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18 Spectral Reduced Homology and Cohomology Theories
18 Spectral Reduced Homology and Cohomology Theories
This section is concerned with the definition of spectral homology and cohomology functors. The ordinary homology group H J X ) can be thought of as an approximation to z,(X). We think of q ( X ) as a classification of n-dimensional “elements” in X . In this case, by an n-dimensional element we mean a continuous image of S“. The groups z , ( X ) are easy to define, and, as we have seen, hard to calculate. A different notion of element is given if we consider elements as represented by imbedded cells. Then S’ x S’ has a 2-dimensional element even though nz(S1 x S’) = 0. The number of cells in a given dimension is not, however, a topological invariant. The ordinary homology in dimemion n is designed to be the classification of certain invariant combinations of the n-dimensional cells (called cycles). The difficulty with homology theory is exactly the opposite to homotopy theory. Homology groups are easy to calculate, but hard to define (in an invariant way). I n Section 20 we shall give a more detailed explanation of what we mean by a cycle and when two cycles are homologous. Our present task is to define certain general functors called homology and cohomology theories on any space in CS*. They will be topological (in fact homotopy) invariants. I n Section 20 we shall show that they correspond to the homology classification of cycles in the case of ordinary homology theory. Our general theoriesI4 will include stable homotopy theory and l 4 Sometimes the theories described here are called extraordinary homology and cohomology to distinguish them from the ordinary theory described above. However, as times goes on, they become less extraordinary.
168
18. Spectral Reduced Homology and Cohomology Theories
169
other functors which have recently become important in algebraic topologyK-theories and cobordism theories. A brief description of these is found in Sections 29 and 30. We begin by defining the notion of a spectrum E = {E,, en} and show that each spectrum gives rise to two sequences of functors I?,,, and I?"'' from CS*. to JLz the first covariant and the second contravariant. These are the spectral homology and cohomology functors.
Definition 18.1 A spectrum E = {E,,, e,,} is a sequence of spaces E, and maps e,: SE,,-+E,,+, for n 2 0 (or equivalently 2:, E,,+RE,,+,) in CS". E is called a suspension spectrum if e,, is a weak homotopy equivalence, for all n sufficiently large and an Q-spectrum if 2, is a weak homotopy equivalence for all n sufficiently large. Examples 1. Let X E CS and define _X by _X, = S'X, and g,,:S ( S n X ) S"+'X to be the natural homeomorphism. Any suspension spectrum is obviously of this form " u p to weak homotopy" where X = & . This spectrum will be written X and So will be abbreviated 8. 2. HTCis given by ( H n ) , = K(n, n), and K(n, n ) -+ QK(n, n + 1) a chosen resolution. -+
(z),,:
Given a spectrum E (we often suppress the spaces E,, and maps e,, from the notation, when it will not lead to confusion), and a space X we will define groups &(X) and I?'"(X) for each integer m.
Definition 18.2 A graded abelian group is a sequence {G,,} of abelian groups, defined for each integer n. A homomorphism f: {G,,} + {G,,'} of graded groups is a sequence {f,}of homomorphisms f,:G,, --+ G,'. One often writes G, for {G,,}. Similar definitions may be made for graded R-modules, or graded sets. Such objects and homomorphisms form a category written Az., AR*, or S, in the cases of graded abelian groups, graded R-modules, and graded sets respectively. Example The sequence G, = .,,(A', *) for n 2 1 and G,, = 0 if n _< 0 is a graded abelian group if n1 is abelian (otherwise i t will be called just a graded group), and the sequence of homotopy groups yields a functor n* : E* + Az*. This sequence is called positively graded since G, = 0 if n < 0. Let C* be a category of spaces with base point and base point preserving mappings.
18. Spectral Reduced Homology and Cohomology Theories
170
Definition 18.3 A reduced homology (cohomology) theory on C* is a from e* to Az* satisfying: covariant (contravariant) functor {Em}({Em}) (A) I f f : X - + Y , write f * : f?,CX) --+ $,( Y ) ( f * : Em(Y)-+ E m ( X ) )for the induced homomorphism. Then i f f - g in e*,f * = g* andf" = g*. (B) There is a natural transformation: (IT: E y X ) + . P + l ( S X ) ) IT: Em(X) Em+l(SX) --+
-
which is an isomorphism. (C) Iff: X+ Y and i: Y -+ Y us C* X , the sequence:
Em(X)
(Ern(X )
f r
4
L
Em(Y)-L Em(Y U f C * X )
Em(
Y)
L
Em(
Y us c
*x>)
is exact in the middle. We now construct, for each spectrum E, functors {Ern} and {Em}which are reduced homology and cohomology theories on (39". These will be called spectral homology and cohomology theories (to distinguish them from theories constructed in other ways). If E = Hrr, these groups will be called the ordinary spectral reduced homology and cohomology theories with coefficients in n. These are classically written fi,(X; n ) and fi"'(X; n). As we shall see, ordinary cohomology agrees with the functor introduced in Section 17 with the same name. If n = Z , this is abbreviated f i m ( X )and f i m ( X ) . Given X E C9*, consider the directed systems *
.
+
...A
ntt +m(X A
Yn
En)
[S"-"X,E,]&
---+
1
nn+ m
+ I(X
A
En + 1)
+ * * *
[ S n + ' - m XE,+,]+..* ,
where the homomorphisms yn and E
n n + m ( X ~E n ) - - - * n n + m + I ( X A
(n 2 4,
A,,are the composites En
(1
A
en)*
A sl)-nn+m+l(XA
En+,)
[P+l-mX, E,+J
[S"-"X, E , ] 2 [ S n + I - m XSE,]% ,
Define E m ( X )= b { n f l + n l ( X A E,,), y,}
and
g m ( X )= ~ ( [ S f ' - " X , E ,A,}. ],
Theorem 18.4 {Ern} is a reduced cohomology theory on CS*. Proof f:X - t Y induces homomorphisms [Sn-my, E , ] L [ S + " Y ,
sEn]%[S"+'-"
y , En+ 1 1
18. Spectral Reduced Homology and Cohomology Theories
171
This induces a homomorphism from the direct sequence for Ern(Y)to the direct sequence for l?"'(X), and hence a homomorphism f *: Ern(Y ) + Ern(X) by 15.12. By the uniqueness assertions, I * = I , and (fg)* =g*f *, Proof of ( A ) Suppose f
-
g in CG*. Then Sn-"'f
N
SnFrng in CG* so
[Sn-rnY, En] [ r r n X En]. ,
( Y r n f ) * = (S-g)*:
-+
1
Hence f * = g * .
