Higher Homotopy Groups: Why Πnpx, X0q, N Ą 1 Is Abelian

Home , Algebraic topology, Homotopy group

Higher Homotopy Groups: Why πnpX, x0q, n ą 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 homotopy, 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 topological 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 complexes, the power of this theorem is striking. By knowing how maps act on higher homotopy groups, one immediately receives a significant amount of information 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 topological spaces. While we won’t discuss the details of homology groups here, they have the nice property that HnpXq “ 0 when n ą 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 following 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 ą 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 operations ¨ 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 topological space pX, τq and give it a binary operation that respects its underlying topological 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 operations, then we immediately get substantial extra structure over our set, namely commutativity. Leveraging this result, if we can show that πnpX, x0q, n ą 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. As in the case of the fundamental group, maps from Sn contain the same data as maps from In which are constant on BIn, so we may consider them interchangeably. n n When working with maps γ : pI , BI q Ñ pX, x0q, the group operation on πnpX, x0q as speciﬁed earlier becomes

1 fp2s1, s2, ..., snq 0 ď s1 ď 2 pf ` gqpsq “ 1 #gp2s1 ´ 1, s2, ..., snq 2 ď s1 ď 1 One can immediately observe, however, that this deﬁnition is arbitrary in the case n ą 1. We could have just as easily chosen to concatenate along a coordinate other than s1.

5 For n ą 1, consider the alternate operation `1 deﬁned by

1 1 fps1, 2s2, ..., snq 0 ď s2 ď 2 pf ` gqpsq “ 1 #gps1, 2s2 ´ 1, ..., snq 2 ď s2 ď 1

Proposition 5. pf ` gq `1 ph ` kq “ pf `1 hq ` pg `1 kq. This is easily seen by simply plugging through the deﬁnitions. It is visually seen by the following picture with ` shown as left/right concatenation and `1 shown as top/bottom concatenation.

Corollary 6. πn is abelian for n ą 1. Proof. The operation `1 is well deﬁned for n ą 1, therefore by Eckmann-Hilton πnpX, x0q is abelian for n ą 1. This approach was nice and clean and a great example of algebra informing topology. However, we could have also approached this problem in such a way that topological considerations would have informed this algebraic result. Con- sider the ﬁgure below which comes from Hatcher’s Algebraic Topology. This diagram gives a topological proof as to why πn should be abelian.

We start with f `g. Because f and g are constant on their boundaries we are able to homotope both to smaller domains where they are now surrounded by an “ocean” of our basepoint x0 (drawn in blue). We are then free to homotope f and g around each other through this “constant ocean” in a way such that they don’t intersect. After fully completing the motion, we then again enlarge the domains and end with g ` f. In contrast to Eckmann-Hilton, this method of proof (once formalized) may feel more immediately intuitive and logical to follow, but in fact, it is entirely equivalent to the Eckmann-Hilton argument! Consider the below clock-diagram. The operations used are ` and `1 and one can easily see how these diagram, which come from Hatcher’s topological argument, produce the same algebraic result as the Eckmann-Hilton argument. In this way, both results are actually the same despite being motivated in diﬀerent ways. This is a rather explicit and satisfying example that highlights the power of algebraic topology. By having both algebraic and topological structures to work with, the analysis of spaces

6 often comes from a blending of the two-tool that allows us use whatever intuition may be more natural to us. We will now move on to our alternate proof which, as should be expected, will leverage the relationship between algebra and topology but in a conceptually diﬀerent way.

3 H-Spaces

Our next approach to the proof that πn, n ą 1 is abelian offers new topological intuition as to why this result should be true as well as a rewarding glance into the actual structure of higher homotopy groups. Before proceeding to the proof, however, we first need to take a quick detour to introduce a structure that can placed on topological spaces that will be fundamental to our understanding of our new approach. This construction will generate spaces known as H-spaces. An H-space is a topological space equipped with a fairly unrestricted algebraic structure, defined as follows.

