Algebraic Topology - Homotopy Groups of Spheres Martin Leslie, University of Queensland

Topology What is topology? The study of shape and continuous maps between them. We are interested in properties like connectivity and number of holes, not in distance. A topologist can’t tell the difference between a coffee cup and a donut!

A piece of string Consider a piece of string. No matter how we wiggle or stretch it it stays the same topologically (we say the two positions are homeomorphic). However if we tie it up to make it into a circle or cut it into two then it is different topologically.

Topological spaces are homeomorphic if they can be deformed to each other just by stretching (compressing, rotating), not by tearing or attaching. Topology is sometimes called rubber sheet geometry. But sometimes it is hard to prove whether 2 spaces are homeomorphic or not. We need more tools. So Poincaré and lots of other smart people invented …

Algebraic Topology We assign to each topological space some kind of algebraic invariant. If 2 spaces have different invariants then we know that they aren’t homeomorphic - but having the same invariant doesn’t necessarily mean that they are homeomorphic.

Fundamental Group For the fundamental group π 1 (X) imagine directed loops inside the space, X, you are interested in. Two loops are homotopic if one can be continuously deformed to the other. The fundamental group is homotopy classes of loops. The group operation is composition: first travel a loop from one class then travel a loop from the next class..

So if we take R 3 : the space around us. Every loop can be deformed to a point so the fundamental group is trivial. On the other hand consider the surface of a torus (donut). A loop that goes around once can’t be deformed to a loop that goes around twice and both can’t be deformed to a point. The fundamental group of a torus is actually Z x Z.

Higher Homotopy Groups In topology call a circle S 1, the one-sphere (lives in R 2). S n is the n-sphere (in R n+1). Another way of considering the fundamental group is as homotopy classes of maps S 1 -> X. To generalise this we consider maps S n -> X. The homotopy classes of these maps forms the homotopy groups π n (X).

These higher homotopy groups are hard to calculate but sometimes have interesting structure.

Recall that S k is a k-sphere and π n is homotopy classes of maps from S n so π n (S k ) is made up of the different ways of mapping an n-sphere onto a k-sphere. With this in mind can see our first result: π n (S k ) = 0 for n < k. This is basically because we have enough ‘wiggle room’ to deform any map to the constant map.

We can also show that π n (S n ) = Z for all n. For n = 1 we can map a circle around itself any number of times (negative is backwards). But above this diagonal we have chaos:

Stable homotopy groups But we do have some stability on the diagonal if you go far enough across. We can prove Freudenthal’s suspension theorem which says π i+n (S n ) = π i+n+1 (S n+1 ) for n > i+1. We call this the i-th stable homotopy group denoted π i s.

The i-th stable homotopy group is one of the fundamental objects in algebraic topology. Complete calculations are known up to about i=60. Once again this seems like chaos. But there are patterns if we consider the p-components: the subgroup of elements of order a power of p.

If we plot the 2-components where n vertical dots mean a Z 2 n factor we start to see some patterns. Look at i = 3. We can see 3 dots so have a Z 8 factor. This makes sense because we have already seen that the π 3 s is Z 24.

The diagonal and horizontal lines give us information about composition of maps. We don’t have enough time to talk much about these diagrams but a look at the 5- components shows that we have both patterns and apparent chaos.

Conclusion This area is still open for research: the last diagram had a few question marks, it hasn’t been proven that these maps exist. Lots of patterns still to be explored.