boundary:
...
11
votes
2answers
379 views
Is $M=\{(x,|x|): x \in (-1, 1)\}$ not a differentiable manifold?
Let $M=\{(x,|x|): x \in (-1, 1)\}$. Then there is an atlas with only one coordinate chart $(M, (x, |x|) \mapsto x)$ for $M$. We don't need any coordinate transformation maps to worry about ...
11
votes
2answers
632 views
concrete examples of tangent bundles of smooth manifolds for standard spaces
I'm having trouble visualizing what the topology/atlas of a tangent bundle $TM$ looks like, for a smooth manifold $M$. I know that
$$dim(TM)=2\cdot dim(M).$$
Do the tangent bundles of the following ...
11
votes
1answer
221 views
Product of spaces is a manifold with boundary. What can be said about the spaces themselves?
Suppose I have two topological spaces $X,Y$ and I know that $X\times Y$ is homeomorphic to a manifold with boundary. Can I conclude that $X$ and $Y$ are manifolds (maybe with boundary)?
If not, ...
10
votes
2answers
455 views
Is the Nash Embedding Theorem a special case of the Whitney Embedding Theorem?
The Whitney Embedding Theorem states that every smooth manifold can be embedded in Euclidean space.
The Nash Embedding Theorem states that every Riemannian manifold can be embedded in Euclidean ...
10
votes
2answers
489 views
Why maximal atlas
This has been on the back of my mind for one whole semester now. Its possible that in my stupidity I am missing out on something simple. But, here goes:
Let $M$ be a topological manifold. Now, even ...
10
votes
4answers
782 views
Reference on Geometric Topology
Geometric topology is more motivated by objects it wants to prove theorems about. Geometric topology is very much motivated by low-dimensional phenomena -- and the very notion of low-dimensional ...
10
votes
1answer
121 views
Proof of $H^k(X,\mathbf k) = H^k(X,\mathbb Z) \otimes \mathbf k$
Let $X$ be a compact manifold and denote $H^k(X,G)$ the $k$-th cohomology group with coefficients in the abelian group $G$.
Using Cech cohomology one can prove that there is a natural isomorphism $ ...
10
votes
1answer
89 views
The “Easiest” non-smoothable manifold
In 1960, Kervaire found the first example of a PL-manifold which does not admit a smooth structure. Since then, I understand that there are many examples of non-smoothable manifolds that can be built. ...
10
votes
2answers
351 views
Motivation behind the definition of a Manifold.
A manifold $M$ of dimension n is a topological space with the following properties:
a) $M$ is Hausdorff
b)$M$ is locally Euclidean of dimension n
c) $M$ has a countable basis of open sets.
Why is ...
10
votes
1answer
328 views
Problem 3-38 in Spivak´s Calculus on Manifolds
This is not homework. Problem 3-38 reads:
Let $A_{n}$ be a closed set contained in $(n,n+1)$. Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for ...
9
votes
2answers
726 views
What is the importance of the Poincaré conjecture?
The Poincaré conjecture is listed as one of the Millennium Prize Problems and has received significant attention from the media a few years ago when Grigori Perelman presented a proof of this ...
9
votes
3answers
215 views
Residual Finiteness of Fundamental Groups of Seifert Fibered Spaces
I'm trying to understand why, if $S$ is a Seifert fibered space, then $\pi_1(S)$ is residually finite. From theorems 12.2 and 11.10 in Hempel's "3-manifolds", we can work with a finite-sheeted ...
9
votes
1answer
183 views
Are these equivalent characterizations of closed manifolds?
Let $M$ be a connected smooth manifold without boundary. Are the following equivalent?
$M$ is compact
$M$ cannot be realized as a proper open subset $M\subset N$ of another connected manifold $N$. ...
9
votes
2answers
145 views
Fiber products of manifolds
Let $\mathsf{Man}$ be the category of smooth manifolds. Denote by $|~|$ the forgetful functor to $\mathsf{Top}$. If $X \to S$ and $Y \to S$ are morphisms in $\mathsf{Man}$, then $X \times_S Y$ exists ...
9
votes
1answer
190 views
intrinsic proof that the grassmannian is a manifold
I was trying to prove that the grassmannian is a manifold without picking bases, is that possible?
Here's what I've got, let's start from projective space.
Take $V$ a vector space of dimension n, and ...
9
votes
1answer
264 views
The only 1-manifolds are $\mathbb R$ and $S^1$
I recall having heard somewhere that the only 1-manifolds (second countable, Hausdorff, connected spaces locally homeomorphic to $\mathbb R$) are $\mathbb R$ and $S^1$. Is this true? If so, is there a ...
