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0, it follows from the homology sequence that : Hm (X) H m (X,F) is a W-isomorphism for every ni> O. Finally, using the isomorphisms k : Hm (B)
.117.(B , B 0), m > 0,
we obtain (A)* — 1?-1 (187)4ti. 8. The generalized Hurewicz theorem
Let be a perfect and weakly complete class. If X is a simply connected space and n > 2 is an integer such that n m (X) e whenever 1
hm :grm (X) H m (X) is a '-isomorphism whenever 0 < ni
Proof. We are going to prove this theorem by induction on n. If n — 2, then this is implied by the usual Hurewicz theorem since no (X) = 0 and n i (X) = O. See (V; 4.4) and (V; Ex. C). Let p > 2 be an integer and assume that the theorem is true for n < 75. Let us prove the theorem for n = p. By the inductive hypothesis, it follows that hm is a W-isomorphism whenever 0 < ni < 75 and is a W-epimorphism for m = p. It remains to prove that hi, is a W-monomorphism and hp+1 is a W-epimorphism. Consider a connective system { (X, r), Pr of the space X as defined in §7. Since X is simply connected, we may assume (X,1) = X. Then (.x, p 1) is a (p 1)-connective fiber space over X with }
p2p3 . 132,„ :
p -1)
as projection. Thus we obtain a commutative rectangle
75- 1) gm
grm (X)
h :a
Hm (X , p
1)
306
X. CLASSES OF ABELIAN GROUPS
where co * , coif are induced by co, and gm, hm are the natural homomorphisms. By the definition of (p - 1)-connective fiber space, co * is an isomorphism for In > p. Since (.x, p - 1) is (p— 1)-connected, it follows from the usual Hurewicz theorem that gp is an isomorphism and gp+i is an epimorphism. p and a ce-epiHence, it suffices to prove that co* is a '-isomorphism for m morphism for ni p + 1. /3 2,33 •••,82,_,, it suffices to prove that, for each r = 2, 3, • • , Since w p - 1, the induced homomorphism,
( Pr)*: ihn (X , r) ± Hm (X , Y - 1) is a '-isomorphism for m p and is a W-epimorphism for ni p. + 1. For this purpose, let B (X, r — 1). Then, B is (r — 1)-connected and we may take a connective system of B with (B, r- 1) B and (B, r) = (X, r). Since B is simply connected and 7(.(B) e whenever 1
This corollary follows from (8.1) by finite induction on m. Therefore, if X is simply connected and W -acyclic, then X is %) -aspherical , i.e. nm (X) for all an > I. In particular, we have the following
The homotopy groups of any simply connected finitely triangulable space are finitely generated. Corollary 8.3.
9. The relative H urewicz theorem
Let V' be a perfect and complete class, X a simply connected space, and X 0 a sirr475ly connected subspace of X. If 7r2 (X, X 0) = 0 and n > 2 is an integer such that m (X, X 0) c V whenever 2
homomorphism
hm :grm (X, X 0) Hm (X, X 0)
is a W-isomorphism whenever 2 < m
Proof. We prove this theorem by induction on n. If n — 2, the theorem is true since H 2 (X, X0 ) = 0 and ho is an isomorphism by (V; Ex. C). It follows from the hypothesis of induction that hm is a V-isomorphikn whenever whenever 0 < ni < n. It remains to 2 < ni < n. Hence Hm (X, X 0) e prove that /In is a W-isomorphism and h. +1 is a W-epimorphism. Pick a point xo E X0 and consider the space of paths Y = [X; X, x0j. Then, Y is a fiber space over X with projection w Y X defined by
DO. THE WHITEHEAD THEOREM
'(0) for each
connected and
E
307
Y. Let Y0 = [X; X 0, x0]. Then Y0 is pathwise
nm (Y 0) nm ,i(X , X o), (m > 1).
Hence, gri ( Y0)
0 and 7cm ( Y0) E W for 2 <m
,(x, 2(0
an
km
c.o * )
7m( Y, Y0)
a*
-› nnt-i(Y0)
km
Hm (X, X 0) ,-- c°1* Hm (Y , Y 0) ° I* -÷ Hm - i (Y o) we obtain hm .= co n 04*-1 gm _10 *co*-1 . Therefore, hm is a W-isomorphism for m = n and is a W-epimorphism for m = n + 1. I X and X 0 are simply connected, 7r2 (X, X 0) = 0, and Hm (X, X 0) is in a perfect and complete class W whenever 2
The theorem (9.1) does not hold if we merely assume that W is perfect and weakly complete. For example, let X=A xB and X0 = A X b, where A and B arc simply connected, B is W-acyclic, and b G B. By making various choices of A, one can see that (9.1) is true for a given class W iff W is perfect and complete. Similarly, (8.1) is true iff W is perfect and weakly complete. 10. The Whitehead theorem Remark.
Let W be a perfect and comfilete class. If X and Y are simply connected spaces, : X 17 is a map such that /* :n2 (X) -±n2 (Y) is an epimorphism, and n > 2 is a given integer, then the following two statements are Theorem 10.1.
equivalent:
(1) /* :yrm (X) -(Y) is a W-isomorphism for ni < n and is a W-epimorphism for ni = n. : Hm (X) H m (Y) is a W-isomorphism for ni < n and is a W-epi(2) morphism for ni n.
Consider the mapping cylinder Zf of the map /. By (I; § 12), both X and Y can be naturally imbedded in Z1 and Y is then a strong deformation retract of 2.1. Thus the map / is decomposed into the composition ri of the inclusion map i: X Zf and a strong deformation retraction Y Zf ---> Y. Since r induces isomorphisms on homotopy and homology groups, (1) and (2) are equivalent respectively to the following two statements: Proof.
:
308
X. CLASSES OF ABELIAN GROUPS
(1') i* :7r.(X) ---)-74„(Z1) is a W-isomorphisan for ni < n and is a W-epimorphism for m n. (2') i4: H(X) 11(Zf) is a W-isoinorph,ism for ni
whenever 2 < ni < n. whenever 2 < m < n. By (9.1), we conclude that (1") and (2") are equivalent. This entails the equivalence of (1) and (2). 1 (1") nni (2.f , X) E (2") Hm (Zf, X) e
EXERCISES A. Examples of Classes of Abelian Groups.
In addition to the classes given in § 2 a,nd § 3. we give the following examples: 1. The class of all abelian groups with power not exceeding a given infinite cardinal number sta . In particular, if xa is the power of the set of natural numbers, this reduces to the class da, of all countable abelian groups. Verify that this class is perfect and weakly complete but not complete. 2. The class of all abelian groups A such that there is an integer N depending on A with Na 0 for every a c A. Verify that this class is perfect and complete but not strongly complete. 3. The class of all abelian groups satisfying the descending chain condition. Verify that this class is perfect and weakly complete but not complete. B. Composed Homomorphisms By using the natural exact sequence S(I, g) of two homomorphisms /:A B and g :B-±C, prove the following six assertions for a given class W: 1. If f and g are W-monomorphisms, then so is gl. 2. If f and g are'-epimorphisms, then so is et. 3. If gf is a W-monomorphism, then so is /. 4. If gf is a W-epimorphism, then so is g. 5. If gl is a W-monomorphism and f is a W-epimorphism, then g is a
W-monomorphism . 6. If gf is a W-epimorphism and g is a W-monomorphism, then / is a W-epimorphism. C. On Perfectness and Completeness
Prove the following assertions: L For any class of abelian groups, the following three statements are equivalent:
EXERCISES
309
(a) W is complete. (b) A e W implies A0BEW for every abelian group B. (c) For any A e W, every finite or infinite direct sum of groups isomorphic to A is in W. . 2. Every strongly complete class is perfect and complete. It is unknown whether there is a class which is not perfect or not weakly complete. D. The C-Generalization of the "Five" Lemma
Prove that the "five" lemma, [E—S; p. 16], remains true modulo a class W. Precisely, if we have two exact sequences each with five terms and five homomorphisms of the groups of the first sequence into the corresponding groups of the second sequence with the commutativity relations being satisfied, and if the four extreme homomorphisms are ''-isomorphisms, then the middle homomorphism is also a 'C-isomorphism. E. The C-Inverse Homomorphism Theorem
Consider a class W and an exact sequence
A. 1 —> A 2 -/-L--> A 3 ±-> A4 ->' A 5 Assume that there exist two homomorphisms gl : A 2 -÷ A 1 , and g4 : A 5 --->A 4 such that the endomorphisms / 1 g 1 and /4g4 are '-isomorphisms. Define a homomorphism h: A 3 + A 5 --)-A 4 by taking h(x,y) — / 3 (x) + g5(y). Prove that h is a ''-isomorphism. F. The Products of C-Equivalent Groups
Let W be a complete class. Prove that, if A and B are 'C-equivalent respectively to A' and B', then A 0 B and Tor(A,B) are W-equivalent respectively to A' 0 B' and Tor(A' , B'). G. C-Exact Sequences
Let W be a class and A, B two subgroups of an abelian group G. We say that A and B are ce-equal if the inclusion homomorphisms A n B --- > A and A n B --> B are ?-isomorphisms. Replacing equality by W-equality, one can define the notion of a W-exact sequence. 1. Establish the elementary properties of W-exact sequences as in [E—S; p. 50]. 2. Generalize the results in (VIII; § 8) to obtain various fundamental W-exact sequences. H. On Induced Homomorphisms
Let X, Y be simply connected spaces, f: X -4- Y a map such that /* : 7 r2 (X) --> n 2 (Y) is an epimorphism. Assume that the homology groups are finitely generated.
