Last Friday, the 18th of October, I talked about homotopy and the fundamental group. As I heard about homotopy type theory, I thought that reviewing what I know of the geometric part could be useful. So I prepared this talk, using Allen Hatcher’s free textbook.
Consider a continuous line on the whiteboard, from point to point , and another continuous line from point to point . Think of a bijection between the two lines: we may think of it as a family of trajectories from a point on the first line to a point of the second line, so that at time we are at the beginning of each trajectory, and at time we are at the end. But we may want to add another requirement: that, at each time , the collection of points that have been reached at time is itself a continuous line on the whiteboard! This is the idea at the base of the geometric concept of homotopy.
Recall that a path in a topological space is a continuous function from the unit interval to . Call pathwise connected if for any two points there exists a path from to , i.e., such that and . If is a path in and is continuous, then is a path in : thus, a continuous image of a pathwise connected space is pathwise connected.The -dimensional unit balls and -dimensional spheres are pathwise connected.
If are paths and , their concatenation (read: “ then ”) is defined by for and for : in other words, we run across both paths in sequence, at twice the speed.
Definition 1. Let and be topological spaces and let be continuous functions. A homotopy from to is a continuous function such that for every and . Two functions are homotopic, written , if there exists a homotopy from to .
It can be easily check that homotopy is an equivalence relation.
Definition 2. A homotopy equivalence between two pathwise connected spaces is a pair of continuous functions , such that and . and are homotopy equivalent if there exists a homotopy equivalence between them.
For example, the zero function and the inclusion form a homotopy equivalence, with and being a homotopy from and . Spaces that are homotopy equivalent to a point are called contractible.
The importance of homotopy equivalence, is that it preserves several algebraic invariants, i.e., algebraic objects associated to topological spaces so that homeomorphic spaces have same invariants. The first such invariant is the fundamental group, which we will define in the next paragraphs.
Let be a path connected space and : consider the family of paths from to . We want to study the homotopy classes of such paths, with an additional restriction on the homotopies to be endpoint-fixing: and for every , i.e., at each time the function must be a path from to , rather than any two arbitrary points of . This restriction on the homotopies makes them congruential with respect to concatenation: if , , and , then . We may thus think about the 2-category whose objects are the points of a topological space, morphisms are the paths from point to point, and morphisms of morphisms are endpoint-fixing homotopies from path to path.
There is more! Although and are not the same, the only difference is in the velocities while running on the different tracks: and it is easy to see how such change of velocity can be done continuously, so that . For the same reason, if is the constant path at , then is clearly homotopic to whatever is; similarly, . (Paths with initial point equal to final point are called loops.) Finally, if we define the reverse of the path by , then and . (Think about the original point having a lazy friend, who stops somewhere to rest and waits for the original point to come back to go back together to .) This justifies
Definition 3. Let be a pathwise connected space and let . The fundamental group of based at is the group whose elements are the homotopy classes of loops at , product is defined by , identity is , and inverse is defined by .
Suppose and form a homotopy equivalence. Let be a loop in based on : then is well defined, and with some computation it turns out to be a group isomorphism. This is the reason why we sometimes talk about , without specifying .
As every loop in in is homotopic to a constant loop via the homotopy , is the trivial group: in this case, when , we say that is simply connected. For , is also simply connected, as intuitively we may always move the path in a bounded region which contains the image of the entire homotopy, and does not contain the origin. On the other hand, the same intuition tells us that it should not be possible to continuously transform a loop in around the origin into a loop in which does not “surround” the origin, without crossing the origin sometimes: this is confirmed by
Theorem 1. .
The proof of Theorem 1 is based on the intuitive idea that we can homomorphically identify the number with the homotopy class of windings based on , counterclockwise if and clockwise if . To prove that such homomorphism is bijective, however, would go beyond the scope of this post. On the other hand, is simply connected for : think to what happens when we tie an elastic rubber band around a tennis ball.
Definition 4. A retraction of a topological space onto a subspace is a continuous function that fixes every point of ; is a retract of if there exists a retraction of onto . A deformation retraction of a topological space onto a subspace is a homotopy from the identity of to a retraction of onto ; is a deformation retract of if there exists a deformation retraction of onto .
is a retract of , a retraction being : it is also a deformation retract, as is a homotopy from the identity of to .
Theorem 2. If is a retract of then is isomorphic to a subgroup of . If is a deformation retract of then is isomorphic to .
It follows from Theorems 1 and 2 that is not a retract of : from which, in turn, follows
Brouwer’s fixed point theorem in dimension 2. Every continuous function from to itself has a fixed point.
Proof: If has no fixed points, then the half-line from through is well defined for every . Let be the intersection of such half line with : then is a continuous function from to such that for every , i.e., a retraction of onto —which is impossible.
There are some ways to compute the fundamental group of a space, given the fundamental groups of other spaces. For example, is isomorphic to the direct product of and , because the loops in based on are precisely the products of the loops in based on and the loops in based on . Another important tool is provided by
van Kampen’s theorem. Let and be pathwise connected spaces such that is pathwise connected and nonempty. Let . Then is the pushout of the diagram , where the arrows are induced by the inclusions.
As an application of van Kampen’s theorem, suppose and are copies of , and that is a single point. Then is free on two generators, because is the pushout of , where is the trivial group that only contains the identity element. More in general, the pushout of is for every two positive integers : thus, the fundamental group of a bouquet of circles—i.e., copies of , joined at a single point —is free on generators.
As a final note, consider a continuous function between two pathwise connected spaces. If is a loop in with base point , then is a loop in with base point : it is also easy to check that and that . From this follows that the function defined by is a group homomorphism: therefore, the application that maps every into the corresponding , is the component on the arrows of a functor from the category of pointed pathwise connected topological spaces with basepoint-preserving continuous function to the category of groups with group homomorphisms, whose component on objects maps every pathwise connected space to its fundamental group.