9
votes
1answer
120 views
When does the quotient of a manifold with boundary become a manifold?
Given a manifold $M$ with boundary $\partial M \neq \varnothing$, when can we form a manifold $\tilde M$ from $M$ by collapsing the boundary? In the examples I've considered it seems like collapsing ...
9
votes
1answer
193 views
uniqueness of the smooth structure on a manifold obtained by gluing
I've just read a proof that
If $M$, $N$ are smooth manifolds with boundary and $f: \partial M\rightarrow \partial N$ is a diffeomorphism then $M \cup_f N$ has a smooth manifold structure such that ...
9
votes
0answers
52 views
Is the fixed point set of an involution on a topological manifold a submanifold?
Let $f:X\to X$ be a homeomorphism of a topological manifold with $f^2=\mathrm{id}$.
Is each connected component of $\{x\in X \mid f(x)=x\}$ a topological submanifold?
9
votes
1answer
484 views
How to apply Stokes' Theorem for manifolds with boundary
Original motivation: How can I apply Stokes' Theorem to the annulus $1 < r < 2$ in $\mathbb{R}^2$?
Concerns:
Since the annulus is a manifold without boundary, it would seem that Stokes' ...
8
votes
1answer
534 views
is triangle a manifold?
Is a triangle (its sides and the region enclosed by its sides) in a 2D Euclidean space $\mathbb{E}^2$ a manifold? I was thinking to use the identity mapping as its charts, but for each point on the ...
8
votes
1answer
838 views
How many dimensions does a circle have?
Is a circle just a line (therefore 1 dimension) or is it a 2-dimensional object because it occupies some surface?
Thanks in advance!
8
votes
2answers
586 views
Question about Singular Homology section in Hatcher
From Hatcher, "Algebraic Topology," Chapter 2, "Singular Homology" section (p. 108-109 in my copy):
Cycles in singular homology are defined algebraically, but they can be given a somewhat more ...
8
votes
1answer
340 views
What are topological manifolds for you?
In this question different people understood different things when talking about topological manifolds. Some argued they they have to be Hausdorff, some that they have to be second countable and some, ...
8
votes
1answer
87 views
Topological manifolds (dimension)
I am taking an introductory course to topology and the professor defined a topological manifold of dimension $n$ if it is hausdorff and if for every point $x$ there exists an open set $U$ around $x$ ...
8
votes
1answer
113 views
When can we recover a manifold when we attach a $2n$-cell to $S^n$?
I have a question related to this one. In my answer I was going to try and say something about the possible manifolds that might arise in this way, i.e. as mapping cones of elements of ...
8
votes
2answers
157 views
How to deal with Homeomorphisms?
I have one doubt that may be too general, I don't know, so sorry if this is not a good place to ask it. I've also seem many other people with the same problem that I have, so I think that if this ...
8
votes
1answer
101 views
Vector Bundle Over Contractible Manifold
The problem comes from Liviu Nicolaescu's book Lectures on the Geometry of Manifolds. He asks the reader to prove that any vector bundle $E$ over $\mathbb{R}^n$ is trivializable. The idea he gives is ...
8
votes
0answers
180 views
Infinite dimensional constant rank theorem
Suppose you have an analytic map $\phi : E \rightarrow \mathbb{C}^n$, where $E$ is a complex Banach space, and such that the rank of $D \phi$ is constant. Is it true then that the set ...
8
votes
0answers
129 views
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 ...
7
votes
5answers
299 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 ...
7
votes
2answers
144 views
Demonstrating the value of abstracting away from elements/subsets to maps
Given a set $S$, here are 5 ways of thinking about elements of $S$, in increasing abstraction:
an actual element, e.g. $s\in S$
an inclusion map, e.g. $i_s:\{s\}\hookrightarrow S$
an ...
7
votes
1answer
271 views
Is there a Möbius torus?
Does the concept of a Möbius torus make sense: taking a cylinder (instead of a rectangle as in the case of the Möbius strip) and twisting it before joining its ends? Or will the resulting twisted ...
7
votes
3answers
213 views
Embedding compact (boundaryless?) n-manifolds in n-dimensional real space
I know the embedding theorems that allow you to embed $n$-manifolds into $\mathbb{R}^k$, provided $k$ is chosen large enough. Here I'm interested in the possibility of taking $k=n$ in the case of ...
7
votes
2answers
97 views
Dimension of de Rham Cohomology groups?
Is there a simple way to prove that the de Rham cohomology groups of a compact manifold $M$ have finite dimension as $\mathbb{R}$-vector spaces?
