*
Smith-Toda Spectrum V (1) exists for all p > 5
Xueguang Zhou
Department of Mathematics, Nankai University
Tianjin 300071, China
1 The main result
In [6], Milnor proved that the dual Steenrod algebra A*phas the following algeb*
*ra structure
A*p= E(ø0, . .,.øn, . .). P (,1, . .,.,n, . .).
where E denotes the exterior algebra and P denotes the polynomial algebra and |*
*øi| =
2pi- 1, |,i| = 2pi- 2, p > 3.
In [8], Smith proved that there exists a spectrum V (n) for 0 6 n 6 2, p > 5*
* such that
as a comodule over the dual Steenrod algebra
H*(V (n), Zp) = E(ø0, . .,.øn)
In [9], Toda proved that there exists a spectrum V (n) for 0 6 n 6 3, p > 7 *
*such that
as a comodule over the dual Steenrod algebra
H*(V (n), Zp) = E(ø0, . .,.øn)
In recent years, the Smith-Toda spectrum V (n) plays an important role in ho*
*motopy
theory. It is natural to ask that whether V (n) exist for n > 4. After long yea*
*rs of hard
work on this subject, we finally proved that V (1) exists for p > 5.
We first introduce some notations. For a graded module A, we use A(n)to deno*
*te the
submodule of A generated by all elements of degree 6 n. Since any connected spe*
*ctrum
________________________________
*Supported by National Natural Science Foundation of China
1
X may be considered as a spectrum, we use Er(X) to denote the r-th space of X*
*. The
following is the main result of this paper.
Main Theorem Let p > 5 be a fixed prime and m be a non-negative integer, then
there exists a spectrum W (m) such that the following properties hold.
(a) As a comodule over the dual Steenrod algebra,
H*(W (m), Zp)(m+1)= E(ø0, . .,.øn, . .).(m+1)
(b) ßn(W (m)) = 0 for n > m+ 1.
(c) W (m) is a ring spectrum with unit.
(d) W (m) is homotopy commutative.
(e) W (m) is homotopy associative.
(f) There exists a map æm : W (m) ! W (m- 1) such that æm is the (m- 1) stag*
*e Post-
nikov decomposition of W (m) and æm is also an H map for m > 1.
(g) There exists a map fm : k(Zp, 1) ! E1(W (m)) such that E1(æ1) O . .O.E1(*
*æm ) O
fm = id: k(Zp, 1) ! k(Zp, 1), where k(Zp, 1) denotes the Eilenberg-Maclane spa*
*ce and
Er(æm ): Er(W (m)) ! Er(W (m- 1)) denotes the natural map induced by æm : W (m)*
* !
W (m- 1).
(h) fm : k(Zp, 1) ! E1(W (m)) is an H map with respect to the loop multiplic*
*ation
structure of E1(W (m)).
In this paper, p is always assumed to be a fixed prime > 5.
Let X be a local finite connected spectrum, then it can be easily seen that *
*there exists
a sequence of subspectrum X(n), n = 1, 2, . .,.such that
(a) X(1) X(2) . . .X(n) . . .X.
(b) X(n)contains no i-cell with i> n.
(c) Every cell represents a cycle modp.
(d) The injection X(n)! X induces an isomorphism from H*(X(n), Zp) to H*(X, *
*Zp)(n)
The above sequence of spectra {X(n)} are called the p standard CW-decomposit*
*ion of
X and X(n)is called the p standard n-skeleton of X. If there is no confusion, w*
*e simply
2
call them the standard decomposition and standard n-skeleton of X. We have that
H*(W (q(n))q(n), Zp) = E(ø0, . .,.øn)
n+1-1
where q(n) = |ø0. .ø.n| = 2p_____p-1- (n+ 1). It is obvious that for p > 5, 3q(*
*n) < |øn+1| =
2pn+1- 1.
We also have that for all q(n) 6 m 6 2pn+1 - 2,
H*(W (m)m , Zp) = E(ø0, . .,.øn)
Therefore, we have
Theorem 2. Let p > 5 be an odd prime, then for all non-negative integers n,*
* V (n)
exists.
Since 3q(n) < (2pn+1- 1), it can be easily seen that the multiplication map *
*of
n+1-2
W (2pn+1 - 2)2p and the homotopy maps for the commutativity and associativi*
*ty of
n+1 + *
* +
W (2pn+1-2)2p respectively map V (n)^V (n), V (n)^V (n)^I , V (n)^V (n)^V (n*
*)^I
into V (n), so we have
Theorem 3. Let p > 5 be an odd prime, then for all integers n > 0, the V (n)*
* constructed
above is a commutative, associative ring spectrum with unit and the natural inj*
*ection
V (n- 1) ! V (n) is an H-map.
Now let V (1) = [06n61 V (n) or limW (n), then it is easy to see that
H*(V (1), Zp) = E(ø0, . .,.øn . .).
Since the injections V (m) ! V (m+ 1) and W (m) ! W (m- 1) are all H-maps, V (1*
*) is
also an ring spectrum with unit. Notice that V (1) is an infinite spectrum and *
*ßm (V (1))
may be nonzero for infinite m's. So we can not prove that V (1) is a commutati*
*ve or
associative ring spectrum.
Let E be an infinite ring spectrum. If homotopy commutativity and associativ*
*ity hold
on any finite subspectrum of E, we say that E is a Q.C.A ring spectrum. We have
3
Theorem 4. Let p > 5 be an odd prime, then V (1) exists and is a Q.C.A ring *
*spectrum
with unit.
Since
H*(K(Zp, 0), Zp) = A*p
= E(ø0, . .,.øn, . .). P (,1, . .,.,n, . .).
= H*(V (1), Zp) H*(BP, Zp)
where BP denotes the Brown-Peterson spectrum and K(Zp, 0) denotes the Eilenberg-
Maclane spectrum. So we have
Theorem 5. Let p > 5 be an odd prime, then
K(Zp, 0) = V (1) ^ BP.
So K(Zp, 0) is decomposable.
Since Theorem 2 to Theorem 5 are all deduced from the main theorem, we need *
*only
prove the main theorem. We shall prove it by induction. Suppose the main theo*
*rem
holds for m, we shall prove that it holds for m+ 1. We use the letter (a)m+1 , *
*. .,.(e)m+1
respectively to denote (a), . .,.(e) hold for m+ 1.
In the literatures, V (n) is constructed from K(Zp, 0) by killing all non øi*
*0. .ø.interms.
The difficulty is that some øi0. .ø.inmay not be killed in this process. It sh*
*ould be
noticed that all øi, i> 0 exist in k(Zp, 1) and all the cohomology operationss *
*Qi are non-
trivial at the fundamental cohomology class of k(Zp, 1). So, we reduce the prob*
*lem of the
existence of V (1) to the problem of lifting the natural map from the stable ho*
*motopy
P -1
type ~k(Zp, 1) to K(Zp, 0) to the Postnikov decomposition W (m) of V (1). I*
*f W (m)
is an associative and commutative ring spectrum with unit, then E(øi0. .ø.in. .*
*).is a
natural subalgebra of H*(W (m), Zp). So V (1) exists. This is the main idea of *
*the proof
of (a) to (e) which is also the first part of our paper.
If W (m) is a commutative and associativbe ring spectrum with multiplicative*
* map
Mm , then it can be easily seen that there exists multiplicative map Mm+1 on W *
*(m+ 1)
4
such that W (m+ 1) ! W (m) is an H-map. However, it can not be easily seen that*
* Mm+1
is commutative and associative. Notice that the difference of the different mul*
*tiplicative
maps is a cohomology class of H*(W (m)^ W (m)), so by studying the relations be*
*tween
D(Mm+1 , ~Mm+1) we prove the existence of a new associative and commutative mul*
*tiplica-
tive map Mm+1 . Since the computation needs a coefficient 1_3, the result hold*
*s only for
p> 5. The above computation is the proof of (f) to (g) and is the second part o*
*f our paper.
By the theory of maps from the stable homotopy type of CW-complex to spectra*
*, we
P -1
reduced the problem of lifting the map from ~k(Zp, 1) to W (m) to the probl*
*em of
lifting the identity map of k(Zp, 1) to a map from k(Zp, 1) to E1(W (m)). The m*
*ain toool
is to use Milnor's construction Br(G) (r = 1, 2) for a topological group G. To *
*study the
relation between Br(k(Zp, 1)) and Br(E1(W (m))), we introduce the minus product*
*, plus
product and semi-mixed product of CW-complexes. Using this construction, we get*
* the
proof of (h). This is the third part of our paper.
The next section introduces some preliminaries and notations used in our pro*
*of. We
use here twice the notion of Ext group of an algebra. Firstly, we use it to pr*
*ove that
k(Zp, 1) ! E1(W (n)) is an H-map. Secondly, we use it to prove that W (m) is ho*
*motopy
associative. So we introduce the basic properties of Ext group of an algebra.
On page 289 of [7], D.C. Ravenal claimed in Theorem 7.5.1 that V (3) does no*
*t exist
for p = 5. His proof depends on the Toda's result ff1fip1= 0. In the appendix, *
*we will show
that all the proofs of the statement ff1fip1= 0 are incorrect. So, the prooof o*
*f Ravenal's
result is also incorrect.
The author wishes here to express his gratitude to J.K. Lin and Q.B. Zheng a*
*nd X.J.
Wang for their helps in our work. The author also wishes to express his gratitu*
*de to D.
Ravenal and J.P. Meyer for their pointing out some errors in our unpublished pr*
*evious
works [11] and [12] about this subject.
2 Notations about spectra
Since any connected spectrum is equivalent to a spectrum, any connected spect*
*rum X
can be expressed in the following form {Er(X), r: Er(X) ! Er+1(X), r> 1} such*
* that
5
(a) Er(X) is the space of Er+1(X) for r> 1. We call Er(X) r-th space of X.
(b) r: Er(X) ! Er+1(X), r> 1 is the adjoint of (Er+1(X)).
Let Y be a CW-complex. We use ~Yto denote the stable type] { Y, 2Y, . .,. n*
*Y, . .}.,
then there is a natural isomorphism between ß[ rY, Er(X)] and ß[Y~, X], r> 1. L*
*et f: Y !
Er(X) be a map, we use -rf~: -rY~ ! X to denote the map of spectra determined*
* by
f.
Let X1, X2 be two connected spectra, f: X1 ! X2 be a map. For r> 1, we use
Er(f): Er(X1) ! Er(X2) to denote the map of space determined by f.
3 Properties of homology groups of algebras
Let M be a commutative and associative graded algebra over Zp with unit. We use*
* M~ to
denote the kernel of the augmentation ö f M . We denote
Cn(M) = M~__._._.~M-z_____"
n copies
We define @n,i: Cn(M) ! Cn-1(M) by that for x1, . .,.xn 2 M~,
@n,i(x1 . . .xn) = x1 . . .xi-1 xixi+1 . . .xn
and define @n: Cn(M) ! Cn-1(M) by @n = n-1i=1(-1)i-1@n,ifor n> 2 and @1 = 0, t*
*hen
C*(M) = {Cn(M), @n, n> 1} is a chain complex. The dual C*(M) = {C*n(M), @*n, n>*
* 1}
is a cochain complex. We call H*,*(C*(M), @*) and H*,*(C*(M), @*) respectively *
*the Tor
and Ext group of M and simply denote them by H*,*(M) and H*,*(M). Notice that
the tensor product from C*m(M) C*n(M) ! C*m+n(M) makes C*(M) a DGA and thus
H*,*(M) is an algebra over Zp and H*,*(M) is a coalgebra over Zp.
For two commutative associative graded algebrs M and N over Zp with unit, the
tensor product (over Zp) algebra M N is also commutative and associative. By *
*using
the tensor product of projective resolutions of M and N, we have that
Proposition 3.1
H*,*(M N) = H*,*(M) H*,*(N)
H*,*(M N) = H*,*(M) H*,*(N)
6
where the tensor products means the tensor product of coalgebras and algebras r*
*espectively.
Let E(ø) be the exterior algebra generated by ø with |ø| an odd number. A d*
*irect
calculation shows that
Proposition 3.2 H*,*(E(ø)) = P (ø)*, where P (ø)* is the dual of the polynom*
*ial algebra
P (ø). P (ø)* has a basis øn = {ø___._.-.øz____", n> 1} with the coalgebra map*
* defined by
n-folds
(øn) = n-1i=1øi øn-i.
Since E(ø0, . .,.øn, . .).= E(ø0) . . .E(øn) . .,.we have
Proposition 3.3 H*,*(E(ø0, . .,.øn, . .).) = P (ø0)* . . .P (øn)*. ...
Let T (x) = P (x)=xp be the truncated polynomial algebra generated by x, a k*
*nown
result is that
Proposition 3.4 H*,*(T (x)) = E(y)* P (z)*, where y is represented by x in*
* C*(M)
and zn in P (z)* is represented by x___xp-1__._.-.xz__xp-1____"in C*(M).
n-folds
It is well-known that the algebra H*(k(Zp, 0), Zp) = E(v) ( 06i<1T (upi)*), *
*where v 2
H1(k(Zp, 1), Zp), upi2 H2pi(k(Zp, 1), Zp), fi(u1) = v (fi denotes the Bockstein*
* operation).
So by the previous propositions, we have
Proposition 3.5
i j
H*,*H*(k(Zp, 1)) = P (y)* 06i<1 (E(yi) P (zi)*)
where yi is represented by upi in C*(H*(k(Zp, 1))) and zniin P (zi)* is represe*
*nted by
upi up-1pi . . .upi up-1pi.
_____________-z____________"
n-folds
4 The case m = 0
We prove that the Main Theorem holds for m = 0. In this case, we take W (0) = K*
*(Zp, 0),
then E1(W (0)) = k(Zp, 1), and we take f0 = id: k(Zp, 1) ! E1(W (0))= k(Zp, 1).*
* It is
7
obvious that (a)0 to (h)0 hold.
