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 usedto study topological spaces by studying how loops behave within them. Higherhomotopy 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 homotopyclasses of based maps γ : pSn, s0q Ñ pX,x0q.

As in the case of the fundamental group, higher homotopy groups are alsogroups.

Proposition 1. πnpX,x0q equipped with the binary operation defined on equiv-alence classes as γ` η “ γ_η ˝ p where p is the quotient map p : Sn Ñ Sn_Sn

which collapses a great circle and γ_η is the map on Sn _ Sn with γ acting onone 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 inHatcher.1 This definition may appear to be a logically sound enough way ofgeneralizing the fundamental group, but one may now naturally ask, “Are theseworthwhile objects to study?”. The answer to this question is a wholeheartedyes.

1Hatcher Algebraic Topology 340.

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Higher homotopy groups have pragmatic uses in characterizing various topo-logical spaces while at the same time offer a satisfying arena to study the naturalintertwining of the fields of algebra and topology.

For an explicit example of a use of higher homotopy groups in studyingspaces, consider the following theorem, known as Whitehead’s Theorem:

Theorem 2 (Whitehead). Let X and Y be connected CW complexes. If a mapf : X Ñ Y induces isomorphisms f˚ : πnpXq Ñ πnpY q for all n, then f is ahomotopy equivalence.

To fully appreciate the theorem, one should have an understanding of CWcomplexes.2 However, even without an in-depth understanding of CW com-plexes, the power of this theorem is striking. By knowing how maps act onhigher homotopy groups, one immediately receives a significant amount of in-formation about the space in question. This is particularly powerful becausemany spaces studied in algebraic topology are homotopy equivalent to a CWcomplex 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 howcomplex these objects can be conceptually, consider the following fact

Fact 1. π3pS2q is non trivial.

This group is generated by what is known as the Hopf fibration4 and theresult 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 consideringthat π2pS

1q 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 ą dimpXq, “capping”their complexity in a sense.5 This is clearly not the case for higher homotopygroups, with the example above acting as a counter example. Therefore, in somesense 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, issurprising.

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.

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This result is perhaps even more surprising after recalling the fact that thisis not true for fundamental groups. Take for example the twice-punctured planeCzt´1, 1u which is homotopy equivalent to a figure eight.

Consider the two loops drawn below in black. When both are placed aroundopposite “holes”, concatenating the loops in different orders produces loopsthat are not based-homotopic. If we take red as going first followed by blue, nomatter how one twists the loops (while preserving the basepoint) it is impossibleto change one loop into the other. A failed attempt at trying to homotopebetween both loops is depicted by the final drawing. This demonstrates thatπ1pCzt´1, 1u, 0q is non-abelian.

The result is made rigorous quite simply by appealing to Seifert-van Kampenwhich gives the explicit result that π1pCzt´1, 1u, 0q – F2, the free group on twogenerators.6 By definition a free group has no relations so, in some sense, thisfundamental group is actually the furthest from abelian you can get.

So while one may anticipate higher homotopy groups to be more complexthan the fundamental group (which they certainly are computationally), theyare in some sense algebraically simpler. This intriguing result is our main goalof proof in this paper.

We will study this result using two different approaches. Our first approachwill use a result known as Eckmann-Hilton and will highlight the practice ofalgebra and topology co-informing each other. Our second approach will beslightly more topological in nature as we introduce H-spaces and Loop-spaceswhich 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.

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2 The Eckmann-Hilton Argument

Our first look at the proof that higher homotopy groups are abelian follows froma rather satisfying look at how algebra and topology co-inform one another. Wefirst introduce the following algebraic theorem that will lead us to our desiredresult.

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 twooperations 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 thisdefinition.

Say we have a set X with a binary operation ¨ which has an identity elementnamed e (i.e. pX, ¨q is a unital magma). Now suppose we give our set anotherbinary operation ˚ : XˆX Ñ X with identity element e˚ where we require that˚ is a homomorphism from the direct product pX ˆX, ¨q to pX, ¨q.7 It’s fairlyeasy to observe that this gives the exact relationship between binary operationsgiven 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 oftopological 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 aunital magma and we give it an additional binary operation which respects theoriginal structure by requiring the new operation be a homomorphism. So, toround 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 tellsus 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 andprogress clockwise by repeatedly using the assumed relation between ¨ and ˚and the fact that e “ e˚. Each step along the clock is a statement of equalityand the right and left sides shows us that a ¨b “ a˚b and b˚a “ b ¨a respectively,

7Note that we don’t require e “ e˚ a priori.

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showing that the operations are equal. The top and bottom respectively giveus that b ¨ a “ a ¨ b and b ˚ a “ a ˚ b, showing that both operations are in factcommutative.

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, namelycommutativity. Leveraging this result, if we can show that πnpX,x0q, n ą 1 hastwo binary structures which satisfy this relationship, we get that it is abelianfor free.

In order to find this operation, it is more fruitful to think of maps from In

which are constant on BIn, the set of elements in In such that one coordinateis 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 onBIn, so we may consider them interchangeably.