310
X. CLASSES OF ABELIAN GROUPS
1. Let eF denote the class of all finite abelian groups, ,F the class of all torsion groups, and G a field of characteristic zero. Prove that the following four statements are equivalent: (a) /* : Hm (X) -->- .11.(17) is an g--isomorphism for In < n and is an g--epimorphism for In -- n. (b) 14t : Hm (X) ---). - H m (Y) is a F-isomorphism for in < n and is a .r-epirnorphism for In — n. (c) /4 : II.(X; G) --->- Hm (Y ; G) is an isomorphism for m
CHAPTER XI
HOMOTOPY GROUPS OF SPHERES 1. Introduction Finally we come to the determination of certain of the homotopy groups of spheres. The calculations are particularly based on the results of the previous two chapters, and, since they are quite technical, we will not attempt to summarize them here. However, in the course of the development, several topics of independent interest appear: Freudenthal's suspension theorem (stated in § 2 and proved in §§ 2-5), pseudo-projective spaces and Stiefel manifolds (§§ 10-11), and the Hopf invariant of a map / : S 212-1 —> Sn ( § 14). We have already seen that if r
2. The suspension theorem Let n> 1 and consider the n-sphere Sn as the equator of the (n + 1)sphere Sn+1 with u and y denoting respectively the north and south poles of S+1. Pick a point so in Sn and consider the space W A(S 11) of loops in Sn+ 1 with so as basic point. There is a natural imbedding i : W described as follows. For each x c Sn , i(x) is the loop in Sn+1 joining so to u, u to x, x to y, and y back to s o , all by shortest geodesic arcs. That i is a homeomorphism of Sn into W is obvious. Furthermore, the loop i(so) is homotopic to the degenerate loop wo e W which maps I into so by means of a natural homotopy; in other words, the points wo and i(sa) of W are connected by a natural path a in W. Hereafter, we shall identify x and i(x) for every x e S. Thus, Sn becomes the subspace i(Sn) of W and 1: Sn ---> W reduces to the inclusion map. For each m > 0, i induces a homomorphism
i* :7c.(Sn , so) -->nm (W , so), the path a induces an isomorphism
o.* m,(W , s 0) 311
m (W , w 0) ,
XI. HOMOTOPY GROUPS OF SPHERES
312
and, according to (IV; 2.2), we have an isomorphism h* nnt(W zoo) nin-f-i(S 71+1, s0) . Composing i * , a* , and h* , we obtain a homomorphism - , so)
:grm (Sn , so)
for each m > 0, called the suspension. One can verify that this definition is equivalent to the more general one given in (V; § 11) for this special case. 2.11. (The Suspension Theorem). The suspension E is an isomorphism if ni < 2n — 1 and is an epimorphism if m 2n — 1. Theorem
Proof. Since a* and h* are isomorphisms, it suffices to prove that i,R is an isomorphism if m < 2n — 1 and is an epimorphism if m = 2n — 1. Thus, according to Whitehead theorem, (X; 10.1), it suffices to prove that the induced homomorphism i#: Iim (Sn) ,Hm(W) is an isomorphism if m <2n — 1 and is an epimorphism if in = 2n — 1. Since Hm (W) Z, if m 0 mod(n), Hm(W)--- 0,
0 mod(n),
if m
by (IX; 13.4), it remains to prove the following Lemma
2.2. i# : H(S)
This lemma will be proved in the next three sections; we conclude this section with one immediate consequence of the suspension theorem. Let U and V denote respectively the north and south hemispheres of Sn+1 , then, using (V; 11.1), we have the following Corollary
2.3. The excision homomorphism
e* :nm (U, Sn)
7rm(S n+1, V)
is an isomorphism whenever 2
1 5n+1 , V).
Since V is contractible to the point so , g is homotopic in (Sn+1, V) to a map h : (En+1, Sn) --> (Sn+1 , so)
which represents a generator of nn+i (Sn+1, so). For any element cc of nm,(Sn, so), choose a map 95: Sm --->Sn which represents a, The map 4 has an extension
(Em+i , Sm)
(End-i Sn).
3. THE
CANONICAL MAP
3 13
Then it can be seen that the composed map l'op represents the element E(a) in an,,,(Sn 41-, so).
3. The canonical map Consider the space of paths X [Sn+1 ; so, Sn+1]. According to (III; § 13), X is a fiber space over Sn+1 with a projection : X --->Sn+1 defined by co(x) = x(1) for every path x e X and with fiber W = Let U and V denote the north and the south hemispheres of Sn+1 respectively and let ,K0 0) -1(Sn) , X tt 0) -1(U) , Xv co _1(v) . We are going to define a map
x : (U x W, Sn x W) -> (X u , X0) which will be called the canonical map. For each point b E U, let y(b) E Xu denote the path joining so to u and then u to b by geodesic arcs. The assignment b ---> y(b) defines a cross-section y: U X. Then x is defined by x(b, i) = /y(b) for each b E U and f e W, where / y(b) denotes the product of the paths f and y(b). As a consequence of the construction, we have cox(b, Lemma 3.1.
b, (b E
e W)
The canonical map x is a homotopy equivalence.
Proof. Let 2: (X e , X 0) ->
x W, Sn X W) be the map defined by
2(x) = (co(x), x • [yco(x)] -1 ),
(x
E Xu).
where 1yco(x)] -1 denotes the reverse of the path yco(x). Then we have x• fyco(x)] -1)
2, (I'Y(b))
(x• [yco(x)] -1).yco(x),
( 1), [17(b)1'[Y(v)] -1).
Hence, n2 and An are both homotopic to the identity maps. I Next, let us define a map :S x W W by taking ,u(b, f) / • i(b) for each b E Sn and fE W, where i : S1Z ---> W denotes the imbedding in § 2. Lemma 31.
y =
The map ,u is homotopic in X v to the map y : S X W X 0
I Sn
definby
X W.
Proof. Intuitively, a homotopy of itt to y is accomplished by "unwinding" the path i(b) to half its original length. More precisely, define a homotopy : Sn -> Xv , (0 < < 1), by taking [ht(b)](s)
ri(x)]
s
(
1
t
), (b E Sn, S E
t
/).
3 14
XI. HOMOTOPY GROUPS OF SPHERES
Then ho = i andh1 = y. Define a homotopy k t : Sn x W X v , (0
4. Wang's isomorphism p * in the present section, we shall construct for each integer q > 0 an isomorphism
p* : H2,(Sn) Hq (W)
Let in follows.
lin÷q(W).
n q + 1. The construction of p * will be made in six steps as
Step 1. Since the south hemisphere V of Sn-o- is contractible to the point so,
an application of the covering homotopy theorem proves that W is a strong deformation retract of X. Hence the inclusion map induces an isomorphism
: Hm(X, W) Re, Hrn,(X, Xv).
Step 2. The inclusion map induces a homomorphisrn :
Let
Hm (xu, X 0) ---> Hm (X,
Du = Sn+1 \y, Dv = Sn+1 \u, D =Du n Dv ; Yu = co -1 (Du) , Yv = co -"(D), Y = Yu
n Y.
Since Yu and Yv are open sets whose union is X, the excision theorem holds and hence the inclusion map induces an isomorphism
Hm ( Yu , Y)
117,1(X, YV) •
Since, U, V, Sn are strong deformation retracts of Du , Dv , D respectively, an application of the covering homotopy theorem proves that Xu, Xv, X0 are strong deformation retracts of Yu , Yv , Y respectively. Hence n is an isomorphism. Step 3. Since the canonical map x is a homotopy equivalence, it induces an isomorphism ff n,(Xu, X0). Ilm (U x W, Sn X W) Step 4. By the Kiinneth theorem, we get an isomorphism
: Hu +i (U, Sn) Hq (W) H in(U X W,S X W).
Step 5. Since X is contractible, we have an isomorphism :
W) Ilm-i(W).
Step 6. Since U is contractible, we have an isomorphism Hu,i(U, Sn)
H(S).
Taking tensor products, we obtain an isomorphism
: .1-4 4. 1 ( U, Sn)
I/6 ( W)
Hu(Sn) H q (W).
5.
RELATION BETWEEN
p * AND
315
Composing these steps, we get an isomorphism
p*
--1-Tpe*0 -1 : H(S) 14(W) Hn +q (W).
This isomorphism p * is the same as the homomorphism p* in Wang's exact sequence (IX; 13.1) for the fibering to : X Sn41-. See also [Wang 21. 5. Relation between p* and 14,v The space W of loops has a continuous multiplication
M:Wx W.-÷W defined in (III; § 11). The total homology group OD
H (W) = H m (W) 112-O
becomes a ring under the Pontrfagin multiplication defined as follows. Let oc E Hp (W) and /3 G 14(W). By the Kiinneth theorem, a and 13 determine a unique element cc X fl of .Hp+q (W X W). The map M induces a homomorphism M * lim (W X W) ---> .H m (W) for every m. Then the Pontrlagin product of a and 16 is defined to be the element ocfl = 11/*(tx X fl) E 1-11)+4. (W). Proposition 5.1.
For every a. E H(S) and 13 e I4(W), we always have P*(oc
fl)
where i# : J-1.(Sn) Hn (W) denotes the homomorphism induced by the imbedding i : Sn W of § 2.
Proof. Consider the diagram
lim (U x W, Sn X W)
Hm (Xtt, X0) -IL Hm (X,
v)
Hm (X , W)
a 10
Rin _1 (Sn
X W) .
Hin-i (X0)
10
Hm-i(W) where o- and 7' are induced by inclusion maps and the homomorphisms a are boundary operators. The rectangules are all commutative and hence (1) By (3.2), we have (2)
xv * = Crit * .
By the Kiinneth theorem, we have an isomorphism X: H(S) 14(W),--zed Hn,q(Sn X W)
XI. HOMOTOPY GROUPS OF SPHERES
316
and a commutative rectangle Hn+q,i (U X W, Su x W)
Hn+i (U, S ) 0 14(W)
Hn ,q (S21 X W).
H(Sn) O Hq (W)
Hence we obtain
00 -1 .
X
(3)
Using (1), (2) and (3), we deduce p* =
= 140-1 = p*X.
==
Then it follows from the definition of la that P*(0c fi) = ra*Z(.7. 0 ,8) = fi /4(0c).
In particular, if q = 0, then Ho(W) is a free cyclic group generated by the element e represented by wo as a 0-cycle of W. For each oc e H(S), we have i44_(oc)
—
e • i 44.(a) = p* (cic
e) .
This proves Lemma 2.2. 6. The triad homotopy groups
Consider the space of paths By (IV; 3.1), we have
T
[W ; Sn, so].
cm (T) = nm , i (W , SI?)
for every m. Hence the homotopy sequence of the pair (W, an exact sequence • --> rc.(T) --> nIn,(Sn)
+i (S
1)
Sn)
gives rise to
n,(T) ---> • • .