5 Comodule H*(W (m), Zp)
From now on, we always assume that the Main Theorem holds for m. In this sectio*
*n, we
prove that H*(W (m), Zp) contains a subalgebra E(ø0, . .,.øn, . .)..
Now fm : k(Zp, 1) ! E1(W (m)) induces a map -1f~m: -1~k(Zp, 1) ! W (m). L*
*et
(f~m)*( -1(upi)) = ø0i. Since æ1. .æ.m -1(f-1m): -1(~k(Zp, 1) ! W (0) is the f*
*undamental
cohomology class of -1~k(Zp, 0), it follows from [6] that (æ1. .æ.m)*(ø0i) = ø*
*i. Since
ø0, . .,.øn, . .g.enerate an subalgebra E(ø0, . .,.øn, . .)., we have
Proposition 5.1
(a) ø00, . .,.ø0n, . .g.enerate an exterior algebra E(ø00, . .,.ø0n, . .)..
(b) H(W (m), Zp)(m+1)= E(ø00, . .,.ø0n, . .).(m+1).
6 The proof of (a)m+1 ,(b)m+1 ,(c)m+1
Let X, Y be two spectra and f: X ! Y be a map. We use C(f) to denote the map co*
*ne
C(X) [ Y . For ff2 Hm (X, Zp), we also use ff to denote the map from X to K(Zp*
*, m)
determined by ff.
Suppose the Main Theorem holds for m, it follows from Proposition 5.1 that t*
*here
exists a set of cohomology classes {ffi2Hm+2 (W (m), zp) | 16 i6 s} such that
(a) < ffi, E(ø00, . .,.ø0n, . .).(m+2)>= 0, 1 6 i 6 s,
where <, > denotes the Kronecker dual product.
(b) ff1, . .,.ffs are lineraly independent.
(c) dim(H*(W (m), Zp)(m+2)=E(ø00, . .,.ø0n, . .).(m+2)) = s
We call ff1, . .,.ffs the Postnikov invariant of W (m + 1) (it is possible t*
*hat s is 0),
then ff1_ . ._.ffs define a map ff: W (m) ! K(Zp, m + 2) _ . ._.K(Zp, m + 2), w*
*e define
_______________-z______________"
s-folds
W (m + 1) = -1C(ff).
8
It can be easily seen that the following is a cofibration sequence
W (m + 1) ! W (m) ! K(Zp, m + 2) _ . ._.K(Zp, m + 2)
_______________-z______________"
s-folds
Let æm+1 : W (m + 1) ! W (m) be the natural injection. It follows from the *
*above
conclusion that
H*(W (m), Zp)(m+2)
= E(ø00, . .,.ø0n, . .).(m+2)
E(ø0, . .,.øn, . .).(m+2)
and ßn(W (m+1)) = 0 for n > m+1 and æm+1 is the (m+1) stage Postnikov decomposi*
*tion
of W (m + 1). So (a)m+1 ,(b)m+1 holds.
Now we prove that W (m + 1) is a ring spectrum with unit.
Let Mm : W (m) ^ W (m) ! W (m) be the multiplication map of W (m) with unit.
Consider the following diagram
jm+1^jm+1
W (m+ 1) ^ W (m+ 1) -! W (m) ^ W (m)
? # Mm #
jm+1 ff
W (m+ 1) -! W (m) -! _s copiesK(Zp, m+ 2)
Since (Mm )*(æm+1 ^ æm+1 )*Hm+2 (W (m+ 1) ^ W (m+ 1), Zp) E(ø0, . .).(m+2)*
*, we have
that (ffi)*(Mm )*(æm+1 ^æm+1 )*Hm+2 (W (m+ 1)^W (m+ 1), Zp) = 0, 06 i6 s, that *
*is, we have
map equality ffOMm O(æm+1 ^æm+1 ) = 0. Therefore, there exists a map M0m+1: W (*
*m+ 1)^
W (m+ 1) ! W (m+ 1) such that æm+1 O M0m+1 = Mm O (æm+1 ^æm+1 ). But we do not
know whether M0m+1is a multiplication with unit. Since Mm |S0^W(m) = id|W(m) a*
*nd
Mm |W(m)^S0 = id|W(m), we have that
æm (M0m+1|S0^W(m+1)- id|W(m+1)) = 0
æm (M0m+1|W(m+1)^S0- id|W(m+1)) = 0
Thus, there exist cohomology classes ff, fi 2 Hm+1 (W (m + 1), ßm+1 (W (m + *
*1)) such
that
9
M0m+1|S0^W(m+1)- id= jff
M0m+1|W(m+1)^S0- id= jfi
where j: K(ßm+1 (W (m+ 1)), m+ 1) ! W (m+ 1) denotes the natural injection. No*
*w we
define
Mm+1 = M0m+1- j(S*0^ ff) - j(fi ^ S*0)
then it is easily seen that Mm+1 is a multiplication with unit and æm : W (m + *
*1) ! W (m)
is an H-map with respect to Mm+1 and Mm . Thus, (c)m+1 holds.
7 Differences of homotopy
First, we construct some spectrum.
Since S2 is a co-H-group, let ffi: S2 ! S2 _ S2 be the cogroup map. Let æ: S*
*2 ! S2 be
the inverse -id of id: S2 ! S2 and ø: S2 _ S2 ! S2 be the map such that the res*
*triction
of it on every summand S2 is the identity map. It is obvious that ø O (id_ æ) O*
* ffi: S2 ! S2
is homotopic to the constant map, so it can be extended to a map L: C(S2) ! S2.
Let X be a spectrum. Since X = 2( -2X) = ( -2X ^ S2), X inherit a co-struct*
*ure
from S2. We also use ~æto denote the map (id)^æ: X= -2X ^S2 ! -2X ^S2= X. The*
*n,
the map ø(id_ ~æ)ffi: X ! X also can be extended to a map (id^ L): -2X ^ C(S2)*
* !
-2X ^ S2 = X.
Let f: X ! Y ba a map of spectra. It can easily be seen that the following d*
*iagram
is commutative
C(X) -L! X
# #
C(Y ) -L! Y
As usual, we use S1+ to denote circle with an added base point *. Notice tha*
*t X ^S1+
and X ^ S1 = X are two defferent spectra.
Proposition 7.1 X ^ S1+ and X _ X are of the same homotopy type.
10
Proof. Notice that X = ( -1X) ^ S1, so we need only prove that space S1 ^ S1*
*+ and
S2 _ S1 are of the same homotopy type. This is a direct checking. *
*Q.E.D.
Now let f, g: X ! Y be two maps of spectra. If there exists a map H: X ^ I+ *
*such
that H|Xx0 = f, H|Xx1 = g, then we say that H is a homotopy from f to g. Let H,*
* H0
be two homotopies from f to g, then H and H0 define a T : X ^ S1+ ! Y as follow*
*s.
T |X^I+1= H
T |X^I+2= H0
where we regard S1 as the quotient space I1[ I2= ~ (I1 = I2 = [0, 1]) by identi*
*fying {0, 1}
of I1 with {0, 1} of I2. Let F : X ! X ^ S1+ be the composite of the natural i*
*njection
from X to X _ X and the map from X _ X to X ^ S1+, then T F is a map from
X to Y . We call T F the difference of H and H0 and denote it by d(H, H0). Hom*
*otopy
extension theory shows that
Proposition 7.2 Let f, g: X ! Y be two maps and H a homotopy from f to g, th*
*en for
any map ff: X ! Y , there exists a homotopy H0 from f to g such that d(H, H0) *
*= ff.
We also call H0 the sum of H and ff and denote it by H + ff = H0.
If X is a CW-complex and Y is a topological group, f, g: X ! Y are two maps *
*and H
and H0are two homotopies from f to g, then we define ~d(H, H0)(x, t) = H0(x, t)*
*(H(x, t))-1.
It is obvious that ~d(H, H0)(x, 0) = f(x)f(x)-1 = y0 = g(x)g(x)-1 = ~d(H, H0)(x*
*, 1) where
y0 denotes the unit of Y . So, ~d(H, H0) can define a map from X ! Y which w*
*e still
denote by ~d(H, H0).
Let X be a finite spectrum and Y be a connected spectra. It can be easily *
*seen
that the problem about the homotopies from Er(X) to Y can be reduced to the pro*
*blem
of homotopies from Er(X) to Er(Y ) for r sufficiently large. Since Er(Y ) is h*
*omotopy
equivalent to a topological group, we can use d~(H, H0) to define d(H, H0) and *
*by this
definition, we have the following proposition
Proposition 7.3 Let X, Y be two spectra, f, g: X ! Y be two maps. H and H0 a*
*re two
11
homotopies from f to g. fi, fi0: X ! Y be two maps, then
d(H + fi, H0+ fi0) = d(H, H0) + fi0- fi.
Proposition 7.4 Let X, Y be two spectra, f, g: X ! Y be two maps. H and H*
*0 are
two homotopies from f to g. Then a necessary and sufficient condition for d(H, *
*H) = 0
is that one of the following condition holds.
(a) H H0 rel X ^ {0, 1}+.
(b) The map T : X ^ S1+ defined above may be extended to a map from X ^ (C(S*
*1))+
to Y .
Let X, Y be two spectra and A be a subpectrum of X. f, g: X ! Y are two maps
such that f|A = g|A. If f g relA, that is, there is a homotopy from f to g su*
*ch that
H(a^ t) = f(a) for all a 2 A and 0 6 t 6 1, we say that H is a stationary homot*
*opy from
f to g relA or f and g are stationary homotopic relA. We use f~: A^ I+ to deno*
*te the
homotopy defined by ~f(a^ t) = f(a) for all a 2 A and 0 6 t 6 1. Let H be a hom*
*otopy
from f to g such that f|A = g|A, then d(H|A^I+, ~f) is defined and is a map fro*
*m A to
Y . If d(H|A^I+, ~f) 0, we say that H is a quasi stationary homotopy from f t*
*o g relA
or f and g are quasi stationary homotopic relA. Then, we have the following pro*
*position
Proposition 7.5 Let f and g be two quasi stationary homotopic map relA, then
(a) f and g are stationary homotopic relA.
(b) The quasi stationary homotopy H from f to g relA is quasi stationary hom*
*otopic
rel X^ 0+ [ X^ 1+ to a stationary homotopy from f to g relA.
Proof. (a) follows from (b). (b) follows from the homotopy extension propert*
*y with
respect to pair X^ I+ , the subspectrum X^ 0+ [ X^ 1+ [ A^ I+ and the map H Q.*
*E.D.
Let H, H0be two quasi stationary homotopy from f to g, then d(H, H0) is defi*
*ned and
is a map from X to Y . It can be easily seen that the following proposition ho*
*lds.
Proposition 7.6 Let H and H0 be two quasi stationary homotopies from f to g *
*relA,
then
12
(a) d(H, H0)| A 0.
(b) Let ff: X ! Y be a map such that ff| A 0, then H +ff is also a quasi *
*stationary
homotopy from f to g relA.
For the convenience of later use, we consider d(H, H0) as a map from X to Y*
* .
Let H be a homotopy from f to f. In what follows, we use d(H) to denote the *
*map
d(H, ~f): X ! Y , where ~fdenotes the stationary homotopy from f to f relX. I*
*t can
be easily seen that for maps f, g: X ! Y and homotopies H, H0 from f to g, we *
*have
d(H, H0) = d(H - H0), where H - H0 denotes the homotopy from f to g defined by
(
H(x^ 2t) 06 t6 1_
(H - H0)(x^ t) = 0 1 2
H (x^ 2(1 - t)) _26t6 1
8 Cochain of differences
Let X, Y be two connected spectra, {X(m), m> 0} be the usual CW decomposition o*
*f X.
Let f, g: X ! Y be two maps and H a homotopy from f|X(m)to g|X(m). As usual, we*
* define
the cochain D(f, g, H) of difference of f and g as follows. Let a be a (m+ 1) c*
*ell in X(m+1),
using the ordinary orientation we have that a^ 0+ [ @(a)^ I+ [ a^ 1+ is a (m+ 1*
*) sphere, so
f|a = f|a^0+, g|a = g|a^1+ and H|@a^I+ define an element D(f, g, H)|a 2 ßm+1 (Y*
* ). Thus,
D(f, g, H)2 Cm+1 (X, ßm+1 (Y )). It is obvious that D(f, g, H) depends on the h*
*omotopy
H.
Now we study the relation between D(f, g, H) and the difference of homotopie*
*s. Let x
be an (m+ 1) cell in X(m+1), then x may be considered as an element in ßm+1 (X(*
*m+1), X(m)).
As usual, we use ~@: ßm+1 (X(m+1), X(m)) ! ßm (X(m)) to denote the boundary ope*
*ration
of relative homotopy group. For x2 Cm+1 (X), we also use ~@(x) to denote the el*
*ements in
ßm (X(m)).
It is obvious that D(f, g, H) is a cocycle. We also use D(f, g, H)* to deno*
*te the
cohomology class and call it the cohomology of difference of f and g with respe*
*ct to H.
Obviously, D(f, g, H)* depends on the homotopy H. In the following, it is assum*
*ed that
pß*(Y ) = 0. We call such a spectrum p spectrum. Let x2 Hm+1 (X, Zp), then x ma*
*y be
considered as a linear combination of (m+ 1) cells. Let H0 be another homotopy*
* from
13
f to g, it may be assumed that ff = d(H, H0). Since on x ^ 0+, x ^ 1+, H and H0
coincide, so D(f, g, H) and D(f, g, H0) differ only on ~@x ^ I+ . It can be eas*
*ily seen that
D(f, g, H0)(x) = D(f, g, H)(x) + ff*(@~(x)). It should be noticed that ~@(x) i*
*s uniquely
determined modp in ßm (X(m)). So, ff*(@~(x)) is also uniquely determined.
