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I am trying to better my understanding on topological manifolds. I have listed some statements based on how I have understood it so far and some questions. Kindly correct me if I am wrong or help me with an answer.

  1. The primary idea is to parametrize curves and surfaces with the condition that (i) nearby points (parameter value) goes to nearby coordinates (on the chart), (ii) Every point (on the manifold) has unique coordinates on the chart (Nakahara). To accomplish this, we take parts of the curve/surface and use multiple charts and describe the whole curve/surface. We do not want to use the implicitization of the curve/surface here. That is, we do not want to work with $$x^2 + y^2 = 1$$ for a circle. Instead we would use the below parametrization. $$\phi^{-1} : \theta \mapsto (cos(\theta), sin(\theta))$$

    Are we doing this because we do not want to work with an enclosing space which contains the object we are actually working on (the circle)? I have read in some place that this allows us to do calculus on charts after parametrization. How exactly does it help? Why can't we do calculus directly on $$x^2 + y^2 = 1$$ ? This is also inside a Euclidean space. Why calculus is difficult here? What is the problem with working along with an enclosing space?

  2. With parametrization, though it does not intuitively mean that there is an enclosing space, but, there is still some external reference we are taking to describe the manifold, right? What advantage does this bring? I am assuming this makes transformations easier, but yet cannot clearly see how (correct me if I'm wrong).

  3. Is it always the case that we first take an implicitized equation describing a manifold, then parametrize it with multiple charts? What is the significance of quotient topology/equivalence relations here? We have the equation $$x^2 + y^2 = 1$$ to describe a circle. But we also say, a circle is a closed interval on $R$ with endpoints "identified". Where does this definition get applied? For instance, when I am making a chart of a manifold, do they have any significance?

  4. I am not able to see how homology groups and homotopy relate to charts. Is it that they are to be studied only on the topology and have no role in charts? That is, we could take a manifold, chart it, do some transformation to another manifold and the resultant manifold should retain the homology/homotpoy properties?

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    $\begingroup$ There is no guarantee that your topological (or differential) manifold starts its life inside some Euclidean space, just as there is no guarantee a vector space starts its life as $\mathbb{R}^n$ or a group is given as a permutation group, for example. Parametrisation is there to say it is locally like a piece of Euclidean space. $\endgroup$ – user10354138 Jun 5 at 16:44
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Question 1.

Your descriptions of charts is too vague and fuzzy for me to tell if you understand it.

A topological manifold is a locally Euclidean topological space.

A space $X$ is locally Euclidean if at every point $X$, there is an open neighborhood $U$ of $x$ and a homeomorphism $\phi:U\to \Bbb{R}^n$ for some $n$ (for convenience we can relax $\phi$ to open embedding). We call $\phi$ a chart for $X$ about $x$.

Subpoint a

Implicit descriptions of manifolds are fine, but not every manifold has such a description. For example, the Mobius strip has no implicit description, since it is not orientable. (At least no smooth implicit description, I'm less familiar with the topological case) The reason for this is that a global implicit description would allow you to define an orientation on the normal bundle, which would in turn yield an orientation on the tangent bundle.

The other problem with working with manifolds embedded in Euclidean space is its not necessarily obvious how to construct or compute the embedding. Such a thing is guaranteed to exist, but it's not necessarily nice. (Also note that there is a distinction between an embedded submanifold and a submanifold given by an implicit description).

Subpoint b

You can't do calculus on topological manifolds. You need a smoothness condition to do calculus on manifolds.

Subpoint c

It's not easy to show that an implicitization gives a manifold in the nonsmooth case. In the smooth case, we have the implicit function theorem and regular level set theorem, but I'm not sure what it looks like in the nonsmooth world.

Subpoint d

Who is Nakahara?

Question 2

Yes, the external references are the Euclidean spaces, $\Bbb{R}^n$. The benefit is that we understand the local structure of Euclidean spaces fairly well, so we can apply that knowledge of the local structure to manifolds. The interesting questions for manifolds are about their global structure.

Not sure what transformations have to do with it?

Questions 3

Question a

No, we don't always begin with an implicitization. See question 1.

Question b

The quotient topology is just a nice way to construct new spaces from old spaces, and if your quotient is reasonable, the quotient of a manifold will still be a manifold. This lets us construct spaces like $\Bbb{R}P^n$ as the quotients of $S^n$ under the antipodal action.

Question c

It's just a useful alternative perspective. $S^1$ is an abstract topological space. There are many ways to describe it, which may be useful at different times.

Question 4

Homology and homotopy groups are mostly independent of the charts. The charts tell you about local behaviour, whereas homology and homotopy groups are about global behaviour.

Not sure what you mean by transformations here.

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