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• 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 5components shows that we have both patterns and apparent chaos.
Higher Homotopy Groups
• In topology call a circle S1, the one-sphere (lives in R2). Sn is the n-sphere (in Rn+1). • Another way of considering the fundamental group is as homotopy classes of maps S1 -> X. • To generalise this we consider maps Sn -> X. The homotopy classes of these maps forms the homotopy groups πn(X).
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 πi+n(Sn) = πi+n+1(Sn+1) for n > i+1. • We call this the i-th stable homotopy group denoted πis.
• 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.
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 tg
• 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.
• We can also show that πn(Sn) = 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:
• So if we take R3: 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.
• 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 Poincaréand 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.
• If we plot the 2-components where n vertical dots mean a Z2n factor we start to see some patterns.
• Look at i = 3. We can see 3 dots so have a Z8 factor. This makes sense because we have already seen that the π3s is Z24.
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!
Fundamental Group
• For the fundamental group π1(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. .
• These higher homotopy groups are hard to calculate but sometimes have interesting structure.
• Recall that Sk is a k-sphere and πn is homotopy classes of maps from Sn so πn(Sk) 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: πn(Sk) = 0 for n < k. This is basically because we have enough ‘wiggle room’ to deform any map to the constant map.