When working with maps γ : pIn, BInq Ñ pX,x0q, the group operation onπnpX,x0q as specified earlier becomes

pf ` gqpsq “

#

fp2s1, s2, ..., snq 0 ď s1 ď12

gp2s1 ´ 1, s2, ..., snq12 ď s1 ď 1

One can immediately observe, however, that this definition is arbitrary inthe case n ą 1. We could have just as easily chosen to concatenate along acoordinate other than s1.

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

pf `1 gqpsq “

#

fps1, 2s2, ..., snq 0 ď s2 ď12

gps1, 2s2 ´ 1, ..., snq12 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 visuallyseen 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-HiltonπnpX,x0q is abelian for n ą 1.

This approach was nice and clean and a great example of algebra informingtopology. However, we could have also approached this problem in such a waythat topological considerations would have informed this algebraic result. Con-sider the figure below which comes from Hatcher’s Algebraic Topology. Thisdiagram 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 areable to homotope both to smaller domains where they are now surrounded byan “ocean” of our basepoint x0 (drawn in blue). We are then free to homotopef and g around each other through this “constant ocean” in a way such thatthey don’t intersect. After fully completing the motion, we then again enlargethe domains and end with g ` f .

In contrast to Eckmann-Hilton, this method of proof (once formalized) mayfeel more immediately intuitive and logical to follow, but in fact, it is entirelyequivalent 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 algebraicresult as the Eckmann-Hilton argument. In this way, both results are actuallythe same despite being motivated in different ways. This is a rather explicit andsatisfying example that highlights the power of algebraic topology. By havingboth algebraic and topological structures to work with, the analysis of spaces

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often comes from a blending of the two-tool that allows us use whatever intuitionmay 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 conceptuallydifferent way.

3 H-Spaces

Our next approach to the proof that πn, n ą 1 is abelian offers new topologicalintuition as to why this result should be true as well as a rewarding glance intothe actual structure of higher homotopy groups.

Before proceeding to the proof, however, we first need to take a quick detourto introduce a structure that can placed on topological spaces that will befundamental to our understanding of our new approach. This construction willgenerate spaces known as H-spaces.

An H-space is a topological space equipped with a fairly unrestricted alge-braic structure, defined as follows.

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

µ : X ˆX Ñ X

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such that the maps x ÞÑ µpx, eq and x ÞÑ µpe, xq are based homotopic to theidentity 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 con-tinuous 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 concernedexclusively with the identity element, may not add significant structure to thespace. However, as we will now see, H-spaces come with very notable built-instructure.

Lemma 7. Let X be an H-space with identity element x0. Then π1pX,x0q isabelian.

This proposition is a very desirable reason to work with H-spaces and willbe important for our overall proof that πnpX,x0q, n ą 1 is abelian. The resultcould be proved multiple ways. One such method is to use a tool that we’vealready constructed – Eckmann-Hilton.

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

Take the map

µ : π1pH, eq ˆ π1pH, eq Ñ π1pH, eq

µpf, gqpxq “ µpfpxq, gpxqq

where µ is defined on representatives. This map is analogous to usual point-wisedefinition of function multiplication on C, which is easily seen if we relax ournotation

pf ¨ gqpxq “ fpxq ¨ gpxq

Note that µ is in fact a map between representative loops because

µpf, gqp0q “ µpfp0q, gp0qq “ µpe, eq “ e

µpf, gqp1q “ µpfp1q, gp1qq “ µpe, eq “ e

where we’ve used the fact that µpe, eq “ e.To show that µ is well-defined we show that

f „ h, g „ k ùñ µpf, gq „ µph, kq

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where „ denotes based homotopy equivalence. The fact that f „ h, g „ k givesus 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 ˆH

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

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

µpf, 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

µpf, 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 show µ is compatible with the standard product onπ1. Let µpf, gq “ fg. Working on class representatives, the fact that

fg ˚ hk “ pf ˚ hqpg ˚ kq

follows simply by unwrapping the definitions.

fg ˚ hkptq “

#

fgp2tq 0 ď t ď 12

hkp2t´ 1q 12 ď t ď 1

“

#

fp2tqgp2tq 0 ď t ď 12

hp2t´ 1qkp2t´ 1q 12 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.

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4 Loop Spaces and Our Result

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

Definition 3 (Loop Space). The loop space of a based space pX,x0q is the spaceΩX Ă CpI,Xq, where ΩX “ tγ P CpI,Xq | γp0q “ γp1q “ x0u and is equippedwith the compact-open topology.

It is important to note that ΩX is itself a based space with basepoint “theconstant loop”, cx0

ptq “ x0, though by convention this is suppressed from ournotation. As formally defined, a loop space ΩX is really just the space of loopsin X which respect the basepoint.

The following proposition shows that loop spaces come with a built in struc-ture now familiar to us.

Proposition 8. Loop spaces are H-spaces, where the multiplication map is de-fined 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 areopen. Continuity then follows immediately because preimages preserve unions.

