In Basics of Topology and Continuous Functions, we discussed how to put an “arrangement” or “organization” on sets using the concept of a topology. We also mentioned the concept of a continuous function, which expresses the idea of a mapping between sets that in some way preserves the arrangement or organization.
What we think of as a “shape” – a circle, a sphere, a cube, or some other complicated shape, like the shape of a living organism – can be thought of as the set of the points that make it up. Using Cartesian coordinates on a plane, for example, a circle with radius is the set of all points whose coordinates are pairs of numbers that satisfy the equation . Similarly, in a three-dimensional space, a sphere with radius is the set of points whose coordinates are triples of numbers that satisfy the equation . The plane itself and the three-dimensional space, in a sense, are also shapes, or geometric objects.
Since shapes are sets, we can put a topology on them. To accomplish this we usually use the concept of a metric, which is a way of expressing an idea of a distance between points. The so-called metric topology then puts an arrangement on this set by grouping together points which are close together into “neighborhoods”, a concept we have also already discussed. We can then think of a continuous function between shapes as a function that sends points close together on one shape to points that are also close together on the other shape.
We also defined a homeomorphism as a continuous function with a continuous inverse. When a function has an inverse, it is also called a bijective function, or a bijection. A bijection between two sets expresses the idea that every element of one set can be paired with exactly one element from the other set, and vice-versa, with no elements of either set left unpaired. If a homeomorphism exists between two topological spaces, we say that the two topological spaces are homeomorphic.
All this means that a homeomorphism can then be thought of as a deformation between two shapes, without any gluing or tearing involved, since gluing or tearing would ruin our arrangement of neighborhoods of points on the shapes. Simply stretching or compressing the shape would change distances between points, but the points would still belong to the same neighborhood. If we had, say, a class of students, and we make them transfer seats to leave a space of one empty chair between each student, the distance between them would change, but if Bob was seated between Alice and Charlie before, he would still be seated in between them after. His closest neighbors would still remain his closest neighbors, although they are a little less closer to each other than they were before.
Related to the notion of homeomorphism are the notions of homotopy and homotopy equivalence. A homotopy can be thought of as a deformation between functions. First one needs an “interval”; formally this is chosen to be the closed interval . This can be thought of as the “time” for the deformation to take place: We have a function which is at the “start time” , at the “end time” and as the “time” runs from to the function is “deformed” continuously from to . We say that there is a homotopy from to , or that and are homotopic.
If we have a continuous function from the topological space to the topological space , and another continuous function from to , we can form the function from to by composition. Similarly we can form we can form the function from to . If the function is homotopic to the identity function (the function that sends every element to itself) on , and if the function is homotopic to the identity function on , then we say that and are homotopy equivalences, and that and are homotopy equivalent.
Topological spaces that are homeomorphic are automatically homotopy equivalent. However, it is possible for topological spaces to be homotopy equivalent but not homeomorphic. Examples of spaces that are homotopy equivalent but not homeomorphic can be found in the references listed at the end of this post.
The continuous functions from to which are homotopic to each other can be considered equivalent (see also the discussion in Modular Arithmetic and Quotient Sets), and they can form equivalence classes. The set of all equivalence classes of continuous functions from to is then denoted .
Of particular importance is the set formed when is the n-dimensional sphere, written . In this notation the ordinary circle is , and the ordinary sphere is . We are used to thinking of the circle as being embedded in a two-dimensional space and the sphere as being embedded in a three-dimensional space, but since we are only looking at the “surface” of these shapes and not their interior, their dimensions are reduced by one from the spaces they are embedded in. There is also the zero-dimensional sphere ; this is just a set with two points. One can see this by considering the equations for the circle and sphere above; we go down one variable and set the zero-dimensional sphere of radius to be the points on the line which satisfy the equation , which gives us the two points and .
After fixing “basepoints” on and , the set forms a group (see Groups). It is called the n-th homotopy group of , and is written . The first homotopy group , , has a special name; it is called the fundamental group of . The fundamental group of a space can be thought of as the group of equivalence classes of loops on which begin and end at the chosen basepoint, with loops considered equivalent if they can be deformed into each other. The identity element of the fundamental group is the equivalence class of the loops on which can be deformed into a point. There is also the zeroth homotopy group, ; although we do not have a special name for this homotopy group, it serves an important role, since it keeps track of how many “pieces” make up our space .
A space for which is the trivial group (the group with only one element, which is the identity element) is called path connected. When a space is path connected, any two points and may be connected by a path, which is a continuous function from the interval to the space for which and . In a way, we may think of a path connected space as being made up of only one “piece”.
A path connected space for which is also the trivial group is called simply connected. A simply connected space is one in which all loops can be deformed into a point. A plane is a simply connected space; however if we punch a “hole” in it it is no longer simply connected. Similarly, the surface of a sphere is also a simply connected space; the surface of a donut, called a torus, however, is not simply connected.
In a path connected space, any two points can be connected by a path. Similarly, in a simply connected space, any two paths with the same endpoints can be connected by a “path of paths”. More on this point of view of homotopy theory can be found in Vladimir Voevodsky’s lecture at the 2002 Clay Mathematics Institute annual meeting which is listed among the references below. The lecture also discusses some basic category theory and algebraic geometry, using fairly intuitive language, without sacrificing the important ideas behind both subjects.
Homotopy theory, which we have barely scratched the surface of in this post, is just one part of the subject called algebraic topology. The name of the subject comes from the use of concepts from abstract algebra, such as groups, to study topological spaces. Other parts of the subject are homology theory and cohomology theory. All three are related to each other and have found applications in other branches of mathematics aside from studying topological spaces. They have been used, for example, to study aspects of linear algebra, via the subjects of homological algebra and algebraic k-theory. In physics, topological concepts applied to the study of phase transitions in condensed matter physics earned the trio of David J. Thouless, F. Duncan M. Haldane, and J. Michael Kosterlitz the 2016 Nobel Prize in Physics. There are also many promising applications of topology to more fundamental aspects of theoretical physics.
More on homotopy theory and algebraic topology can be found in the books Algebraic Topology by Allen Hatcher and More Concise Algebraic Topology by J. P. May, both freely and legally available online and also listed among the references below.