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Homotopy theory begins with the homotopy groups π n (X), which are the natural higher-dimensional analogs of the fundamental group. These higher homotopy groups have certain formal similarities with homology

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Homotopy theory begins with the homotopy groups π n (X), which are the natural higher-dimensional analogs of the fundamental group. These higher homotopy groups have certain formal similarities with homology groups. For example, π n (X) turns out to be always abelian for n 2, and there are relative homotopy groups fitting into a long exact sequence just like the long exact sequence of homology groups. However, the higher homotopy groups are much harder to compute than either homology groups or the fundamental group, due to the fact that neither the excision property for homology nor van Kampen s theorem forπ 1 holds for higher homotopy groups. In spite of these computational difficulties, homotopy groups are of great theoretical significance. One reason for this is Whitehead s theorem that a map between CW complexes which induces isomorphisms on all homotopy groups is a homotopy equivalence. The stronger statement that two CW complexes with isomorphic homotopy groups are homotopy equivalent is usually false, however. One of the rare cases when a CW complex does have its homotopy type uniquely determined by its homotopy groups is when it has just a single nontrivial homotopy group. Such spaces, known as Eilenberg MacLane spaces, turn out to play a fundamental role in algebraic topology for a variety of reasons. Perhaps the most important is their close connection with cohomology: Cohomology classes in a CW complex correspond bijectively with homotopy classes of maps from the complex into an Eilenberg MacLane space. 338 Chapter 4 Homotopy Theory Thus cohomology has a strictly homotopy-theoretic interpretation, and there is an analogous but more subtle homotopy-theoretic interpretation of homology, explained in 4.F. A more elementary and direct connection between homotopy and homology is the Hurewicz theorem, asserting that the first nonzero homotopy group π n (X) of a simply-connected spacex is isomorphic to the first nonzero homology group H n (X). This result, along with its relative version, is one of the cornerstones of algebraic topology. Though the excision property does not always hold for homotopy groups, in some important special cases there is a range of dimensions in which it does hold. This leads to the idea of stable homotopy groups, the beginning of stable homotopy theory. Perhaps the major unsolved problem in algebraic topology is the computation of the stable homotopy groups of spheres. Near the end of 4.2 we give some tables of known calculations that show quite clearly the complexity of the problem. Included in 4.2 is a brief introduction to fiber bundles, which generalize covering spaces and play a somewhat analogous role for higher homotopy groups. It would easily be possible to devote a whole book to the subject of fiber bundles, even the special case of vector bundles, but here we use fiber bundles only to provide a few basic examples and to motivate their more flexible homotopy-theoretic generalization, fibrations, which play a large role in 4.3. Among other things, fibrations allow one to describe, in theory at least, how the homotopy type of an arbitrary CW complex is built up from its homotopy groups by an inductive procedure of forming twisted products of Eilenberg MacLane spaces. This is the notion of a Postnikov tower. In favorable cases, including all simply-connected CW complexes, the additional data beyond homotopy groups needed to determine a homotopy type can also be described, in the form of a sequence of cohomology classes called thek invariants of a space. If these are all zero, the space is homotopy equivalent to a product of Eilenberg MacLane spaces, and otherwise not. Unfortunately the k invariants are cohomology classes