This is essentially the suspension sequence of the triad (Sn+1 ; U, V), n. (T) being essentially" the triad homotopy group nm+2(Sli+1 ; U, V). See (V; §§ 10-11).
Because of this exact sequence, it is desirable to determine the triad homotopy groups nn,(T). The following lemma is an immediate consequence of the suspension theorem (2.1).
6.t nm (T) = 0 for every rit < 2n —2. To determine the higher homotopy groups of T, let us study the space of paths Q [W ; Sn, TV] Lemma
which is of the same homotopy type as Sn. Consider the projection co : Q ->W defined by co(a) = a(1) for every a E Q; then Q becomes a fiber space over W with fiber co - i(so) T.
7.
FINITENESS OF HIGHER HOMOTOPY GROUPS
Since Hm (W) O whenever 0 <m < n and .NT) -,0 <m <2n — 1, we have by (IX; 14.1) an exact sequence
H3n-2(T)
••
Hm+1(Q) -÷lim+,1(W) -±ffm(T) -÷11m(Q)
317
whenever
• -->H1(W) --->0 -
Since Q is of the same homotopy type as Sn, we have Hm+i (Q) 0 - Hm (Q) for every ni > n. This implies that Hm (T) Hm+1 (W) whenever n < ni < 3n —2. Hence, we deduce the following Lemma 6.2.
H 21 (T) c Z and Hm(T) -= 0 whenever 2n —1 < ni < 3n —2.
Choose a map f : S 271-1 T which represents a generator of the free cyclic group 7C 2 _ 1 (T) I/21, ( T). Then f induces an isomorphism
Then, by (6.1) and (6.2), it follows that fit : lim (S 2n -1) Hm(T) for every m <3n 2. An application of Whitehead's theorem proves that the induced homomorphism /* :2.rm (s27/-1) , 21.(T) is an isomorphism if m < 3n — 3 and is an epimorphism if m Hence we have proved the following
3n — 3.
is isomorphic to nm (S 292-1) for every m < 3n — 3 and n3n _8 (T) is isomorphic to a quotient group of n3 ...3 (S27"). Proposition 6.3. 7C m(T)
7. Finiteness of higher homotopy groups of odd-dimensional spheres
In the present section, we are concerned with an odd-dimensional sphere S. Since the homo topy groups of the 1-sphere 51- are completely computed in (IV; § 2), we may assume that n > 3. Consider an n-connective fiber space X over Sn with a projection : By definition,
= 0, (m < n), co * :7cm (X)
nm,(Sn), (m > n).
1). Then it follows that the fiber F is a space of the homotopy type (Z, n By (X; 8.3), z(S) is a finitely generated abelian group for every m. An application of the generalized Hurewicz theorem proves that Hm (X) is finitely generated for every m. Let K be a field of characteristic zero. Since n — 1 is even, it follows from (IX; Ex. G) that the cohomology algebra H*(F; K) is isomorphic to a polynomial algebra over K generated by an element of degree n — 1. Let a e Hn -l(F; K) be a generator of II* (F ; K). Consider Wang's cohomology sequence (IX; 13.2): • • -->Hin(X; K) Hm (F ; K) -› Hm --92-0-(F ;K) -÷ Hm+1 (X ; K)
-,
XI. HOMOTOPY GROUPS OF SPHERES
318
where p* is a derivation since n is odd. Since fin(X; K) — 0=.- Hu-1 (X; K), p* sends Hn -1 (F ; K) isomorphically onto ll"°(F ; K). Therefore, p*(oc) is a non-zero element of H°(F ; K)r:.-; K. Now let us prove that p* :
HP(92-1) (F ;
K) F.', H(7?-4 (n-1)(F ; K)
for every positive integer p. In fact, the vector space HP(n-1)(F ; K) over K admits oc29 as a basis. Since p* is a derivation, we have p * (kOCP) — pkp*(a)ccP -1, (k e K).
Hence p* is an isomorphism for every m > O. Then an exactness argument proves that .Fini(X; K) ,- 0 for every m> O. Since Hm (X) is finitely generated and K is of characteristic zero, this implies that Hm(X) is finite for every m > O. An application of the generalized Hurewicz theorem (X; 8.1) proves that 7Cm(X) is finite for every m. Thus, we have proved the following Theorem 7.1.
If Sn is an odd-dimensional sphere and m > n, then nn2,(Sn)
is finite.
8. The iterated suspension The natural imbedding Sn+1 c A(Sn4-2) of § 2 induces an imbedding i : A(Sn-o-) --->A 2(Sn+2).
Composing with the natural imbedding i:Sn--›-A(Sn+ 1), we obtain an imbedding k — fi : Sn -)-11_ 2 (Sn 4-2).
For each rn, k induces a homomorphism k * :n.(Sn, so) -÷7r,m (A 2 (Sn 4-2), so).
As in § 2, we have a natural isomorphism 1* : nm (Ap(sn+2) , so ) R.e nm+2 (Sn-1-2 , Proposition 8.1.
1 *k * is equal to the iterated suspension E2.
Proof. By § 2, there is an isomorphism :7rtn (A(s2H 3.) , so -
)
76
. +I (sn+i , so) .
Similarly, there are isomorphisms ,. p : 76.(A2(st14-2) , so ) s.. , jrnt+1 (A(Sn+2) , so) , y : nra+1 (A (Sn+2) , so) rrtz," n (sn+2, so ) . Then l* = yfl and k * = . The proposition is a consequence of the cornmutativity of the diagram: i* grm (Sn, so) ---> ani (A(Sn 4-1), so) l '—'--)- 7-cm (A2(Sn4-2) , so)
la am +1 (91+1
So)
? -->. 7Cm+ 1(11(Sn+2 ) , So)
I
Y
ani+2(sn+ 2 so). 1 ,
The following proposition is an immediate consequence of (8.1) and (2.1),
9.
THE P-PRIMARY COMPONENTS OF nnt(S 3)
319
homomorphism k * is an isomorphism if m < 2n 1 I. and is an epimorphism if m = 2n If we study the p-primary components instead of the whole homotopy groups, then we can deduce more detailed information from the iterated PrOposition 8.2. The
—
—
suspension E 2. n > 3 be an odd integer, p a prime number, and the class of all finite abelian groups of order prime to p. Then the iterated suspension E2 :m(S) „grm+2(914.2) Theorem 8.3. Let
-isomorphism if m < p(n + 1) — 3 and is a W-epimorphism if m P(n + 1 ) 3. Proof. According to (8.1), it suffices to prove the theorem for the homomorphisms k*:am(Sn) ___>. 7cm (A2(Sni-2) ) is a
indUced by the natural imbedding h : Sn A 2 (Sn+2). Let K be a field of characteri3tic p. Then k induces the homomorphisms k# : Hm(A 2 (Sn+2); K) Hm(Sn K). By Whitehead theorem (X; 10.1) and (X; Ex. H2), it suffices to show that klt is an isomorphism for every m <(n + 1) 3. By (8.2) k * is an isomorphism if m <2n — 1 and is an epimorphism if 2n — 1. An application of (X; 10.1) and (X; Ex. H2) proves that klt is an isomorphism for m <2n — 1. By (IX; Ex. F6 and F7), we have —
Hm(A 2 (Sn+2) ; K) = 0, (n < m < (n + 1) — 3). Since n >3, it follows that k* is an isomorphism for every m < p (n
1)-3. I
If n > 3 is an odd integer, p a prime number, and m
the corollary by induction on n. When n = 3, there is nothing to prove. Assume that q > 5 is an odd integer and the corollary is true for every odd integer n with 3
—
–
-
—
9. The p-primary components of 01m (S3)
The corollary (8.4) reveals the importance of finding the p-primary components of the homotopy groups of the 3-sphere Sn.
XL HOMOTOPY GROUPS OF SPHERES
320
Consider a 3-connective fiber space X over S3 with a projection w : X -> S3 and fi ber F which is a space of homotopy type (Z, 2).
The integral homology groups of X are as follows: Hm(X) — 0 if m is odd; H2 (X) is cyclic o/ order n for every n > O. Thus, the first few homology groups are: Z, 0, 0, 0, z2, 0, z3, 0, Z4, 0, Z 5,• • • Lemma 9.1.
Proof. Consider Wang's cohomology sequence (IX; 13.2) : Hm(X) Hm(F)
••
H -2(F) Hm+1-(X) • -
where p* defines a derivation of H* (F) . By (IX; Ex. G), H* (F) is isomorphic to a polynomial algebra over the ring of integers generated by an element of degree 2. Since nm (X) — 0 for m < 3, we have Hm(X) = 0 for m < 3. Hence we obtain
p* : H2(F) H°(F) = Z.
Let oc denote element of H2 (F) with p* (or.) = 1. Then oc generates the algebra nocn-1. H*(F) and p*(an) Let m — 2n with n > 2. The sequence becomes H2 (X) H 2 (F)
H22(F) H2111-1(X)
Since H2 (F) is free cyclic with ocn as generator and p*(ccn) na.n -1 , it follows that p* is a monomorphisrn and its cokernel is cyclic of order n. Then the exactness implies that H2 (X) — 0, 212n +1 (X) Z . The lemma follows from this and the duality between homology and cohomology. I
I/ p is a prime number, then the p-primary component of nm (S 3) is 0 if m < 2p and is zp if m = 2p. Theorem 9.2.
Proof. Let V denôte the class of all finite abelian groups of order prime to p. By (9.1), we have Hm (X) c V whenever 0 <m < 2. An application of the generalized Hurewicz theorem proves that 7cm (X) c V whenever 0 < m < 2p and 7.1. 2p (X) is '-isomorphic to Zp. Since X is a 3-connective fiber space over S3, we have 7r7n (S3) 7rm(X) for each ni> 3. This implies the theorem. I // n > 3 is an odd integer and 1) a prime number, then the p-primary component of am (Sn) is 0 if ni < n ± 2p — 3 and is Zp M n 2p — 3. Corollary 9.3.
