Higher Homotopy Groups: Why πnpX; x0q; n ¡ 1 is Abelian Nicholas Holfester 1 Introduction and Motivation The fundamental group, π1pX; x0q, is a powerful tool of algebraic topology used to study topological spaces by studying how loops behave within them. Higher homotopy groups, which will be our main object of study, are the natural gen- eralization of the fundamental group. While π1pX; x0q considers how loops can live in a space up to based ho- motopy, higher homotopy groups more generally consider how closed surfaces (maps from the n-sphere) can be mapped into spaces. To be explicit Definition 1. A higher homotopy group, πnpX; x0q, is the set of based homotopy n classes of based maps γ : pS ; s0q Ñ pX; x0q. As in the case of the fundamental group, higher homotopy groups are also groups. Proposition 1. πnpX; x0q equipped with the binary operation defined on equiv- n n n alence classes as γ ` η “ γ_η ˝ p where p is the quotient map p : S Ñ S _ S n n which collapses a great circle and γ_η is the map on S _ S with γ acting on one sphere and η acting on the other. The group operation can be depicted visually as The proof that this operation is well defined and satisfies the group ax- ioms follows closely to the proof that π1pX; x0q is a group and can be found in Hatcher.1 This definition may appear to be a logically sound enough way of generalizing the fundamental group, but one may now naturally ask, \Are these worthwhile objects to study?". The answer to this question is a wholehearted yes. 1Hatcher Algebraic Topology 340. 1 Higher homotopy groups have pragmatic uses in characterizing various topo- logical spaces while at the same time offer a satisfying arena to study the natural intertwining of the fields of algebra and topology. For an explicit example of a use of higher homotopy groups in studying spaces, consider the following theorem, known as Whitehead's Theorem: Theorem 2 (Whitehead). Let X and Y be connected CW complexes. If a map f : X Ñ Y induces isomorphisms f˚ : πnpXq Ñ πnpY q for all n, then f is a homotopy equivalence. To fully appreciate the theorem, one should have an understanding of CW complexes.2 However, even without an in-depth understanding of CW com- plexes, the power of this theorem is striking. By knowing how maps act on higher homotopy groups, one immediately receives a significant amount of in- formation about the space in question. This is particularly powerful because many spaces studied in algebraic topology are homotopy equivalent to a CW complex so therefore this theorem has wide applicability, highlighting the prac- tical importance of understanding higher homotopy groups.3 Despite their seemingly simple definition, higher homotopy are often com- plicated objects that can be extremely difficult to compute. For a taste of how complex these objects can be conceptually, consider the following fact 2 Fact 1. π3pS q is non trivial. This group is generated by what is known as the Hopf fibration4 and the result shows that there is a way of wrapping S3 around S2 such that it is not null- homotopic, a fact that may be rather un-intuitive at first thought considering 1 that π2pS q is trivial. The complexity of higher homotopy groups is also highlighted by comparing them with homology groups HnpXq, another algebraic tool for studying topo- logical spaces. While we won't discuss the details of homology groups here, they have the nice property that HnpXq “ 0 when n ¡ dimpXq, \capping" their complexity in a sense.5 This is clearly not the case for higher homotopy groups, with the example above acting as a counter example. Therefore, in some sense higher homotopy groups capture more complexity than homology groups (which are already fairly complex in their own right). Because of this apparent complexity of higher homotopy groups, the fol- lowing theorem, which will be our primarily goal to prove in this paper, is surprising. Theorem 3. Let pX; x0q be a based space with basepoint x0. For n ¡ 1, πnpX; x0q is abelian. 2Hatcher offers a good discussion of CW complexes early on in the book Algebraic Topology, Chapter 0. 3For a non-example, look up \Postnikov Towers". 4See Hatcher Algebraic Topology 380, example 4.51. 5See Hatcher Algebraic Topology Chapter 3 for a full discussion of homology. 