LetV 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 openset 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 bydefinition pf ˚ gqpKq Ă U . This means that

fp2tq P U, t P K X

„

0,1

2

and that

gp2t´ 1q P U, t P K X

„

1

2, 1

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Equivalently, this means that

f P V1p2K X r0, 1s, Uq

g P V2p2K ´ 1X 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 andcontinuous images of compact sets are compact. V1 and V2 are therefore basicopen sets and therefore

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

and hence open. Therefore µ is continuous.

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

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

F pγ, tqpsq “

#

x0 0 ď x ď t2

γp 2s´t2´t q

t2 ď x ď 1

Similarly, for µp¨, eq we can define a homotopy

Gpγ, tqpsq “

#

γp 2st q 0 ď x ď t`12

x0t`12 ď x ď 1

Observe that these are both continuous following a similar proof to that µ iscontinuous with the boundary 1

2 replaced with t2 . F and G are our desired

homotopiesF1 “ 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 bothconceptually and computationally. One sees in the study of the fundamentalgroup that the behavior of loops in a topological space contain a lot of informa-tion about that space. By constructing a space now made of these loops, we inpractice encode information about X in ΩX.

This somewhat vague discussion will be made precise in a moment by a lattertheorem, but first, a few preliminary definitions.

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

Remark 2. In the case that X “ Sn then xSn, 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 definedas the quotient SX “ pX ˆ IpX ˆ t0uqqpX ˆ t1uq.

This is the space created by “elongating” the topological space X into acylinder and then collapsing the top of the cylinder to one point and the bottomto another point.

The set xSX, Y y will be a very important object of study moving forwardbecause it encodes information about loops in Y . Consider the case whereX “ S1. If we have a map γ : SX Ñ Y , then we get a pre-quotiented mapfrom the cylinder γ : X ˆ I Ñ Y which consists of loops γt : X Ñ Y as wellas loops γps, ¨q : I Ñ Y . This plethora of information about loops encoded inxSX, Y y will translate into knowledge of the loop space ΩY , as we will see injust a moment.

Inspired by the fact that πnpX,x0q is a group, we would like xSX, Y y tobe a group by extending the binary operation on xX,Y y. However, in order tomeaningfully define this, we need a basepoint. Unfortunately, SX has no uniqueway for us to extend the basepoint from X, which has its own suspended subsetin SX (demonstrated visually for S1 below). To resolve this problem, we simplyquotient 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 suspensionof X and is defined by SXptx0u 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 thereduced suspension of S1, where the red line follows the basepoint. It also givesa 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 homeomorphismsare generalizations of those drawn below.

With these constructions out of the way, we can now formally show why wehave bothered introducing the the notion of a reduced suspension. The followingtheorem is the last piece of the puzzle needed in our ultimate proof.

8Hatcher Algebraic Topology 394-395.

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Theorem 11. Let ΣX be the reduced suspension of X and ΩK be the loop spaceof 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

where γ “ γ ˝ p, with p the quotient map p : X ˆ I Ñ ΣX. The map γ inducesa map

φ : X Ñ ΩK

φpxq “ γpx, ¨q

where φpxq P ΩK because

φpxqp0q “ γpx, 0q “ pγ ˝ pqpx, 0q “ γpx0q “ k0

φpxqp1q “ γpx, 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

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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 mapis 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 mapφ : X Ñ ΩK

there is an induced mapφ : X ˆ I Ñ K

φpx, tq “ pφpxqqptq

Observe that φ is constant on the set tpx, tq | t “ 0 or t “ 1 or x “ x0u wherex0 is the basepoint in X because

φpx0, tq “ pφpx0qqptq “ ck0ptq “ k0, @t

φpx, 0q “ pφpxqqp0q “ k0

φpx, 1q “ pφpxqqp1q “ k0

where ck0is 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 asimilar 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 bijectivehomomorphism and hence an isomorphism.

9The general continuity statement is proven in Curio 4.

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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 7the fundamental group of an H-space is abelian. Therefore π1pΩXq is abelianwhich means that π2pX,x0q is abelian. Because we chose an arbitrary pX,x0qto 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 everypX,x0q. We know that πn`1pX,x0q “ πnpΩXq again by the prior corollary. Byour inductive step, πnpΩXq is abelian and therefore πn`1pX,x0q is abelian forevery 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 anexample to see what really just happened in these past few arguments. Letstake for example π1pΩS

2q. It will be helpful to visualize what this space is.Taking intuition from our above proof, each point on S1 associates with a basedloop in S2. Visually, because S1 Ă S2 it’s convenient to picture our generatingcircle in S2.

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We’ll visualize this using the above diagram. We first embed a red circlewithin S2. For each point in the circle, we associate a loop in S2. We do soby having points further from the basepoint of the embedded S1 correspond toloops in S2 which travel further from the corresponding basepoint in S2 in thefashion drawn above. In practice, when we look at the continuum of S1, theseloops fully cover S2. By construction this is a visualization of a “loop” in theloop space ΩS2, with each blue loop in S2 corresponding to a point on the loopin 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 preserv-ing 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 canbe compressed and expanded by homotopy, but cannot have additional loopsadded because π1pS

1q – Z. Therefore, we observe that xΣS1, S2y – xS1,ΩS2y,the exact content of the theorem above.

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