—
Proof. Since p > 2, we have It + 2p — 3
10. PSEUDO-PROJECTIVE SPACES
321
10. Pseudo-projective spaces
If we adjoin to the n-sphere Sn an (n 1)-cell En+1 by means of a map = Sn -÷ Sn of degree h as in (I; § 7), we obtain a space p7; -Fi
p
which is called a pseudo-projective space, [A-H; p. 266]. We shall assume that h > 0; in this case, the homology groups of P are Ho (P) Rz, Z, H(P) Hm(P)
0,
Lemma 10.1. For every m < 2n 0 -->nn,(Sn)
—
n.
in 0 0, M
1, we have an exact sequence
Zh, 7Cm,(P) --->
Tor(nm,_1 (Sn), Zh,)
O.
Proof. The map 0 extends to a map y : En+1 ---> P in an obvious way. We
obtain a commutative diagram: 0
• • • ---> 7'471+1 (P ,Sn) ---> 7Cm(S n) —>7Cm(P)
(Sn)
-÷ • • •
A
F*
V*
nn,÷1 (En-4 ,Sn)
>
9S*
V*
nm (En-o-,Sn) L
Cm(S)
7rin (Sn )
where the top row is the homotopy sequence of the pair (P, Sn) and 0 * : 7Cm(S) nin,(S n)
: 7(m(Er6 +1,
Sn) -÷ 7rm (P , Sn)
are induced by the maps 0 and y. Since 0 is of degree h, we have 96* (a) = ha for each a E m (Sm) whenever ni < — 1; on the other hand, y* is an isomorphism for every m < 2n 1. See Ex. A and Ex. B at the end of the chapter. Hence we may replace the homotopy sequence of (P, Sn) by the exact sequence 7cm ( 5 n)
nm,(Sn) -›-am (P) -nm1 (S)
nm _ 1(Sn)
for every m < 2n I. Since the kernel and the cokernel of 0* : 7Vm(S) nm(Sn) are isomorphic to Tor(nm (Sn), ZA) and nm (Sn) h respectively, the exactness of this sequence implies the lemma. I Let X denote a 3-connective fiber space over S3 with projection w : X -÷ 53. Since the p-primary component of 7r2 (X) is cyclic of order p, there exists a map / : S 22) X which represents a generator [f] of this p-primary component of n2p (X). Consider the pseudo-projective space P = . Then 521) c P. Since pm = 0,1 can be extended to a map g : P X. Composing with w : X S 3, we get a map 2C = co g: P S3 which induces the homomorphisms X* :
(P)
m(S3).
322
XI. HOMOTOPY GROUPS OF SPHERES
is a monomorphism for m < 4p —1 and sends 7Cm(P) onto the p-primary component of nin (S 3) for m < 4p — 1. Lemma 10.2. Z *
Proof'. It suffices to prove the lemma for the induced homomorphisms g* :grin (P)
m (X).
It follows from the generalized Hurewicz theorem (X; 8.1) that TCm(P) is a p-primary group for every m. Hence g* sends nm (P) into the p-primary component of nm (X). Let V denote the class of all finite abelian groups of order prime to p. It remains to prove that g* is a V-isomorphism for in < 4p —1 and is a W-epimorphism for m . 4p — 1. From the construction of g, one can see that g#: H2p (P),--.-d, H 2p (X). Then, by (9.1), glt : Hm (P) -÷ Hm (X) is a V-isomorphism for m < 4 5 . An application of the Whitehead theorem (X; 10.1) completes the proof. I This lemma reveals the importance of finding the homotopy groups of p , D2p+i -L
P
'
If p is a prime number and m <4 — 2, then nm (P) --- 0 if m is different from 2, 4p — 3, and 4 5 — 2, while: Theorem 10.3.
71 222(P)
wd
ZP;
r4 _3 (P)
no -2 (P)R-.%
Zp ,
if p > 2.
Proof. According to Hurewicz's theorem, we have 7Cm(P) = 0 for each m < 25 and no (P) F.-, Zp . Applying (10.1) with n . 2, h . p, and m --‹ 4 5 — 3, we obtain an exact sequence 0 --)-nm (S2P) 0 Zp -9-cm(P) ---> Tor(nm _1 (S 22), Zp) ----> O.
By the suspension theorem (2.1), we have nm (S 24))r-z-./, nm _.1 (S 2P -1) for every m < 4p — 3. Since 2p —1 > 3, it follows from (9.3) that the p5-primary component of nm _..1,(S 2P -1) is 0 if m < 4p — 3 and is Z1, if in --= 4p — 3. Hence we obtain: 7rm(P) = 0, (2 5
By (10.2), the 75-primary component of nn,(S 3) is 0 whenever 2p <m < 4p —3 and is Zp if m — 4p — 3. Then, by (8.4), we deduce that, if n >3 is odd, the p-primary component of nm (Sn) is 0 whenever n + 2 75 — 3 < m
II. STIEFEL MANIFOLDS
323
If p is a prime number, then the p-primary component of am (S3) is 0 if 2p <m < 4p — 3 and is zp is m 4p — 3. 11 p > 2, then the p-primary component of -2(S3) is 4. Corollary 10.4.
If n > 3 is an odd integer and p a prime number, then the p-primary component of nm (Sn) is 0 if n + 2p — 3 < m < n + 4p — 6 and that of nn+42)---6(S n) is 0 or Z. Corollary 10.5.
11. Stiefel manifolds Let n > 4 be an even integer and consider the Stiefel manifold V — of all unit tangent vectors on Sn, (III; Ex. G). Then, V is simply connected and its homology groups are as follows: 110(V)
Z, H
1 (V)
Z2, 1/222-1(V)
Z
and all other homology groups are zero, {Stiefel 1, 2] and ES; p. 1321 Since V is the tangent bundle of Sn, it is a fiber space over Sn with a projection co: V -÷S and fibers homeomorphic to Sn ---1. According to (V; § 6), this fibering gives an exact sequence • • • -4- nm(V)
r,
nni (Sn.)
This exact sequence is usually used to deduce properties of the homotopy groups of V. Here, on the contrary, it will be used to study the groups 7c,n (Sn). Let um denote the generator of 7m(S) represented by the identity map. Consider the following part of the sequence: a)* ( V) —
nn(Sn)
el*
*
nn-1(V)
O.
By Hurewicz's theorem, n_ 1 (V) Z 2. Since r* is an epimorphism, the exactness of the sequence implies that the image of 4 is a subgroup of an _1 (Sn -') of index 2. Hence we deduce that 4 (it.) = 2un ...1 . In fact, it is known that d* (un) 2un _ 1 , but we shall not need this refinement. The structure of the homomorphism 4 with m> n is described by the following X is a fiber space over Sn with a pathwise connected fiber F, then the homomorphism 4 in the exact ho-motopy sequence Lemma 11.1. If
• • • 7tm(X) ---> 7rm(S n)
d*
ni—i (F) -±
—1 (X) -÷ • • •
sends the suspension E(a) of any element cc E nra _1 (Sn -1) into the composition d(u) 0 a. If a map k :SI" F represents 4 (74,0 , then k induces a homomorphism k * :nnt _1 (Sn -1) 7r,1 (F) which gives k(oc) = d(u) 0 oc for each oc in
324
XI, HOMOTOPY GROUPS OF SPHERES
(Sn-1). Hence the relation stated in the lemma means that the following triangle is commutative: grm _i (Sn -1)
g-c,,(Sn)
Proof. The projection co : X Sn induces an isomorphism co * : m(X, F) 7rm(S 4). By definition, d* is co;' followed by the boundary homomorphism : grm (X , F) --> 7Cm -1 (F). A representative map h : F of d(u) can be constructed as follows. Sn-1) so) Let h: (Es, be a map which represents U. It follows from (Sa, the covering homotopy theorem that there exists a map H: (En, Sn -1) —> (X, F) such that coll. h. Then the restriction k H 1S 21-1 represents d(u). Let a e74m _1(Sn -1-). It remains to prove k(oc) d * E (a). Let 95: Sm --1 ---> Sn --1 represent oc; then k * (a) is represented by k.95. Extend ç6 to a map i : (Em, Sm -1) (En, S"). By § 2, hp represents E (cc). Since cofhp =p, Eli) I Sm --1 represents d* E (a). I HT represents co *---1- E (cc) and hence k5t. Next, let us study the homotopy groups of the Stiefel manifold V Vrt+1,2. Using the generalized Hurewicz theorem, we can deduce that: (1) grm (V) is finitely generated. 0 if m< n — 1. (2) nm (V) Z 2. (3) 7rn-1(V) (4) grm (V) is a finite 2-primary group whenever n — 1 < ni < 2n — I. (5) 2._1 (V) is isomorphic to the direct sum of Z and a finite 2-primary
group. Lemma 11.2. 1 denotes the class of all finite 2-primary groups, then there exists a map q : S 211-1 --> V such that the induced hamomorphism * ani (S. 2n -1) _÷ anz (v)
is a W-isomorphisni for every m. Proof. By (5), there is a map q : S 271-1 --> V which represents a free element oc of 21 2n-i(V) such that the free cyclic subgroup generated by oc is of index some power of 2. Since the natural homomorphism of gr2n _1 (V) into H21 _i (V) is a W-isomorphism, it follows that the induced homomorphism Hm (V) is a W -isomorphism for m 2n — 1 and hence for every m. An application of Whitehead's theorem (X; 10.1) proves the lemma. I We list the following additional results on the homotopy groups of V; these are immediate consequences of (10.2). (6) grm (V) is finite if ni> 2n — 1.
13. THE P-PRIMARY COMPONENTS OF HOMOTOPY GROUPS
325
(7) If p is an odd prime number, then the 5-primary component of arm (V) is isomorphic to that of n m (S2n --1).
