Topology help please!

Not really my area, and it's possible some of what I'm going to say will be wrong/unhelpful, but here's a few comments / questions:

(1), (2) I'm not familiar with "#ed", and I doubt I'll be able to google it. Is there a plain text equivalent?
(3) It's the surface of the sphere that's represented by S^2. I hope you can see how (conceptually) the surface of a sphere can be considered to be 2-dimensional.
(4) What's the formula you're using? If it's the V-E+F one, then to use it on a surface you need to triangularize the surface, that is, draw lines on it to divide it into triangles (with the vertices being dots). Which is basically a graph. https://en.wikipedia.org/wiki/Graph_(discrete_mathematics)

Sorry, # is the operation for combining two surfaces together, and no I can't really see that, spheres are 3D?
I don't know what triangularise means, but that's the formula. I don't see what the graph does or tells me, it just seems random what is drawn does that make sense?

If you're on the surface of a sphere, and you want to move a small distance (while staying on the surface), your movement is effectively constrained to two dimensions (hence maps of local regions are effectively flat and 2-dimensional). Or more globally (pun intended), your position on the sphere can be described by 2 values (latitude + longitude). This is very different from the entire sphere, where you need 3 coordinates to describe position and/or movement.

The Euler formula originally came from actual polyhedra (cube, octahedron, etc). You can smoothly distort the edges of a cube/octahedron/etc. so you end up with edges that lie on the surface of a sphere (although the edges won't be straight). I can't comment/help with whether the graphs you have make any sense unless you post them.

Well, I tried.

1: What actually is the real projective plane, and how can it be made from a Mobius strip and a disc? How can another surface be #ed with a plane?

2: When #ing surfaces together, you have to cut away discs. One of the main rules of topology is that you can't break surfaces. Why are you doing that here, and what happens to the discs?

3: A sphere (3D) is represented by S², as if it were a 2-dimensional object.

4: What is the Euler characteristic? I know the formula, but then my lecturer starting drawing lines and dots on the shapes and calling them graphs?

When you have a mathematical object, for example, a topological space, one of the things you might be interested in is creating new mathematical objects out of it. One of the classic ways of doing this is to take the object and impose an equivalence relation on it; your new mathematical object is then the old object, but with equivalent points identified. In the case of topological spaces, we often talk about the points identified under the equivalence relation as being “glued” together.

The real projective plane is an example of a topological space obtain in this way. Start off with the good old two-dimensional sphere (only the surface of it). Apply an equivalence relation that identifies antipodal points (or “glue” together points that are opposite each other). The resulting topological space is the real projective plane.

But that’s not the only way of thinking about the projective place, from a topological point of view. Another way of making new topological spaces out of old ones is to start “gluing” together a couple of them (that is, apply an equivalence relation that identifies a subset of one space with a subspace of another). Here, if you identify the boundary of a disc with the boundary of a mobius strip (“gluing” them together), it turns out you get a topological space that is homoeomorphic to the projective plane.

The “#” notation, called the “connected sum” is a shorthand for a particular type of equivalence relation on a pair of topological spaces – in the case of topological surfaces, you cut out a disc from each of the spaces, and identify the resulting circles with each other – and you get a new topological space.



Indeed, one of the main concerns in topology is about all the things that stay the same when you apply homeomorphisms. We consider two topological spaces to be the “same” when they are homeomorphic. However, that’s not the end of the subject; as I suggested above, if you have a bunch of topological spaces, one of the things you should be interested in is making new spaces out of the ones you have. Equivalence relations (and in particular the # operation, which can be expressed as an equivalence relation) are one of the main ways of doing this.

The connected sum is particularly important as you can run things in reverse with it: if I’m given a particularly complicated topological space, can I express it as a connected sum of simpler topological spaces? This has been one of the major driving forces in topology in the last century, culminating in the proof of the Poincare conjecture!



One of the big intellectual steps in topology (and geometry) is to start treating topological spaces as intrinsic objects – that is, in themselves, without any relationship to spaces that they might be embedded in. So your 2-sphere in everyday life you think of as a curved surface embedded in 3-D Euclidean space. But what if we think about it in the abstract? What characterises it? How can we construct it abstractly? Take two unit discs and identify their boundaries. That’s what the sphere “is” as far as a topologist is concerned – what characterises it is that it looks locally like a 2-D surface – it just has this property of “joining up” with itself so it wraps around. And the point is this – you can do topology (and geometry) in this way – only refer to the object itself, ignoring the particular way in which the object is initially represented (in this case as a surface embedded in 3-D space).



It’s a topological invariant of graphs (the lined and dots you refer to) drawn on a surface. That means that it is invariant under any transformation of the surface/graph that is a homeomorphism. It turns out that the Euler characteristic of a graph drawn on a surface depends only on the topological characteristics of the surface (whether it’s a sphere or a doughnut etc etc). And, as with many such things, it can be generalized way beyond the case of surfaces!

Thank you so so much that has really helped I don't really get some of the words (antipodal, homeomorphic etc.) and I have one question.
In the first answer you mentioned equivalence relations. What do these have to do with topology? We haven't really linked group theory with topology, which is annoying as they are closely linked but my lecturer hasn't shown how. I need the how & why that's how my brain works.
Thanks for the help!!

An equivalence relation is a way of saying "these things are equivalent". In topological terms, it's how you "glue" points together. For example, you can take a sheet of paper and glue the left/right edges to make a cylinder. Formally, you'd set up an equivalence relation specifying exactly how each point on the left is "glued" to a point on the right.

If instead you glued the point on the left edge a distance x from the top of the sheet to the point on the right edge a distance x from the *bottom*, you'd end up with a mobius strip.

You might find this link helpful: https://mathcircle.berkeley.edu/sites/default/files/handouts/2018/topology_of_surfaces_2018_0.pdf

Edit: note equivalence relations don't have to have anything to do with group theory.