What is the Topology of a Tangent Bundle?

[MattRob]

Hello!

I'm trying to teach myself some mathematics, and I want to see if I understand this concept correctly from what I've been reading.
(And just to be clear, this isn't part of any coursework, so I assume it doesn't go under that section for that reason?)

So, essentially, although there's a lot more to it, just in a quick summary, would it transfer the general idea correctly to say that a Topological Space is a set of points X, where each of the points has a function f(x)? Like, for example, a Newtonian gravitational function of location,

$f(x) = \frac{dv}{dt} = \frac{GM}{x^2}$

If this were coupled with an open set of locations, would it be a Topological Space?

What if it's more generalized to 3-dimensional space, as,

$F(x) = \frac{dv}{dt} = \frac{GM}{(x^2 + y^2 + z^2)}$

When would that be a Topological Space?


And also, one question about what Topological Space isn't: If there were a portion of space where each point were assigned a vector for Newtonian gravity, would that be a Vector Bundle instead?

 

[disregardthat]

A topological space is a set X with a topology. A topology on X is a set of subsets of X which we call the open subsets of X. The topology must satisfy certain properties: Both X and the empty set must be open, any union of open sets must be open, and any finite intersection of open sets must be open.

The standard topology on $\mathbb{R}^n$ is defined by the euclidean metric on $\mathbb{R}^n$. The norm $||\cdot ||$of a vector in $\mathbb{R}^n$ defines a metric by letting the distance between two vectors x and y be $||x-y||$. So we may consider the open balls of $\mathbb{R}^n$: If $\epsilon > 0$, and $z \in \mathbb{R}^n$, the open ball $B_z(\epsilon)$ is the set of vectors with distance less than $\epsilon$ to z. We call these open balls open sets of $\mathbb{R}^n$, and let the topology be the sets which may be written as a union of such balls (and we include the empty set).

With two topological spaces X and Y we may talk of continuous functions. A function $f : X \to Y$ is continuous if for every open set U of Y, $f^{-1}(U)$ is an open set of X.

If each point in a subset of $\mathbb{R}^n$ was assigned a vector in $\mathbb{R}^m$, you are probably talking about a vector field, which is a function $F : \mathbb{R}^n \to \mathbb{R}^m$, usually properties like continuity and differentiability is assumed.

 

[homeomorphic]

More intuitively, it's a set together with enough structure that you know which functions on it are continuous. This idea only really makes sense if you know the theorem from real analysis that a function is continuous in the usual sense if and only if the pre-images of open sets are open. So, you are taking the idea of continuous functions and generalizing that to sets besides R^n by specifying which sets are open.

This all becomes more meaningful when you have a bigger repertoire of examples to draw from, such as the original concept of topology, which was the topological classification of surfaces (this answers the question of which surfaces can be continuously deformed into one another). Another idea is to think of a collection of mathematical objects as a space. So, for example, you could consider the set of all triangles up to congruence. You might have functions from that set of triangles to the real numbers and maybe you would like some definition of what it would mean for such a function to be continuous (intuitively, if two triangles are close to being congruent, the corresponding function values should be close, but the question is how to define it precisely).

An example of a vector bundle would be the tangent bundle to a surface in R^3. At each point, you don't just have ONE vector, you have a whole tangent plane. Roughly speaking, the tangent bundle is simply the union of all the tangent spaces at different points (this could be thought of as a subset of R^6, which is a copy of R^3 to hold the surface, crossed with a copy of R^3 to hold the tangent plane, so that you don't have to worry about different tangent planes intersecting because they have different basepoints and are thus considered disjoint). Vector bundles are built out of topological spaces: a total space, which would be this union of tangent planes, and a base space, which would be the surface itself. There is also a mapping, the bundle projection, from the total space to the space base which maps the points of each tangent plane to the point that the plane is tangent to. Your example of a gravitational field (or a vector field, in general) is an example of a what's called a section of a vector bundle.

As with topological spaces, the motivation for the vector bundle concept requires a bigger repertoire of examples, and there's more to the actual definition than what I have described. It turns out that one of the key points is how all the tangent spaces at the different points fit together, and I have left that part out of my description.

 

[bolbteppa]

Topology is nothing but a mathematical way to talk about "nearness" and "farness" with sets and the points in those sets. The way to define a notion of nearness on a set X of points is by taking a set of subsets of X and using 'unions' and 'intersections' to establish notions nearness and farness. If you ever look at a textbook:

An open set topology is a way to talk about nearness in terms of sets (i.e. on a global scale).
A neighbourhood topology is a way to talk about nearness in terms of points (i.e. on a local scale, but it has to work for each element in the underlying set).
A closed set topology is way to talk about farness globally in terms of sets.
A limit point is a way to talk about farness locally in terms of neighbourhoods.
(c.f. Kelley - General Topology, here etc...)