Proof' of ( B ) Replace S X by X A S' and note that the natural homeomorphisms X A Sk = ( X A S ' ) A Sk-' induce (unlabeled) natural isomorphisms E
[XA s"-",En]
[(XA S')A
1%
[XA
s-"A sl, EnA s]'-
s " - m - ' , ~ m ] ~ [ (SX ' ) AA
I_
s-"-l
A
[XA so-"+' &+,I
I= 9
S ' ) A s"-m,E,+l]
s',E,A S']'C")'"(XA
Now the diagram commutes, and both horizontal composites are I , , as occurring in the direct limit for Ern(X)and Ern+'(XA S'), it follows that the limits are naturally isomorphic. I
-
-
Proof of ( C ) By Exercise 21, Section 14, the sequence
[&Yrn( Y u C * X ) ,En] ZII
[ Y u C*X, an-rnEn] f
i*
i*
-
'f [rrn Y, En]
?I1
[Sn-"'X,En]
f*
[ Y, an-rnEn]
211
[ X , t2n-rnEn]
is exact at the middle. By 15.13 we are done.
Theorem 18.5 {Em)is a reduced homology theory on CG*. Proof To define,f,, consider the commutative diagram
15.12 provides a map f * : &,(A') that 1, = 1 and (fg)* =f * g * .
-+
Em(Y ) and the uniqueness assertions imply
Proofof(A) This follows as before since iff ( f A
-
g,f
A
1
N
g
A
1)+=(g~ I ) + : r n + r n ( X ~ E , ) - , ~ ~ n + m (EY)h*
1 and hence
I
18. Spectral Reduced Homology and Cohomology Theories
172
Proof of ( B ) We define (r as follows. For any space X , define
C: 7tn(X)+ 71,,+1(s1A X) by X{@
= 11 A
O}. The diagram
z nn+rn(XA En)-nn+m+l(S'
A
X A En>
commutes, hence C induces a natural transformation Q : Em(X)-+ Ern+,(S1 A X)z g , + l ( S X ) , where the isomorphism is given by the natural transposition homeomorphism T,: S' A X X A S' = S X . -+
The proof that
0
is an isomorphism will depend on a lemma.
Lemma 18.6 The diagram nn(W
n,,(Xh
r
b
s') ' '
?T,,(S'h
x)
n,+l(S'
A X A
s')
commutes up to sign. Proof Ts,,: S' A S" + s" A S' is a homeomorphism and hence, after identification of the spaces involved with spheres, is homotopic to 1. Thus the upper triangle commutes up to sign since forf: S" + X ,
s'
commutes. Let f : S" + S'
commutes.
A
A
Uf)
S"-S'
X.Then
A
x
173
18. Spectral Reduced Homology and Cohomology Theories
Now T,, = E(a) for some a: S' (1 A f) a A f . Consequently
+ S'
with { a } = f l . Thus (T,,
A
1)
0
N
-
We now complete the proof of B. By 18.6 the diagram x n,,+,,,(X A En) n,,+,,,+1(S1A X A E n )
n,,+,,,+,(X A EnA S')
En+,)
xn+m+l(XA
E
I:
nn+,+z(S1A X
nn+,+2(S1 A
A
EnA S ' )
X A En+,)
commutes up to sign. If .(a) = 0, there is an x E n,+,(X A E,) such that x represents a and Cx = 0. Consequently Ex = 0 so CI = 0. If a E ,!?m+l(S1 A X ) there is an X E TC,,+,+~(S~ A X A E,,) which represents a. But y,(x) = X((1 A e,) * TxAE,)*(x), so a E Image (r. I To prove ( C ) , we need some lemmas.
Lemma 18.7 Letf: X nn +
E(kerf*)
Y
-
-+ Y. Then
C*X)
in the diagram
(PY)*
nn+ 1 ( S X )
= Im(P,)*.
Proof Let a : s" + Xrepresent an element of n,(X) and H : C*S"+ Y be an extension of fa. Define /I: s " + l = SS" + Y us C * X by
(
(a(@, 2t - l), P(e,t> = H ( e , 1 - zt),
+ It I1
oI tI 4
for 0 E S". A homotopy R : Sa "pr /I is given by 0
R(e,t , s)
=
(a(O), (2t - s)/(2 - 8 ) )
(*,
s/2 I tI 1 0I tI s/2.
I
174
18. Spectral Reduced Homology and Cohomology Theories
Lemma 18.8 The diagram
C*YU/C*X
Pr
JC*X
,yy
sx commutes up to homotopy. Proof At time t we will pinch a piece of size t off of C* Y and size 1 --t off of C * X (see Fig. 18.1): 1-f
f
Figure 18.1
This is a well-defined homotopy H : (C*Y uf C * X ) x Z - t S Y between Pc*x and
(-Sf>O P C * Y
I E Ern(Y ) with i*(cc) = 0. There exists x E nm+,,(Y A
9
P r o o f o f ( C ) Let c( with i*(x) = 0 E n,+,,(( Y uf C * X ) A En)such that x represents 18.7 to the diagram
n,+,( Y A En)
-
rm+n((
Y
c(.
U/
En)
Applying
C * X ) A En)
175
18. Spectral Reduced Homology and Cohomology Theories
(which commutes by 18.8), one finds an element y E 7cn+,+,(S(X A En)) such A En+,) is a that (Sf)&) = E(x). Consequently, (I A en)&) E 7cn+m+1(X representative for an element p E ,!?JX) such thatf*(B) = a. This proves C and completes the proof of 18.5. I and {Em}be reduced homology and Theorem 18.9 Letf: X + Y. Let {Em} cohomology theories. Then there are exact sequences
-: -
... + Em(X ) 2Em(Y )
it
Em(Y u c*X )
Em- 1( X ) -+
f
... + E m ( X ) L E m ( Y )
Em(
Y Vf C * X )
Am-
1
where i: Y + Y ufC*X is the inclusion, A, = 0 - l ( p y ) * 0, where CT is the suspension isomorphism.
*
-.