Let `: (X(m)) ! Y be a map. We define _(`) 2 Hm+1 (X, ßm+1 (Y )) by _(`)(*
*x) =
`*(@~(x)) 2 ßm+1 (Y )) for x 2 Hm+1 (X, Zp).
Let G be the subgroup of Hm+1 (X, ßm+1 (Y )) generated by all _(`) with `: *
*X(m) ! Y .
We define D(f, g)* to be the set {D(f, g, H)*} with H taken over all the homoto*
*pies from
f to g. It can be easily seen that D(f, g)* is a coset modG. We have
Proposition 8.1 Let D(f, g)* be as defined above, then
(a) D(f, g)* is uniquely defined by the homotopy classes of f and g.
(b) A necessary and sufficient condition for f and g to be m+ 1 homotopy is *
*that
D(f, g)* = 0 modG.
Let A be a subspectrum of X and f, g: X ! Y be two maps such that f|A = g|A.*
* Let
H be a quasi stationary homotopy from f|X(m)to g|X(m)relA(m), then it can be ea*
*sily seen
that D(f, g, H)|A(m)= 0. So D(f, g, H) 2 Hm+1 (X, A, ßm+1 (Y )) = Hm+1 (X=A, ßm*
*+1 (Y )).
Let f: X ! Y be a map of spectra, H be a homotopy from f|X(m) to f|X(m). It *
*can
be easily seen that D(f, f, H)|a = d(H)*(@~(a)) for any (m+ 1) cell a in X. If *
*d(H) 0,
then D(f, f, H)|a = 0, so we have the following proposition
Propostion 8.2 Let f and H be as above. If d(H) = 0, then D(f, f, H) = 0.
Let A be a subspectrum of X, f|A = g|A, and H be a quasi stationary homotopy
from f|X(m) to g|X(m) relA(m), then it can be easily seen that D(f, g, H)|A(m) *
*= 0, so
D(f, g, H) 2 Hm+1 (X, A, ßm+1 (Y )) = Hm+1 (X=A, ßm+1 (Y )).
9 W (m+ 1) is a commutative ring spectrum
For any spectrum W , we use T : W ^ W ! W ^ W to denote the map switching the t*
*wo
factors. Then, the statement that Mm+1 is homotopy commutative is equivalent to*
* that
Mm+1 OT = Mm+1 . Since S0^W (m+ 1) = W (m+ 1)^S0 = W (m+ 1), Mm+1 T |S0^W(m+1) =
14
Mm+1 |S0^W(m+1) = id and Mm+1 T |W(m+1)^S0 = Mm+1 |W(m+1)^S0 = id. So we have
Mm+1 T |W(m+1)^S0_S0^W(m+1)= Mm+1 |W(m+1)^S0_S0^W(m+1).
In the following part of this section, we use X to denote W (m+ 1) ^ W (m+ 1*
*), A to
denote the subspectrum W (m+ 1) ^ S0 _ S0 ^ W (m+ 1). Then, (d)m+1 is included *
*by a
stronger proposition.
Proposition 9.1 Suppose (a) to (c) hold for m+ 1 and (d)m holds, then there *
*exists a
multiplication Mm+1 : X ! W (m+ 1) such that
(a) Mm+1 has a unit.
(b) Mm+1 and Mm+1 T are quasi stationary homotopic relA.
Using the standard CW decomposition, we have that W (m+ 1)(m+2) is obtained *
*by
killing (m+ 2) dimensional cohomology classes ff1, . .,.ffs. So W (m+ 1)(m+1)= *
*W (m)(m+1).
It follows from the induction hypothesis that there exists a multiplication wit*
*h unit
M~m+1 : X ! W (m+ 1) and a quasi staionary homotopy H from M~m+1|X(m)to M~m+1T *
*|X(m)
relA(m). So D(M~m+1 , ~Mm+1T, H) is defined. In order to prove Proposition 9.1,*
* we need
only prove that D(M~m+1 , ~Mm+1T, H) = 0.
First, we state some T-properties of homotopy. Let X, Y be two spectra, f, g*
*, h: X !
Y be maps with H1 a homotopy from f to g and H2 a homotopy from g to h. We use
H1+ H2 to denote homotopy defined by
(
H1(x ^ (2t)+) 06 t6 1_
(H1+ H2)(x ^ t+) = + 1 2 x 2 X
H2(x ^ (2t - 1) ) _26t6 1
We use -H to denote the homotopy from g to f defined by (-H)(x ^ t+) = H(x ^
(1 - t)+). Suppose H: (W (m+ 1) ^ W (m+ 1))(m)^ I+ ! W (m+ 1) be a homotopy from
M~m+1 to M~m+1T , then H O ~Tis also a homotopy from M~m+1 to M~m+1T , where T~*
* =
T ^ (id): (W (m+ 1) ^ W (m+ 1))(m)^ I+ ! (W (m+ 1) ^ W (m+ 1))(m)^ I+ . We have*
* the
following proposition.
Proposition 9.2 There exists a quasi stationary homotopy H from M~|X(m)to M~*
*m+1T |X(m)
relA(m) such that d(H, -HT~) 0.
15
Proof. Let H~ be a homotopy from M~m+1|X(m) to M~m+1T |X(m). In general, w*
*e do
not know whether d(H~, -H~T ) = 0. However, d(H~, -H~T ) is a map from (W (m+ *
*1) ^
W (m+ 1))(m)to W (m+ 1), so
d(H~, -H~T~) O ( T ) = d(H~T~, ~H~T)c= d(H~T~, -H~).
It can be easily seen that d(H~T~, -H~) are obtained from D(H~, -H~T ) by movin*
*g the point
e0 in S1 to e1. Now the honotopy class of d(H~, -H~T ) is independent of the ch*
*oice of e0
and e1. It can be easily seen that H can be extended to a quasi stationary homo*
*topy relA
from M to MT on X. We define a new quasi stationary homotopy H from M~m+1|X(m) *
*to
M~m+1T |X(m) relA(m) by H = ~H+ 1_d(H~, -H~T~), then it is obvious that d(H, -H*
*T~) 0.
2
Q.E.D.
Proposition 9.3 Let H be a quasi stationary homotopy from M~m+1 to M~m+1 T *
*and
x 2 Hm+1 (W (m+ 1)^W (M+ 1), Zp), D(M~m+1 , ~Mm+1T, H)*.x = -D(M~m+1 , ~Mm+1T, *
*H)*.
T*(x), then it is equivalent to the statement
D(M~m+1 , ~Mm+1T, H)* = -T *(D(M~m+1 , ~Mm+1T, H)*).
Proof. It follows by definition of D(M~m+1 , ~Mm+1T, H) and Proposition 9.1.*
* Q.E.D.
Now we prove Propostion 9.1. For a general M~m+1: (W (m+ 1) ^ W (m+ 1))(m+1*
*) !
W (m+ 1) with unit, we do not know whether D(M~m+1 , ~Mm+1T, H) = 0. However, w*
*e can
define a new multiplication Mm+1 with unit by
1
Mm+1 = M~m+1 - j O __D(M~m+1 , ~Mm+1T, H)
2
i j
where j: K ßm+1 (W (m+ 1)), m+ 1 ! W (m+ 1) denotes the natural injection. It *
*should
be noticed that W (m+ 1) is a p spectrum and ßm+1 (W (m+ 1)) is a Zp vector spa*
*ce, that
is, 1_22Zp, so 1_2D(M~m+1 , ~Mm+1T, H) is defined. It is obvious that
D(Mm+1 , Mm+1 T, H)
= D(M~m+1 , ~Mm+1T, H) - D(M~m+1 , ~Mm+1T, H)
= 0
16
10 W (m+ 1) is an associative ring spectrum
In this section, we always assume Y = W (m+ 1), X = Y ^Y ^Y and A = S0^ Y ^Y [
Y ^S0^ Y [ Y ^Y ^S0.
Since S0^ S0^ Y = Y . S0^ Y ^S0 = Y , Y ^S0^ S0 = Y and that Mm+1 is a multi*
*plica-
tion with unit, so
Mm+1 (id^Mm+1 )|S0^S0^Y [S0^Y ^S0[Y ^S0^S0
= Mm+1 (Mm+1 ^id)|S0^S0^Y [S0^Y ^S0[Y ^S0^S0
Similarly, we have Mm+1 (id^Mm+1 )|S0^Y ^Y= Mm+1 |Y ^Y= Mm+1 (Mm+1 ^id)|S0^Y ^Y*
*. So
we have that Mm+1 (id^Mm+1 )|A = Mm+1 (Mm+1 ^id)|A. Then, (e)m+1 is reduced to*
* a
more stronger proposition.
Proposition 10.1 Suppose (a)m+1 to (d)m+1 hold and (e)m holds, then there e*
*xists a
quasi stationary homotopy relA from Mm+1 (id^Mm+1 ) to Mm+1 (Mm+1 ^id).
Proof. We prove by induction. It is obvious that Proposition 10.1 holds fo*
*r m= 0.
Since in this case W (1) = W (0 = K(Zp, 0), we may suppose that m> 0. Let ~Mm+1*
*: Y ^Y !
Y be a commutative multiplication with unit. By the induction hypothesis, there*
* is a quasi
stationary homotopy ~Hfrom M~m+1(id^M~m+1 )|X(m)to Mm+1 (Mm+1 ^id)|X(m)relA(m).*
* So,
D(M~m+1 (id^M~m+1 ), Mm+1 (Mm+1 ^id), ~H) is defined. For simplicity, we use xy*
* to denote
M~m+1 (x^ y) and analogously x(yz) to denote M~m+1(x^ ~Mm+1(y^ z)) and so on. *
*Then
D(M~m+1 (id^M~m+1 ), Mm+1 (Mm+1 ^id), ~H) can be expressed as D(x(yz), z(xy), ~*
*H). Be-
fore the proof of Proposition 10.1, we state the relation between D(x(yz), z(xy*
*), ~H) and
the action of the group Z6 on the factors of X.
Let P, T : X ! X be the maps defined by P (x^ y^ z) = y^ z^ x, T (x^ y^ z) =*
* z^ y^ x,
x, y, z 2 Y . ø: Y ^Y ! Y ^Y be the map defined by ø(x^ y) = y^ x. Let G be the*
* quasi
stationary homotopy from xy to yx relS0^ Y [ Y ^S0. In what follows, we use x,*
* y, z
respectively to denote the element of the first, second and third factor Y of *
*X. Since
i j
(xy)^ z ^ idis a map from X^ I+ to Y ^Y ^I+ , so G O ((xy)^ z)^ idis a homotop*
*y from
(xy)z to z(xy) on X(m). Let
i j
J1 = H + GO ((xy)^ z)^ id
17
i j
J2 = HO(P ^id) + GO ((xy)^ z)^ id
i j
J3 = HO(P 2^id) + GO ((xy)^ z)^ id
, then J1 is a homotopy from x(yz)|X(m)to z(xy)|X(m)and J2 is a homotopy from z*
*(xy)|X(m)
to y(xz)|X(m)and J3 is a homotopy from y(xz)|X(m)to x(yz)|X(m). Notice that G i*
*s defined
on Y ^Y ^I+ , so we have the following proposition.
Proposition 10.2 D(x(yz), (xy)z, H) = D(x(yz), z(xy), J1).
It is obvios that J1+ J2+ J3 is a homotopy from x(yz) to x(yz), we have the *
*following
proposition.
Proposition 10.3 J1+ J2+ J3 is a quasi stationary homotopy relA(m).
Proof. Since (S0^ Y ^Y )(m)= (Y ^Y )(m), we have
J1 + J2 + J3|(S0^Y ^Y )(m)^I+
= (M~m+1 +G+ M~m+1+ ~Mm+1+ GO(ø^ id)+ M~m+1)|Y ^Y ^I+
= (G+ GO(ø^ id))|Y ^Y ^I+
It should be noticed that the symbol M~m+1 in the above equality is the stat*
*ionary
homotopy from M~m+1 to M~m+1 relY ^Y .
Now by Proposition 9.1, G + GO(ø^ id): S0^ Y ^Y ^I+ ! Y is a stationary homo*
*topy
from yz to yz rel(S0^ S0^ Y ^I+ [S0^ Y ^S0^ I+ ), by the same argument as above*
* we have
0^Y ^I+ 0 0 *
* +
that (J1+ J2+ J3)|Y ^S is a stationary homotopy from xz to xz rel(S ^ S ^ *
*Y ^I [
0^I+
S0^ Y ^S0^ I+ ) and that (J1+ J2+ J3)|Y ^Y ^S is a stationary homotopy from x*
*y to xy
rel(Y ^S0^ S0^ I+ [ S0^ Y ^S0^ I+ ). Therefore, the sum of above mentioned homo*
*topies
satisfies that d(J1+ J2+ J3|A(m)) 0 and J1+ J2+ J3 is a quasi stationary homo*
*topy from
x(yz) to x(yz) relA(m)on X(m).
Since (P ^id) induces a map from X^ I+ to itself and (P ^id) also induces a *
*map from
X to itself, we also use ~Pto denote the map induced by (P ^id). We have the f*
*ollowing
proposition.
18
Proposition 10.4 The following diagram is homotopy commutative.
d(J1+J2+J3)
(X(m)) -! Y
# ~P # id
d(J1+J2+J3)
(X(m)) -! Y
Proof. Since d(J1+J2+J3)OP~is obtained from d(J1+J2+J3) by removing the point
e of S1+ in X^ S1+, we have d(J1 + J2 + J3)OP~ d(J1 + J2 + J3). We use ~Tto d*
*enote
the map X ! X to denote the map induced by the map T ^id: X^ I+ ! X^ I+ . It *
*is
obvious that J1O(ø^ id): X^ I+ ! Y is a homotopy from z(yx) to x(yz), so J1+ J1*
*O(ø^ id)
is also a homotopy from x(yz) to x(yz). By the same argument as above, we have
Proposition 10.5 The following diagram is homotopy commutative.
d(J1+J1O(fi^id))
(X(m)) - ! Y
# ~T # id
d(J1+J1O(fi^id))
(X(m)) - ! Y
In what follows, for any homotopy H: X(m)^I+ ! Y from x(yz) to z(xy), we alw*
*ays
use J1, J2, J3 to denote the homotopy defined above. We have the following prop*
*osition
Proposition 10.6 There exists a homotopy H from x(yz) to z(xy) on X(m) such *
*that
(a) d(J1 + J2 + J2) 0.