Deﬁnition 2. An H-space is a pair ppX, eq, µq, with pX, eq a based space and µ a continuous multiplication map

µ : X ˆ X Ñ X

7 such that the maps x ÞÑ µpx, eq and x ÞÑ µpe, xq are based homotopic to the identity through pX, eq Ñ pX, eq. Remark 1. For any H-space ppX, eq, µq, µpe, eq “ e. Keeping in mind that an H-space is really a based space with added structure, we will immediately become less formal and refer to the “H-space ppX, eq, µq” as the “H-space X” when no confusion will result. A familiar example of an H-spaces are topological groups, which have a continuous multiplication map and an identity element such that µpx, eq “ µpe, xq “ x are the identity (and hence homotopic to it). When compared with the more complex definition of topological groups, one may anticipate that the weaker definition of an H-space, which is concerned exclusively with the identity element, may not add significant structure to the space. However, as we will now see, H-spaces come with very notable built-in structure.

Lemma 7. Let X be an H-space with identity element x0. Then π1pX, x0q is abelian. This proposition is a very desirable reason to work with H-spaces and will be important for our overall proof that πnpX, x0q, n ą 1 is abelian. The result could be proved multiple ways. One such method is to use a tool that we’ve already constructed – Eckmann-Hilton.

Proof. In order to use Eckmann-Hilton, we need to show that π1pH, eq has a second binary operation compatible with the usual one. Our H-space comes with a built in multiplication map, µ, which we will want to leverage. Take the map

µ¯ : π1pH, eq ˆ π1pH, eq Ñ π1pH, eq µ¯pf, gqpxq “ µpfpxq, gpxqq whereµ ¯ is deﬁned on representatives. This map is analogous to usual point-wise deﬁnition of function multiplication on C, which is easily seen if we relax our notation pf ¨ gqpxq “ fpxq ¨ gpxq Note thatµ ¯ is in fact a map between representative loops because

µ¯pf, gqp0q “ µpfp0q, gp0qq “ µpe, eq “ e µ¯pf, gqp1q “ µpfp1q, gp1qq “ µpe, eq “ e where we’ve used the fact that µpe, eq “ e. To show thatµ ¯ is well-deﬁned we show that

f „ h, g „ k ùñ µ¯pf, gq „ µ¯ph, kq

8 where „ denotes based homotopy equivalence. The fact that f „ h, g „ k gives us based homotopies F and G. Using these we build the map

FG : I ˆ I Ñ H

FG “ µ ˝ pFˆGq where FˆG is the map

FˆG : I ˆ I Ñ H ˆ H

FˆGpx, tq “ pF px, tq,Gpx, tqq which is continuous on the product topology, so FG is continuous. Furthermore FGpx, 0q “ µpF px, 0q,Gpx, 0qq “ µpfpxq, gpxqq FGpx, 1q “ µpF px, 1q,Gpx, 1qq “ µphpxq, kpxqq

Thereforeµ ¯ is a well-deﬁned binary operation on π1pH, eq. Observe thatµ ¯ also has the constant loop as it’s identity. Start with

µ¯pf, ceqp¨q “ µpfp¨q, eq

Recall that the induced map

µe : H Ñ H µpxq “ µpx, eq is homotopic to the identity by deﬁnition of an H-space. Observing that µpfp¨q, eq “ µe ˝ f, this implies

µ¯pf, ceqp¨q “ µpfp¨q, eq “ µe ˝ f „ f

Therefore ce is a right identity. A similar argument shows it is a left identity.

With this done, we now showµ ¯ is compatible with the standard product on π1. Letµ ¯pf, gq “ fg. Working on class representatives, the fact that

fg ˚ hk “ pf ˚ hqpg ˚ kq follows simply by unwrapping the deﬁnitions.

1 1 fgp2tq 0 ď t ď 2 fp2tqgp2tq 0 ď t ď 2 fg ˚ hkptq “ 1 “ 1 #hkp2t ´ 1q 2 ď t ď 1 #hp2t ´ 1qkp2t ´ 1q 2 ď t ď 1 “ pf ˚ hqpg ˚ kqptq

Finally, applying Eckmann-Hilton, we have π1pH, eq is abelian.