Ern-' ( X ) 4-
0
* * *
( p y ) * , and Am-'
=
0
Corollary 18.10 Let ( X , A ) have the AHEP. Then there are exact sequences
-
.' * + EJA) LE,(X>%
Em(X/A)--%I?m-l(A)-+ * .
i*
- * *
vm-1
tEm(A)cEm(X)~~rn(X/A)-~m-'(A)t...
where i: A -+ X is the inclusion, p A : X + X / A is the quotient map, V, = Am 0 ( p C . A ) L 1 , and V" = ( p p A ) * Am. The corollary follows from the theorem since p C e A :X u C*A -+ X / A is a homotopy equivalence by Exercise 20, Section 16. I The theorem follows from: 0
Proposition 18.11 The following diagram commutes up to homotopy, where the last two horizontal arrows are the inclusion into the mapping cone of the previous map, and the vertical arrows are homotopy equivalences:
x --&Y
-
\ \ jpcb'.-.cb*~ c,*
Yu,C*X 2 C*YU,C*X
sx
- Sf
+ SY Proof The statements about the right-hand triangle follow from those about the left-hand one by replacing f:X - t Y by i: Y + Y u C*X. p C d yis a homotopy equivalence by Exercise 20, Section 16. The middle triangle commutes by 18.8. I
,
18.11 implies that the sequence
E,(X)-fi, Em(Y)- i. Em(Y u / c*X)=
Em(SX)-
(-Sf)*
E,(SX)
18. Spectral Reduced Homology and Cohomology Theories
176
is exact. (-Sf)* may be replaced by (Sf)* since - 1 : S X -+ S X is a homotopy equivalence and
(Sf)*
Em(Sx)
Em(S Y)
(Sf)
4 &,(sY)
&CSx>
- =
commutes. Similarly in cohomology we have an exact sequence
zEm( Y )zEm( Y u C*X)
Pyx)
f
(PY)*
EJyS Y )
Ern(SX)
Piecing these together and using the natural isomorphism 18.9 I
0,
one proves
Proposition 18.12 The sequences of 18.9 are natural, i.e., a commutative square f X-Y
induces commutative ladders f*
**'-Em(x)---Em(Y)-Em(Y
...
EyX)
i*
Uf
- -
l(=l)*
f*
f,
... c--E m ( X ' )
Em( Y ) l(azl*
i*
Em(
~j,)*
- I -
Y us C * X )
l(u)*
E m ( y') c--E m (
AIM c*x)-Em-l(x)-***
Y
Am-I
Em- '(X)
*
a
(Ul)'
Am-1
+Em-
US'C*X')
(xl)
*
..
where a : Y uf C*X-+ Y u C*X' is constructed from q and a2 in the obvious way. f,
Proof
This follows immediately from the various naturality results about
I?*, E*, and c, I
The sequences of 18.9 are called the long exact sequences of the homology and cohomology theories. They are infinite in both directions.
177
18. Spectral Reduced Homology and Cohomology Theories
Proposition 18.13 Suppose E is an &spectrum and Xis a CW c0mp1ex.l~ There is a natural isomorphism R m ( X )Z [ X , Em].
\ E
Proof Consider the diagram
An
[S"-"X, En]
(b).
where Z:, E n - t Q E , + , is the adjoint to en: SE,+E,,+, and 3 is the 1-1 correspondence of 8.24. It is easy to verify that the diagram commutes, and since (en)* and $ are 1-1 correspondences 1, is a 1-1 correspondence. Hence
,!?"(A') z [Sn-mX,En] z [ X , CY"'mEn] [ X , Em]. I Proposition 18.14 For any spectrum E, gm(Sk) z &Sm).
-
Proof By property (B), it is sufficient to show that ,!?,,,(So) E &"(So). These are both the direct limit of the sequence
...
-
\
nn+m(En)
nn+m+l(En+l)
-
* * .
/A
I
nn + m + l(SEn)
This group is called the mth homotopy group of the spectrum E and is sometimes written n,(E). It is also called the group of coefficients for the theories .& and g*.
Theorem 18.15 Suppose X assume16:
=
u Xu
=
lim Xu has the weak topology and
(a) For all a, p, E A , there exists 6 E A such that X , n X , = X,. (b) F o r a l l a ~ A , { p ~ A ] p r c c } i s f i n i t e ( p I a Xi f, fc X u } . Then E m ( hXu) z
l?!,,,(X,).
l 5 The hypothesis that X i s a CW complex is used in our proof, but may be dropped if each En is a CW complex since it is known [SO]that in this case RE, is the homotopy type of a CW complex (see the proof). l6 Compare to 15.9.
178
18. Spectral Reduced Homology and Cohomology Theories
Proof The inclusions i,: X , + X induce homomorphisms (in)*: Em(X,) + Em(X). These are compatible with the homomorphisms (i,,)* : Em(X,) +
Em(Xp)and hence induce a homomorphism Z: & l gm(Xu) + Em(X).Suppose I(x) = 0. x must have a representative v E nm+,(XuA En)such that (iu)*(u) = O E ~ , + , ( X AEn).Let f : Sm+"-+X , A En represent v and H : i,f * be a homotopy. Since X A En = (h X,) A En 3 u ( X , A En), we may apply 15.10 and conclude that there exists 1 E A with H(S"+" x I ) c X , A E n . Thus (iaP)*(u) = 0 so x = 0. Hence I is 1-1. Let x E Em(X). Choose v E ~ , + , ( X AEn) to represent x and f: Sm+,-+X A En to represent v. As before f ( S m + , )c X , A En for some a. So u E (i,)*(n,+,(X, A E,)). Hence Zis onto. I N
Theorem 18.16 Em(Xl v Eyxl v
* * * * * '
v X,) E Em(X1)0. . * 0Em(X,), v X,) z Em(Xl) 0 * * .0 E,(X,).
The decomposition is given by the induced homomorphisms from the maps i k : X k + X l v . * . vX k a n d p k : X l v . . . v X , + X k f o r l I k < n . Proof We first do the case n = 2. Observe that since C* Y is contractible in CQ*. pc*y: X v C* Y - + X is a homotopy equivalence. Consequently, the sequence
Em(Y)-
(id*
-
Em(X v Y )5 Em(X)
is exact in the middle. Since the diagram
Em(Y)
OZL
EJX v Y )
(PI).
Em(W
commutes, (iz)*is a monomorphism, (p,)* is an epimorphism, and we may apply 11.1 1 to prove the splitting. If n > 2, we apply induction, observing that XI v . . . v A', = ( X , v ... v v X , . The case of cohomology is similar. I
M , was introduced in 15.3. One Given R-modules M u , the direct sum M a to be the set of all sequences similarly defines the direct product {x,} with x, E M a . Addition and scalar multiplication are coordinatewise. Note that @ , , A M a c A Mu and if A is finite, they are equal.
nuE
naEA
18. Spectral Reduced Homology and Cohomology Theories
-
179
VaeA
Let (X,, *J E CG* be indexed by a set A . By X , we will mean the quotient space A'=/*, *p If A is finite, this is the one-point union X,, v * - . v Xan.