in rather complicated spaces in general, so this is not a practical way of classifying homotopy types, but it is useful for various more theoretical purposes. This chapter is arranged so that it begins with purely homotopy-theoretic notions, largely independent of homology and cohomology theory, whose roles gradually increase in later sections of the chapter. It should therefore be possible to read a good portion of this chapter immediately after reading Chapter 1, with just an occasional glimpse at Chapter 2 for algebraic definitions, particularly the notion of an exact sequence which is just as important in homotopy theory as in homology and cohomology theory. Homotopy Groups Section Perhaps the simplest noncontractible spaces are spheres, so to get a glimpse of the subtlety inherent in homotopy groups let us look at some of the calculations of the groupsπ i (S n ) that have been made. A small sample is shown in the table below, extracted from [Toda 1962]. π i (S n ) i n 1 Z Z Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 Z 2 Z 2 Z Z Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 Z 2 Z 2 Z Z Z 2 Z 2 Z Z 12 Z 2 Z 2 Z 2 Z 2 Z 24 Z 3 Z 15 Z Z Z 2 Z 2 Z 24 Z 2 Z 2 Z 2 Z Z Z 2 Z 2 Z 24 0 Z Z Z Z 2 Z 2 Z Z Z 2 Z 2 Z 24 0 This is an intriguing mixture of pattern and chaos. The most obvious feature is the large region of zeros below the diagonal, and indeed π i (S n )=0 for all i n as we show in Corollary 4.9. There is also the sequence of zeros in the first row, suggesting that π i (S 1 )=0 for all i 1. This too is a fairly elementary fact, a special case of Proposition 4.1, following easily from covering space theory. The coincidences in the second and third rows can hardly be overlooked. These are the case n=1 of isomorphismsπ i (S 2n ) π i 1 (S 2n 1 ) π i (S 4n 1 ) that hold for n=1, 2, 4 and all i. The next case n=2 says that each entry in the fourth row is the product of the entry diagonally above it to the left and the entry three units below it. Actually, these isomorphismsπ i (S 2n ) π i 1 (S 2n 1 ) π i (S 4n 1 ) hold for all n if one factors out 2 torsion, the elements of order a power of 2. This is a theorem of James that will be proved in [SSAT]. The next regular feature in the table is the sequence ofz s down the diagonal. This is an illustration of the Hurewicz theorem, which asserts that for a simply-connected spacex, the first nonzero homotopy groupπ n (X) is isomorphic to the first nonzero homology group H n (X). One may observe that all the groups above the diagonal are finite except for π 3 (S 2 ), π 7 (S 4 ), and π 11 (S 6 ). In 4.B we use cup products in cohomology to show that π 4k 1 (S 2k ) contains a Z direct summand for all k 1. It is a theorem of Serre proved in [SSAT] that π i (S n ) is finite for i n except for π 4k 1 (S 2k ), which is the direct sum of Z with a finite group. So all the complexity of the homotopy groups of spheres resides in finite abelian groups. The problem thus reduces to computing the p torsion in π i (S n ) for each primep. 340 Chapter 4 Homotopy Theory An especially interesting feature of the table is that along each diagonal the groups π n+k (S n ) with k fixed and varying n eventually become independent of n for large enough n. This stability property is the Freudenthal suspension theorem, proved in 4.2 where we give more extensive tables of these stable homotopy groups of spheres. Definitions and Basic Constructions Let I n be the n dimensional unit cube, the product of n copies of the interval [0, 1]. The boundary I n of I n is the subspace consisting of points with at least one coordinate equal to 0 or 1. For a space X with basepoint x 0 X, define π n (X,x 0 ) to be the set of homotopy classes of maps f :(I n, I n ) (X,x 0 ), where homotopies f t are required to satisfy f t ( I n )=x 0 for all t. The definition extends to the case n=0 by taking I 0 to be a point and I 0 to be empty, so π 0 (X,x 0 ) is just the set of path-components of X. When n 2, a sum operation in π n (X,x 0 ), generalizing the