# Tagged Questions

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

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### 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 ...
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### CW complexes and manifolds

What is the strongest known theorem (if any) that classify which manifolds can be built as CW complexes? Thank you guys. This is just a question I thought of during class, obviously not homework...
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### 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 $\mathbb{R}^n$. ...
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### 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 ...
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### Smooth surfaces that isn't the zero-set of $f(x,y,z)$

The zero-set of any smooth function $f(x,y,z)$ with a non-vanishing gradient is a smooth surface. I was wondering if the reverse is true: is every smooth surface in $E^3$ the zero-set of some smooth ...
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### Orientation of hypersurface

Some books on mean curvature flow (e.g. Mantegazza, Ecker) state that an embedded hypersurface in $\mathbb{R}^{k+1}$ is orientable (Mantegazza page 3, Ecker page 110). In other words, they assume the ...
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### Good introductory book on Calculus on Manifolds

I have already taken up to Multivariable Calculus, Linear Algebra and Diff Eq. I want to learn Calculus on Manifolds by myself, could you recommend a good introductory book on this subject? Should I ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### Definition of “a topological manifold with corners”.

How can we define a topological manifold with corners and its corners? Then, do we use "invariance of domain" to define corners, as we really need this theorem in order to define "boundaries of a ...
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### 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. ...
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### Global sections of a tensor product of vector bundles on a smooth manifold

This question is similar to Conditions such that taking global sections of line bundles commutes with tensor product? and Tensor product of invertible sheaves except that I am concerned with ...
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### map of arbitrary degree from compact oriented manifold into sphere

This is a question from a qualifying exam. Let $X$ be a compact, oriented $n$-dimensional manifold. Show that for any $k \in \mathbb{Z}$, there exists a continuous map $f: X \to S^n$ of degree $k$. I ...
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### Tensors constructed out of metric other than the Riemann curvature tensor

Let $(M,g_{ab})$ be a Riemannian (or Pseudo-Riemannian) manifold and let us define a tensor field as 'something' that transforms in an appropriate way under coordinate transformations. (This is how ...
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### Does there exist a $1$-form $\alpha$ with $d\alpha = \omega$?

Let $\omega := dx \wedge dy$ denote the standard area form on $\mathbb{R}^2$. As the question title suggests, does there exist a $1$-form $\alpha$ with $d\alpha = \omega$?
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### Functionally structured spaces and manifolds

The question requires some definitions that I have listed below for your convenience. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups. On a ...
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### Is there a retraction of a non-orientable manifold to its boundary?

It's easy to show using Stokes theorem that a compact orientable manifold with boundary cannot retract to its boundary, by choosing a volume form. But for the non-orientable case I don't know if this ...
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### Non-orientable 1-dimensional (non-hausdorff) manifold

Is there any nice example of a 1-dimensional non-hausdorff manifold that is not oriented? I have tried the line with two origins, but maybe something more exotic is needed?
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### 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 ...
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### How to show that the unit sphere is a topological manifold?

Sorry for this basic question, but I´m not sure of something. I want to see one example. The definition of a n-manifold is a Hausdorff space, such that each point has an open neighborhood homeomorphic ...
I know that a way of proving Poincare lemma is to use the homotopy invariance and contractibility of the Euclidean space. Is there is a way of doing it directly (without using the contractibility of $\... 4answers 1k views ### What is the codimension of matrices of rank$r$as a manifold? I'm reading through G&P's Differential Topology book, but I hit a wall at the end of section 4. There is a result stating The set$X=\{A\in M_{m\times n}(\mathbb{R}):\mathrm{rk}(A)=r\}$is a ... 2answers 243 views ### Codimension 1 homology represented by Embedded Submanifold I'm looking for a reference for the following statement: Given an oriented manifold$M$and a class$\xi \in H_{n-1}(M)$, there is some embedded oriented submanifold$F^+ \hookrightarrow M$such that ... 1answer 771 views ### Showing$[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]$is a manifold with a boundary. Can you please tell me if my proof is correct: The definition for manifold with boundary: ... 2answers 325 views ### On the smooth structure of$\mathbb{R}P^n$in Milnor's book on characteristic classes. This question concerns problem 1-B in the book of Milnor and Stasheff part a. They first define the set$F:=\{f:\mathbb{R}P^n\rightarrow\mathbb{R} \mid \text{$f\circ q$ is smooth}\}$where$q:\mathbb{...
Let G be a finitely generated abelian group and M a compact manifold, I want to prove that $H_r(M,G)$ is finitely generated for $r\ge 0$. First I was thinking if I could do induction over $r$ ...