12. Finiteness of higher homotopy groups of even-dimensional spheres is an even-dimensional sphere and m is an integer such that m> n and m 2n 1, then nni(Sn) is finite and n21 (Sn) is isomorphic to the direct sum of Z and a finite group. Theorem 12.1. // Sn
—
Since 7r,n (S 2)r,-;_, 7c.(S 3) for every m> 2, the theorem is true for n . 2. Hence we may assume n >4 and apply the results of § 11. If ni> n and m 2n — 1, then both am (V) and mi (Sn') are finite. Therefore, the exactness of the sequence nm (V) -+7 m (S) -÷ n 2,_ 1 (Sn -1) implies that 7m(S) is finite. To study the critical case m . 2n — 1, let W denote the class of all finite abelian groups. It follows from the exact sequence that Proof.
w* : 7r271,-1( 17) -›- n2n-1(S n) is a W-isomorphism. This implies that n 2._1 (Sn) is isomorphic to the direct sum of Z and a finite group. I
13. The p-primary components of homotopy groups of even-dimensional spheres In the present section, we are concerned *ith the p-primary components of the homotopy groups nm (Sn) of an even-dimensional sphere S. Since nm (S 2) -,-.., nm (S3) for every m > 3, we may restrict ourselves to the case n _> 4 and apply the results of § 11. Consider the following part of the exact sequence appearing in § 11: amil.(S n) Ll* nni(Sn-1) '''-÷ nnt(V) a ' --' 7m(S) -cL'L' Using the homomorphism co * and the suspension /, we define a homomorphism r :7(T7) ± 7l m _ 1 (Sn --1) --) - 7r,(Sn) by setting Iloc, ig) — co* (a) ± /(16) for each a e 7rm (V) and ,8 en._1 (Sn A ). Lemma 13.1. 1/ W
denotes the class of all finite 2-primary groups, then P is a
'-isomorphism. According to (11.1), the suspension E followed by d* is the induced endomorphism k * on nni-i (S n-1) of a map k : Sn -1- --> Sn -1 of degree d — ± 2. By Ex. A6 at the end of the chapter, k * is a W-automorphism. Hence the lemma follows as a consequence of (X; Ex. E). 1 Proof.
denotes the class of all finite 2-primary groups, then the homotopy group am (Sn) of an even-dimensional sphere Sn is (g-isomorphic to the direct sum of grm (S 2n -1) and nni_1(Sn-1). Theorem 13.2. If W
XI. HOMOTOPY GROUPS OF SPHERES
326
Proof. Since yon (S 2) 7(.(S 3) for every m > 3, the theorem holds for n = 2. If n > 4, then (13.2) is a direct consequence of (13.1) and (11.2). I The importance of (12.1) and (13.2) is that the calculation of the homotopy groups of an even-dimensional sphere, except for their 2-primary components, reduces to that of the homotopy groups of odd-dimensional spheres. Precisely, we have the following Corollary 13.3.
If n is even and p an odd prime, then the p-primary com-
ponent of 7m(S) is isomorphic to the direct sum of those of ani(S 291-1) and
14. The Hopf invariant In order to strengthen (13.2), we propose to present Serre's version of the notion of Hopf invariant. Consider the n-sphere Sn and a given point so E sm. Let .§2n A(Sn) denote the space of loops in Sn with s o as basic point. Consider the natural iso-
morphism
:g2n-2(nn) n2n-i(S n)
and the natural homomorphism h gr2n - 2(Qn) -->H2n-On) Z.
Let f: S2n-1 -->Sn be a given map representing an element [f] e7 27,1 (Sn). The Hopf invariant of f is the integer 11(f) uniquely determined by hi-1 ([f]) = H(i)u2 ,
where u 2 denotes the generator p* 2 (1) of H2._2 (0n) determined by the homomorphisms p* in (IX; 13.1). For other definitions of Hopf invariant, see Ex. C at the end of the chapter. When n is odd, H(f) is always zero; when n is even, there exists a map f with H(f) = 2; if n 2, 4 or 8, then there exists a map f with H(f) = 1, namely the Hopf maps of (III; § 5). See [Hopf 2] and [S; p. 113]. Theorem 14.1. Let n be an even integer and f S 2n -1 -÷Sn
a map with Hof
invariant II(f) = I? 0 O. Let V denote the class of all finite abelian groups of order dividing some power of k. If Xf:
m .1(S n-1) + nm (S 2
) --)-nm (Sn)
is the homomorphism defined by Xf(a , P)
E(00
f(fi)
x enm-1( 9"),
e 7tnb(S 2n-1),
then Xf is a V-isomorphism for every m > 1.
Proof. The theorem is obvious if n = 2. Hence we assume n > 4. The map / defines a map g: S22n -1 nn in the obvious way and g induces a homomorphism g* : .H*(S2n) H*([22n-1)
14. THE HOPF INVARIANT
327
of the cohomology algebras with integral coefficients. According to (IX; Ex. H), 1-/*(S2n) admits a homogeneous basis { ai} with dim (ai) = i(n —1) for each i — 0, 1 , • • • such that ao = 1, (a1) 2 = 0, (a 2)P = (5I)a 2p, a 1ot 2p
a2pa1
a 2p, 1 .
Similarly, H*(S221z -1) admits a homogeneous basis {bi} with dim (bi) i(2n — 2) for each i = 0, 1,» such that 1)0 = 1, (b 1)P
Since H(/)
(N)b p .
k, it follows that g*(a2) = kb,. Then we get (p!)g*(a2p) = g*(a229) = kP(bi)P
and therefore g*(a 2p) = kPbp. Since obviously g*(a2p .+1 ) = 0, it follows that g* is completely determined. Now let i : Sa -1 --)..f2n denote the natural imbedding in § 2 and consider the induced homomorphism i* • H(Q) ->11*(Sn -1).
By (2.2), e = i* (al) is a generator of Hn --1 (Sn -1). By means of the multiplication in Qn, we may define a map ,95 : Sn -1 X s22n---1 „Du by setting 0(x, w) = i(x) g(w) for each x c Sn -1- and %V E S2 211-1 . By (IX; Ex. H), H*(Sn -1 x .02n -1) is naturally isomorphic to the tensor product H*(Sn -1) H*(D2n -1). This enables us to determine the induced homomorphism
as follows:
y51* : .H*(Qn) 11*(Sn -1 X Q27 1) 1954*(a 2p) 964 (a2P+i)
hP•I 0 bp,
0*(ai.)95#(a2p)
kPe
bp.
D271-1) is a monomorphism and its Hence, 0,4* Han (pn) wiz cokernel is finite of order equal to a power of k for every dimension m. By duality, this implies that 0.# : Hin,(Sn-1 >< syn.-1) Hm(Q) is a f-isomorphism for every m. An application of the Whitehead theorem shows that the induced homomorphism
4)* grnz(Sn-1
X 122n-1)
„nin (Qn)
is a ce-isomorphism for every In. The group 7Cm(.91-1 X Q2n -1) is isomorphic to the direct sum 7Cm (Sn -1) ani (D2n-1‘) ; on the other hand, nm(Q) n.+1 (Sn). Then, the construction of çb shows that O. * reduces to the homomorphism Z1. I I/ ,11(1) = ± 1, then ,Cf is an isomorphism and hence the suspension E :nm _1 (Sn -1) -->nm,(Sn) is a rnonomorphism for every ni> 1. Corollary 14.2.
Because of the existence of a map f with 11 (f) = 2, (14.1) implies (13.2).
XL HOMOTOPY GROUPS OF SPHERES
328-
15. The groups an+1 (S ") and an+2 (S")
Since an,(51) -= 0 for each m> 1 and 7r.(52) nm (S 3) for every m> 2, we may assume n > 3. Theorem 15.1.
n. +1(Sn) is cyclic of order 2 for every n > 3.
Proof. Let X denote a 3-connective fiber space over S3. Then, by (9.1), we have Z 2. n 4(S3) rk-,, TC 4 (x) c .ff 4 (X)
By the suspension theorem (2.1), we deduce • • • tr -' n +1(Sn) ''Jj • • • • n,5(54) Hence n 1 (S) Z 2 for every n > 3. 1 One constructs the generator of 1 n+1(S9i) as follows. Let us consider the Hopf map p : .s3 5 2 defined in (III; § 5). According to (V; § 6), p represents the generator of n3(52). Since the suspension E n3(52) ->r4 (S 3) is an epimorphism, the suspended map Ep : S4 --> S 3 represents the generator of 714(53). Then the generator of n(Sn) is represented by the (n 2)-times iterated suspension En -2P of the Hopf map p. 7r4(S3)
Theorem 15.2.
n,,, 2( 5n) is cyclic of order 2 for every n > 3.
Proof. Applying (10.1) with n -= 4, h
sequence
2, m 5, we obtain an exact
0 -)-Z 2 (23) Z 2 > 5 (P ) -> Tor(Z , Z 2) -> O.
Since Z20 Z 2 W, Z 2 and Tor(Z , Z 2) = 0, this implies r5 (P) Z 2. Then, by (10.2), the 2-primary component of 'r5 (S 3) is isomorphic to Z 2. Since the p-primary component of 2 r 5 (53) is 0 if p > 2 by (9.2), it follows that n 5 (53) Z2. Since the Hopf map 57 -*S is of Hopf invariant 1, we may apply (14.2) with n .= 4 and ni 6. Hence, (2.1) and (14.2) imply that E :n5 (S 3) 7c6 (S 4). Thus, n6 (S4) c Z 2. Finally, by (2.1). we deduce 7C 6(S 4 )
277(.55)
•••
n +2(Sn)
•••
Hence 7C n + 2 (S n ) Pe Z 2 for every n > 3. 1 To obtain the generator for 7rn + 2(S n) , let i :S4 -÷ .11 denote the imbedding given by the definition of P. Then, by the proof of (10.1), the generator of n 5 (.11) is represented by the composition of i and E275: 55 -->- 54 • Composing with the map X: P -÷S3 in § 10, we obtain a representative map Xi E2p for the generator of n 5 (S 3). This implies that Xi represents the generator of 7c4 (S 3) and hence is homotopic to Ep :s4 -3.53. Therefore, the generator of n 5 (.5 3) is represented by 9, Ei5 E275 s5 where 75 : S 3 52 denotes the Hopf map. Then it follows that the generator of 7rn + 2 (Sn) is represented by the (n 3)-times iterated suspension En --3q of q for every n > 3.
16. THE GROUPS nn4.3(Sn) Corollary 15.3. n 3 (S 2)
Z, 7C 4(S2)
Z 2, 5 (S2)
329
Z2.
By (V; § 6), the generators of these cyclic groups are represented by respectively the maps
P s3
P
EP S 4 ÷ S2 , P 0 EP 0 E 2P s5 s2 -
-
16. The groups r +3(Sn) Theorem 16.1.
8 (S3)
Z12 .
Proof, Applying (10.1) with n — 4, h
sequence
0
Z
—
2, m = 6, we obtain an exact
Z -± 6 (4 ---> T (2 2 , Z2) -÷ 0.
Since Z20 Z 2 Z 2 and T or(Z 2 , Z 2) c Z 2, 7C6(11 is isomorphic to an extension of Z 2 by Z 2 and hence has 4 elements. Hence, by (10.2), the 2-primary component of 7T6 (S3) has 4 elements. By (9.2), the 3-primary component of 7(6 (S3) is isomorphic to Z3 and the p5-primary component of 6 (S 3) is 0 for every prime > 3. It follows that gr6 (S3) has 12 elements and hence is isomorphic to either Z 12 or Z Z6. Suppose that n 6 (S 3) Z6 . Let X denote a 5-connective fiber space Z over S3, then H6 (X) g6 (X) .7r6 (S3) Z ± Z6, and it follows from the universal coefficient theorem [E-S; p. 161] that H6 (X; Z2 ) i Horn VI 03 (X) ; Z 2)
Z
Z2.
This contradicts to (IX; Ex. 1); hence, we conclude that 6(S 3) Z 12'2. I Examination of the first paragraph of the proof reveals that the composition of the maps SG --)- S 5 — z2P-> S4 Sa represents an element of 7c6 (.53) of order 2. A generator of 7C6 (S 3) is represented by the characteristic map : S 6 -9- S 3 of the fiber bundle sp(2) over S 6 with S(l) as fiber, [Borel and Serre 1; p. 442]. For the definition of the characteristic map, see [S; p. 97]. Corollary 16.2. n6 (52)
Z12 .
A generator of a6 (S 2) is represented by the composed map Theorem 16.3. 7r7 (S 4 )
Z
:56 52.
Z 12.
Proof. Let us denote by q; S 7 --- S 4 the Hopf map in (III; § 5). Since H(q) = 1, we may apply (14.2) with n = 4, na = 7, and / = q. Thus, we obtain an isomorphism zq 2.r 7(54) 6(S3) + 7r ( 5 7 )
Since 7c7 (S 7) Z and n 6 (S 3) Z 12 , this proves the theorem, I From the preceding proof, it follows that q represents the generator of
XI. HOMOTOPY GROUPS OF SPHERES
330
the free component Z of :r7 (S4) and the suspended map E$ : S7 --> S 4 represents an element of order 12 which generates the torsion component Z 12 of 7C7 (S4). Theorem 16.4.
r+3(S)
Z24 i/ n > 5 .
By the suspension theorem (2.1), E maps n7 (S 4) onto n8(S 5). According to Ex. D at the end of the chapter, the kernel of E is the free cyclic subgroup of n7 (S 4) generated by the Whitehead product [e, e], where e denotes the generator of 7c4 (S4) represented by the identity map on S. On the other hand, it follows from a theorem on characteristic maps, [S: p. 121 1 , that 2 [q] e E [$] [e, e] Proof.
where e =± I depends on the conventions of orientation. Hence, in 7C8(S 5), we have
E2 [$]
e2E
[q].
This implies that n8 (.35) is isomorphic to 224 with I [q] as a generator. Finally, by the suspension theorem (2.1), we deduce 7r9 (S 6)
7c8(S 5)
••
7C714.3(Sn) rr:-5 • • • .
for every n > 5. 1 Obviously, a generator of n 11+3 (Sn), n > 5, is represented by the (n — 4)times iterated suspension En -4q : Sn 4-3 Sn of the Hopf map q : S7 -÷ S4 .
Hence 741+3 (Sn)
Z24
17. The groups arn+4(S ) Theorem 17.1..7r7 (S3)
Z2.
By (9.2), the p-primary component of n7 (S 3) is 0 for every prime > 3. By (10.4), the 3-primary component of 7(7 (S 3) is also O. Hence 7(7(S3) is a 2-primary group. Next, consider a 6-connective fiber space X over S. Then 7c7 (S 3) H7 (X) and hence we have 7C7 (X) Proof.
Hom (7c7 (S 3), Z 2) r.s H7 (X; Z2).
By (IX; Ex. 14), H7(X; Z 2) Z 2 . This implies that 7(7 (S3) is isomorphic to a cyclic group Zq with q 24 , h > 1. If h> 1, every homomorphism of a7 (.53) into Z 2 can be factored into n7 (S 3)
4 -÷ Z2.
Then it follows from the exact sequence in (VIII; Ex. 17) that Sea = 0 for every element a c H7 (X ; Z 2). This contradicts (IX; Ex. 14). Hence It = 1 and
7(7 (S 3)
c
According to [Hilton 2; p. 549], the two maps 0 E475 S7 S 3, E P 0 q S 7 S3
17. THE GROUPS n224.4(S-4)
331
are both essential. Hence they are homotopic and represent the non-zero element of n7 (S 3). Corollary 17.2. gr7 (S 2)
Z2.
The non-zero element of
7(7 (5' 2)
is represented by the homotopic maps
PO OI4P:S7.-÷ S2, PO EPOq: 51 .-S2Theorem 17.3. Proof.
a8(S4) Z 2 Z2.
As in the proof of (16.3), we obtain an isomorphism :
7 (S3)
+ 768(.5 7)
a8 (S 4).
Since n7 (S3) Z 2 and n8(S7) Z 2 , the theorem is proved. 1 The group 7r8 (S4) is generated by two elements cc and /3 of order 2. cc is represented by the hornotopic maps (e 0 E4p)
st, E (E
p
s4
q) S
and le is represented by q (j) Egp : S8 S4. Theorem 17.4. n9(S5) Z 2 . Proof.
Consider the following part of the suspension sequence in § 6:
n8 (S 4) -÷ 9(S5) -÷ 7(7(T ) 7(8(S 5) -± 0 n7(-54) As mentioned in the proof of (16.4), the kernel of E :7c7 (S 4) -÷7c8(S 5) is a free cyclic group. By (6.3), 7C.7 (7" ) Z. It follows from the exactness of the sequence that s6 :n7 (T) --->-7r7 (S 4) is a monomorphism and hence : 7c8 (S 4) -->-vr9(S5) is an epimorphism. Since 7 8 (T) 7C8 (S 7) r,■". Z 2 , the kernel of E : n8 (S 4) -± 7C 9 (S 5) contains at most two elements. On the other hand, consider the element oc of 7(8 (5.4) represented by E (e C Vp) : S 8 -÷ S 4. In 7c9(S5), we have grs(T)
X(cc) E 2 [4:] 0 E6 [P] ( 82 E[q]) 0 (E 6 [P]) = (e:E[q]) 0 (2 D[P])
0Hence the kernel of E :7c8 (S4) -->- 7c9 (S 5) consists of exactly two elements, namely, 0 and a. This implies that n9 (S 5) fr.,/ Z 2 . I The non-zero element of 7( 9 (S 5) is E(p) represented by the map E(q0E 6 5) : S 9 S 5. On the other hand, the Whitehead product [e, e] of the generator e of (S 5) is also non-zero and hence [e, e] = E(p) [Serre 3; p. 230]. Theorem 17.5.7c. +4 (Sn) ---- 0
n > 6.
Proof. By the suspension theorem (2.1), E maps n9(S 5) onto n 10 (S8). According to the delicate suspension theorem in Ex. D at the end of the chapter, E [e, e] O. Hence we obtain n10(S 6) 0. Finally, by (2.1), we
deduce Hence
7V10(S)nu(S
grn +4(S n) =
7) r•`-'
0 for every n > 6. 1
• R-1, grn+4(Sn)
•••
332
XI. HOMOTOPY GROUPS OF SPHERES
18. The groups 7c,, + ,(S"), 5 <
r<
15
In this final section of the book, we will list the groups an+r(S) for the cases r ---- 5, 6, 7, 8. For more detailed information, see [Serre 5, 6]. H. Toda has computed the groups nn+r(S) for 9 < r < 15. We will not list his results here; the interested reader should refer to [Toda 1,2] with recent corrections given in [Toda 3]. y
.-_-_-_. 5. 7 c7 (S 2) ,:s.,, Z 2 7C F AS 3) R., Z 2 9(S) ',-+t), Z 2 + Z 2 TC 10 (35) r■-f, Z 2 i (S6) re.-5 Z 7Ci nni-r(S 79 = 0, (n > 7).
7' ------
6. 7c8(S 2) ,-,..,_, Z 2
Z3 aio (S 4) ,--..,, Z 24 TC 2(S 3)r;-::',
+
Z2
2,
r .= _-
Z
7.
Z2
7C 3 (S 2) Ç--:-',
7 r 1,,(S 3)
is.-,-, Z1 5
gr ii (S4) R,-, Z15 i2 (S5) ss., Z30 n
Z60 gri4(S7)-^41-, Z 120
n 15 (58)
(At,-
Z + Z120 Z240,
r -- -,--
(n > 9).
8. Z15
n10(S2) 7r11(S 3)rr
j-, Z2 Z2
gr 13 (S5) ,,,,-,
Z2
,-...5 Z24 + Z2 n15(S 7),,..--3, Z 2 + Z 2 +Z 2 7C 16 (S8) g--.d, Z2+ Z 2 +Z 2 + Z2 gr n (S9),s--,., Z 2 +Z2 +Z 3 a 14 (S 6)
Z 2 + Z2,
(n > 1 0) .