(b) d(J1 + J1 O (T ^ id)) = 0.
(c) J1 + J2 + J3 is also a quasi homotopy from x(yz) to x(yz) relA(m).
Proof. For a general homotopy H~ from x(yz) to (xy)z on X(m), we do not know
whether d(J1+ J2+ J2) 0. Since p> 5 and Y is a ring spectrum with unit, ß( X(*
*m), Y )
is also a Zp-vector space. Thus, 1_3d(J1+ J2+ J2) is also a homotopy class from*
* X(m) to Y .
We define a new homotopy H0 = ~H- 1_3d(J1 + J2 + J2). Let J01, J02, J03be the h*
*omotopies
defined above for H0, then
d(J01+ J02+ J03)
i 1 1 1 *
* j
= d J1 - __d(J1 + J2 + J3) + J2 - __P *d(J1 + J2 + J3) + J3 - __P *2d(J1 + *
*J2 + J3)
3 3 3
1i j
= d(J1 + J2 + J3) - __d(J1 + J2 + J3) + d(J1 + J2 + J3) + d(J1 + J2 + J3)
3
= 0
19
Then we set H = H~ - 1_2d(H~ + ~H(T ^id)). By the same argument as above, we h*
*ave
d(J1 + J1O(T ^id)) 0. It is also easy to check that d(J1+ J2+ J3) 0.
It is obvious that J1+ J2+ J3 is also a quasi stationary homotopy from x(yz)*
* to x(yz)
relA(m). It follows from Proposition 7.6 that
Proposition 10.7
(a) D(x(yz), z(xy), J1)+ D(z(xy), y(zx), J2)+ D(y(zx), x(yz), J3) = 0
(b) Let a 2 Hm+1 (X, Zp), then
D(x(yz), z(xy), J1)|a+ D(x(yz), z(xy), J1)|T*(a)= 0
It is obvious that (a) is equivalent to (1 + P *+ P 2*)D(x(yz), z(xy), J1) =*
* 0 and (b) is
equivalent to (1 + T *)D(x(yz), z(xy), J1) = 0.
It can be easily seen that all the homotopy from x(yz) to (xy)z on X(m)just *
*mentioned
above are quasi stationary homotopy relA(m). So we have
D(x(yz), z(xy), J1)|H*(A,Zp)
= D(x(yz), z(xy), H)|H*(A,Zp)
= 0
since A ! Z ! (Y=S0)^ (Y=S0)^ (Y=S0) is a cofibration sequence. By the collapse*
*d exact
sequence mentioned above we have D(x(yz), z(xy), J1) H*((Y=S0)^ (Y=S0)^ (Y=S0*
*),
ßm+1 (Y )). Since ßm+1 (Y ) is a Zp-vector space, we define ffin: Cn,*(H*(Y, ß*
*m+1 (Y )) !
Cn+1,*(H*(Y, ßm+1 (Y )) and D(x(yz), z(xy), J1) 2 C3,m-2(H*(Y ), ßm+1 (Y )), so*
* we can de-
fine ffi3D(x(yz), z(xy), J1) and have the following proposition.
Proposition 10.8 ffi3D(x(yz), z(xy), J1) = 0.
Let W = Y ^Y ^Y ^Y and 3,1, 3,2, 3,3, 3,4: W ! X be the map defined by
3,1(a^ b^ c^ d)= (ab^ c^ d)
3,2(a^ b^ c^ d)= (a^ bc^ d)
20
3,3(a^ b^ c^ d)= (a^ b^ cd)
3,4(a^ b^ c^ d)= (da^ b^ c)
where a, b, c, d respectively denotes the element in the first, second,third an*
*d fourth factors
of W . Then ffi3 = ( *3,1- *3,2+ *3,3),
Let a(): W ! W (m+ 1) be the homotopy class of the map a(b(cd)). Since x(yz)
(xy)z holds on X(m)and W (m+ 1) is homotopy commutative, we have a(b(cd)) a((*
*bc)d)
a(b0(c0d0)) (b0(c0d0))a holds on W (m+1). Since ßn(W (m+ 1)) = 0 for n > m+ 1*
* > 0, we
have a(b(cd)) a(b0(c0d0)) (b0(c0d0))a on W for any permutation (b0, c0, d0)*
* of (b, c, d).
Therefore, a(b0(c0d0)) and (b0(c0d0))a both belong to the homotopy class a[ ].
In the folowing part of this section, we use S to denote ffi3(D(x(yz), z(xy)*
*, J1)). Then
S
= ( *3,1- *3,2+ *3,3)D(x(yz), z(xy), J1)
= D((ab)(cd), d[ ], J1O( 3,1^id))
-D(a[ ], d[ ], J1O( 3,2^id))
+D((a[ ], (ab)(cd), J1O( 3,3^id))
For simplicity, we omit the homotopies in D since there is no confusion. So *
*S may be
expressed as D((ab)(cd), d[ ]) + D(d[ ], a[ ]) + D((a[ ], (ab)(cd)).
We use Q to denote the expression D(x(yz), z(xy))+D(z(xy), y(zx))+D(y(zx), x*
*(yz)),
then Q = 0. So we have *3,i(Q) = 0 for 16 i6 4. So we have
D((ab)(cd), d[ ]) + D(d[ ], c[ ]) + D(c[ ], (ab)(cd))
= *3,1(Q) = 0
D(a[ ], d[ ]) + D(d[ ], (bc)(ad)) + D((bc)(ad), a[ ])
= *3,2(Q) = 0
D(a[ ], (ab)(cd)) + D((ab)(cd), b[ ]) + D(b[ ], a[ ])
= *3,3(Q) = 0
D((bc)(ad), c[ ]) + D(c[ ], b[ ]) + D(b[ ], (bc)(ad))
= *3,4(Q) = 0
21
Let L: W ! W be the map defined by L(a^ b^ c^ d) = (b^ c^ d^ a), where a, b*
*, c, d
respectively denotes the elements of the first, second, third and fourth factor*
*s of W .
Then,
L*(S) = D((bc)(da), c[ ]) + D(c[ ], d[ ]) + D(d[ ], (bc)(da))
L2*(S) = D((ab)(cd), b[ ]) + D(b[ ], c[ ]) + D(c[ ], (ab)(cd))
L3*(S) = D((bc)(da), a[ ]) + D(a[ ], b[ ]) + D(b[ ], (bc)(da))
It is a directing that
S + L2*(S)
= L*(S) + L3*(S)
= D(c[ ]c, d[ ]) + D(d[ ], a[ ]) + D(a[ ], b[ ]) + D(b[ ], c[ ])
i X3 ji *
* j
( *3,i) - *3,4D(x(yz), z(xy)) + D(z(xy), y(zx)) + D(y(zx), x(yz))
i=1
= L*(S) + L2*(S) + L3*(S) - S
So we have
L*(S) + L2*(S) + L3*(S) - S
= L1*(S) + L3*(S) + L2*(S) - S
= S + L2*(S) + L2*(S) - S
= 2L2*(S) = 0
Since p> 5, ßm+1 (W (m+ 1)) is a Zp-vector space, we have L2*(S) = 0. Therefor*
*e, S =
L2*L2*(S) = 0. Thus, Proposition 10.8 is proved. Q.*
*E.D.
Now we state the relation between the changing of multiplication Mm+1 and t*
*he
cohomology class D(x(yz), z(xy)). Let u 2 Hm+1 (Y ^Y, ßm+1 (Y )), u . Hm+1 (S0*
*^ Y [
Y ^S0, Zp) = 0, that is, u 2 Hm+1 (Y ^Y=S0^ Y [ Y ^S0, ßm+1 (Y )), then u repre*
*sents
a map from (Y ^Y ) to the fibre K(ßm+1 (Y ), m+ 1). Let j: K(ßm+1 (Y ), m+ 1) !*
* Y be the
natural injection, then we define Mm+1 : Y ^Y ! Y by M0m+1= M~m+1+ jOu: Y ^Y ! *
*Y .
22
Since u . Hm+1 (S0^ Y [ Y ^S0, Zp) = 0, M0m+1is a multiplication with unit. If *
*ø*(u) = u,
where ø: Y ^Y ! Y is the map defined by ø(x^ y) = y^ x, since M~m+1 is commut*
*a-
tive, then it can be easily seen that the new multiplication M0m+1is also commu*
*tative.
It should be noticed that M0m+1|(Y ^Y )(m)= M~m+1 |(Y ^Y )(m), so the value of *
*multiplica-
tion M0m+1only differs from M~m+1 on the m+ 1 cell on Y ^Y . To avoid confusio*
*n, we
use D(x(yz), z(xy), ~Mm+1) to denote the cochain of difference with respect to *
*the same
homotopy on W (m). We have the following
Proposition 10.9 D(x(yz), z(xy), M0m+1)- D(x(yz), z(xy), ~Mm+1) = -ffi2(u*
*),
where ffi2: C2,*(H*(Y, Zp), ßm+1 (Y )) ! C3,*(H*(Y, Zp), ßm+1 (Y )) is the cobo*
*undary opera-
tion defined in section 2. So the cohomology class D(x(yz), z(xy), M0m+1) is in*
*dependent
of the choice of M~m+1.
Let T : X ! X be the map defined by T (x^ y^ z) = (z^ y^ x), then we have t*
*he
following
Proposition 10.10 Let u 2 Hm+1 (Y, ßm+1 (Y )), then ffi2ø*(u) = -T *ffi2(u).
Proof. Let ff, fi, fl 2 H*(Y=S0, Zp), |ff|+ |fi|+ |fl| = m+ 1, then
ø*@2(ff^ fi^ fl)
= (-1)|ff||fifl|fifl^ ff - (-1)|fffi||fl|fl^ fffi
@2ø*(ff^ fi^ fl)
= (-1)|ff||fl|@2(fl^ fi^ ff)
= (-1)|ff||fl|(fl^ fiff - flfi^ ff)
= (-1)|ff||fl|((-1)|ff||fi|fl^ fffi - (-1)|fi||fl|fifl^ f*
*f)
= -ø*@2(ff^ fi^ fl)
Therefore, ø*@2 = -@2ø*. Dually, we have ffi2ø*(u) = -T *ffi2(u). *
* Q.E.D.
Now we prove Proposition 10.1. First, we prove that there exists a multiplic*
*ation Mm+1
such that D(x(yz), z(xy), Mm+1 ) = 0. Since H*(Y, Zp)(m+2)= E(ø0, . .,.øn, . .)*
*.(m+2), we
23
i j(m+2) i j(m+2)
have H*,*H*(Y, Zp) = H*,*E(ø0, . .,.øn, . .). . Notice that
i j __ __
H*,*E(ø0, . .,.øn, . .). = P(ø0) . . .P(øn) . . .
We divide the proof into the following cases.
If there is no integers i, j, k> 0 such that m+ 1 = (2pi-1)+ (2pj- 1)+ (2pk-*
* 1), then
H3,m-2(E(ø0, . .,.øn, . .)., ßm+1 (Y )) = 0. So D(x(yz), z(xy), ~Mm+1)* 0 in *
*C*(H*(Y, Zp),
ßm+1 (Y )) = C*(E(ø0, . .,.øn, . .)., ßm+1 (Y )) and there is u 2 Hm+1 (Y=S0^ Y*
*=S0, ßm+1 (Y ))
such that D(x(yz), z(xy), ~Mm+1) = ffi2(u), then ø*(1_2(u + ø*(u)) = 1_2(u + ø**
*(u)). We define
a new multiplication Mm+1 : Y ^Y ! Y by Mm+1 = M~m+1+ jO 1_2(u+ ø*(u)), then it*
* follows
easily that Mm+1 is also commutative. We have
D(x(yz), z(xy), Mm+1 ) - D(x(yz), z(xy), ~Mm+1)
1 1 *
= -ffi2(__u+ __ø (u))
2 2
1 i * j
= -__ ffi(u)+ ffi(ø (u))
2
i 1 j i1 j
= - __D(x(yz), z(xy), ~Mm+1) - __T *D(x(yz), z(xy), ~Mm+1)
2 2
i 1 j i1 j
= - __D(x(yz), z(xy), ~Mm+1) - __D(x(yz), z(xy), ~Mm+1)
2 2
= -D(x(yz), z(xy), Mm+1 )
Therefore, we have D(x(yz), z(xy), Mm+1 ) = 0
If there exist integers i, j, k> 0 such that m+ 1 = (2pi-1)+ (2pj- 1)+ (2pk-*
* 1). For
__
any homogeneous element u = øi1. .ø.ik2 E (ø0, . .,.øn, . .)., we define the we*
*ight of u to
__
be w(u) = k. Let u1 . . .ul be any homogeneous element of l copiesE(ø0, . .,*
*.øn, . .).,
we define w(u1 . . .ul) = li=1w(ui). We use U(l, k) to denote the subvector *
*space of
__
l copiesE(ø0, . .,.øn, . .).spanned by all homogeneous elements with weight k.*
* Let U(k) =
06l<1U(l, k). It is obvious that U(l, k) = 0 for l> k, that is, w(u1 . . .ul)*
*> l. It can be
easily seen that U(k) is a subcomplex of C*(H(Y ), Zp). Now that H*,*(E(ø0, . .*
*,.øn, . .).=
__ __
P (ø0) . . .P(øn) . .,.we have the following proposition.
