For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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69
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5answers
2k 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 $\mathbb{R}^n$. ...
48
votes
7answers
5k 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) ...
46
votes
1answer
1k 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 $$(f\circ g)(x):=g(f(x)),$$ and with $e(x)=x$, the inverible functions even seem to form ...
41
votes
3answers
4k 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 ...
34
votes
3answers
528 views

Does every bijection of $\mathbb{Z}^2$ extend to a homeomorphism of $\mathbb{R}^2$?

Given a bijection $f\colon \mathbb{Z}^2 \to \mathbb{Z}^2$, does there always exist a homeomorphism $h\colon\mathbb{R}^2\to\mathbb{R}^2$ that agrees with $f$ on $\mathbb{Z}^2$? I don't see any ...
29
votes
3answers
1k 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- ...
28
votes
2answers
985 views

Do all continuous real-valued functions determine the topology?

Let $X$ be a topology space. If I know all the continuous functions from X to $\mathbb R$, will the topology on $X$ be determined? I know the $\mathbb R$ here is, somewhat, artificial. So if this is ...
27
votes
2answers
1k 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
4k views

Under what conditions the quotient space of a manifold is a manifold?

There are many operations we can do with topological spaces that when we apply to topological manifolds gives us back topological manifolds. The disjoint union and the product are examples of that. ...
23
votes
1answer
502 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 ...
22
votes
3answers
551 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 ...
21
votes
1answer
241 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 ...
20
votes
2answers
958 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. ...
20
votes
3answers
1k 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 $$\dim(TM)=2\dim(M).$$ Do the tangent bundles of the following ...
19
votes
2answers
928 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 ...
19
votes
2answers
2k 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. ...
19
votes
1answer
474 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 ...
18
votes
2answers
1k 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 ...
18
votes
4answers
2k 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 value $p\in N$ of $\phi$, the level set $\phi^{-1}(p)\subset M$ is a ...
18
votes
3answers
433 views

Is the tangent bundle of $S^2 \times S^1$ trivial or not?

As the question title suggests, is the tangent bundle of $S^2 \times S^1$ trivial or not? Progress: I suspect yes. If I could construct three independent vector fields, I would be done. But I'm not ...
18
votes
2answers
533 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 characteristic....
17
votes
2answers
556 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 have ...
17
votes
1answer
254 views

Reconstructing a manifold from critical points

I am teaching theoretical calculus this semester, and on the last discussion section we were discussing critical points of functions. I explained the idea of Morse theory, and a student of mine asked ...
16
votes
2answers
3k 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 space....
16
votes
1answer
482 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?
16
votes
2answers
374 views

How equivalent are the homeomorphism and diffeomorphism groups of 3-manifolds?

Every topological 3-manifold carries a smooth structure, unique up to diffeomorphism. In addition, "up to homotopy", the maps between them are the same: every continuous map is homotopic to a smooth ...
16
votes
1answer
469 views

Is there any holomorphic version of the tubular neighborhood theorem?

This question arised when I was studying Beauville's book 'Complex Algebraic Surfaces'. Castelnuovo's theorem says that a smooth rational curve $E$ on an algebraic surface $S$ is an exceptional ...
16
votes
0answers
810 views

When is a fibration a fiber bundle?

In this question I am using Wiki's definitions for fibration and fiber bundle. I want to be general in asking my question, but I am mostly interested in smooth compact manifolds and smooth fibrations ...
15
votes
7answers
766 views

Moebius band not homeomorphic to Cylinder.

I have been trying to think of a rather basic way of proving this, but it seems a bit elusive. In the case with boundary (looking at them as quotients in $ [0,1] \times [0,1] $), they can be ...
15
votes
1answer
273 views

Intuition for Exotic $\mathbb R^4$'s

Today one of my professors told me that $\mathbb R^4$ admits uncountably many non-diffeomorphic differential structures. When I asked him whether there's an intuitive reason to expect a result like ...
15
votes
1answer
452 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
15
votes
1answer
289 views

$C^{k}$-manifolds: how and why?

First of all, I have a specific question. Suppose $M$ is an $m$-dimensional $C^k$-manifold, for $1 \leq k < \infty$. Is the tangent space to a point defined as the space of $C^k$ derivations on the ...
14
votes
2answers
2k 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 ...
14
votes
3answers
458 views

Is every self-homeomorphism homotopic to a diffeomorphism?

Given a smooth manifold $M$, is every homeomorphism $M \to M$ homotopic to a diffeomorphism? Hirsch's "Differential Topology" has a proof that every $C^1$ diffeomorphism of smooth manifolds is ...
14
votes
2answers
649 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 ...
14
votes
2answers
319 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 $x$ ...
14
votes
1answer
845 views

Is Zorn's lemma required to prove the existence of a maximal atlas on a manifold?

In the definition of smooth manifolds, complex manifolds, and similar constructions, one starts by defining a property on neighborhoods in the space, specifying how they relate on overlapping ...
14
votes
2answers
1k 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 ...
14
votes
1answer
766 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: ...
14
votes
1answer
424 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 ...
14
votes
1answer
207 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
4answers
2k 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 ...
13
votes
1answer
153 views

Is a maximal open simply connected subset $U$ of a manifold $M$, necessarily dense?

There is a short argument using Zorn's lemma and the compactness of $[0,1]$, that shows every manifold must have maximal open simply connected subspaces. However, I am wondering if it is necessarily ...
12
votes
4answers
1k views

Why do we need Hausdorff-ness in definition of topological manifold?

Suppose $M^n$ is a topological manifold, then $M^n$ locally looks like $\mathbb{R}^n$. $M^n$ is locally Hausdorff, since $\mathbb{R}^n$ is Hausdorff and Hausdorff-ness is a topological invariant. ...
12
votes
1answer
633 views

Second Stiefel-Whitney Class of a 3 Manifold

This is exercise 12.4 in Characteristic Classes by Milnor and Stasheff. The essential content of the exercise is to show that $w_2(TM)=0$, where $M$ is a closed, oriented 3-manifold, $TM$ its tangent ...
12
votes
1answer
827 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 $k>0$. ...
12
votes
2answers
143 views

How do we construct an associated bundle $V_{n, q}(\omega)$ over $B$ with typical fiber $V_{n - q}(F)$?

Let $V_{n - q}(\mathbb{C}^n)$ denote the complex Stiefel manifold consisting of all complex $(n - q)$-frames in $\mathbb{C}^n$, where $0 \le q < n$. This manifold is $2q$-connected, and$$\pi_{2q + ...
12
votes
3answers
2k views

Inducing orientations on boundary manifolds

Given a $k$-manifold $M$, such that $\partial M$ is a $(k-1)$-manifold, there is a standard way in which $\partial M$ inherits the orientation of $M$. So if $M$ is oriented by the form field $\omega$, ...
12
votes
1answer
621 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 ...
12
votes
2answers
666 views

geometric interpretation of Lie bracket

On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written: We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to which the integral ...