EXERCISES
333
EXERCISES A. The distributive laws
Let ot c nn,(Sn, so) and /3 c 7c.(X, x0). If cc and /3 are represented by the maps f: (Ent, Sm-i) --->- (Sa, so), g: (Sn, so) ---> (X,, xo) respectively, that the composed map gf represents an element y of 7c.(X, xo). Prove: 1.The element y depends only on the elements cc, /3 and will be called the composition 18 0 OL of oc and p. ' 2. The right distributive law. For a given p enn(X, x0), the assignment a -+j9 0 a defines a homomorphism. In fact, this is the induced homomorphism g* . 3. The left distributive law. If X is an H-space with xo as homotopy unit or if a is the suspension E(6) of some element 6 e n.,_ 1 (Sn ---1-, so), then the assignment )8 -- > p 0 a defines a homomorphism. {S; p. 122]. In particular, let (X, x o) ---- (Sn, so). Consider a map g: (Sn, so) -÷ (Sn, so) of degree d and. study its induced homomorphism g* : nm,(Sn, so) -÷7c.(Sn, so). Prove: 4. g * (a.) — da for every oc C nin(Sn, so) if n — 1, 3, 7 or if in <2n — I. 5. If m — 3 and n — 2, then g* («) — d'a for every cc en 3 (S2, so). [Hopf 1]. 6. If V denotes the class of all finite abelian groups of order dividing a power of d, then g* is a V-automorphism for every in > 0 and n > 3. B. On relative (n ± 1)-cells
Let (X, A) be a relative (n + 1)-cell obtained by adjoining E 1 +1 to A by means of a map g: Sn --> A. Then g has an extension f; (En-0 - , Sn) --* (X, A) defined by /(x) — x for every x c En-o- \ S n = x- \ A. This map / is called the characteristic map of (X, A). If we identify A to a -single point, we obtain an (n + 1)-sphere Sn -fl- as quotient space with projection h: X ---› 5n4-1 • Choose s o e Sn and let xo — f() G A. Use so and xo as basic points of the homotopy groups. Verify that the rectangle .7r,-,4 (En4-1 , Sn) -IL+ 7r.(X
10
, A)
h*
7rm- i (Sn)
is commutative, i.e. h*/* — E O. Prove 1. If E is a monomorphism, so is /*. If E is an epimorphism, so is h*.
XI. HOMOTOPY
334
GROUPS OF SPHERES
2. If m < 2n, then /* is a monomorphism, h * is an epimorphism, and .(X, A) decomposes into the direct sum of hn(/) * and Ker(h* ). 3. If A is y-connected for some Y -<, n, then /* is an epimorphism whenever in < n + Y. See [J. H. C. Whitehead 6; p. 14] and [Hilton 1; p. 464]. As an application of these results, consider the relative (p + q)-cell (X, A) with X — SP X Sq and A = SP v Sq. By 2 and (V; 3.1), we obtain natural isomorphisms
grm (SP x
Sq, SP v Sq) ,:-' 7( ffi,4, 1 (SP+q -1 ) + KeY(h * ),
7Cm_4 (SP v Sq)R-27Cm_ i ( SP) + 7r,m— i (Sq) +
for every m < min (p,q) — 2.
nm1 (SP41-1) + Ker(h*)
2 75 + 2q —2. Finally, by 3, Ker(h *) = 0
if m
+q±
C. Other definitions of the Hopf invariant
In § 14, we gave Serre's definition of the Hopf invariant H(f) of a given map / : S 2n --1 ---> Sn, n >2. There are a few different but equivalent definitions given by various authors listed below. Each of these definitions has a natural generalization but the various generalized Hopf invariants are not necessarily equivalent. 1. Hopi's definition. By using a homotopy if necessary, we may assume that / is a simplicial map relative to some triangulations. Choose a pair of distinct interior points, u, y of n-simplexes of S. Then /-1-(u), / -1-(v) are disjoint (n —1)-manifolds in S 2n --1 ; and II(/) is the linking number of the (n — 1)-cycles /-4 (u) and /-4-(y). See [Hopi 2] and [S; p. 113]. This definition can be generalized to manifolds. 2. Steenrod' s definition. Let M denote the mapping cylinder of / with snc .111 and 5 2 n--1 M. The integral cohomology algebra I/*(M, S 2n-1) has a homogeneous basis { a, b} with dim (a) = n and dim (b) = 2n. Then, H(/) is the integer defined by a2 — H(f)b. See [Steenrod 3; p. 983] and [Serre 2; p. 286]. This definition can be generalized to the maps / : Sn+i -1 -->. Sn as follows. In this case, H*(M, Sn+"; Z2) has a homogeneous basis { a, b } with dim (a) — n and dim (b) = n + i. Then, H(f) E Z 2 is defined by Sqi a —
H(f)b. 3. White/wad's definition. Identifying the equator Sn -4- of Sn to a single point, we obtain a quotient space sn y sn with projection g : Sn ---)- Sn v S. The composed map d represents an element [gf] of 7c2n _1(Sn y Sn). Then H(f) is obtained by projecting [glj into the direct summand 7t2 _1 (S2-1) Z of 2 _1 (S v Sn). Since grm (sn
v sn),..,...., n.(sn) + gr.(sn) +
grm(S 21) + KeY(h *)
for every in <4n —4, this definition can be generalized to the maps
EXERCISES
335
1 : Sm -> Sn for m <. 4n —4 in an obvious way. See [G. W. Whitehead 2 3 and [Hilton 1]. Thus, for each m < 4n —4, we obtain a Hop/ homomorphism H :m(S) _>(52n-1)
Among the properties of this generalization of Hopf invariants, verify the following few: (1) If or. E nnitSn), M < 3n —3, and :83. , r/3 2 E 7Cn(X) , then.
(P1 + P2) 0 a --- Pi 0 tx + P2 0 tx + [Pp PO 0 H(a). 1) with m <4n —4, then H E(a) — O. (iii) If n is odd and In < 4n —4, then 211(a) = 0 for every a enm (Sn). (iv) If cc E(Sn) with nt <4n —4 and E(a) = 0, then H(a) -, 0 if n is odd H(a) e 2grm (S 2n-1) if n is even. 4. Hilton's definition. If we identify the subset Sn y sn of the product (11) If cc E m -»i(S
space sn x sn to a single point so , we obtain a 2n-sphere space with projection h: (Sn x S, 5m v S n ) -> 52m , so ) .
52m
as quotient
(
For each ni> 0, define a homomorphism
H* :alm (Sn) ->grm ,i(S 2n) by taking H* to be composition of the sequence
7c.,,(Sn)
--> 7c. (Sn y Sn) -1---> 7Cm 4. I (Sn x sn, sn y S m) 1-22-' -›-
where g* , h* are induced homomorphisms and p denotes the projection of nm (Sn y Sn) onto its direct summand 7Vin 1_1(S n X Sn Sn y S n ) . Prove the following relations: (i) H* . EH whenever m < 4n — 4. [Hilton I; p. 473] and [Hilton 3; p. 166]. (ii) If a egn (sq), p e7rm (Sn), then ,
H* (cx OP) — 1/* (a) 0 Z(P) + Zg(a) 0 INC() 0 11* (13)(iii) If n is odd and cc C nni (Sn), then 2H*(a) -- O. (iv) If n is even, cc Em(S) is of odd order, and H*(a) ,---- 0, then cc = E(fl) for some p E grm-i(Sn-1). D. The delicate suspension theorem The suspension theorem (2.1) is the easy part of Freudenthal's result and is usually called the crude suspension theorem. The delicate part of Freudenthal's result has been slightly strengthened to the following form, [G. W. Whitehead 2] : 1. The image of E : 7r2 (Sm) -->n2n+I (Sn+1) is the subgroup of z2n-Fi(S n+1) consisting of the elements of Hopf invariant zero.
336
XI. HOMOTOPY GROUPS OF SPHERES
2. The kernel of E :7r27,..1 (S/j) --)-7r27,(Sn4-1) is the cyclic subgroup of ar21 (Sn) generated by the Whitehead product [e, e], where e denotes the generator of nn (Sn) represented by the identity map. If n is even, [e, e] has Hopf invariant 2 and therefore has infinite order. If n is odd, 2[e, e] = 0, and [e, e] . 0 iff there is an element of gr 2,„±1 (Sni-1) with Hopf invariant 1.
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C. 1. Simpiicia/ spaces, nuclei and m-groups. Proc. London Math. Soc. (2), 45 (1939),
WHITEHEAD, J. H.
243-327. 2. On adding relations to homotopy groups. Ann. of Math., 42 (1941), 409-428. 3. On the groups nr(1771,m) and sphere bundles. Proc. London Math. Soc. (2), 48 (1944), 243-291. Corrigendum, 49 (1947), 478-481. 4. Combinatorial homolopy I, II. Bull. Amer. Math. Soc., 55 (1949), 213-245, 453-496. 5. On the realizability of homotopy groups. Ann. of Math., 50 (1949), 261-263. 6. Note on suspension. Quart. J. Math., Oxford (2), 1 (1950), 9-22. 7. A certain exact sequence. Ann. of Math., 52 (1950), 51-110. 8. On the theory of obstructions. Ann. of Math., 54 (1951), 66-84. WOJDYSLAWSKI, 1. Raractes
M.
absolus et hyperespaces des continus. Fund. Math., 32 (1939), 184-192.