UaEA
Theorem 18.17 Em(VaEA X,) E 0 Em(Xa). If ( X u , *,) are CW complexes" and {E,,, e,,} is an a-spectrum, J!?~(V,~~X,) iz naEAEm(Xa). Proof Let X = V d l E A X Oand L consider all subspaces X,, ,..., v Xan of X . This satisfies the hypothesis of 18.15 so X,, v Em(X)z lim Em(X,, v * . . v Xan) '
,.
z l i l l l J ! ? m ( X a , ) ~ . . . ~ ~ m ( ~E a . o>, E A & ( x ~ ) .
In the cohomology case, the hypothesis implies that X = VasA Xu is a CW complex so we may apply 18.13. Let i,: Xu -+ Y be the inclusion, and consider the homomorphisms (ia)*: gm(X) -+ gm(Xa).These induce a homomorphism I: Em(X)+ n a E A E m ( X e )Consider . now the diagram
~mm--n.Em(Xa) I
t
t
In the bottom row I is a 1-1 correspondence, for given
-
{h}where f,: X ,
--t
Emone easily constructs 8 X -+ Em with f I x, =h.Iff [ x. * for each a ; the homotopies Ha: X , x I+ Em define a homotopy H : X x I - r Em by
HI x, I = Ha since they are base point preserving homotopies. Hence I is an isomorphism. I
s"
-m
The conclusion of 18.17 is sometimes called the wedge axiom. Consider now the sphere spectrum 5. In this case the homology group ( X ) is given by the direct limit E
s") -+?T,+,,+~(XA
?Tm+,(XA
Sn+')-+*''.
This is also written nmS(X),and is the functor defined at the end of Section 16. 18.15 implies that nms((X)= 0 if m < 0 and X i s a CW complex. Similarly one defines the stable cohomotopy groups zSm(X)as the cohomology theory associated with this spectrum, i.e., the direct limit of the sequence
[ p - m x , 91:
[Sn-m+lX,
F+'] -+ . '. *
These groups can be nonzero for negative values of rn when applied to CW complexes. For example, ns'(SO)z 7c4(S3) E Z , (see 27.19). l7
See footnote 15.
180
18. Spectral Reduced Homology and Cohomology Theories
Exercises
1. Give a detailed proof that the diagrams in the conclusion of 18.12 commute. 2. Give a detailed proof of the isomorphism
o
lim
*
o 4 n < X a , > aoE AErnCXa)
in the proof of 18.17.
3. Define h: z i ( X )-t n:(X) to be the injection homomorphism from the first group of the direct system for n:(X) into the limit. Show that if X is simply connected, the following are equivalent: (a) zi(X) = 0 for i < n ; (b) n F ( X ) = 0 for i < n; they imply that h : ni(X)+ n:(X) is an isomorphism if 1 I i < 2n - 1 and ontoifi=2n-1. 4. Let q,: En +En' be defined for each n 2 No such that the diagrams
commute. Such a sequence {q,} will be called a map of spectra of degree k. Show that spectra form a category with this definition of morphism. Show that if q = {q,}: E -t E' is a map of spectra of degree k, it induces natural homomorphisms of homology and cohomology theories q : R,(X)
--f
4!?;-k(X),
q : Ern(X)
prn+k(X)
for all m commuting with the suspension isomorphism; i.e., q(a(x)) = a(q(x)). Such a transformation is called a stable homology or cohomology operation. (Exercise 11; Exercise 1, Section 22; Section 27) 5. A spectrum E and the corresponding theories are called connective if En is (n - 1)-connected. Show by taking connected covering spaces that given any spectrum En there is a connective spectrum En and a mapping v,: En-t En of spectra such that (qrn)+ is an isomorphism in n ifor i > 0. 6. Let P be a one-point topological space. Show that ,!!,,,(P) = 0 = gm(P) for any functors Ernand i?"' satisfying axioms (A)-(D). 7. Let f: X - t Y be a map in CQ* and suppose f * in CQ*. Show that f* = 0 =f*. N
18. Spectral Reduced Homology and Cohomology Theories
181
8. Let E be a spectrum and define W,, to be the telescope construction (Exercise 1, Section 15) on the system
Show that b ( X ) = [(X, *), ( W,,, *)] for X a compact Hausdorff space. Show that { W,,, w,,}is an R-spectrum where G,,: W,,4 R W,,,, is given on the kth term of the telescope by
9. Show that Ern(-%')r h E r n ( K where ) the limit is taken over all compact subsets K of X containing *.
10. Let cp: &*(X)+&*'(X) be a natural transformation such that cp: &*(So) + E,'(So) is an isomorphism. Show that cp: &'(X) 3 l?"'(X) is an
isomorphism for each CW complex X . Prove an analogue for cohomology if Xis a finite complex. (30.25)
11. Using Exercise 4, construct for each homomorphism cp: x + p coefficient transformations c,:
R,,(X;n) 3 fl,,(X;p ) ,
ce: Pyx;).
+P
( X ;p)
and show, using these, that homology and cohomology are covariant functions of the coefficient groups. (Exercise 18, Section 23) 12. Given a spectrum E and an abelian group G define EG = (EG,,,eG,,} by EG,, = M(G, 1)
A
En-,
and (eG),, = 1
A
e n - , :M(G, 1)
A
A
S ' + M ( G , 1)
A
En.
We write &"(A'; G ) for z * ( X ) and E * ( X ; G) for %*(X). Show that if E = H , these definitions agree with the ordinary definitions if X is a CW complex.'* (Hint: Use Exercise lo.) (Exercise 13; Exercise 11, Section 22)
-
13. Let 0 + G -% H II. J + 0 be a short exact sequence of abelian groups. Construct natural long exact sequences
It is actually only necessary to assume that X i s well pointed by 21.7.