Proposition 10.11
(a) Hl,*(H*(E(ø0, . .,.øn, . .).)) = Hl,**,*(U(l)).
24
(b) Hl,*n,*(U(l)) = 0 for n< l.
We also use U(l, k)* to denote the dual of U(l, k), U(k)* to denote the dual*
* of U(k)
and (øi1. .ø.il)* to denote the dual of øi1. .ø.il).
Then D(x(yz), z(xy), ~Mm+1) may be written in the form u+ v with w(u) = 3 and
w(v) > 3. It follows from Proposition 10.11 that ffi3(u) = ffi3(v) = 0. Since*
* v is a co-
__ __
cycle, there exists ~v2 E (ø0. .ø.n) E(ø0. .ø.n), w(~v) > 3 such that ffi2(~v*
*) = v. Then,
since T *D(x(yz), z(xy), ~Mm+1) = -D(x(yz), z(xy), ~Mm+1), we have T *(u) = -u,*
*T *(v) =
-v. So 1_2ffi(~v+øv~) = 1_2v+ 1_2T *(v) = 1_2v+ 1_2v = v. Now we define a *
*new multiplica-
tion M~0m+1by M~0m+1+j(1_2(~v+øv~)). It can be easily seen that D(x(yz), z(xy)*
*, ~M0m+1) -
D(x(yz), z(xy), ~Mm+1) = v, so D(x(yz), z(xy), ~M0m+1) = u. Thus, u can be expr*
*essed in
the following form ~(i0, j0, k0)ø*i0ø*j0ø*k0, where ~(i0, j0, k0) 2 Zp and the*
* set {i0, j0, k0, } =
{i, j, k} We also divide the proof into three cases
(a) i= j= k
(b) i= j, j6=k.
(c) i, j, k are mutually different.
Case (a). Since i= j= k, U(3, 3)(m+1) contains only one m+ 1 cell øi^øi^øi,*
* we have
P*(øi^øi^øi) = (øi^øi^øi). Since D(x(yz), z(xy), ~M0m+1)|(a+p*+p2*)(fii^fii^fi*
*i)= 0, we have
3D(x(yz), z(xy), ~M0m+1)|fii^fii^fii= 0. Therefore, D(x(yz), z(xy), ~M0m+1)|fii*
*^fii^fii= 0.
Case (b). Since i= j, j6=k, U(3, 3)(m+1)contains three m+ 1 cells øk^ øi^øi,*
* øi^øk^ øiand
øi^øi^øk, we have P*(øk^ øi^øi) = (øi^øi^øk), P*2(øk^ øi^øi) = (øi^øk^ øi), T*(*
*øk^ øi^øi) =
-(øi^øi^øk). We define a new multiplication map Mm+1 by
ji * * * *j
Mm+1 = M~0m+1+__~i,i,køi^ (øi^øk) + ~k,i,i(øi^øk) ^øi
2
where ~i,i,k= ~k,i,iand ~i,i,k+~i,k,i+~k,i,i= 0. It can be easily seen that Mm+*
*1 is commu-
tative and
D(x(yz), z(xy), ~M0m+1)|fik^fii^fii
= D(x(yz), z(xy), ~M0m+1)|fii^fik^fii
= D(x(yz), z(xy), ~M0m+1)|fii^fii^fik
= 0
25
and so D(x(yz), z(xy), ~M0m+1) = 0.
Case (c). Since i, j, k are mutually different, there are six cells
øi^øj^øk, øj^øk^ øi,øk^ øi^øj,
øk^ øj^øi,øi^øk^ øj,øj^øi^øk.
It follows from Proposition 10.5 that
D(x(yz), z(xy), ~M0m+1)(øi^øj^øk+ øj^øk^ øi+øk^ øi^øj)
= ~i,j,k+~j,k,i+~k,i,j= 0
Since T *D(x(yz), z(xy), M0m+1) = -T *D(x(yz), z(xy), M0m+1) and T*(øi^øj^*
*øk) =
-(øk^ øj^øi), T*(øj^øk^ øi) = -(øi^øk^ øj), T*(øk^ øi^øj) = -(øj^øi^øk), we als*
*o have
~i,j,k= -~k,j,iand ~j,k,i= -~i,k,j,. We now define a new multiplication Mm+1 by
j i * * * * * * * * *
** j
Mm+1 = M0m+1+__ ~i,j,k((øi(øjøk) + (øjøk) øi)) + ~k,i,j((økøi) øj+ øj(økø*
*i))
2
It is a direct checking that Mm+1 is also commutative and D(x(yz), z(xy), Mm*
*+1 ) = 0.
Therefore, Mm+1 is associative.
Since D(x(yz), z(xy), H) = D(x(yz), z(xy), Mm+1 ) = 0, the stationary homoto*
*py H
from x(yz) to (xy)z relA(m) on X(m) can be extended to a stationary homotopy fr*
*om
x(yz) to (xy)z relA on X^ I+ . So Proposition 10.1 is proved.
11 The proof of (g)m+1 .
In what follows, we will prove that æm : k(Zp, 1) ! E1(W (m)) can be lifted to *
*a map
æm+1 : k(Zp, 1) ! E1(W (m)).
Let ( -1E1)*(ff1), . .,.( -1E1)*(ffs) 2 Hm+1 (E1(W (m)), Zp) be the image of*
* ff1, . .,.
ffs in Hm+3 (E1(W (m), Zp). Since E1(W (m+ 1), Zp) is the fibre space obtained *
*by killing
the cohomology classes ( -1E1)*(ff1), . .,.( -1E1)*(ffs), it is obvious that ( *
*-1E1)*(ff1),
. .,.( -1E1)*(ffs) are additive cohomology classes in H*(E1(W (m), Zp). By the *
*induction
hypothesis (h)m , fm : k(Zp, 1) ! E1(W (m)) is an H-map, so f*m( -1E1)*(ff1), .*
* .,.
f*m( -1E1)*(ffs) are additive cohomology classes. We have H*(k(Zp, 1), Zp) = E*
*(ff)
P (fi(ff)), so the only additive cohomology classes in H*(k(Zp, 1), Zp) are mul*
*tiples of
26
n n
ff, fi(ff) and (fi(ff))p . Therefore, if m + 3 = 2p , n = 1, 2, . .,.by the as*
*sumption
of ff1, . .,.ffs, ffiø0n = 0, 16 i6 s. Since ø0nis the image of upn in H*(W (*
*m), Zp), it
follows that ( -1E1)*(ffi)(fm )*(H2pn(k(Xp, 1), Zp)) = 0, so f*m( -1E1)*(ff1) =*
* 0 ,. .,.
f*m( -1E1)*(ffs) = 0 and fm can be lifted to a map fm+1 : k(Zp, 1) ! E1(W (m+ 1*
*)).
If m+ 3 6= 2pn, n = 1, 2, . .,.it is obvious that f*m( -1E1(ffi)) = 0, 16 i6*
* s. So fm can
also be lifted to a map fm+1 : k(Zp, 1) ! E1(W (m+ 1)).
12 Semi-product of CW complexes.
In order to prove (h)m+1 , we introduce the notion of semi-product of CW comple*
*xes.
Let X be a CW complex, as usual we use X to denote the suspension X ^ S1. S*
*ince
S1 = I={0, 1}, any point of X can be expressed as x ^ t, 06 t6 1. Let x02X be *
*the base
point of X. It should be noticed that X^ 0 [ x0^I [ X^ 1 collapse to the base p*
*oint of X.
Let Y be another CW complex, as usual we use X#Y to denote Join(X, Y ), the *
*space
obtained by all the segment joining the point of X to the point of Y . All the*
* point
of X#Y can be expressed as x ^ t ^ y with x2 X, y2 Y , 06 t6 1. We define two*
* maps
`1, `2: X#Y ! (X x Y ) as follows `1(x ^ y ^ t) = (x, y) ^ t, `2(x ^ y ^ t) = *
*(x, y) ^ (1- t).
In general, `1 and `2 are not homotopic.
It can be easily seen that X#Y = (X ^ Y ). We have the following proposition
Proposition 11.1 (i) _ (j) _ `1, (i) _ (j) _ `2: X _ Y _ X#Y ! (X x Y*
* ) are
homotopy quivalences, where i and j are respectively the natural injections fro*
*m X and
Y to X x Y .
!
We define semi-product X x Y and X x Y of X and Y as follows.
X x Y = {(x ^ r, y ^ s) | x2 X, y2 Y, 06 r6 s6 1} Xx Y
!
X x Y = {(x ^ r, y ^ s) | x2 X, y2 Y, 06 s6 r6 1} Xx Y
!
We call X x Y the minus product of X and Y and X x Y the plus product of X a*
*nd
Y . It can be easily seen that
!
(1) X x Y = (X x Y ) [ (X x Y ).
27
!
(2) (X x Y ) = (X x Y ) \ (X x Y ).
We have the following proposition
Proposition 11.2
(1) X x Y = C(`2).
!
(2) X x Y = C(`1).
Proof. The proof is straightfowward. We define two homotopy equivalences _1 *
*and
!
_2 respectively from C(`1) and C(`2) to X x Y and X x Y by
!
_1((x ^ t ^ y), l)= (x ^ ((1- l)t + l), g ^ (1- l)t) 2 X x Y
!
_2((x ^ t ^ y), l)= (x ^ ((1- t)(1 - l), g ^ ((1- t)(1- l) + l)) 2 X x Y
x 2 X, y 2 Y 06 t6 1, 06 l6 1.
Q.E.D.
Let Z be another CW complex, since (X x Y )#Z = ((X x Y ) ^ Z) = ( X _ Y _
(X ^ Y ) ^ Z = (X ^ Z) _ (Y ^ Z) _ (X ^ Y ) ^ Z, we have
!
(X x Y ) x Z
((X x Y ) x Z) [ C((X x Y )#Z)
= ( X _ Y _ (X ^ Y ) _ Z _ X ^ Z _ Y ^ Z _ (X ^ Y ) ^ Z)
[C ( (X ^ Z) _ (Y ^ Z) _ (X ^ Y ) ^ Z)
= X _ Y _ Z _ ( (X ^ Z) [ C( X ^ Z))
_( (X ^ Z) [ C( X ^ Z)) _ C ( (X ^ Y ) ^ Z ^ C((X ^ Y ) ^ Z))(I)
Similarly, we also have
!
X x (Y x Z)
= X _ Y _ Z _ ( (X ^ Y ) [ C( X ^ Y ))
_( (X ^ Z) [ C( X ^ Z)) _ C ( (X ^ (Y ^ Z)) ^ C(X ^ (Y ^ Z))) (II)
Since there is a natural topological isomorphism between (X x (Y x Z)) and
((X x Y ) x Z) such that ( (X ^ (Y ^ Z)) = ((X ^ Y ) ^ Z), we may identify
28
!
the subspace (X x (Y x Z)) of X x (Y x Z) with the subspace ((X x Y ) x Z) of
!
(X x Y ) x Z). With this identification ~, we define
! !
W (X, Y, Z) = X x (Y x Z) [ (X x Y ) x Z)= ~ .
It follows from (I) and (II) and the following equalities
(X ^ Z)
i j
= X ^ Z [ C X ^ Z [ C X ^ Z)
(X ^ Y ^ Z)
i j
= (X ^ Y ) ^ Z [ C (X ^ Y ) ^ Z [ C( (X ^ (Y ^ Z))
we have the following proposition
Proposition 11.3 Let X, Y, Z be three CW complexes, then W (X, Y, Z) and ( X*
*) _
i j
( Y ) _ ( Z) _ (X ^ Z) _ (X ^ Y ^ Z) are of the same homotopy type.
In what follows, we use _: (X ^ Y ^ Z) ! W (X, Y, Z) to denote the natural
injection. We call W (X, Y, Z) the semi-product of X, Y and Z.
13 The Milnor construction of topological group.
Let G be a topological group. J. Milnor introduce the notion Br(G), r> 0, to st*
*udy the
relation between G and its classifying space BG. In what follows, we need only*
* B1(G)
and B2(G).
As we know, B1(G) = G. We have a map M~: G#G ! G defined as follows.
M~(g1 ^ t ^ g2) = (g1g2 ^ t) 2 G (g1g2 is the product of G), then B2(G) is the*
* mapping
cone C(M~) = G [ C(G#G) of M~.
! !
Since G x G = C(`1) = (G x G) [ C(G#G), we may define "+: G x G ! B2(G) by
"+| (GxG) = M: (G x G) ! G
"+|C(G#G) = id: C(G#G) ! C(G#G) B2(G)
29
where M is the product of G and define
! !
j1: G x (G x G) ! G x G
! !
j2: (G x G) x G ! G x G
by that for any g1, g2, g3 2 G and 06 s6 r6 1,
j1(g1 ^ r, (g2, g3) ^ s)
= (g1 ^ r, g2g3 ^ s)
j2((g1, g2) ^ r, g3 ^ s)
= (g1g2 ^ r, g3 ^ s)
Since G is associative, we have that
"+j1 (G, (G, G)) = "+j1 (G, G, G) = "+j2 ((G, G), G)
and so "+j1 and "+j2 define a map j: W (G, G, G) ! B2(G) by
j|Gx!x(GxG)= "+j1
j|(GxG)!xG= "+j2
Let T be a topological group. t0 2 is the unit of T . G is the loop space *
* (T ) of
T with base point t0. The multiplication map M of G inherits from that of T , t*
*hat is,
M(~1, ~2)(t) = ~1(t)~2(t) for any ~1, ~2 2 (T ). It is well known that M and t*
*he loop
multiplication of G are homotopic. We define oe1: B1(G) = (G) ! T , oe2: B2(G)*
* ! T as
follows. For any ~ 2 (T ), g1, g2 2 G, and 06 t, l6 1,
oe1(~ ^ t) = ~(t)
oe2| G = oe1
oe2((g1 ^ t ^ g2), l)
= g1((1- l)t + l)g2((1- l)t)
It can be easily seen that oe2 is an extension of oe1.
