Undergrad How to think of S1 (circle) abstractly

[vancouver_water]

Im having a bit of trouble when it comes to what the abstract object S1 actually is. Often in a book they will mention a parametrization of the circle in the complex or real plane. But this requires embedding the circle in Euclidian space. How should one think of the object S1 without thinking of it as embedded in another space? I suppose it is much more difficult to actually calculate anything, such as the fundamental group, without a parametrization. Thoughts?

 

[jedishrfu]

Can you provide some context here? What course? What book? What problem?

When I think of parametrization, I think of a variable t and a vector function in 3D space <f(t), g(t), h(t)> that traces out a curve as t varies along a range of values.

 

[suremarc]

Well first you'd have to describe the circle as a topological space -- for example, you can construct it as the quotient space $\mathbb{R/Z}$, or as the one-point compactification of $\mathbb{R}$. Then you construct an atlas for the space. That way, you have an atlas for all spaces homeomorphic to your choice of construction, and the transition functions are preserved by homeomorphisms.

 

[fresh_42]

... or as unitary group $U(1,\mathbb{C})$ or orthogonal group $SO(2,\mathbb{R})$.

The central question will be: what do you mean by circle? Sure, the first thing that comes to mind is a graph produced with a compass and that requires a Euclidean sheet of something. I do not think that you can avoid it. A circle occurs simply too often and always embedded: orbits, design of something, graph. Even its linguistic use as in "running around in circles" requires a floor to run on. You can also object that the examples above, the topological spaces, come along with a parameterization, but then the answer is: you will have to describe it somehow, and such a description is necessarily closely related to an embedded version.

 

[suremarc]

... or as unitary group $U(1,\mathbb{C})$ or orthogonal group $SO(2,\mathbb{R})$.

D'oh. I guess the choice of description doesn't matter, if in the end you're looking at intrinsic properties.

 

[fresh_42]

D'oh. I guess the choice of description doesn't matter, if in the end you're looking at intrinsic properties.

Sure. My point was a different one: any description, definition if you like, necessarily carries all information of all other descriptions among which the parameterization as well as the embeddings are. So however a circle will be defined, the thingy with the compass is automatically also defined - at least, if we want to have the equidistance to its center or equivalent metric properties, in contrast to topological deformations. Even your descriptions as $\mathbb{R}/\mathbb{Z}$ or $\mathbb{P}(1,\mathbb{R})$ carry those symmetries, either by $\mathbb{Z}$ or by the Riemann projection. The word "circle" automatically includes all possible descriptions, one cannot strip it off, only disregard it.

 

[lavinia]

D'oh. I guess the choice of description doesn't matter, if in the end you're looking at intrinsic properties.

As others have pointed out, the description of the circle depends on how you view it. As $R/Z$ it is a group. As the 1 point compactification of the real line it is a topological space. It may also be seen as a Lie group or as a differenttable manifold. In each case the abstract definition is different.

As @fresh_42 said in post #4, the question is what do you mean by a circle. In mathematics, objects are defined through structures - e.g. a group law or a topology - and an idea of how to compare two structures - e.g. a group homomorphism or a continuous map. Objects together with a method of comparison are called a category. Two objects are considered identical in that category if there is an isomorphism - an invertible comparison - between them. So in the category whose objects are abelian groups and whose comparisons - abstractly called morphisms in category theory - are group homomorphisms, two objects are the same if there is group isomorphism between them.

One defines an isomorphism between the group $R/Z$ and the unit complex numbers by exponentiating an arbitrary representative of each coset. So the coset $x+Z$ is mapped to $e^{ix}$. This map shows that $R/Z$ and the unit complex numbers are the same in the category of abelian groups and group homomorphisms.

To show that the unit complex numbers and the 1 point compactification of the real line are the same in the category of topological spaces and continuous maps one would need to find a homeomorphism between them.

Just to drive the point home that morphisms are necessary, the seven dimensional sphere as a topological manifold can represent different differentiable manfolds. The same topological seven sphere can be given the structure of a differentiable manifold in more than one way. This means that while the underlying topological manifolds are homeomorphic they are not diffeomorphic.

Im having a bit of trouble when it comes to what the abstract object S1 actually is. Often in a book they will mention a parametrization of the circle in the complex or real plane. But this requires embedding the circle in Euclidian space. How should one think of the object S1 without thinking of it as embedded in another space? I suppose it is much more difficult to actually calculate anything, such as the fundamental group, without a parametrization. Thoughts?

If you think if the circle as the quotient $R/Z$ of two topological groups $R$ and $Z$ then you can calculate the fundamental group using theorems on covering spaces. I don't see why one would need an explicit parameterization.

If you describe the circle as two overlapping intervals whose intersection is two disjoint intervals, then. you can calculate its homology groups using a Meyer-Vietoris sequence. No parameterization is needed.

 

[WWGD]

As many pointed out, the circle has many different types of structure, albeit with some inter-connections, so it seems wise to choose which type of structure to address. Same goes for most Mathematical entities, e.g., Real numbers are a complete metric/ordered, separable (densely-)space, a field, a vector space, etc. You may think of the analogy with a person who may be, e.g., a parent, a son, a professional, etc.

 

[mathwonk]

I believe a circle is characterized intrinsically as the unique compact connected one dimensional real manifold. But just try computing anything at all about it from only that description.