This structure will be very important to later discussion.

9 4 Loop Spaces and Our Result

With our discussion of H-spaces out of the way, we can now move on to look at a particular family of spaces that will both stand at the heart of our proof as well as aid our understanding of how higher homotopy groups behave. These spaces are called “Loop Spaces”.

Definition 3 (Loop Space). The loop space of a based space pX, x0q is the space ΩX Ă CpI,Xq, where ΩX “ tγ P CpI,Xq | γp0q “ γp1q “ x0u and is equipped with the compact-open topology. It is important to note that ΩX is itself a based space with basepoint “the constant loop”, cx0 ptq “ x0, though by convention this is suppressed from our notation. As formally defined, a loop space ΩX is really just the space of loops in X which respect the basepoint. The following proposition shows that loop spaces come with a built in structure now familiar to us. Proposition 8. Loop spaces are H-spaces, where the multiplication map is defined by function concatenation.

The proof of this fact is fairly straight forward. Proof. First we show that

µ :ΩX ˆ ΩX Ñ ΩX µpf, gq “ f ˚ g is continuous. We do so by proving that the preimage of basic open sets are open. Continuity then follows immediately because preimages preserve unions. Let V pK,Uq “ V˜ pK,Uq X ΩX where V˜ pK,Uq is a basic open set in CpI,Xq, meaning V pK,Uq is a basic open set in ΩX. By deﬁnition

µ´1pV pK,Uqq “ tpf, gq P ΩX ˆ ΩX | f ˚ g P V pK,Uqu with K Ă I compact and U open. Take pf, gq P µ´1pV pK,Uqq. Then by deﬁnition pf ˚ gqpKq Ă U. This means that

1 fp2tq P U, t P K X 0, 2 „  and that 1 gp2t ´ 1q P U, t P K X , 1 2 „ 

10 Equivalently, this means that

f P V1p2K X r0, 1s,Uq

g P V2p2K ´ 1 X r0, 1s,Uq where

2K “ t2x | x P Ku 2K ´ 1 “ t2x ´ 1 | x P Ku which are compact because multiplication and subtraction are continuous and continuous images of compact sets are compact. V1 and V2 are therefore basic open sets and therefore

´1 pf, gq P V1 ˆ V2 Ă V pK,Uq ùñ µ pV pK,Uqq is locally open and hence open. Therefore µ is continuous.

Now we show this concatenation has the desired homotopy property on the identity which we take as the constant loop. More formally, we now prove that µpe, ¨q and µp¨, eq are based-homotopic to the identity through e, with e the constant loop e : I Ñ pX, x0q, epyq “ x0 @y P I.

Consider the homotopy F :ΩX ˆ I Ñ ΩX deﬁned by

t x0 0 ď x ď 2 F pγ, tqpsq “ 2s´t t #γp 2´t q 2 ď x ď 1 Similarly, for µp¨, eq we can deﬁne a homotopy

2s t`1 γp t q 0 ď x ď 2 Gpγ, tqpsq “ t`1 #x0 2 ď x ď 1 Observe that these are both continuous following a similar proof to that µ is 1 t continuous with the boundary 2 replaced with 2 . F and G are our desired homotopies F1 “ e ˚ ¨,F0 “ Id and G0 “ ¨ ˚ e, G1 “ Id Therefore ΩX is an H-space. While in practice, the compact-open topology may be tricky to work with, loops spaces are a particularly fruitful way to study topological spaces both conceptually and computationally. One sees in the study of the fundamental group that the behavior of loops in a topological space contain a lot of information about that space. By constructing a space now made of these loops, we in practice encode information about X in ΩX. This somewhat vague discussion will be made precise in a moment by a latter theorem, but ﬁrst, a few preliminary deﬁnitions.

