Do you really mean S¹?


May 6, 2011

This is a follow up post to my previous post on \(\mathbb{R}^n\) . Mathematicians will often write \(S^1\) without being clear of the context and structure associated with it.

To a Euclidean geometer \(S^1\) means a circle – a maximal set of points equidistant from a given point. All circles are equivalent in the sense that they can be made equal by a translation and a scaling about the centre.

In a two dimensional inner product space \(S^1\) typically means the set of all points with norm \(1\). A circle is more generally the set of all points with norm \(r\) for some real number \(r\) and is related to \(S^1\) by a scaling transformation.

To a group theorist \(S^1\) would mean the one dimensional orthogonal group – the group of all transformations in the plane.

To a complex analyst \(S^1\) could mean either the set of points in \(\mathbb{C}\) with length 1 or it could be the group of linear transformations associated with multiplication by elements of this set. (These are quite different – in the set \(1\) has no special meaning but in the group it corresponds to the identity).

To a topologist \(S^1\) means anything homeomorphic to a Euclidean circle – so includes ellipses, polygons, simple closed curves,…

To a differential geometer \(S^1\) means anything diffeomorphic to a Euclidean circle, which doesn’t include “most” things a topologist means.

A set theorist wouldn’t call it \(S^1\) , but to her it would be any set with the same cardinality as the real numbers.

This is a major theme of category theory – it’s not only the objects that matter but also the maps that preserve them – whether it be affine transformations, orthogonal transformations, group homomorphisms, group homomorphisms, homeomorphisms, diffeomorphisms or bijections. So if you must write \(S^1\) to represent a structure at least make sure it is clear which category you are working in – i.e. which maps preserve the structure.