 

[WWGD]

I believe a circle is characterized intrinsically as the unique compact connected one dimensional real manifold. But just try computing anything at all about it from only that description.

An interval?

 

[mathwonk]

"manifold" means always manifold without boundary. In math at least, the terms are either manifold, or manifold with boundary.

this seems to be standard:
https://en.wikipedia.org/wiki/Manifold

perhaps it is one the many differences in language between mathematicians and physicists.

 

[WWGD]

"manifold" means always manifold without boundary. In math at least, the terms are either manifold, or manifold with boundary.

this seems to be standard:
https://en.wikipedia.org/wiki/Manifold

perhaps it is one the many differences in language between mathematicians and physicists.

No, it's not, my bad, actually.

 

[lavinia]

I believe a circle is characterized intrinsically as the unique compact connected one dimensional real manifold. But just try computing anything at all about it from only that description.

How do you compute anything from calling the circle the unit length complex numbers? Don't you need a parameter or something else to do something with that description?

 

[fresh_42]

How do you compute anything from calling the circle the unit length complex numbers? Don't you need a parameter or something else to do something with that description?

With the unit length we already have equidistance to the origin, which is pretty close to the original definition of a circle. The compass then yields the parameterization.

 

[mathwonk]

thank you putting me on notice lavinia. well i guess actually if you know the circle is the unique one dimensional compact connected manifold, you can compute everything about it quite easily. I.e. one can observe that the real locus of points satisfying x^2 + y^2 = 1 is such a manifold, and hence by uniqueness must be the circle. then you can compute its universal cover, and anything else you want, such as the fundamental group.../

I was actually thinking of the challenge of proving that uniqueness, but even that is perhaps easy enough. To be honest I just was recalling that it had seemed clever several decades ago in milnor's litte book. I have not thought of it since. So perhaps I meant that it might be interesting to begin just from the fact that one has a compact connected one manifold, and try to compute the fundamental group (without knowing uniqueness).

I suppose one should begin by finding a covering of it by R.

And @WWGD: I have to come off my high horse of speaking so confidently about what happens "in mathemetics", with only wikipedia as my authority, since the great Milnor gives exactly your usage on page 55, line 9, of his wonderful book "Topology from the differentiable viewpoint", where he states there are exactly 4 connected "one manifolds", namely the circle, (0,1), [0,1], and [0,1). You can't be in better company than that. So thank you very much for that clarification.

 

[WWGD]

And @WWGD: I have to come off my high horse of speaking so confidently about what happens "in mathemetics", with only wikipedia as my authority, since the great Milnor gives exactly your usage on page 55, line 9, of his wonderful book "Topology from the differentiable viewpoint", where he states there are exactly 4 connected "one manifolds", namely the circle, (0,1), [0,1], and [0,1). You can't be in better company than that. So thank you very much for that clarification.

No, actually, seriously, I was embarrassed for my inability to remember something like this. My bad, Wonk :).

 

[lavinia]

thank you putting me on notice lavinia. well i guess actually if you know the circle is the unique one dimensional compact connected manifold, you can compute everything about it quite easily. I.e. one can observe that the real locus of points satisfying x^2 + y^2 = 1 is such a manifold, and hence by uniqueness must be the circle. then you can compute its universal cover, and anything else you want, such as the fundamental group.../

I was actually thinking of the challenge of proving that uniqueness, but even that is perhaps easy enough. To be honest I just was recalling that it had seemed clever several decades ago in milnor's litte book. I have not thought of it since. So perhaps I meant that it might be interesting to begin just from the fact that one has a compact connected one manifold, and try to compute the fundamental group (without knowing uniqueness).

The OP was asking how one could work with a definition of the the circle without a parameterization or an embedding in the plane.

I wanted to show that one can construct the circle in different ways that allow computation without parameterization. But defining the circle as the unique closed 1 manifold does not construct it. So while it is non-parametric, I didn't think of it as an alternative to the question.

The 1 point compactification of the real line constructs the circle as a topological space and this allows computation of its homology and fundamental group without parameters or embedding.

The $R/Z$ definition is implicitly a parameterization but still gets the circle out of the plane and into the category of topological groups.

The definition as the unique closed 1 manifold brought out the cool point that there are definitions that are not constructive but rather follow from existence theorems and uniqueness theorems. In those cases, it may well be more difficult or impossible to do computations.

 

[mathwonk]

lets see: just knowing it is a compact one manifold, means that there is a finite cover by intervals, and using connectedness and shrinking the intervals until they overlap "simply", one should be able to conclude there is a cover by R, hence represents the circle as R/Z., one of your definitions, so maybe these are not so far apart... ?

 

[lavinia]

lets see: just knowing it is a compact one manifold, means that there is a finite cover by intervals, and using connectedness and shrinking the intervals until they overlap "simply", one should be able to conclude there is a cover by R, hence represents the circle as R/Z., one of your definitions, so maybe these are not so far apart... ?

Intuitively that would seem to work. Cover the circle by charts chosen to be homeomorphs of open intervals then adjust a finite subcover so that each pair intersects in a common subinterval or has empty intersection. Then by taking unions reduce this to two intervals whose intersection is two disjoint subintervals. Then use Meyer-Vietoris for homology and Van Kampen's for the fundamental group. The $R/Z$ route would seem to work also.