11 Deﬁnition 4. Let pX, x0q, pY, y0q be two based spaces. We write xX,Y y for the set of based homotopy classes of based maps from X to Y .

n n Remark 2. In the case that X “ S then xS ,Y y “ πnpY, y0q. This follows from the deﬁnition of higher homotopy groups.

Definition 5. Let X be a space. SX is called the suspension of X and is defined as the quotient SX “ pX ˆ I{pX ˆ t0uqq{pX ˆ t1uq. This is the space created by “elongating” the topological space X into a cylinder and then collapsing the top of the cylinder to one point and the bottom to another point. The set xSX,Y y will be a very important object of study moving forward because it encodes information about loops in Y . Consider the case where X “ S1. If we have a map γ : SX Ñ Y , then we get a pre-quotiented map from the cylinderγ ¯ : X ˆ I Ñ Y which consists of loopsγ ¯t : X Ñ Y as well as loopsγ ¯ps, ¨q : I Ñ Y . This plethora of information about loops encoded in xSX,Y y will translate into knowledge of the loop space ΩY , as we will see in just a moment. Inspired by the fact that πnpX, x0q is a group, we would like xSX,Y y to be a group by extending the binary operation on xX,Y y. However, in order to meaningfully define this, we need a basepoint. Unfortunately, SX has no unique way for us to extend the basepoint from X, which has its own suspended subset in SX (demonstrated visually for S1 below). To resolve this problem, we simply quotient out by the suspension of x0. This gives us the following definition.

Deﬁnition 6. Let pX, x0q be a based space. ΣX is called the reduced suspension of X and is deﬁned by SX{ptx0u ˆ Iq. As desired, we can easily show xΣX,Ky is a group.

Proposition 9. xΣX,Ky is a group. A discussion of why this is true can be found in Hatcher.8

The following ﬁgure gives a visualization of both the suspension and the reduced suspension of S1, where the red line follows the basepoint. It also gives a visual demonstration of a more general fact.

Proposition 10. SSn – ΣSn – Sn`1 for n ě 0. While we leave this result formally unproved, the desired homeomorphisms are generalizations of those drawn below. With these constructions out of the way, we can now formally show why we have bothered introducing the the notion of a reduced suspension. The following theorem is the last piece of the puzzle needed in our ultimate proof.

8Hatcher Algebraic Topology 394-395.

12 Theorem 11. Let ΣX be the reduced suspension of X and ΩK be the loop space of K, then xΣX,Ky – xX, ΩKy Proof. Take γ P rγs P xΣX,Ky. The map γ determines a map

γ¯ : X ˆ I Ñ K whereγ ¯ “ γ ˝ p, with p the quotient map p : X ˆ I Ñ ΣX. The mapγ ¯ induces a map φ : X Ñ ΩK φpxq “ γ¯px, ¨q where φpxq P ΩK because

φpxqp0q “ γ¯px, 0q “ pγ ˝ pqpx, 0q “ γpx0q “ k0

φpxqp1q “ γ¯px, 1q “ pγ ˝ pqpx, 1q “ γpx0q “ k0 where x0 is the basepoint in ΣX and we’ve used that γ is a based map. Therefore, we have a map xΣX,Ky Ñ xX, ΩKy deﬁned on representatives by

γ ÞÑ pγ ˝ pqpŸ, ¨q where Ÿ denotes the X-input. Observe that this map is well-deﬁned. Take η „ γ. For each map we get the respectively induced maps

pγ ˝ pqpŸ, ¨q pη ˝ pqpŸ, ¨q

13 The composition of based-homotopic functions is based-homotopic, therefore

γ ˝ p „ η ˝ p Denote the based homotopy by F : X ˆ I ˆ I Ñ K. This will induce a map F¯ : X ˆ I Ñ ΩK F¯px, tq “ F px, t, ¨q that is continuous and easily seen to be a based homotopy. Therefore the map is well deﬁned.9

