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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570 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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182 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 ...
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988 views

Boundaries of Manifolds Necessarily Orientable

Let $M$ be a smooth manifold (not necessarily orientable) and let $N=\partial M$. Is $N$ necessarily orientable? I have no particular reason to believe that this is the case, but I wasn't able to ...
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798 views

Why are differential forms more important than symmetric tensors?

In differential geometry, differential forms are totally anti-symmetric tensors and play an important role. I am led to wonder why do we not study totally symmetric tensors as much as forms. What ...
3answers
365 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 ...
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963 views

A Banach Manifold with a Riemannian Metric?

Given an infinite dimensional manifold modeled on a Banach space, what does it mean for it to have a Riemannian metric? Does it necessarily mean that it is actually a Hilbert manifold? My ...
1answer
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Why are Banach manifolds not so popular?

Why are Banach and Frechet manifolds studied not even remotely as much as Euclidean manifolds? I assume like many other mathematical subjects, theory of manifolds has been developed much more than the ...
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Prerequisite for Petersen's Riemannian Geometry

A professor recently told me that if I can cover the chapters on curvature in Petersen's Riemannian geometry book (linked here) within the next few months then I can work on something with him. ...
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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 ...
1answer
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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 $N$. ...
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129 views

Detecting compactness from the ring of smooth functions

Given a smooth manifold $M$, is there some ring-theoretic property (preferably not mentioning $M$) such that $C^{\infty}(M)$ has this property if and only if $M$ is compact?
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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 ...
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Algebraic varieties in $\mathbb{C}^n$ cannot have interior points

I know that the zero-set of a non-zero polynomial in $\mathbb{C}[x_1,...,x_n]$ can not have interior points, but I'm trying to find a proof that doesn't require a knowledge of complex analysis like ...
5answers
6k views

Simpler definition of manifold

I'm new to topology but I must understand how it works to progress in my research. First, can anybody point me to a document that introduces topology in a "gentle" way. What pre-requisites do I need ...
4answers
388 views

Transition functions in manifold

Following is given as the definition of a manifold in a book which I am reading. Let $\{U_{\alpha}:\alpha\in I\}$ be an open covering of a Hausdorff topological space $X$ and $\phi_{\alpha}$ be ...
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Spin manifold and the second Stiefel-Whitney class

We know that: Spin structures will exist if and only if the second Stiefel-Whitney class $w_2(M)\in H^2(M,\mathbb Z/2)$ of $M$ vanishes. Can someone use simple words and logic to show why the ...
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For $n$ even, antipodal map of $S^n$ is homotopic to reflection and has degree $-1$?

How do I see that for $n$ even, the antipodal map of $S^n$ is homotopic to the reflection$$r(x_1, \dots, x_{n+1}) = (-x_1, x_2, \dots, x_{n+1}),$$and therefore has degree $-1$? Thanks for your time. ...
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723 views

The Heisenberg manifold

I am interested in the Heisenberg manifold, which is the quotient of the real Heisenberg group by the discrete Heisenberg (sub)group. It is a $3$-manifold which may be viewed as a circle bundle over ...
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The reason behind the definition of manifold

I was going thorough the definition of a manifold and needless to say it wasn't something that I could digest at one go. Then I saw the following Quora link and Qiaochu's illustrative answer. It was ...
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399 views

What hyperbolic space *really* looks like

There are several models of hyperbolic space that are embedded in Euclidean space. For example, the following image depicts the Beltrami-Klein model of a hyperbolic plane: where geodesics are ...
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705 views

Each point of $M$ has a smooth coordinate neighborhood in which the coordinate frame is orthonormal iff $g$ is flat

Let $(M,g)$ be a Riemannian manifold. Then I want to show that these are equivalent: (i) Each point of $M$ has a smooth coordinate neighborhood in which the coordinate frame is orthonormal. (...
1answer
67 views

Can intersection of two manifolds be $xy=0$

I know that if two manifolds intersect transversally then their intersection is a manifold. But I was trying to construct an example where the intersection is not a manifold. But I still do not see ...
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Can (singular) homology classes always be represented by images of closed manifolds?

My intuition tells me that if $A \in H_2(M;\mathbf Z)$, then $A$ can be represented by a map $\Sigma \to M$, where $\Sigma$ is a closed (= compact boundaryless) surface, i.e., the connected sum of ...
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Cancellation in topological product

I was wondering whether $M\times \mathbb{R}$ is homeomorphic to $N\times \mathbb{R}$ implies $M$ is homeomorphic to $N$, where let us say $M,N$ are smooth manifolds. (They are certainly homotopy ...
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Open subsets in a manifold as submanifold of the same dimension?

An open set in an $n$-manifold is clearly a submanifold of the same dimension as its containing manifold (see open manifolds). Now, given an $n$-manifold $M$, is it true that a set, to be the ...
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$K$-theory of smooth manifolds: continuous vs. smooth vector bundles

Suppose I have a smooth manifold $M$, and want to consider the $K$-theory $K^0(M)$. I could define this in the usual way (by taking the Grothendieck group of the monoid of equivalence classes of ...
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What is this manifold?

As picture below ,it is a Mobius band with a cylinder crossing it .Let it be $\Omega$ . Obviously , $\partial \Omega$ is a circle. Now , what is $\Omega/\partial \Omega$ ( I mean glue the boundary to ...
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Do the singular matrices form a topological manifold

So the definition of manifold I'm using is that of a topological manifold (a topological space with an atlas of homeomorphisms to $\mathbb{R}^n$). I have two related questions: Is the set of ...
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209 views

What kind of tools do we have to detect when a manifold is a product of other manifolds?

What sort of tools are out there that can detect when a manifold is a product of other manifolds? For example, comparing the homology of the circle to the torus, the homology of the torus gets more "...