composition operation in π 1, is defined by (f+g)(s 1,s 2,,s n )= { f(2s1,s 2,,s n ), s 1 [0, 1 / 2 ] g(2s 1 1,s 2,,s n ), s 1 [ 1 / 2, 1] It is evident that this sum is well-defined on homotopy classes. Since only the first coordinate is involved in the sum operation, the same arguments as forπ 1 show that π n (X,x 0 ) is a group, with identity element the constant map sending I n to x 0 and with inverses given by f(s 1,s 2,,s n )=f(1 s 1,s 2,,s n ). The additive notation for the group operation is used becauseπ n (X,x 0 ) is abelian forn 2. Namely,f+g g+f via the homotopy indicated in the following figures. The homotopy begins by shrinking the domains of f and g to smaller subcubes of I n, with the region outside these subcubes mapping to the basepoint. After this has been done, there is room to slide the two subcubes around anywhere in I n as long as they stay disjoint, so if n 2 they can be slid past each other, interchanging their positions. Then to finish the homotopy, the domains of f and g can be enlarged back to their original size. If one likes, the whole process can be done using just the coordinates s 1 and s 2, keeping the other coordinates fixed. Maps (I n, I n ) (X,x 0 ) are the same as maps of the quotient I n / I n =S n tox taking the basepoints 0 = I n / I n tox 0. This means that we can also viewπ n (X,x 0 ) as homotopy classes of maps(s n,s 0 ) (X,x 0 ), where homotopies are through maps Homotopy Groups Section of the same form(s n,s 0 ) (X,x 0 ). In this interpretation ofπ n (X,x 0 ), the sumf+g is the composition S n c S n S n f g X where c collapses the equator S n 1 in S n to a point and we choose the basepoint s 0 to lie in this S n 1. We will show next that if X is path-connected, different choices of the basepoint x 0 always produce isomorphic groups π n (X,x 0 ), just as for π 1, so one is justified in writing π n (X) for π n (X,x 0 ) in these cases. Given a path γ :I X from x 0 =γ(0) to another basepoint x 1 =γ(1), we may associate to each map f :(I n, I n ) (X,x 1 ) a new map γf :(I n, I n ) (X,x 0 ) by shrinking the domain off to a smaller concentric cube in I n, then inserting the path γ on each radial segment in the shell between this smaller cube and I n. When n = 1 the map γf is the composition of the three paths γ, f, and the inverse of γ, so the notationγf conflicts with the notation for composition of paths. Since we are mainly interested in the cases n 1, we leave it to the reader to make the necessary notational adjustments when n = 1. A homotopy ofγ orf through maps fixing I or I n, respectively, yields a homotopy of γf through maps (I n, I n ) (X,x 0 ). Here are three other basic properties: (1) γ(f+g) γf+γg. (2) (γη)f γ(ηf). (3) 1f f, where 1 denotes the constant path. The homotopies in (2) and (3) are obvious. For (1), we first deform f and g to be constant on the right and left halves of I n, respectively, producing maps we may call f + 0 and 0 + g, then we excise a progressively wider symmetric middle slab of γ(f+ 0)+γ(0+g) until it becomes γ(f+g): An explicit formula for this homotopy is { ( ) γ(f+ 0) (2 t)s1,s h t (s 1,s 2,,s n )= 2,,s n, s1 [0, 1 / 2 ] γ(0+g) ( ) (2 t)s 1 +t 1,s 2,,s n, s1 [ 1 / 2, 1] Thus we have γ(f+g) γ(f+ 0)+γ(0+g) γf+γg. If we define a change-of-basepoint transformationβ γ :π n (X,x 1 ) π n (X,x 0 ) by β γ ([f]) = [γf], then (1) shows that β γ is a homomorphism, while (2) and (3) imply that β γ is an isomorphism with inverse β γ where γ is the inverse path of γ, 342 Chapter 4 Homotopy Theory γ(s)=γ(1 s). Thus ifx is path-connected, different choices of basepointx 0 yield isomorphic groupsπ n (X,x 0 ), which may then be written simply as π n (X). Now let us restrict attention to loopsγ at the basepointx 0. Sinceβ γη =β γ β η, the association [γ] β γ defines a homomorphism from π 1 (X,x 0 ) to Aut(π n (X,x 0 )), the group of automorphisms of π n (X,x 0 ). This is called the action of π 1 on π n, each element of π 1 acting as an automorphism [f] [γf] of π n. When n = 1 this is the action of π 1 on itself