182
18. Spectral Reduced Homology and Cohomology Theories
This is called the Bockstein sequence and p is called the Bockstein homomorphism. Show that if E = H , c = c,, and d = c$. (Exercise 8, Section 21 ; 27.15) 14. A spectrum E is called properly convergent if en: SE, -,E n + , is a (2n + 1)-isomorphism for each n 2 0. Show that if E,, is well pointed for each n and E is properly convergent, E,, is (n - 1)-connected. Show that if E is an R-spectrum and E, is connected for each n, E is properly convergent. (Exercise 12, Section 22; 27.5)
This section is concerned with the definition of spectral homology and cohomology functors. The ordinary homology group H J X ) can be thought of as an approximation to z,(X). We think of q ( X ) as a classification of n-dimensional “elements” in X . In this case, by an n-dimensional element we mean a continuous image of S“. The groups z , ( X ) are easy to define, and, as we have seen, hard to calculate. A different notion of element is given if we consider elements as represented by imbedded cells. Then S’ x S’ has a 2-dimensional element even though nz(S1 x S’) = 0. The number of cells in a given dimension is not, however, a topological invariant. The ordinary homology in dimemion n is designed to be the classification of certain invariant combinations of the n-dimensional cells (called cycles). The difficulty with homology theory is exactly the opposite to homotopy theory. Homology groups are easy to calculate, but hard to define (in an invariant way). I n Section 20 we shall give a more detailed explanation of what we mean by a cycle and when two cycles are homologous. Our present task is to define certain general functors called homology and cohomology theories on any space in CS*. They will be topological (in fact homotopy) invariants. I n Section 20 we shall show that they correspond to the homology classification of cycles in the case of ordinary homology theory. Our general theoriesI4 will include stable homotopy theory and l 4 Sometimes the theories described here are called extraordinary homology and cohomology to distinguish them from the ordinary theory described above. However, as times goes on, they become less extraordinary.
168
18. Spectral Reduced Homology and Cohomology Theories
169
other functors which have recently become important in algebraic topologyK-theories and cobordism theories. A brief description of these is found in Sections 29 and 30. We begin by defining the notion of a spectrum E = {E,, en} and show that each spectrum gives rise to two sequences of functors I?,,, and I?"'' from CS*. to JLz the first covariant and the second contravariant. These are the spectral homology and cohomology functors.
Definition 18.1 A spectrum E = {E,,, e,,} is a sequence of spaces E, and maps e,: SE,,-+E,,+, for n 2 0 (or equivalently 2:, E,,+RE,,+,) in CS". E is called a suspension spectrum if e,, is a weak homotopy equivalence, for all n sufficiently large and an Q-spectrum if 2, is a weak homotopy equivalence for all n sufficiently large. Examples 1. Let X E CS and define _X by _X, = S'X, and g,,:S ( S n X ) S"+'X to be the natural homeomorphism. Any suspension spectrum is obviously of this form " u p to weak homotopy" where X = & . This spectrum will be written X and So will be abbreviated 8. 2. HTCis given by ( H n ) , = K(n, n), and K(n, n ) -+ QK(n, n + 1) a chosen resolution. -+
(z),,:
Given a spectrum E (we often suppress the spaces E,, and maps e,, from the notation, when it will not lead to confusion), and a space X we will define groups &(X) and I?'"(X) for each integer m.
Definition 18.2 A graded abelian group is a sequence {G,,} of abelian groups, defined for each integer n. A homomorphism f: {G,,} + {G,,'} of graded groups is a sequence {f,}of homomorphisms f,:G,, --+ G,'. One often writes G, for {G,,}. Similar definitions may be made for graded R-modules, or graded sets. Such objects and homomorphisms form a category written Az., AR*, or S, in the cases of graded abelian groups, graded R-modules, and graded sets respectively. Example The sequence G, = .,,(A', *) for n 2 1 and G,, = 0 if n _< 0 is a graded abelian group if n1 is abelian (otherwise i t will be called just a graded group), and the sequence of homotopy groups yields a functor n* : E* + Az*. This sequence is called positively graded since G, = 0 if n < 0. Let C* be a category of spaces with base point and base point preserving mappings.
18. Spectral Reduced Homology and Cohomology Theories
170
Definition 18.3 A reduced homology (cohomology) theory on C* is a from e* to Az* satisfying: covariant (contravariant) functor {Em}({Em}) (A) I f f : X - + Y , write f * : f?,CX) --+ $,( Y ) ( f * : Em(Y)-+ E m ( X ) )for the induced homomorphism. Then i f f - g in e*,f * = g* andf" = g*. (B) There is a natural transformation: (IT: E y X ) + . P + l ( S X ) ) IT: Em(X) Em+l(SX) --+
-
which is an isomorphism. (C) Iff: X+ Y and i: Y -+ Y us C* X , the sequence:
Em(X)
(Ern(X )
f r
4
L
Em(Y)-L Em(Y U f C * X )
Em(
Y)
L
Em(
Y us c
*x>)
is exact in the middle. We now construct, for each spectrum E, functors {Ern} and {Em}which are reduced homology and cohomology theories on (39". These will be called spectral homology and cohomology theories (to distinguish them from theories constructed in other ways). If E = Hrr, these groups will be called the ordinary spectral reduced homology and cohomology theories with coefficients in n. These are classically written fi,(X; n ) and fi"'(X; n). As we shall see, ordinary cohomology agrees with the functor introduced in Section 17 with the same name. If n = Z , this is abbreviated f i m ( X )and f i m ( X ) . Given X E C9*, consider the directed systems *
.
+
...A
ntt +m(X A
Yn
En)
[S"-"X,E,]&
---+
1
nn+ m
+ I(X
A
En + 1)
+ * * *
[ S n + ' - m XE,+,]+..* ,
where the homomorphisms yn and E
n n + m ( X ~E n ) - - - * n n + m + I ( X A
(n 2 4,
A,,are the composites En
(1
A
en)*
A sl)-nn+m+l(XA
En+,)
[P+l-mX, E,+J
[S"-"X, E , ] 2 [ S n + I - m XSE,]% ,
Define E m ( X )= b { n f l + n l ( X A E,,), y,}
and
g m ( X )= ~ ( [ S f ' - " X , E ,A,}. ],
Theorem 18.4 {Ern} is a reduced cohomology theory on CS*. Proof f:X - t Y induces homomorphisms [Sn-my, E , ] L [ S + " Y ,
sEn]%[S"+'-"
y , En+ 1 1
18. Spectral Reduced Homology and Cohomology Theories
171
This induces a homomorphism from the direct sequence for Ern(Y)to the direct sequence for l?"'(X), and hence a homomorphism f *: Ern(Y ) + Ern(X) by 15.12. By the uniqueness assertions, I * = I , and (fg)* =g*f *, Proof of ( A ) Suppose f
-
g in CG*. Then Sn-"'f
N
SnFrng in CG* so
[Sn-rnY, En] [ r r n X En]. ,
( Y r n f ) * = (S-g)*:
-+
1
Hence f * = g * .