Index A Absolute neighborhood retract, 26, 29-32, 59, 184, 198 Absolute retract, 26, 29 Admissible transformation, 135-136
B Basic point, 125, 137 Betti number, 28, 277 Bigraded exact couple, 234-236 Bigraded group, 231 Borsuk extension theorem, 29 fibering theorem, 103 hornotopy extension theorem, 31 Boundary operator, 112, 159 Bridge theorems, 59, 195 Brouwer fixed-point theorem, 4 Bundle property of a map, 65, 99 Bundle space, 65
C Y-notions, 298 Y-acyclic, 303 Y-aspherical, 306 Y-epimorphism, 299 Y-equal, 309 f-equivalence, 299 Y-monomorphism, 299 Y-null, 299 Characteristic class, 88 Characteristic element of a complex, 193 of a map, 189 of a pair of maps, 186 Classes of abelian groups, 297-310 complete, 300, 308 perfect, 300, 308 strongly complete, 300 weakly complete, 300 Classification problem, 13, 16, 187, 198 Classification theorem of covering spaces, 96 Hopf, 53, 59 primary, 191
343
Closed surface, 28 fundamental group of, 58 Coboundary operator, 208,214 Cohomology algebra of a space, 294 of a space of loops, 295 of (Z, n), 295 Cohomology group cubical singular, 261-262 of (n, n) 199 Cohomotopy group, 205-228 Cohomotopy sequence, 2 14-2 16 set, 205, 213 Cohrntpy Complex contractible, 200 CW-, 193, 254 Eilenberg-MacLane, 203 d-, 246 6-, 246 semi-simplicial, 140-142 simplicial, 140 singular, 195 Cone over a space, 20, 28 partial, 22 Contraction, 12 Covering homotopy, 62 Covering homotopy extension property, 62 absolute, 62 polyhedral, 62, 63 Covering ho-motopy property, 24, 37, 62, 63, 70, 99 absolute, 62, 98 polyhedral, 62, 63, 99 Covering homotopy theorem, 62, 66, 314 Covering map property, 43 Covering path property, 36 Covering space, 89-97 generalized, 104 of a torus, 105 regular, 92 universal, 91, 96 Covering theorem, 91 Covering transformation, 93 Cross-section, 70, 73, 99, 313 Curve, 103
INDEX
344 D Deformation, 12 cochain, 179 homeomorph, 103 obstruction theory of, 197 problem, 22 retract, 16, 33 Degree base, 266 complementary, 235 fiber, 266 of a map, 12, 38, 52, 57, 60 of homogeneity, 231 primary, 235 total, 235 Derived triplet, 111 Difference cochain, 179 Differential group, 229-231 derived group of, 230 filtered, 239, 244 filtered graded, 245-248 Differential operator, 229, 232 Dimension, 266 Direct sum theorems, 150-152
Eilenberg extension theorem, 180 subcomplex, 45, 174 Eilenberg-MacLane complex, 203 Equivalence theorem for covering spaces, 91 for homotopy systems, 135 Euler characteristic, 277 Evaluation, 74, 76 Exact couple, 232 associated, 263-266 bigraded, 234-236 cohomology, 251 • cohornotopy, 254 derived couple of, 232, 266-269 homology, 234, 251 hornotopy, 251, 252 of a bundle space, 251 regular, 236-238 Exact sequence, 115 Gysin's, 280-281 natural, 299 truncated, 284 Wang's, 282- 284 Exactness property, 115 Exactness theorem, 136 Excision theorem, 208
Extension, 1 Extension index, 176 Extension problem, 1, 2, 15, 20, 71, 198 Extension property, 28-29 Extension theorem Borsuk, 29 Eilenberg, 180 Hopf, 53, 59 primary, 190
Fiber, 62, 262 Fiber map, 71, Fiber space, 62-106 homotopy groups of, 152-154 n-connective, 156, 304, 320 sliced, 97 spectral sequence of, 259-296, 300-304 Fibering, 62 induced, 72 n-connective, 155 of spheres, 66, 100 Fibering property, 118 Fibering theorem for homotopy groups, 136 for mapping spaces, 83-84 Filtered d-group, 244-245 associated graded group of, 244 exact couple associated with, 245 spectral sequence of, 249 Filtered graded d-group, 245-248 Filtraction, 244-246, 262 Five lemma, 309 Freudenthal's suspension, 162, 227, 311 Fundamental exact sequence, 240-242 Fundamental group, 39-47, 193 as a group of operators, 130 influence on homology and cohomology, 201-202, 288-290 Fundamental homotopy lemma, 186
G Gamma functor, 255-256 Graded group, 231, 237, 238-239 associated, 244 Grassmann manifold, 101 Group bi graded., 231 Bruschlinsky, 47-52 differential, 299-231 fundamental, 39-47, 57, 130, 193, 201, 288
INDEX graded, 231 homology, 44, 261
homotopy, 107-164
re-, 201 p-primary, 298 torsion, 297, 310 Gysin's exact sequence, 280-281 H Homology group, 44, 261 cubical singular, 261 of (n, n), 199 of a group, 199
Homotopy, 11 addition theorem, 164-166 class, 13, 16 connecting maps f and g, 11, 15 index, 183 invariant, 18 partial, 13 relative, 15 unit, 81 Homotopy equivalence, 117 Horno-topy extension property, 13-15,
30-34 absolute, 13, 31 covering, 62 neighborhood, 30 Hom.otopy extension theorem, 30-31 Homotopy group, 107-164 absolute, 107-110 abstract, 128, 137 of adjunction spaces, 168 of covering spaces, 154 of fiber spaces, 152-154 of H-spaces, 139 of spheres, 311-336 relative, 110-112 relations with cohornotopy groups, 224-226 triad, 160, 316 Homotopy property, 117 Homotopy sequence of a fibering, 152 of a triad, 160-161 of a triple, 159 of a triplet, 115, 120, 136 Homotopy system, 119 axioms of, 120 equivalence theorem for, 135 equivalent, 121 group structures for, 123 inductive construction of, 135
345
properties of, 136 uniqueness theorem for, 121 Homotopy theorem Hopf, 53 primary, 191 Homotopy type, 17, 198 Hopf classification theorem, 53, 59 extension theorem, 53, 59 fiberings of spheres, 66 homotopy theorem, 53 invariant, 326-327, 334-336 map, 66 theorems, 52-56, 59, 255
H-space, 81 Hurewicz theorem, 57, 148, 166, 253 generalized 305 relative, 306 I Induced transformation of a map on cohomotopy groups, 206, 213 on fundamental groups, 42 on homotopy groups, 113, 125
K Kan complex, 141-142 Klein bottle, 28 Kiinneth relations, 202 Kuratowski 's imbedding, 27
L Leray, 229, 249 Lifting, 24, 69, 72, 86 Lifting problem, 24, 70, 72 Local system of groups, 129 simple, 131 Loop, 36, 311 degenerate, 40 equivalent, 39 product of, 39, 79 representative, 40 reverse, 40 space of, 79 -82, 295
M Map, 1 algebraically trivial, 67-69 association, 75 canonical, 313
INDEX
346
cellular, 172 characteristic, 170, 333 combined, 7 deformable into a subset, 16, 22, 24, 197 derived, 114 essential, 16 exponential, 35 fiber, 71 homotopic, 11, 15 inclusion, 2 inessential, 16 n-extensible, 175, 194-196 n-hornotopic, 182, 194-196 n-normal, 197 null-homotopic, 16, 33 of surfaces, 106 of exact couples, 242-243 of filtered d-groups, 244 of torus into projective plane, 106 on topological products, 102 partial, 1 suspended, 60 Map excision theorem, 207 Mapping, 1 Mapping cylinder, 18 partial, 21, 31 Mapping space, 73-78, 101-102 evaluation of, 74, 76 exponential law for, 77 fibering theorem for, 83-84 induced map in, 85 subbasic sets of, 73 Maximal cycle theorem, 293 Möbius strip, 27
Natural correspondence, 119 Natural equivalence, 122 Natural homomorphism, 44, 48, 287, 305 Natural projection, 9, 21, 143
0 Obstruction, 175-204, 227 cocycle of a -map, 177, 195 cohomology class, 180, 184 of a homotopy, 183 of a map, 177 primary, 188-190 set, 181, 185 One-point union of two spaces, 145
Pair binorm al, 208 n-coconnected, 226 n-simple, 197 Path, 36 component, 78 equivalent, 41 space of, 78-79, 313, 316 Path lifting property, 82-83, 98-99 Poincaré group, 40 polynomial, 277-280 Primary component, 298, 319, 325
Realizability theorem, 169 Regular couple, 236-238 a-, 236 6-, 237 graded groups of, 238-239 two-term condition for, 250 Relative homeomorphism, 10 Relative homotopy, 15 addition theorem, 166 class, 16 Relative homotopy group, 110-112, 137 Relative n-cell, 11, 333 Restriction of a map, 1 Retract, 5, 25 absolute (AR), 26, 29 absolute neighborhood (ANR), 26, 29-32, 59, 194, 216 deformation, 16, 33 neighborhood, 26 strong deformation, 16, 32, 33 Retraction, 5 Retraction problem, 5
Semi-simplicial complex, 140-142 complete, 141 degeneracy operators in a, 141 fundamental group of a, 193 of a group, 199 topological realization of, 170 Singular complex, 45, 196 admissible subcomplex of, 172 first Eilenberg subcomplex of, 45, 174 Singular n-cube, 259 degenerate, 260 faces of, 259
INDEX
of degeneracy q, 262 weight of, 263 Singular simplex, 45, 196 Space adjunction, 9-11, 31 base, 61, 65 binormal, 14 bundle, 65, 286 connective system of, 304 contractible, 12 covering, 89-97 director, 65 dominating, 32 fiber, 62 filtered, 234, 248 generalized covering, 104 H-, 81 homogeneous, 99 homotopically equivalent, 17 locally contractible, 32 locally pathwise connected, 43 locally simply connected, 93 mapping, 73-78 n-coconnected, 210, 226 n-connected, 57, 148, 166, 210 n-connective, 155 n-simple, 131-134 of curves, 103 of homotopy type (a, n), 168, 198 orbit, 9, 200, 286 pathwise connected, 41, 78 pseudo-projective, 321-323 quotient, 9, 99 real projective, 321-323 regular covering, 92 semi-locally simply connected, 93 simply connected, 42 solid, 2, 26 topological sum of, 10 total, 61 totally pathwise disconnected, 89 universal covering, 91 Spectral cohomology sequence, 292 Spectral homology sequence, 271 Spectral sequence associated with exact couple, 233 limit group of, 234 of a fiber space, 259-285 of a regular covering space, 285-287 of filtered d-groups, 249 Sphere as homogenous space, 100 connective fiber space over, 296
347
finite groups operating freely 290-291 homotopy groups of, 311-336 Hopf's fiberings of, 66 Steenrod square, 258 Stiefel manifold, 100-101, 323-325 Suspension, 163, 257, 312, 335 iterated, 318 theorem, 311, 312, 335 Suspension sequence of a triad, 163
on,
Tietze's extension theorem, 26 Topological identification, 8, 27 Topology admissible, 102 compact-open, 73 identification, 9 of uniform convergence, 102 quotient, 9 weak, 200 Whitehead, 170 Torus, 28 Transgression, 256-258, 284, 293 Triad, 78, 160 generalized, 78 homotopy groups of, 160, 316 Triple, 214-216, 219-222 binormal, 214 cohomotopy sequence of, 214 Homotopy sequence of, 159 Triplet, 110 derived, 111 hornotopy sequence of, 115, 120, 136 Two-term condition, 241 Li Uniqueness theorem for homotopy, 121 Unit n-simplex, 7 Universal coefficient theorem, 202, 329 Universal covering space, 91
Wang exact sequence, 282-284, 320 isomorphism, 314 Weight, 244-245 Whitehead exact sequence, 253 product, 138-139, 330, 336 theorem, 167, 307 topology, 170 Wojdyslawski's theorem, 27