by inner automorphisms. When n 1, the action makes the abelian group π n (X,x 0 ) into a module over the group ring Z[π 1 (X,x 0 )]. Elements of Z[π 1 ] are finite sums i n i γ i with n i Z and γ i π 1, multiplication being defined by distributivity and the multiplication inπ 1. The module structure on π n is given by ( i n i γ ) i α= i n i (γ i α) for α π n. For brevity one sometimes says π n is a π 1 module rather than a Z[π 1 ] module. In the literature, a space with trivial π 1 action on π n is called n simple, and simple means n simple for all n. In this book we will call a space abelian if it has trivial action ofπ 1 on all homotopy groupsπ n, since whenn=1 this is the condition that π 1 be abelian. This terminology is consistent with a long-established usage of the term nilpotent to refer to spaces with nilpotent π 1 and nilpotent action of π 1 on all higher homotopy groups; see [Hilton, Mislin, & Roitberg 1975]. An important class of abelian spaces is H spaces, as we show in Example 4A.3. We next observe that π n is a functor. Namely, a map ϕ :(X,x 0 ) (Y,y 0 ) induces ϕ :π n (X,x 0 ) π n (Y,y 0 ) defined by ϕ ([f])=[ϕf]. It is immediate from the definitions that ϕ is well-defined and a homomorphism for n 1. The functor properties (ϕψ) = ϕ ψ and 11 = 11 are also evident, as is the fact that if ϕ t :(X,x 0 ) (Y,y 0 ) is a homotopy then ϕ 0 =ϕ 1. In particular, a homotopy equivalence(x,x 0 ) (Y,y 0 ) in the basepointed sense induces isomorphisms on all homotopy groupsπ n. This is true even if basepoints are not required to be stationary during homotopies. We showed this for π 1 in Proposition 1.18, and the generalization to highern s is an exercise at the end of this section. Homotopy groups behave very nicely with respect to covering spaces: Proposition 4.1. A covering space projection p :( X, x 0 ) (X,x 0 ) induces isomorphisms p :π n ( X, x 0 ) π n (X,x 0 ) for all n 2. Proof: For surjectivity of p we apply the lifting criterion in Proposition 1.33, which implies that every map(s n,s 0 ) (X,x 0 ) lifts to ( X, x 0 ) provided that n 2 so that S n is simply-connected. Injectivity of p is immediate from the covering homotopy property, just as in Proposition 1.31 which treated the case n=1. In particular, π n (X,x 0 )=0 for n 2 whenever X has a contractible universal cover. This applies for example tos 1, so we obtain the first row of the table of homotopy groups of spheres shown earlier. More generally, the n dimensional torus T n, Homotopy Groups Section the product of n circles, has universal cover R n, so π i (T n ) = 0 for i 1. This is in marked contrast to the homology groups H i (T n ) which are nonzero for all i n. Spaces with π n = 0 for all n 2 are sometimes called aspherical. The behavior of homotopy groups with respect to products is very simple: Proposition 4.2. For a product α X α of an arbitrary collection of path-connected spacesx α there are isomorphisms π n ( α X ) α α π n (X α ) for all n. Proof: A map f :Y α X α is the same thing as a collection of maps f α :Y X α. Taking Y to be S n and S n I gives the result. Very useful generalizations of the homotopy groups π n (X,x 0 ) are the relative homotopy groupsπ n (X,A,x 0 ) for a pair(x,a) with a basepoint x 0 A. To define these, regardi n 1 as the face ofi n with the last coordinates n = 0 and letj n 1 be the closure of I n I n 1, the union of the remaining faces of I n. Then π n (X,A,x 0 ) for n 1 is defined to be the set of homotopy classes of maps(i n, I n,j n 1 ) (X,A,x 0 ), with homotopies through maps of the same form. There does not seem to be a completely satisfactory way of definingπ 0 (X,A,x 0 ), so we shall leave this undefined (but see the exercises for one possible definition). Note thatπ n (X,x 0,x 0 )=π n (X,x 0 ), so absolute homotopy groups are a special case of relative homotopy groups. A sum operation is defined inπ n (X,A,x 0 ) by the same formulas as forπ n (X,x 0 ), except that the coordinate s n now plays a special role and is no longer available for the sum operation. Thusπ n (X,A,x 0 ) is a group forn 2, and this group is abelian for n 3. For n=1 we have I 1 =[0, 1], I 0 ={0}, and J 0 ={1}, so π 1 (X,A,x 0 ) is the set of homotopy classes of paths in X from a varying point in A to the fixed basepoint x 0 A. In general this is not a group in any natural way. Just as elements of π n (X,x 0 ) can be regarded as homotopy classes of maps (S n,s 0 ) (X,x 0 ), there is an alternative definition of π n (X,A,x 0 ) as the set of homotopy classes of maps (D n,s n 1,s 0 ) (X,A,x 0 ), since collapsing J n 1 to a point converts(i n, I n,j n 1 ) into(d n,s n 1,s 0 ). From this viewpoint, addition is done via the map c :D n D n D n collapsing D n 1 D n to a point. A useful and conceptually enlightening reformulation of what it means for an element of π n (X,A,x 0 ) to be trivial is given by the following compression criterion: A map f :(D n,s n 1,s 0 ) (X,A,x 0 ) represents zero in π n (X,A,x 0 ) iff it is homotopic rels n 1 to a map with image contained in A. For if we have such a homotopy to a map g, then [f] = [g] in π n (X,A,x 0 ), and [g] = 0 via the homotopy obtained by composing g with a deformation retraction of D n onto s 0. Conversely, if [f]=0 via a homotopyf :D n I X, then by restricting F to a family of n disks in D n I starting with D n {0} and ending with the disk D n {1} S n 1 I, all the disks in the family having the same boundary, then we get a homotopy from f to a map into A, stationary on S n 1. 344 Chapter 4 Homotopy Theory A map ϕ :(X,A,x 0 ) (Y,B,y 0 ) induces maps ϕ :π n (X,A,x 0 ) π n (Y,B,y 0 ) which are homomorphisms for n 2 and have properties analogous to those in the absolute case: (ϕψ) = ϕ ψ, 11 = 11, and ϕ = ψ if ϕ ψ through maps (X,A,x 0 ) (Y,B,y 0 ). Probably the most useful feature of the relative groups π n (X,A,x 0 ) is that they fit into a long exact sequence π n (A,x 0 ) i π n (X,x 0 ) j π n (X,A,x 0 ) π n 1 (A,x 0 ) π 0 (X,x 0 ) Here i and j are the inclusions (A,x 0 ) (X,x 0 ) and (X,x 0,x 0 ) (X,A,x 0 ). The map comes from restricting maps(i n, I n,j n 1 ) (X,A,x 0 ) toi n 1, or by restricting maps (D n,s n 1,s 0 ) (X,A,x 0 ) to S n 1. The map, called the boundary map, is a homomorphism when n 1. Theorem 4.3. This sequence is exact. Near the end of the sequence, where group structures are not defined, exactness still makes sense: The image of one map is the kernel of the next, those elements mapping to the homotopy class of the constant map. Proof: With only a little more effort we can derive the long exact sequence of a triple (X,A,B,x 0 ) with x 0 B A X : π n (A,B,x 0 ) i π n (X,B,x 0 ) j π n (X,A,x 0 ) π n 1 (A,B,x 0 ) π 1 (X,A,x 0 ) WhenB=x 0 this reduces to the exact sequence for the pair(x,a,x 0 ), though the latter sequence continues on two more steps toπ 0 (X,x 0 ). The verification of exactness at these last two steps is left as a simple exercise. Exactness at π n (X,B,x 0 ): First note that the composition j i is zero since every map (I n, I n,j n 1 ) (A,B,x 0 ) represents zero in π n (X,A,x 0 ) by the compression criterion. To see that Kerj Imi, letf :(I n, I n,j n 1 ) (X,B,x 0 ) represent zero in π n (X,A,x 0 ). Then by the compression criterion again, f is homotopic rel I n to a map with image in A, hence the class [f] π n (X,B,x 0 ) is in the image of i. Exactness at π n (X,A,x 0 ): The composition j is zero since the restriction of a map (I n, I n,j n 1 ) (X,B,x 0 ) to I n 1 has image lying in B, and hence represents zero in π n 1 (A,B,x 0 ). Conversely, suppose the restriction of f :(I n, I n,j n 1 ) (X,A,x 0 ) to I n 1 represents zero in π n 1 (A,B,x 0 ). Then f I n 1 is homotopic to a map with image in B via a homotopy F :I n 1 I A rel I n 1. We can tack F onto f to get a new map (I n, I n,j n 1 ) (X,B,x 0 ) which, as a map (I n, I n,j n 1 ) (X,A,x 0 ), is homotopic to f by the homotopy that tacks on increasingly longer initial segments of F. So [f] Imj. Homotopy Groups Section Exactness at π n (A,B,x 0 ): The compositioni is zero since

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