Proof' of ( B ) Replace S X by X A S' and note that the natural homeomorphisms X A Sk = ( X A S ' ) A Sk-' induce (unlabeled) natural isomorphisms E
[XA s"-",En]
[(XA S')A
1%
[XA
s-"A sl, EnA s]'-
s " - m - ' , ~ m ] ~ [ (SX ' ) AA
I_
s-"-l
A
[XA so-"+' &+,I
I= 9
S ' ) A s"-m,E,+l]
s',E,A S']'C")'"(XA
Now the diagram commutes, and both horizontal composites are I , , as occurring in the direct limit for Ern(X)and Ern+'(XA S'), it follows that the limits are naturally isomorphic. I
-
-
Proof of ( C ) By Exercise 21, Section 14, the sequence
[&Yrn( Y u C * X ) ,En] ZII
[ Y u C*X, an-rnEn] f
i*
i*
-
'f [rrn Y, En]
?I1
[Sn-"'X,En]
f*
[ Y, an-rnEn]
211
[ X , t2n-rnEn]
is exact at the middle. By 15.13 we are done.
Theorem 18.5 {Em)is a reduced homology theory on CG*. Proof To define,f,, consider the commutative diagram
15.12 provides a map f * : &,(A') that 1, = 1 and (fg)* =f * g * .
-+
Em(Y ) and the uniqueness assertions imply
Proofof(A) This follows as before since iff ( f A
-
g,f
A
1
N
g
A
1)+=(g~ I ) + : r n + r n ( X ~ E , ) - , ~ ~ n + m (EY)h*
1 and hence
I
18. Spectral Reduced Homology and Cohomology Theories
172
Proof of ( B ) We define (r as follows. For any space X , define
C: 7tn(X)+ 71,,+1(s1A X) by X{@
= 11 A
O}. The diagram
z nn+rn(XA En)-nn+m+l(S'
A
X A En>
commutes, hence C induces a natural transformation Q : Em(X)-+ Ern+,(S1 A X)z g , + l ( S X ) , where the isomorphism is given by the natural transposition homeomorphism T,: S' A X X A S' = S X . -+
The proof that
0
is an isomorphism will depend on a lemma.
Lemma 18.6 The diagram nn(W
n,,(Xh
r
b
s') ' '
?T,,(S'h
x)
n,+l(S'
A X A
s')
commutes up to sign. Proof Ts,,: S' A S" + s" A S' is a homeomorphism and hence, after identification of the spaces involved with spheres, is homotopic to 1. Thus the upper triangle commutes up to sign since forf: S" + X ,
s'
commutes. Let f : S" + S'
commutes.
A
A
Uf)
S"-S'
X.Then
A
x
173
18. Spectral Reduced Homology and Cohomology Theories
Now T,, = E(a) for some a: S' (1 A f) a A f . Consequently
+ S'
with { a } = f l . Thus (T,,
A
1)
0
N
-
We now complete the proof of B. By 18.6 the diagram x n,,+,,,(X A En) n,,+,,,+1(S1A X A E n )
n,,+,,,+,(X A EnA S')
En+,)
xn+m+l(XA
E
I:
nn+,+z(S1A X
nn+,+2(S1 A
A
EnA S ' )
X A En+,)
commutes up to sign. If .(a) = 0, there is an x E n,+,(X A E,) such that x represents a and Cx = 0. Consequently Ex = 0 so CI = 0. If a E ,!?m+l(S1 A X ) there is an X E TC,,+,+~(S~ A X A E,,) which represents a. But y,(x) = X((1 A e,) * TxAE,)*(x), so a E Image (r. I To prove ( C ) , we need some lemmas.
Lemma 18.7 Letf: X nn +
E(kerf*)
Y
-
-+ Y. Then
C*X)
in the diagram
(PY)*
nn+ 1 ( S X )
= Im(P,)*.
Proof Let a : s" + Xrepresent an element of n,(X) and H : C*S"+ Y be an extension of fa. Define /I: s " + l = SS" + Y us C * X by
(
(a(@, 2t - l), P(e,t> = H ( e , 1 - zt),
+ It I1
oI tI 4
for 0 E S". A homotopy R : Sa "pr /I is given by 0
R(e,t , s)
=
(a(O), (2t - s)/(2 - 8 ) )
(*,
s/2 I tI 1 0I tI s/2.
I
174
18. Spectral Reduced Homology and Cohomology Theories
Lemma 18.8 The diagram
C*YU/C*X
Pr
JC*X
,yy
sx commutes up to homotopy. Proof At time t we will pinch a piece of size t off of C* Y and size 1 --t off of C * X (see Fig. 18.1): 1-f
f
Figure 18.1
This is a well-defined homotopy H : (C*Y uf C * X ) x Z - t S Y between Pc*x and
(-Sf>O P C * Y
I E Ern(Y ) with i*(cc) = 0. There exists x E nm+,,(Y A
9
P r o o f o f ( C ) Let c( with i*(x) = 0 E n,+,,(( Y uf C * X ) A En)such that x represents 18.7 to the diagram
n,+,( Y A En)
-
rm+n((
Y
c(.
U/
En)
Applying
C * X ) A En)
175
18. Spectral Reduced Homology and Cohomology Theories
(which commutes by 18.8), one finds an element y E 7cn+,+,(S(X A En)) such A En+,) is a that (Sf)&) = E(x). Consequently, (I A en)&) E 7cn+m+1(X representative for an element p E ,!?JX) such thatf*(B) = a. This proves C and completes the proof of 18.5. I and {Em}be reduced homology and Theorem 18.9 Letf: X + Y. Let {Em} cohomology theories. Then there are exact sequences
-: -
... + Em(X ) 2Em(Y )
it
Em(Y u c*X )
Em- 1( X ) -+
f
... + E m ( X ) L E m ( Y )
Em(
Y Vf C * X )
Am-
1
where i: Y + Y ufC*X is the inclusion, A, = 0 - l ( p y ) * 0, where CT is the suspension isomorphism.
*
-.