Now that we have a well defined map, name it Ψ, defined as above Ψ: xΣX,Ky Ñ xX, ΩKy Ψprγsq “ rpγ ˝ pqpŸ, ¨qs where Ÿ is the X input. We now show Ψ is an isomorphism. First, Ψ is a homomorphism. This is because Ψpγ ˚ ηq “ ppγ ˚ ηq ˝ pqpŸ, ¨q “ ppγ ˝ pq ˚ pη ˝ pqqpŸ, ¨q “ pγ ˝ pqpŸ, ¨q ˚ pη ˝ pqpŸ, ¨q “ ΨpγqΨpηq by unwrapping the definition of concatenation. Second, Ψ is a bijection. We show this by constructing an inverse. Given a based map φ : X Ñ ΩK there is an induced map φ¯ : X ˆ I Ñ K φ¯px, tq “ pφpxqqptq ¯ Observe that φ is constant on the set tpx, tq | t “ 0 or t “ 1 or x “ x0u where x0 is the basepoint in X because ¯ φpx0, tq “ pφpx0qqptq “ ck0 ptq “ k0, @t ¯ φpx, 0q “ pφpxqqp0q “ k0 ¯ φpx, 1q “ pφpxqqp1q “ k0 where ck0 is the constant loop in ΩK and we used φ is a based loop. We therefore get an induced map from ΣX Ñ K. Define Θ: xX, ΩKy Ñ xΣX,Ky where Θpηq is the map induced by the processes described above. Following a similar analysis to above, we would find that the map Θ is a well-defined. The maps Θ and Ψ are easily seen to be inverses. Therefore Ψ is a bijective homomorphism and hence an isomorphism.

9The general continuity statement is proven in Curio 4.

14 As an immediate consequence we get the following corollary.

Corollary 12. πn`1pX, x0q “ πnpΩXq. Proof. Combining theorem 11 with proposition 10, we get xΣSn,Xy “ xSn, ΩXy ùñ xSn`1,Xy “ xSn, ΩXy

ùñ πn`1pX, x0q “ πnpΩXq

Finally, this gives us our desired result.

Corollary 13. πnpX, x0q is abelian for n ą 1. Proof. The result follows by induction. Take a based space pX, x0q. By the prior corollary π2pX, x0q “ π1pΩXq. Recall that ΩX is a loop space which means it is also an H-space. By lemma 7 the fundamental group of an H-space is abelian. Therefore π1pΩXq is abelian which means that π2pX, x0q is abelian. Because we chose an arbitrary pX, x0q to work with, this shows π2 is abelian for all based spaces. For our inductive step assume πnpX, x0q is abelian with n ą 1 for every pX, x0q. We know that πn`1pX, x0q “ πnpΩXq again by the prior corollary. By our inductive step, πnpΩXq is abelian and therefore πn`1pX, x0q is abelian for every based space. By induction, πn is abelian for every n ą 1. The end of this argument came together very quickly, so lets look at an example to see what really just happened in these past few arguments. Lets 2 take for example π1pΩS q. It will be helpful to visualize what this space is. Taking intuition from our above proof, each point on S1 associates with a based loop in S2. Visually, because S1 Ă S2 it’s convenient to picture our generating circle in S2.

15 We’ll visualize this using the above diagram. We ﬁrst embed a red circle within S2. For each point in the circle, we associate a loop in S2. We do so by having points further from the basepoint of the embedded S1 correspond to loops in S2 which travel further from the corresponding basepoint in S2 in the fashion drawn above. In practice, when we look at the continuum of S1, these loops fully cover S2. By construction this is a visualization of a “loop” in the loop space ΩS2, with each blue loop in S2 corresponding to a point on the loop in ΩS2. If this drawing looks familiar, it’s because it is the exact same the homeomor- phism from ΣS1 Ñ S2 that we drew earlier. This means that basepoint preserving maps from ΣS1 Ñ S2 are exactly the same thing as maps from S1 Ñ ΩS2. Intuitively, this will hold for based homotopy classes as well because a circle can be compressed and expanded by homotopy, but cannot have additional loops 1 1 2 1 2 added because π1pS q – Z. Therefore, we observe that xΣS ,S y – xS , ΩS y, the exact content of the theorem above.