30
Now, let _: (G^G^G) ! W (G, G, G) be as defined in the last part of the pr*
*evious
section, then
Proposition 11.4 (oe2j_)* = 0: H*( (G ^ G ^ G), Zp) ! H*(T, Zp).
In fact, Proposition 11.4 is the law of associativity for the relation betwe*
*en the ho-
mology group of loop space and its classifying space.
Since we only concern about the homology group of product of spaces, we use *
*here
Serr's cubic singular homology theory. Before the proof of Proposition 11.4, we*
* introduce
the semi-product of singular cubes.
Let X, Y be two topological spaces, l, m positive integers, ff: Il ! X, fi: *
*Im ! Y are
!
respectively l, m cubes in X and Y . We use ff x fi to denote the (l+ m+ 2) sin*
*gular cube
defined as follows. Let A = (0, 0), B = (1, 0), C = (1, 1) and oe: I2 ! 4ABC be*
* the map
that leaves the segment AB and BC invariant (we regard 4ABC as a subspace of I2)
and sends the segment from (1, 1) to (0, 1) linearly to CA and collapse the seg*
*ment from
(0, 1) to (0, 0) to point A. It is required that oe is a topological map from t*
*he interior of
I2 to the interior of 4ABC. Suppose oe(r, s) = (r0, s0) 2 4ABC, then 1> r0>s0>0*
*, and we
define
!
(ff x fi)(x, y, r, s)
!
= (ff(x) ^ r0, fi(y) ^ s0) 2 X x Y
! !
So ff x fi is a (l+ m+ 2) cube of X x Y . We call it semi-product of ff and fi.
For chains a = ~iffi and b = ~jfij, where ffi and fij are respectively sin*
*gular cubes
! !
in X and Y , we define a x b = ~i~jffix fij.
Let fl: In ! Z be a cube in Z, we define the semi-mixed product W (ff, fi, *
*fl) by
! !
W (ff, fi, fl) = ff x (fi x fl)- (ff x fi) x fl
We must point out that W (ff, fi, fl) is not a cub but a (l+ m+ 2) chain in *
*W (X, Y, Z).
We can similarly define the semi-mixed product of chains in X, Y and Z.
Let a, b, c be respectively the l, m, n-cycles modp in X, Y, Z. It can be ea*
*sily seen that
W (a, b, c) is also a cycle modp in W (X, Y, Z) and any homology class in _*H*(*
* X ^
Y ^ Z, Zp) can be expressed in the form ~W (a, b, c)*, where as usual, we use *
*W (a, b, c)*
31
to denote the homology class containing W (a, b, c).
Now we prove Proposition 11.4. Let ff, fi, fl be respectively l, m, n singul*
*ar cubes in G.
We define a singular (l+ m+ n+ 3) singular cube D(ff, fi, fl) by D(ff, fi, fl)(*
*x, y, z, r, s, t) =
ff(x)(r0)fi(y)(r0t+ s0(1- t))fl(z)(s0) 2 T , where r, s, t2 I, x2 Il, y2 Im ,*
* z2 In, oe(r, s) =
(r0, s0). Specifically, we have that D(ff, fi, fl)(x, y, z, r, s, 0) = ff(x)(r0*
*)fi(y)(s0)fl(z)(s0), and
that
D(ff, fi, fl)(x, y, z, r, s, 1) = ff(x)(r0)fi(y)(r0)fl(z)(s0). Thus, D(ff, fi, *
*fl) is a homotopy from
! !
oe2j1(ff x (fi x fl)) to oe2j2((ff x fi) x fl).
Let a, b, c be as above, it follows from the construction of D(a, b, c) that*
* @(D(a, b, c)) =
(oe2)*j*W (a, b, c)+ some degerated chains. Since in singular cubic chain compl*
*ex of topo-
logical space, degenerated chains = 0, we have (oe2)*j*W (a, b, c) = 0. Since a*
*ny homology
class in _*H*( G ^ G ^ G) can be expressed in the form ~W (a, b, c)*, we have*
* that
(oe2j_*)* = 0 Q.E.*
*D.
Let ": G ! g0 be the augmentation map. As usual, we use ~H(G, Zp) to denote*
* the
subgroup "-1*(0). We also denote group homomorphisms induced by M respectively *
*by
M*: ~H*(G, Zp) ~H*(G, Zp) ! ~H*(G, Zp) and M*: ~H*(G, Zp) ! ~H*(G, Zp) ~H*(*
*G, Zp).
~M fi ff
Consider the following sequence G#G -! G -! B2(G) -! (G#G) -! where ø
denotes the natural injection and oe denotes the natural map from B2(G) = G[C(*
*G#G)
to (G#G) = G ^ G. Since it is a cofibration, we have
Proposition 11.5
(a)
H*(B2(G), Zp)
i j
= ~H*( G, Zp)=M* (H~*(G, Zp) ~H(G, Zp)) + M-1*(0)
H*(B2(G), Zp)
i j i j
= H~*(G, Zp) ~H*(G, Zp) =M* H*( G, Zp) + M*-1(0)
(b) oej_ = @3: G ^ G ^ G ! (G ^ G) = (G#G), where @3 is the boundary map
in the chain complex C*(H*(G, Zp)).
By this proposition and the definition of H*,*(H*(G), Zp), we have the follo*
*wing propo-
32
sition
Propoition 11.6
(a) ø*H*( G, Zp) = H1,*(H*(G, Zp)).
(b) 0 ! j*_*~H( (G ^ G ^ G), Zp)) ø*(H*( G, Zp)) ! ~H(B2(G), Zp) !
H2,*(H*(G, Zp)) ! 0 is an exact sequence.
Let Y be a space, f, g: Y ! G be maps. Since G is a topological group, we ca*
*n define
fg: Y ! G by fg(y) = f(y)g(y). We denote f, g: (fg): Y ! G as usual. Since
G is a cogroup, we can define f _ g: Y ! G. We have the following proposit*
*ion
Proposition 11.7 ø( f _ g) ø( (fg)).
Proof. Let d: Y ! Y xY be the diagonal map, then fg is defined to be M O(f x*
*g)Od.
Since (Y x Y ) = Y _ Y _ Y #Y , we have ø (fg) = ø( f _ g + (f#g)). Since
B2(G) = G [ C(Y #Y ), so (f#g) = 0. Thus ø( f _ g) = ø( (fg)). Q.E.D.
Let X, Y be two homotopy commutative H-space. For any x1, x2 2 X and y1, y2 *
*2 Y ,
we use x1x2 and y1y2 to denote the their products of the H-spaces. Let f: X ! Y*
* be an
H-map, then there exists a homotopy H from f(x1)f(x2) to f(x1x2). We define a m*
*ap
B2(f, H): B2(X) ! B2(Y ) as follows.
(
f(x1) ^ (2t- 1) ^ f(x2)1_6t6 1
B2(f, H)(x1 ^ t ^ x2)= 2 1
H(x1, x2, 1- 2t) 06 t6 _2
B2(f, H)| X = (f)
It can be easily seen that B2(f, H) is uniquely determined by f and H. In w*
*hat
follows, we simply use B2(f) to denote B2(f, H).
14 The proof of (h)m+1 for m+ 2 6= 2pi+ 2pj- 2, 2pi, 2pi- 1.
Since W (m) is an infinite loop space, E1(W (m)) = (E2(W (m)), E2(W (m+ 1)) *
* =
(E3(W (m+ 1)), E3(W (m+ 1)) = (E4(W (m+ 1)). We may assume that E4(W (m+ 1))
is a simplicial complex and E3(W (m+ 1)) is the Milnor's simplicial loop space *
*defined in
33
[5] and is thus a topological group. Therefore, E2(W (m+ 1)) = (E3(W (m+ 1)) *
*inher-
its a multiplication map from that of E3(W (m+ 1)) (see section 11) and E1(W (m*
*)) =
(E2(W (m)) also inherits a multiplication map from that of E2(W (m)), that is,*
* they are
both topological groups. So all the conclusions in section 11 hold for G = E1(W*
* (m)) and
T = E2(W (m)).
For two H-spaces X, Y , we use X x Y to denote the H-space whose product is*
* in-
duced by those of each factors. Suppose (a)m+1 to (g)m+1 hold, we will prove (*
*h)m+1 ,
that is, fm+1 : k(Zp, 1) ! E1(W (m+ 1)) is an H-map. We reduce the problem to *
*the
property of H*(k(Zp, 1), Zp). Let gm : C(m) ! W (m) be the universal covering o*
*f W (m),
then E1(gm ): E1(C(m)) ! E1(W (m)) is also the universal covering of E1(W (m)).*
* It
can be easily seen that fm x E1(gm ): k(Zp, 1) x E1(C(m)) ! E1(W (m)) and fm+1 x
E1(gm+1 ): k(Zp, 1)xE1(C(m+ 1)) ! E1(W (m+ 1)) are both homotopy equivalences. *
*Since
fm is an H-map, so is fm x E1(gm ). Thus, the statement that fm+1 is an H-map i*
*s equiv-
alent to the statement that fm+1 x E1(gm+1 ) is an H-map. Since B2(G) = C(M~) w*
*here
M~: G#G ! G is induced by M: G x G ! G, the homotopy type of B2(G) is uniquely
determined by that of M. We may assume that E1(W (m)) = k(Zp, 1) x E1(C(m)), th*
*en
fm : k(Zp, 1) ! E1(W (m)) induces a map B2(fm ): B2(k(Zp, 1)) ! B2(E1(W (m))). *
*It can
be easily seen that the following diagram is commutative.
fm^fm^fm
k(Zp, 1) ^ k(Zp, 1) ^ k(Zp, 1) -! E1(W (m)) ^ E1(W (m)) ^ E1(W (m))
j_ # # j_
B2(fm)
B2(k(Zp, 1)) -! B2(E1(W (m)))
We have the following proposition
Proposition 12.1 The following statements are equivalent.
(a) fm+1 : k(Zp, 1) ! E1(W (m+ 1)) is an H-map.
(b) oe2B2(fm ): B2(k(Zp, 1)) ! E2(W (m)) can be lifted to a map from B2(k(Zp*
*, 1)) to
E2(W (m+ 1)).
(c) B2(fm )*oe*2E2( 2ffi) = 0, 16 i6 s, where ff1, . .,.ffs 2 Hm+2 (W (m), Z*
*p)) are the
Postnikov invariants and E2( 2ffi) are the image of ffi in Hm+4 (E2(W (m)), Zp).
Proof. First, we prove that (a) implies (b). Suppose that (a) holds, then B2*
*(fm+1 ):
34
B2(k(Zp, 1)) ! B2(E(W (m+ 1))) exists. It is obvious that E2(æm+1 )*B2(fm+1 ) =*
* oe2B2(fm ),
where æm+1 : W (m+ 1) ! W (m) is the map in (c) and E2(æm+1 ): E2(W (m+ 1)) ! E*
*2(W (m))
is the map induced by æm+1 . So (b) holds.
Now we prove that (b) implies (a). Let M0: k(Zp, 1) x k(Zp, 1) ! k(Zp, 1) b*
*e the
multiplication of k(Zp, 1) and M: E1(W (m+ 1)) x E1(W (m+ 1)) ! E1(W (m+ 1)) be*
* the
multiplication map inheriting from E2(W (m+ 1)), we will prove that the followi*
*ng diagram
is homotopy commutative.
fm+1xfm+1
k(Zp, 1) x k(Zp, 1) - ! E1(W (m + 1)) x E1(W (m + 1))
# M0 # M
fm+1
k(Zp, 1) - ! E1(W (m+ 1))
We have oe2Oø O M O (fm+1 xfm+1 ) = oe2O( fm+1 _ fm+1 ) = oe1O( fm+1 _ fm+1 *
*) and
i j
M0 = (id_id_M~0): k(Zp, 1)xk(Zp, 1) = k(Zp, 1)_ k(Zp, 1)_k(Zp, 1)#k(Zp, 1)*
* !
k(Zp, 1), so ( fm+1 )( M0) = fm+1 _ fm+1 _ ( fm+1 )( M~0). Since oe2B2(fm ) *
*can be
lifted to a map from B2(k(Zp, 1)) to E2(W (m+ 1)) and so øM~0 = 0 in B2(k(Zp, 1*
*)). Thus,
in E2(W (m+ 1)), ø( fm+1 )(M~0) = 0. So we have
oe1( fm+1 )( M0)
= oe1( fm+1 _ fm+1 )
= oe2øM (fm+1 x fm+1 )
Since E1(W (m+ 1)) = E2(W (m+ 1)), we have fm+1 OM = MO (fm+1 x fm+1 ). So
fm+1 is an H-map, that is, (b) implies (a).
Since (b) and (c) are equivalent, (a), (b) and (c) are all equivalent. *
* Q.E.D.
So to prove (h)m+1 , we need only prove B2(fm )*oe*2( 2E)*(ffi) = 0.
According to Proposition 11.5, we discuss the problem in four cases;
(a) m+ 1 6= 2pi+2pj- 2, 2pi, 2pi-1 for any i, j> 0.
(b) m+ 1 = 2pi+2pj- 2 for some i, j> 0.
(c) m+ 1 = 2pi-1 for some i> 0.
(d) m+ 1 = 2pi for some i> 0.