Ern-' ( X ) 4-
0
* * *
( p y ) * , and Am-'
=
0
Corollary 18.10 Let ( X , A ) have the AHEP. Then there are exact sequences
-
.' * + EJA) LE,(X>%
Em(X/A)--%I?m-l(A)-+ * .
i*
- * *
vm-1
tEm(A)cEm(X)~~rn(X/A)-~m-'(A)t...
where i: A -+ X is the inclusion, p A : X + X / A is the quotient map, V, = Am 0 ( p C . A ) L 1 , and V" = ( p p A ) * Am. The corollary follows from the theorem since p C e A :X u C*A -+ X / A is a homotopy equivalence by Exercise 20, Section 16. I The theorem follows from: 0
Proposition 18.11 The following diagram commutes up to homotopy, where the last two horizontal arrows are the inclusion into the mapping cone of the previous map, and the vertical arrows are homotopy equivalences:
x --&Y
-
\ \ jpcb'.-.cb*~ c,*
Yu,C*X 2 C*YU,C*X
sx
- Sf
+ SY Proof The statements about the right-hand triangle follow from those about the left-hand one by replacing f:X - t Y by i: Y + Y u C*X. p C d yis a homotopy equivalence by Exercise 20, Section 16. The middle triangle commutes by 18.8. I
,
18.11 implies that the sequence
E,(X)-fi, Em(Y)- i. Em(Y u / c*X)=
Em(SX)-
(-Sf)*
E,(SX)
18. Spectral Reduced Homology and Cohomology Theories
176
is exact. (-Sf)* may be replaced by (Sf)* since - 1 : S X -+ S X is a homotopy equivalence and
(Sf)*
Em(Sx)
Em(S Y)
(Sf)
4 &,(sY)
&CSx>
- =
commutes. Similarly in cohomology we have an exact sequence
zEm( Y )zEm( Y u C*X)
Pyx)
f
(PY)*
EJyS Y )
Ern(SX)
Piecing these together and using the natural isomorphism 18.9 I
0,
one proves
Proposition 18.12 The sequences of 18.9 are natural, i.e., a commutative square f X-Y
induces commutative ladders f*
**'-Em(x)---Em(Y)-Em(Y
...
EyX)
i*
Uf
- -
l(=l)*
f*
f,
... c--E m ( X ' )
Em( Y ) l(azl*
i*
Em(
~j,)*
- I -
Y us C * X )
l(u)*
E m ( y') c--E m (
AIM c*x)-Em-l(x)-***
Y
Am-I
Em- '(X)
*
a
(Ul)'
Am-1
+Em-
US'C*X')
(xl)
*
..
where a : Y uf C*X-+ Y u C*X' is constructed from q and a2 in the obvious way. f,
Proof
This follows immediately from the various naturality results about
I?*, E*, and c, I
The sequences of 18.9 are called the long exact sequences of the homology and cohomology theories. They are infinite in both directions.
177
18. Spectral Reduced Homology and Cohomology Theories
Proposition 18.13 Suppose E is an &spectrum and Xis a CW c0mp1ex.l~ There is a natural isomorphism R m ( X )Z [ X , Em].
\ E
Proof Consider the diagram
An
[S"-"X, En]
(b).
where Z:, E n - t Q E , + , is the adjoint to en: SE,+E,,+, and 3 is the 1-1 correspondence of 8.24. It is easy to verify that the diagram commutes, and since (en)* and $ are 1-1 correspondences 1, is a 1-1 correspondence. Hence
,!?"(A') z [Sn-mX,En] z [ X , CY"'mEn] [ X , Em]. I Proposition 18.14 For any spectrum E, gm(Sk) z &Sm).
-
Proof By property (B), it is sufficient to show that ,!?,,,(So) E &"(So). These are both the direct limit of the sequence
...
-
\
nn+m(En)
nn+m+l(En+l)
-
* * .
/A
I
nn + m + l(SEn)
This group is called the mth homotopy group of the spectrum E and is sometimes written n,(E). It is also called the group of coefficients for the theories .& and g*.
Theorem 18.15 Suppose X assume16:
=
u Xu
=
lim Xu has the weak topology and
(a) For all a, p, E A , there exists 6 E A such that X , n X , = X,. (b) F o r a l l a ~ A , { p ~ A ] p r c c } i s f i n i t e ( p I a Xi f, fc X u } . Then E m ( hXu) z
l?!,,,(X,).
l 5 The hypothesis that X i s a CW complex is used in our proof, but may be dropped if each En is a CW complex since it is known [SO]that in this case RE, is the homotopy type of a CW complex (see the proof). l6 Compare to 15.9.
178
18. Spectral Reduced Homology and Cohomology Theories
Proof The inclusions i,: X , + X induce homomorphisms (in)*: Em(X,) + Em(X). These are compatible with the homomorphisms (i,,)* : Em(X,) +
Em(Xp)and hence induce a homomorphism Z: & l gm(Xu) + Em(X).Suppose I(x) = 0. x must have a representative v E nm+,(XuA En)such that (iu)*(u) = O E ~ , + , ( X AEn).Let f : Sm+"-+X , A En represent v and H : i,f * be a homotopy. Since X A En = (h X,) A En 3 u ( X , A En), we may apply 15.10 and conclude that there exists 1 E A with H(S"+" x I ) c X , A E n . Thus (iaP)*(u) = 0 so x = 0. Hence I is 1-1. Let x E Em(X). Choose v E ~ , + , ( X AEn) to represent x and f: Sm+,-+X A En to represent v. As before f ( S m + , )c X , A En for some a. So u E (i,)*(n,+,(X, A E,)). Hence Zis onto. I N
Theorem 18.16 Em(Xl v Eyxl v
* * * * * '
v X,) E Em(X1)0. . * 0Em(X,), v X,) z Em(Xl) 0 * * .0 E,(X,).
The decomposition is given by the induced homomorphisms from the maps i k : X k + X l v . * . vX k a n d p k : X l v . . . v X , + X k f o r l I k < n . Proof We first do the case n = 2. Observe that since C* Y is contractible in CQ*. pc*y: X v C* Y - + X is a homotopy equivalence. Consequently, the sequence
Em(Y)-
(id*
-
Em(X v Y )5 Em(X)
is exact in the middle. Since the diagram
Em(Y)
OZL
EJX v Y )
(PI).
Em(W
commutes, (iz)*is a monomorphism, (p,)* is an epimorphism, and we may apply 11.1 1 to prove the splitting. If n > 2, we apply induction, observing that XI v . . . v A', = ( X , v ... v v X , . The case of cohomology is similar. I
M , was introduced in 15.3. One Given R-modules M u , the direct sum M a to be the set of all sequences similarly defines the direct product {x,} with x, E M a . Addition and scalar multiplication are coordinatewise. Note that @ , , A M a c A Mu and if A is finite, they are equal.
nuE
naEA
18. Spectral Reduced Homology and Cohomology Theories
-
179
VaeA
Let (X,, *J E CG* be indexed by a set A . By X , we will mean the quotient space A'=/*, *p If A is finite, this is the one-point union X,, v * - . v Xan.
