The First Homotopy

OVERVIEW
In this essay we will discuss The First Homotopy Group and covering spaces. We will sketch out how a covering space can be used to compute a second group: The Fundamental Group of the Circle.
In topology we consider mappings from one space to another. If there is a continuous transformation from one mapping function to the other, the maps are homotopic. The paths which these homotopic functions map into the space will be homotopic paths. So, a homotopy can be thought of as a continuous deformation of one path into another. The Fundamental Group, or First Homotopy Group, is a tool which can be used to distinguish different types of spaces by considering what types of homotopic paths with a single base-point are possible in a given space. A path with a base-point is one which begins and ends at the same point. The equivalence classes of this group are homotopy classes, so any paths homotopic to each other are representative of the same equivalence class. The operation of this group can be thought of as a combining of equivalence classes of paths, so one type of path is followed by another. For example, if one equivalence class is a homotopy class for loops around one specific hole in a two-holed plane and another equivalence class is the homotopy class for loops around the other hole, then the result of the binary operation on these two equivalence classes will be the homotopy class for loops that go around both holes. So, any representative paths from their respective equivalence classes will, when brought together under the defining operation, join to form a new loop representative of the equivalence class of the combined loop: one type of loop followed by the other. The identity of this group is the equivalence class of trivial paths, paths which can be contracted to the base-point. Clearly, the issue of whether a certain type of path is possible in a certain type of space is a matter which hinges upon characteristics of that space. For example, all loops with the same base-point in R^2 are homotopic. This is not the case if we puncture R^2.
Covering spaces can be used to calculate fundamental groups. Let’s say you have a continuous, onto function from one space to another. An open set in the space to which your function maps is evenly covered if the inverse image of that open set can be expressed as a union of disjoint sets. These can be thought of as pancakes, each of which have the same size and shape as the open set. The function maps all these pancakes onto the open set. The function’s mapping of any one of these pancakes onto the open set is homeomorphic. A homeomorphism is a continuous function between two topological spaces that has a continuous inverse. Homeomorphisms are mappings which preserve all topological properties of a given space. If every point in the whole space into which the function maps is a point in an open set which is evenly covered, then the function is a covering map and the space from which it maps is a covering space.
In the case of the circle, the Real number line can serve as a covering space, and the covering map is the function that maps the Real Line to the circle. This covering space can be used to compute the Fundamental Group of the Circle, The Infinite Cyclic Group; that is, we can show that the Fundamental Group of the Circle is isomorphic to the Additive Group of Integers. To do this we require something called a ‘lifting’. Let’s say we have a function from some space to the circle. To say that that function has a lifting to the Reals means that if the covering map from the Real Line to the circle is composed of the lifting of a given mapping to a path on the circle, the composition will be equal to that path on the circle. So, if f maps one loop around the circle, the lifting of f will map a path from o to 1 on the real line. If f is homotopic, the lifting of f will be homotopic in R, which is path connected; that is any two points can be connected by a path, so the path moves unimpeded. Let’s say f maps to one wind around the circle at a given rate, and g maps to one wind around the circle at different rate. It follows that f and g are homotopic, and their associated liftings are as well. Moreover, if we have homotopic paths, we have equivalence classes, equivalence classes for winds around the circle and equivalence classes for paths of liftings on the Real Line.
The equivalence classes for the circle are the homotopy classes for loops, or winds. For any number of winds around the circle, there is an equivalence class. The binary operation of this group is a ‘followed by’ operation: a loop of a certain number of winds is followed by another. When the equivalence class for three winds enters into the operation with the equivalence class for two winds the result is the equivalence class for five winds. The identity of this group is the equivalence class for paths that are homotopic to the mapping into the base-point on the circle (1,0). For example, a path which winds some partial distance around the circle and then winds back to the base-point would be a representative path. The path of the lift associated with this representative path would move from 0 towards 1 and then move back to 0. It should be understood that the inverse of the covering map takes us from a loop on the circle to a lifting on the Real Line. The loop on the circle is homeomorphic to the lifting on the Real Line. The inverse of the covering map gives us a homeomorphism between the equivalence class for the identity on the circle and the equivalence class for paths that map to 0 on the Real Line. Basically, we have an isomorphism between the Group for the Circle and the Additive Integers. The equivalence class for n number of winds can be mapped to the equivalence class for integer n. If n winds are counterclockwise, that maps to a negative integer, and the equivalence class for the idenity maps to equivalence class for 0. It can be shown that this is one to one, onto and homomorphic.

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