Now we prove the first case (a). In this case, m+ 1 6= 2pi+2pj- 2, 2pi, 2pi-*
*1 for any
i, j> 0, then it follows from Proposition 11.6 that H2*,m+2(H*(k(Zp, 1))) = 0. *
*So we have
35
Hm+2 (B2(k(Zp, 1), Zp) = j*_*H*( k(Zp, 1) ^ k(Zp, 1) ^ k(Zp, 1), Zp). Therefor*
*e,
oe2*B2(fm )*(Hm+2 (B2(k(Zp, 1), Zp)
= oe2*j*_* (fm ^ fm ^ fm )*H*( k(Zp, 1) ^ k(Zp, 1) ^ k(Zp, 1), Zp)
= 0
i j
So we have oe2* Hm+2 (B2(k(Zp, 1), Zp)) = 0. That is, B*2((fm )*oe*2( 2E2)*(f*
*fi) = 0. It
follows from Proposition 12.1 that fm+1 : k(Zp, 1) ! E1(W (m+ 1)) is an H-map.
Now we prove the case (c). In this case, Hm+4 (B2(k(Zp, 1)), Zp) = j*_*Hm+4 *
*(k(Zp, 1)^
k(Zp, 1)^k(Zp, 1), Zp)+ø*(Hm+1 ( k(Zp, 1), Zp)). Since fm+1 exists, ø*(B2(fm ))*
**( 2E2)*(ffi)
= 0. By the same argument as above and the reason that oe2*j*_*( fm ^ fm ^ fm *
*)* = 0,
we have that B*2(fm )oe*2( 2E2)*(ffi) = 0 and so fm+1 : k(Zp, 1) ! E1(W (m+ 1))*
* is also an
H-map.
15 The proof of (h)m+1 for m+ 1 = 2pi + 2pj - 2.
Since for i 6= j, upi upj- upj upi is a generater of H2,*(H*(k(Zp, 1), Zp)), *
*to prove
(h)m+1 we need only prove that upi upj- upj upiis the product of some homolog*
*ical
classes in H*(E2(W (m)), Zp). First, we deduce some properties of Pontrjagin pr*
*oduct in
H*(E2(W (m)), Zp).
First, we define a map from E1(W (m)) x E1(W (m)) to E2(W (m)).
Let X be an arcwise connected Hausdorff space. A, B X are two closed arcw*
*ise
connected subspaces of X such that X = A[ B. Let C = A\ B and T (A, B) = Ax {0}*
* [
Bx {1} [ Cx I Xx I. We define æ: T (A, B) ! X by
æ(a, 0) = aa 2 A
æ(b, 1) = bb 2 B
æ(c, t) = cc 2 C 0 6 t 6 1
It can be easily seen that æ is a weak homotopy equivalence and a homotopy e*
*quiva-
lence if X is a CW complex and A, B are sub CW complexes of X.
Let Y be another Hausdorff space and f: A ! Y and f: B ! Y be two maps, then
f|C and g|C are two maps from C to Y . If f|C g|C and H is the homotopy from *
*f|C to
36
g|C, then we define a map s(f, g, H): T (A, B) ! Y as follows.
s(f, g, H)(a, 0) = f(a)a 2 A
s(f, g, H)(b, 1) = g(b)b 2 B
s(f, g, H)(c, t) = H(c,ct)2 C 0 6 t 6 1
! !
Let X be a topological group. Then Xx X = Xx X [ Xx X, Xx X \ Xx X =
!
(Xx X) = (Xx X) ^ S1. We define `1: Xx X ! B2(X) and `2: Xx X ! B2(X) by that
for any x, y 2 X,
`1(x ^ r, y ^ s) = (x ^ r, y ^ s) 2 B2(X)r > s
`1(x ^ r, y ^ s) = (xy ^ s) 2 X 2 B2(X)r = s
`2(x ^ r, y ^ s) = (y ^ s, x ^ r) 2 B2(X)r < s
`2(x ^ r, y ^ s) = (yx ^ s) 2 X 2 B2(X)r = s
Suppose that X is a homotopy commutative topological group and H is a homoto*
*py
from (Xx X)x I to X between the maps (xy)^idto (yx)^id, then s(`1, `2, H) is *
*defined
!
and is a map from T (Xx X, Xx X) to B2(X). If X is of the same homotopy type wi*
*th a
!
CW complex, then T (Xx X, Xx X) and B2(X) are both of the same homotopy type wi*
*th
!
CW complexes. Thus, T (Xx X, Xx X) and ( Xx Y ) are of the same homotopy type.
Therefore, s(`1, `2, H) determine a homotopy class of maps from Xx Y to B2(X).
Let T be a homotopy commutative topological group. X = (T ) be the loop spa*
*ce of
T with the end point the unit e of T . The multiplication of X is induced from *
*that of T
as we defined previously.
Let ~H: T xT xI ! T be a homotopy from t1t2 to t2t1, t1, t2 2 T and H: Xx Xx*
* I ! X
is the homotopy induced by ~H, that is, H(~1, ~2, t)(s) = ~H(~1(s), ~2(s), t) f*
*or 06 s, t6 1.
!
Let æ: T ((Xx X, Xx X) ! Xx Y be the map defined above and oe: X = (T )*
* !
T be the map defined by oe(~ ^ s) = ~(s). We have the following proposition
Proposition 14.1 The following homotopy relation holds.
oe2 O s(`1, `2, H) M O (oex oe) O æ
! j
T (Xx X, Xx X) -! Xx X
# oex oe
# s(`1, `2, H) T xT
# M
B2(X) -ff2! T
37
where oe2 is as defined in section 12 and M is the product map of T and æ is th*
*e homotopy
equivalence defined in the beginning part of this section.
! !
Proof. Since T (Xx X, Xx X) = Xx Xx {0}\Xx Xx {1}\ (Xx Y )x I, we construct
!
the required homotopy piecewisely on the three component Xx Xx {0}, Xx Xx {1} a*
*nd
(Xx Y )x I.
It is obvious that for x, y 2 X and 06 s6 r6 1,
i j
oe2s(`1, `2, H ^ id) (x ^ r, y ^ s)x {0}
= oe2`1(x ^ r, y ^ s)
= x(r)y(s)
i j
= M(oex oe)æ (x ^ r, y ^ s)x {0}
!
So we define ~H: Xx Xx {0}x I ! T by ~H(x ^ r, y ^ s, 0, t) = x(r)y(s), 06 r6 s*
*6 1, 06 t6 1.
i j
Now we have oe2s(`1, `2, H) (x ^ r, y ^ s)x {1} = y(s)x(r) and M(oex oe)æ(x ^ *
*r, y ^ s) =
x(r)y(s), so we can define H~: Xx Xx {1}x I ! T for x, y 2 X, 06 r6 s6 1, 06 t6*
* 1 by
H~(x ^ r, y ^ s, 1, t) = ~H(x(r), y(s), 1- t). Now on (Xx Y ) = (X!xX) \ (Xx X*
*) we have
H~(x ^ r, y ^ s, 0, t)
= x(r)y(s)
H~(x ^ r, y ^ s, 1, t)
= H~(x(r), y(s), 1- t)
i j
oe2s(`1, `2, H) (x ^ r, y ^ t), u
= H~(x(r), y(s), u)
= M(oex oe)æ(x ^ r, y ^ s, u)
= x(r)y(s)
These maps can define a map from (Xx Y )x @(Ix I) to T . It can be easily s*
*een that
the map can be extended to a map ~Hfrom (Xx Y )x (Ix I) to T . So oe2Os(`1, `2*
*, H^id)
M O (oex oe) O æ. Q.*
*E.D.
Since æ is a homotopy equivalence, Proposition 14.1 says that the product M(*
*oex oe)
can be determined by oe2s(`1, `2, H).
38
Let X, W be two homotopy commutative H-space, f: X ! W be an H-map. We now
study that under what conditions the following diagram
! !
T (Xx X, Xx X) - ! T (W xW, W xW )
Xx X W x W
# #
B2(X) - ! B2(W )
is homotopy commutative.
! ! !
Since T (Xx X, Xx X) = (Xx X)x 0[ (Xx X)x I [(Xx X)x 1, any point of T (Xx X,
Xx X) can be written in the form (x1^s^ x2^t)x l, where x1, x22X and if l= 0, t*
*hen
06 t6 s6 1 and if l= 1, then 06 s6 t6 1 and if 0< l< 1, then 06 s= t6 1. Thus,*
* we can define
! !
T (f): T (Xx X, Xx X) ! T (W xW, W xW ) by
T ((x1^s^ x2^t)x l) = (f(x1)^ s^ f(x2)^ t)x l).
Since X, W are homotopy commutative, there exist a homotopy H from x1x2 to x*
*2x1
and a homotopy ~~Hfrom u1u2 to u2u1. Since f is an H-map, there exists a homoto*
*py ~H
from f(x1)f(x2) to f(x1x2). So both fH + ~Hand H~+ ~~Hare homotopies from f(x1x*
*2)
to f(x2)f(x1). Thus d(fH + ~H, ~H+ ~~H) is defined. If d(fH + ~H, ~H+ ~~H) 0,*
* we say
that f is a strong H-map with respect to H, ~H, ~~H. According to the construc*
*tion of
!
T (Xx X, Xx X) and T (f), we can easily obtain the following result.
Proposition 14.2 If f is a strong H-map with respect to H, ~H, ~~H, then the*
* following
diagram is homotopy commutative;
! T(f) !
T (Xx X, Xx X) -! T (W xW, W xW )
# #
B2(f)
B2(X) -! B2(W )
Now we study under what conditions f can be a strong H-map.
Let ø: Xx X ! Xx X be the map defined by ø(x1, x2) = (x2, x1). We also use ø*
* to
denote the similar map from W xW to W . If d(H, -H(ø^ id)) 0, we say that H i*
*s a
ø-homotopy from x1x2 to x2x1. An H-space X is a p-H-space if pß[Q, X] = 0 for a*
*ny CW
complex Q. It is obvious that Er(W (m)) is a p-H-space for r > 1. We have the f*
*ollowing
proposition
39
Proposition 14.3 Let X be a homotopy commutative p-H-space, then there exist*
*s a
ø-homotopy from x1x2 to x2x1.
Proof. Let H~ be n homotopy from x1x2 to x2x1, we define a new homotopy H by
H = ~H- d(H~, (-H~)(ø^ id)). It must be pointed out that (-H~)(ø^ id) is also a*
* homotopy
from x1x2 to x2x1. So d(H~, -H~(ø^ id)) is defined and is a map from (Xx X) to*
* X. Since
X is a p-H-space, 1_2d(H~, (-H~)(ø^ id)) is defined . Therefore, we define a ne*
*w homotopy
H = ~H- 1_2d(H~, (-H~)(ø^ id)). It is obvious that H is a ø-homotopy. *
* Q.E.D.
Now we have
Propotition 14.4 Let W be a p-H-space and H be a ø-homotopy from x1x2 to x2x*
*1 and
H~~be a ø-homotopy from u1u2 to u2u1 and f: X ! W be an H-map, then there exist*
*s an
homotopy H~ from f(x1)f(x2) to f(x1x2) such that f is a strong H map with respe*
*ct to
H, ~H, ~~H.
Proof. Let ~Hbe a homotopy from f(x1)f(x2) to f(x1x2). In general, d(fH + ~H*
*, ~H+
H~~) 6 0. We define a homotopy f(x1)f(x2) to f(x1x2) by
~H= ~H- 1_d(fH + ~H, ~H+ ~~H).
2
It can be easily seen by simple calculation that d(fH + ~H, ~H+ ~~H) 0. So f *
*is a strong
H-map with respect to H, ~H, ~~H. Q*
*.E.D.
Apply the above theory to fm : X= k(Zp, 1) ! W =E1(W (m))= E2(W (m)), then *
*any
homotopy from x1x2 to x2x1 is a ø-homotopy. Let ~~Hbe a homotopy from u1u2 to u*
*2u1
(u1, u2 2 W ) inheriting from a ø-homotopy from t1t2 to t2t1 with t1, t2 2 E2(W*
* (m)).
Then, ~~His also an ø-homotopy. So we have
Proposition 14.5 Let X = k(Zp, 1), W = E1(W (m)), then there exists a homoto*
*py H~
from f(x1)f(x2) to f(x1x2) such that the following diagram is homotopy commutat*
*ive;
! T(fm) !
T (Xx X, Xx X) -! T (W xW, W xW )
# s(`1, `2, H) # s(`1, `2, ~~H)
B2(fm,H)
B2(X) -! B2(W )
40
By Propotition 14.1 to Proposition 14.5, we have the following proposition
Proposition 14.6
i j
oe2*O B2(fm )* O s*(`1, `2, H)(æ-1)* ( upi)x ( upj)
i j i j
= oe1*( fm )*(upi) . oe1*( fm )*(upj)
!
where æ: T (k(Zp, 1)x k(Zp, 1), k(Zp, 1)x k(Zp, 1)) ! ( k(Zp, 1))x ( k(Zp, 1)) *
*denotes the
map defined at the beginning of this section and . denotes the Pontrjagin produ*
*ct of ho-
mology group in E2(W (m)).
The following is the proof of (h)m+1 for m+ 1 = 2pi+2pj- 2.
Since ( -2E2)*(ffi) = oe*1 -3E*3(ffi) = 0, 16 i6 s, where oe1: E2(W (m)) ! *
*E3(W (m+ 1))
i j i j
denotes the adjoint map, we have oe1*( fm )*(upi) . oe1*( fm )*(upj) E*2( 2f*
*fi) = 0. By
the same argument as above, we have (B2(fm )*oe*2E*2( 2ffi)) = 0. Therefore, fm*
*+1 is an
H-map for m+ 1 = 2pi+2pj- 2.
16 The N (Zp, 0) action on spectra.
Before we prove (h)m+1 , we state some properties of N(Zp, 0) action on spectra*
*. In what
follows, we always use N(Zp, 0) to denote the Moore spectrum and n(Zp, 1) to de*
*note the
Moore space, then N(Zp, 0) = -1~n(Zp, 1)
For X = W (m), fm : k(Zp, 1) ! E1(W (m)), and f~mdenote the map from ~k(Zp, *
*1)
to E~1(W (m)), then E1f~mis a map from ~k(Zp, 1) to W (m) and E2~oe2is a map f*
*rom
B~2(~k(Zp, 1)) to 2W (m).