2 This result is perhaps even more surprising after recalling the fact that this is not true for fundamental groups. Take for example the twice-punctured plane Czt´1; 1u which is homotopy equivalent to a figure eight. Consider the two loops drawn below in black. When both are placed around opposite \holes", concatenating the loops in different orders produces loops that are not based-homotopic. If we take red as going first followed by blue, no matter how one twists the loops (while preserving the basepoint) it is impossible to change one loop into the other. A failed attempt at trying to homotope between both loops is depicted by the final drawing. This demonstrates that π1pCzt´1; 1u; 0q is non-abelian. The result is made rigorous quite simply by appealing to Seifert-van Kampen which gives the explicit result that π1pCzt´1; 1u; 0q – F2, the free group on two generators.6 By definition a free group has no relations so, in some sense, this fundamental group is actually the furthest from abelian you can get. So while one may anticipate higher homotopy groups to be more complex than the fundamental group (which they certainly are computationally), they are in some sense algebraically simpler. This intriguing result is our main goal of proof in this paper. We will study this result using two different approaches. Our first approach will use a result known as Eckmann-Hilton and will highlight the practice of algebra and topology co-informing each other. Our second approach will be slightly more topological in nature as we introduce H-spaces and Loop-spaces which are useful general tools of study that will also provide additional \geo- metric" intuition for what is actually happening in the proof. 6This was a result proved in class. 3 2 The Eckmann-Hilton Argument Our first look at the proof that higher homotopy groups are abelian follows from a rather satisfying look at how algebra and topology co-inform one another. We first introduce the following algebraic theorem that will lead us to our desired result. Theorem 4 (Eckmann-Hilton). Let X be a set equipped with two binary op- erations ¨ and ˚ such that ¨ has identity e and ˚ has identity e˚ and the two operations satisfy the following relation pa ¨ bq ˚ pc ¨ dq “ pa ˚ cq ¨ pb ˚ dq a; b; c; d P X then ¨ and ˚ are commutative and are in fact the same operation. Before we see the proof, lets take a moment to see whats going on in this definition. Say we have a set X with a binary operation ¨ which has an identity element named e (i.e. pX; ¨q is a unital magma). Now suppose we give our set another binary operation ˚ : X ˆX Ñ X with identity element e˚ where we require that ˚ is a homomorphism from the direct product pX ˆ X; ¨q to pX; ¨q.7 It's fairly easy to observe that this gives the exact relationship between binary operations given in the statement of Eckmann-Hilton. For ease, we temporarily denote a ˚ b explicitly as µpa; bq. Then µppa; cq ¨ pb; dqq “ µpa; cq ¨ µpb; dq ðñ pa ¨ bq ˚ pc ¨ dq “ pa ˚ cq ¨ pb ˚ dq This construction is analogous to the perhaps more familiar construction of topological groups. When building topological groups, we start with a topologi- cal space pX; τq and give it a binary operation that respects its underlying topo- logical structure by requiring it be continuous. In the same way, here we have a unital magma and we give it an additional binary operation which respects the original structure by requiring the new operation be a homomorphism. So, to round out the analogy, rather than having a group within a topological space, we have a unital magma within a unital magma. We now look at the actual proof of the statement. Proof. We first show that both identity elements are the same. Take pe ¨ e˚q ˚ pe˚ ¨ eq “ pe ˚ e˚q ¨ pe˚ ˚ eq ùñ e˚ ˚ e˚ “ e ¨ e ùñ e˚ “ e We can now complete the proof using the following clock diagram, which tells us that a ¨ b “ b ¨ a and that a ¨ b “ a ˚ b. To read the diagram, start at the top with the element pe˚ ˚ aq ¨ pb ˚ e˚q and progress clockwise by repeatedly using the assumed relation between ¨ and ˚ and the fact that e “ e˚. Each step along the clock is a statement of equality and the right and left sides shows us that a¨b “ a˚b and b˚a “ b¨a respectively, 7 Note that we don't require e “ e˚ a priori. 4 showing that the operations are equal. The top and bottom respectively give us that b ¨ a “ a ¨ b and b ˚ a “ a ˚ b, showing that both operations are in fact commutative. This result tells us that if we have a set with two compatible binary opera- tions, then we immediately get substantial extra structure over our set, namely commutativity. Leveraging this result, if we can show that πnpX; x0q; n ¡ 1 has two binary structures which satisfy this relationship, we get that it is abelian for free. In order to find this operation, it is more fruitful to think of maps from In n which are constant on BI , the set of elements in In such that one coordinate is 0 or 1, rather than maps from Sn.