UaEA
Theorem 18.17 Em(VaEA X,) E 0 Em(Xa). If ( X u , *,) are CW complexes" and {E,,, e,,} is an a-spectrum, J!?~(V,~~X,) iz naEAEm(Xa). Proof Let X = V d l E A X Oand L consider all subspaces X,, ,..., v Xan of X . This satisfies the hypothesis of 18.15 so X,, v Em(X)z lim Em(X,, v * . . v Xan) '
,.
z l i l l l J ! ? m ( X a , ) ~ . . . ~ ~ m ( ~E a . o>, E A & ( x ~ ) .
In the cohomology case, the hypothesis implies that X = VasA Xu is a CW complex so we may apply 18.13. Let i,: Xu -+ Y be the inclusion, and consider the homomorphisms (ia)*: gm(X) -+ gm(Xa).These induce a homomorphism I: Em(X)+ n a E A E m ( X e )Consider . now the diagram
~mm--n.Em(Xa) I
t
t
In the bottom row I is a 1-1 correspondence, for given
-
{h}where f,: X ,
--t
Emone easily constructs 8 X -+ Em with f I x, =h.Iff [ x. * for each a ; the homotopies Ha: X , x I+ Em define a homotopy H : X x I - r Em by
HI x, I = Ha since they are base point preserving homotopies. Hence I is an isomorphism. I
s"
-m
The conclusion of 18.17 is sometimes called the wedge axiom. Consider now the sphere spectrum 5. In this case the homology group ( X ) is given by the direct limit E
s") -+?T,+,,+~(XA
?Tm+,(XA
Sn+')-+*''.
This is also written nmS(X),and is the functor defined at the end of Section 16. 18.15 implies that nms((X)= 0 if m < 0 and X i s a CW complex. Similarly one defines the stable cohomotopy groups zSm(X)as the cohomology theory associated with this spectrum, i.e., the direct limit of the sequence
[ p - m x , 91:
[Sn-m+lX,
F+'] -+ . '. *
These groups can be nonzero for negative values of rn when applied to CW complexes. For example, ns'(SO)z 7c4(S3) E Z , (see 27.19). l7
See footnote 15.
180
18. Spectral Reduced Homology and Cohomology Theories
Exercises
1. Give a detailed proof that the diagrams in the conclusion of 18.12 commute. 2. Give a detailed proof of the isomorphism
o
lim
*
o 4 n < X a , > aoE AErnCXa)
in the proof of 18.17.
3. Define h: z i ( X )-t n:(X) to be the injection homomorphism from the first group of the direct system for n:(X) into the limit. Show that if X is simply connected, the following are equivalent: (a) zi(X) = 0 for i < n ; (b) n F ( X ) = 0 for i < n; they imply that h : ni(X)+ n:(X) is an isomorphism if 1 I i < 2n - 1 and ontoifi=2n-1. 4. Let q,: En +En' be defined for each n 2 No such that the diagrams
commute. Such a sequence {q,} will be called a map of spectra of degree k. Show that spectra form a category with this definition of morphism. Show that if q = {q,}: E -t E' is a map of spectra of degree k, it induces natural homomorphisms of homology and cohomology theories q : R,(X)
--f
4!?;-k(X),
q : Ern(X)
prn+k(X)
for all m commuting with the suspension isomorphism; i.e., q(a(x)) = a(q(x)). Such a transformation is called a stable homology or cohomology operation. (Exercise 11; Exercise 1, Section 22; Section 27) 5. A spectrum E and the corresponding theories are called connective if En is (n - 1)-connected. Show by taking connected covering spaces that given any spectrum En there is a connective spectrum En and a mapping v,: En-t En of spectra such that (qrn)+ is an isomorphism in n ifor i > 0. 6. Let P be a one-point topological space. Show that ,!!,,,(P) = 0 = gm(P) for any functors Ernand i?"' satisfying axioms (A)-(D). 7. Let f: X - t Y be a map in CQ* and suppose f * in CQ*. Show that f* = 0 =f*. N
18. Spectral Reduced Homology and Cohomology Theories
181
8. Let E be a spectrum and define W,, to be the telescope construction (Exercise 1, Section 15) on the system
Show that b ( X ) = [(X, *), ( W,,, *)] for X a compact Hausdorff space. Show that { W,,, w,,}is an R-spectrum where G,,: W,,4 R W,,,, is given on the kth term of the telescope by
9. Show that Ern(-%')r h E r n ( K where ) the limit is taken over all compact subsets K of X containing *.
10. Let cp: &*(X)+&*'(X) be a natural transformation such that cp: &*(So) + E,'(So) is an isomorphism. Show that cp: &'(X) 3 l?"'(X) is an
isomorphism for each CW complex X . Prove an analogue for cohomology if Xis a finite complex. (30.25)
11. Using Exercise 4, construct for each homomorphism cp: x + p coefficient transformations c,:
R,,(X;n) 3 fl,,(X;p ) ,
ce: Pyx;).
+P
( X ;p)
and show, using these, that homology and cohomology are covariant functions of the coefficient groups. (Exercise 18, Section 23) 12. Given a spectrum E and an abelian group G define EG = (EG,,,eG,,} by EG,, = M(G, 1)
A
En-,
and (eG),, = 1
A
e n - , :M(G, 1)
A
A
S ' + M ( G , 1)
A
En.
We write &"(A'; G ) for z * ( X ) and E * ( X ; G) for %*(X). Show that if E = H , these definitions agree with the ordinary definitions if X is a CW complex.'* (Hint: Use Exercise lo.) (Exercise 13; Exercise 11, Section 22)
-
13. Let 0 + G -% H II. J + 0 be a short exact sequence of abelian groups. Construct natural long exact sequences
It is actually only necessary to assume that X i s well pointed by 21.7.
182
18. Spectral Reduced Homology and Cohomology Theories
This is called the Bockstein sequence and p is called the Bockstein homomorphism. Show that if E = H , c = c,, and d = c$. (Exercise 8, Section 21 ; 27.15) 14. A spectrum E is called properly convergent if en: SE, -,E n + , is a (2n + 1)-isomorphism for each n 2 0. Show that if E,, is well pointed for each n and E is properly convergent, E,, is (n - 1)-connected. Show that if E is an R-spectrum and E, is connected for each n, E is properly convergent. (Exercise 12, Section 22; 27.5)