You can use whatever you like to construct a structure encoding nearness and farness (in other words, to construct a set of subsets on which a notion of nearness can be defined), e.g. sets/points/relations/functions/... so long as the resulting set of subsets of elements of the underlying set allows us to use any of the other topological notions.

As an example, take https://www.math.ucdavis.edu/~gooding/talks/Pointless.pdf of an open-set topology

as this is a well-defined topology, we can say certain things are nearer to others, e.g. a is nearer to b than it is to c, a is nearer to c than it is to d, a is far away from c but is is further away from d than it is from c, in terms of being near or far away from c & d, a is indistinguishable from b (topological indistinguishability, motivating the separation axioms).

I could have defined that topology by specifying the subsets of S = {a,b,c,d} to be the ones in T, or I could have used a function f : S ---> M between sets such that the inverse images of elements of M result in the collection T of subsets, i.e. the idea of continuity of functions naturally leads us to notions of nearness on the underlying set. In real analysis you use the extra idea of a "metric" to establish notions of distance on the underlying sets, then functions respecting what the metrics do are continuous (your gravity example is using this notion).

So, essentially, although there's a lot more to it, just in a quick summary, would it transfer the general idea correctly to say that a Topological Space is a set of points X, where each of the points has a function f(x)? Like, for example, a Newtonian gravitational function of location,

$f(x) = \frac{dv}{dt} = \frac{GM}{x^2}$

If this were coupled with an open set of locations, would it be a Topological Space?

In $f(x) = \frac{dv}{dt} = \frac{GM}{x^2}$ you are working with a function of the form $f \ : \mathbb{R}-\{0\} \ \rightarrow \mathbb{R}^+$ which is a continuous function between two (metric) topological spaces $(\mathbb{R} - \{0\},d)$ and $(\mathbb{R}-^+,d)$, you see the ideas of nearness and farness are already encoded in the picture implicitly when the function is well-defined and continuous...

And also, one question about what Topological Space isn't: If there were a portion of space where each point were assigned a vector for Newtonian gravity, would that be a Vector Bundle instead?

In a physicsey way I guess you could define a vector bundle that way, how do you apply notions of nearness and farness to an example like this though? Furthermore, how do you do it in a physically relevant sense?

 

[homeomorphic]

In a physicsey way I guess you could define a vector bundle that way, how do you apply notions of nearness and farness to an example like this though? Furthermore, how do you do it in a physically relevant sense?

But a vector bundle would include all possible vectors based at different points, not just a single vector at each point. A single vector at each point (vector field) is a section of a vector bundle, not a vector bundle.

Two tangent vectors in the tangent bundle are "close" if they are based at nearby points and locally, you need a way to compare vectors based at different points, as well. Basically, you need some form of local parallel transport from one basepoint to another (formally, a local product structure, but you could also view it as a locally defined "flat connection", the way I'm describing). Locally, your choice of how to do this turns out not to affect the topology, but globally (if the space has some sort of non-trivial topology), there could be many ways of doing this, resulting in distinct vector bundles. Because the topology of the plane is sort of trivial, in that case, there's only one possible vector bundle over it, the trivial bundle, which, topologically, is just good old R^4 with the usual projection to R^2.

 

[bolbteppa]

Fair enough, I was just thinking in terms of the gravity example and skewing the definitions to semi-apply.

Two tangent vectors in the tangent bundle are "close" if they are based at nearby points and locally, you need a way to compare vectors based at different points, as well. .

I've never seen the topology of a fiber bundle defined in this way, how would you define open sets in a fiber bundle along these lines?

 

[homeomorphic]

I've never seen the topology of a fiber bundle defined in this way, how would you define open sets in a fiber bundle along these lines?

It's more an intuitive interpretation of a definition than a real definition, since you asked about nearness and far-ness in this context. Roger Penrose talks about it similarly in one of his books.

Slightly more formally, you can identify a piece of the tangent bundle over a chart with the chart cross R^n, using tangent vectors to the coordinate directions to math up the tangent plane with R^n. This defines the topology locally, and you can sort of patch things together to get a topology (or manifold structure) on the whole tangent bundle. So, locally, you are transferring the product topology from the chart cross R^n to the piece of the tangent bundle lying over your coordinate neighborhood. A product structure can be thought of as a flat connection because, for a product, you get (basepoint, vector) as the points and it makes sense to vary the base point while keeping the vector the same (which is a kind of parallel transport). In general, on a surface (or manifold), it doesn't make sense initial to do that, so you have to come up a way to do that consistently (and just locally, not globally).