Recent content by lavinia

After continuous deformation it is a regular 2d torus. I had trouble describing the situation in a clear way even though the picture is clear, so I left it as a problem. If I knew how to post drawings this would be easier to see but I don't. My apologies. Here is another stab at it. Given a...

Another way to think about the linking of two fibers in the Hopf fibration is to look at the topology of their complement in S^3. Take for instance, the complement of the equator(the unit circle in the xy-plane) and the fiber that passes through the north pole of S^3. Since the north pole is...

yes I just wanted to emphasiize that by sliding the linked pair of fibers along the rays emanating from the north pole of the three sphere the linked fibers are continuously moved into R^3. Stereograaphic projection by itself without this sliding is just a mapping and is not a movement of the...

Here is a way of doing it using the Hopf fibration. Slide the two linked fibers along the rays emananting from the north pole of the 3 sphere until they reach R^3. Then raise one circle up into R^4 so that its fourth cooridinate is not zero. (This all occurs in R^4)

Right. BTW: Your answer proves that the Klein bottle can be embedded in R^4. So for two linked fibers in 3 the 3 sphere, is the worry that they would have to crossover each other just to get outnof the 3 sphere into R^4?

OK. So how would you move two linked loops in R^3 off of each other into R^4 without them crossing over each other? How would you move a point in the northern hemisphere of a sphere in R^3 to the southern hemisphere without crossing the equator if the point is allowed to move anywhere in R^3...

As an example, consider the equator of the three sphere, the circle in S^3 whose third and fourth coordinates are both zero. It divides the equatorial 2 sphere, the sphere whose fourth coordinate is zero, into two hemispheres and it is the boundary of both. Any fiber of the Hopf fibration...

What are your thoughts on this?

@Hornbein Intuitively, in Euclidean 3 space, two non-intersecting closed loops are linked if they cannot be separated without crossing over each other even if they can be bent or stretched in trying to do so. Similarly, on the 3 sphere two fibers are linked if they cannot be separated on the...

A way to construct topological manifolds from closed subsets of Euclidean space is through triangulations. For example, one can imagine triangulating a surface then cutting out each triangular region to obtain a set of solid triangles. Each solid triangle is homeomorphic to a closed disk in the...

@mathwonk Thank you for this wonderful reply. It inspires me to enter into this incrediblle world.

@cianfa72 This thought came up while thinking about your thread. A topological 7 spere is two 7 disks that are glued together along their 6 sphere boundaries. Any such gluing creates a topological 7 sphere. But the gluing map still matters since the topological 7 sphere can have more than one...

In the category of topological spaces and homeomorphisms, one can define those manifolds which are made from gluing two disks along their boundaries. It turns out that they are all homeomorphic to the standard sphere. Once this is proved, the topological sphere can be defined as two disks pasted...

@cianfa72 Your thoughts brought to mind a few considerations that I thought would be of interest to you. You start out by saying that the definition of the 2-sphere is the quotient space of two disks with their boundary circles glued together. To me this definition is incomplete because it...

I thought it might be relevant to describe the topology of the real line without any use of the Euclidean metric or any other metric for that matter and without any algebraic structure. A little web research revealed several ways to characterize the topology of the real line or equivalently any...