It may be assumed that W (m) =-lim! -rE~r(W (m)) and -rE~r(W (m)) m = 0, 1,*
* . . .
are all CW subspectra of W (m).
It can be easily seen that the following diagram is commutative.
E1(f~m) 2
~k(Zp, 1) -! W (m)
# ~ø # id
B~2(k(Zp, 1)) -~ff2! 2W (m)
where ø denotes the natural injection.
41
Since W (m) is a ring spectrum with unit, ß0(W (m)) = Zp, ß1(W (m)) = 0, N(Z*
*p, 0) =
V (0), N(Zp, 0) may be considered naturally as a subspectrum of W (m). Let Mm :*
* W (m)^
W (m) ! W (m) be the multiplication of W (m), then P (m) = Mm |N(Zp,0)^W(m)is a
map from N(Zp, 0) ^ W (m) to W (m). In general, any map S: N(Zp, 0) ^ X ! X that
satisfies S|S0^X = S|X = id|X is called an action of N(Zp, 0) on X. The map P (*
*m) is an
action of N(Zp, 0) on W (m) which we call the natural action of N(Zp, 0) induce*
*d by the
multiplication Mm .
For the space k(Zp, 1). Since k(Zp, 1) is a topological group, let d > 1, w*
*e use
ed: k(Zp, 1) ! k(Zp, 1) to denote the map defined by ed(x) = xd, x 2 k(Zp, 1). *
*It follows
from Proposition 11.6 that in the space B2(k(Zp, 1)), we have ø(d . id k(Zp,1))*
* ø( ed).
Now in k(Zp, 1), ep 0, so we have ø(p . id k(Zp,1)) 0 in B2(k(Zp, 1)). *
*So the
map ø: k(Zp, 1) ! B2(k(Zp, 1)) can be extended to a map S: n(Zp, 1) ^ k(Zp, 1)*
* !
B2(k(Zp, 1)) (n(Zp, 1) is the Moore space), then we have the following sequence*
* of maps
~n(Zp, 1) ^ ~k(Zp, 1) -! N(Zp, 0) ^ ~E1(W (m)) -!
N(Zp, 0) ^ W (m)= 2N(Zp, 0) ^ W (m) -!
~S E2 2
2W (m) ^ ~n(Zp, 1) ^ ~k(Zp, 1) -! ~B2(k(Zp, 1)) -! W (m)
So we have the following diagram
F^E1Of~m 2
~n(Zp, 1) ^ ~k(Zp, 1)-! N(Zp, 0) ^ W (m)
# ~S # 2P (m)
~B2(k(Zp, 1)) E2~ff2~B2(fm)-! 2W (m)
where F : ~n(Zp, 1) ! N(Zp, 0) is the natural injection. Then, we have the fo*
*llowing
proposition
Proposition 15.1 Suppose (a)m to (h)m holds for m, then there exists a commu*
*tative
associative multiplication M~m: W (m) ^ W (m) ! W (m) such that the above diagr*
*am is
homotopy commutative for the natural N(Zp, 0) action P (m) induced by M~m.
Proof. We prove it by induction. If m = 0, then in this case, W (0) = k(Zp, *
*0) and
the conclusion follows easily from the fact that 2P (0) O (F ^E1f0) and EB~2(f*
*0)S~induce
42
the same homomorphisms on the first non-zero homology group modp. Suppose that *
*the
Proposition holds for m, we will prove it for m+ 1.
Before we prove it, we introduce the notations which will be used later. Let*
* X be a
spectrum, A be an Abellian group, ff: X ! K(A, m+ 2) be a map. Let P (X, ff) de*
*note
the map cone C(X) [ K(Z, m+ 2). Then, we have the following cofibration sequence
j ff
-1P (X, ff) -! X -! K(A, m+ 2)
where æ is the natural identification map by collapsing K(A, m+ 2) to the base*
* point.
Let B be a CW subspectrum of X, oe: B ! X be the natural injection, ~: X ! X*
*=B
be the natural identification map. Suppose that there exists a map ~ff: X=B ! K*
*(A, m+ 2)
such that ff = ~ff~, we define a map J(~ff): B ! P (X, ff) as follows. J(~ff)(*
*b^ t) = (b^ t) 2
C(X) P (X, ff) for 06 t6 1. It should be noticed that J(~ff)(b^ 0) = J(~ff)(*
*b^ 1) = base
point. So J(~ff) define a map from B to P (X, ff)
Let ff0, ff00: X=B ! K(A, m+ 2) be two maps such that ff0~ ff00~, then P (*
*X, ff0~)
and P (X, ff00~) are of the same homotopy type. There exists a homotopy equiva*
*lence
l: P (X, ff0~) ! P (X, ff00~) such that the following diagrams are commutative
1) P (X, ff0~)-!l P (X, ff00~)
# æ
æ X
2) K(A, m+ 2) q
# q
P (X, ff0~)-!l P (X, ff00~)
where q: K(A, m+ 2) ! P (X, ff0~), P (X, ff00~) denotes the natural injection*
*s of the
fibre.
From the conclusion that ff0~ and ff00~ denote the same cohomology class if *
*and only
if ff0-ff002 im Hm+2 ( B, A) we have the following proposition.
Proposition 15.2 J(ff0)-J(ff00) q< ff0-ff00>, where denotes the *
*cohomology
class in Hm+2 ( B, A) which is map to ff0- ff00by the map X=B ! B.
43
Let ff 2 Hm+2 (X, A), then the homotopy type of P (X, ff) is uniquely determ*
*ined by
the cohomology class ff. Suppose that oe*(ff) = 0 and J: X ! P (X, ff) be a ma*
*p such
that (æ) O J (oe). It follows from Proposition 15.2 that
Proposition 15.3 There exists a ff02 Hm+2 (X=B, A) such that ff ff0~ and J*
*(ff0)
J: B ! P (X, ff0~) = P (X, ff).
To prove Proposition 15.1, we must define a new map homotopic to P (m). Let
P (m): N(Zp, 0) ^ W (m) ! W (m) be the N(Zp, 0) action induced by Mm . Let
______ i j
W (m) = W (m) [ N(Zp, 0)^ W (m) ^ I+
i j i *
* j
where N(Zp, 0)^ W (m) ^0+ is identified with the subspectrum P (m) N(Zp, 0)^ W*
* (m) ,
that is, W~(m) is the mapping cylinder of P (m).
In what follows, we set A = ßm+1 (W (m+ 1)) = Zp . . .Zp, ff = ff1+ . .+.f*
*fs 2
______-z_____"
s-copies____
Hm+2 (W (m), A). Since W (m) is a deformation retract of W (m) , we may also s*
*et ff 2
______ ______
Hm+2 (W (m) , A) and ff|W(m) = ff. Let P~(m): N(Zp, 0) ^ W (m) ! W (m) be the*
* map
______
defined by P~(n ^ u) = n^ u^ 1 2 W (m) for n 2 N(Zp, 0), u 2 W (m). Let _: W (m*
*) !
______ _____
W (m) be the natural injection. It is obvious that _P (m) P (m). Now, W (m*
*+ 1) =
_________ ______
-1(P (W (m), ff)), we may define W (m+ 1) = -1(P (W (m) , ff)). It is obvio*
*us that
_________
W (m+ 1) is a deformation retract of W (m+ 1). It follows from the definition o*
*f W (m+ 1)
that P (m+ 1): N(Zp, 0)^ W (m+ 1) = N(Zp, 0)^ P (W (m), ff) ! W (m+ 1) = P (*
*W (m),
ff) may be constructed as follows. For any n 2 N(Zp, 0), w 2 W (m),
P (m+ 1)((n^ w)^ t) = P (m)(n^ w)^ (2t- 1) 2 C(W (m)) P (W (m), ff)1_26t*
*6 1
P (m+ 1)((n^ w)^ t) = H(n^ w)^ 2t 2 K(A, m+ 2) 06 t6 1_2
where H: N(Zp, 0) ^ W (m) ^ I+ is a homotopy from the composed map MO (id|N(Zp,*
*0)^ff):
N(Zp, 0) ^ W (m) ! N(Zp, 0) ^ K(A, m+ 2) ! K(A, m+ 2) to the composed map ffP (*
*m):
N(Zp, 0)^W (m) ! K(A, m+ 2), where M denotes the natural N(Zp, 0) action of K(A*
*, m+ 2)
and P (m+ 1)|N(Zp,0)^K(A,m+2)= M.
_________
Now we define a map P~(m+ 1): N(Zp, 0) ^ W (m+ 1) ! W (m+ 1) related to
P (m+ 1).
44
Let G: N(Zp, 0) ^ W (m) ^ I+ be the homotopy from _P (m) to P~(m) defined by
______
G(n^w ^s) = (n^w ^s) 2 N(Zp, 0)^W (m)^I+ W (m), 06 s6 1. We define P~(m+ 1)
as follows. For any n 2 N(Zp, 0), w 2 W (m),
______ ______
P~(m+ 1)((n^ w)^ t) = ~P(m)(n^ w)^ (2t- 1) 2 C(W (m) ) P (W (m) ,1ff)_26t*
*6 1
P~(m+ 1)((n^ w)^ t) = ff(G(n^ w^ 4t- 1)) 2 K(A, m+ 2) 1_46t6 1_2
P~(m+ 1)((n^ w)^ t) = H(n^ w^ 4t) 2 K(A, m+ 2) 06 t6 1_4
We also define P~(m+ 1)|N(Zp,0)^K(A,m+2)= M.
It can be easily seen that the following proposition holds
______
Proposition 15.4 ~_ P (m+ 1) P~(m+ 1), where _~: P (W (m), ff) ! P (W (m)*
* , ff) =
__________
W (m + 1) denotes the natural homotopy equivalence.
Now we prove proposition 15.1. We prove it by induction. If m = 0, then in t*
*his case,
W (0) = K(Zp, 0) and the conclusion follows easily from the fact that 2P (0) O*
* (F ^E1f0)
and Eoe2S induces the same homomorphisms on the first non-zero homology group m*
*odp.
Suppose that the proposition holds for m 6= 2pi, i = 0, 1, . .,.we shall prove *
*that it
holds for m+ 1. Since fm+1 : k(Zp, 1) ! E1(W (m+ 1)) is an H-map, oe2: B2(k(Zp*
*, 1)) !
E2(W (m+ 1)) is defined. For simplicity, we use J to denote the map ~E2Ooe2: ~B*
*2(k(Zp, 1)) !
2W (m+ 1). Since H*(N(Zp, 0)^B~2(k(Zp, 1)), A) = H*(B~2(k(Zp, 1)), A)+ H*(B~2(*
*k(Zp, 1)),
A) ~ø0and ff . 2H*(N(Zp, 0) ^ ~B2(k(Zp, 1), A) = 0, B~2(k(Zp, 1)) = S0 ^ ~B2(*
*k(Zp, 1))
*
* ______
N(Zp, 0)^B~2(k(Zp, 1)) and B2(k(Zp, 1))^ I+ [N(Zp, 0)^ 1+ is a CW subspectrum o*
*f W (m),
______ i
it follows from Proposition 15.2 that there exists a map ff0: 2W (m) = B2(k(Zp*
*, 1))^ I+ [
j
N(Zp, 0)^ 1+ ! K(A, m+ 2) such that J J(ff0~) where ~ denotes the natural map
______ ______ i j
from 2W (m) to 2W (m) = B2(k(Zp, 1))^ I+ [ N(Zp, 0)^ 1+ .
By induction hypothesis and homotopy extension properties with respect to th*
*e pair
N(Zp, 0) ^ ( W (m), ~k(Zp, 1)), it may be assumed that 2P (m)|N(Zp,0)^~k(Zp,1)*
*= E2~oe2~S.
Since ~ø: ~k(Zp, 1) ! B2(k(Zp, 1)) is the natural injection, we have E1f~m*
*+1= -1~øJ =
-1~øJ(ff0~). Therefore, E1f~m+1~k(Zp, 1) is on the spectrum J(ff0)(B~2(k(Zp, 1*
*))). It follows
from the construction of P (m+ 1) and ~P(m+ 1) just mentioned.
45
For t = 0, 1_2, n 2 N(Zp, 0), b 2 B~2(k(Zp, 0)), we have P~(m+ 1)(n ^ b ^ t)*
* = the
base point of K(A, m+ 1), so P~(m+ 1) induces a map oe from N~(Zp, 0) ^ ~B2(k(*
*Zp, 0))
to K(A, m+ 2). It follows from the construction of P (m+ 1) and P~(m+ 1), we h*
*ave
~P(m+ 1)|N(Z = E ~oe~S+ ø*(oe). Since m 6= 2pi, ø* = 0: Hm+2 (N(*
*Z , 0) ^
p,0)^~k(Zp,1) 2 2 *
* p
B2(k(Zp, 0)), A) ! Hm+2 (N(Zp, 0) ^ ~k(Zp, 1), A), so we have P~(m+ 1)|N(Zp,0)^*
*~k(Zp,1)=
E2~oe2~S. So Proposition 15.1 holds.
If m+ 1 = 2pi, by the induction hypothesis we have E2~oe2~SPm (F ^ ~E1~fm) o*
*n 2W (m),
so E2~oe2~Sand 2Pm+1 (F ^E~1~fm+1) are m+ 2 homotopic. Thus, we can define the*
* cocycle of
difference D(E2~oe2~S, 2Pm+1 (F ^ ~E1~fm+1)) 2 Hm+3 (~n(Zp, 1) ^ ~k(Zp, 1), ßm*
*+1 (W (m+ 1))).
Since Pm+1 |S)^W(m+1) = id, S|S1^k(Zp,1)= S| k(Zp,1)= ø: k(Zp, 1) ! B2(k(Zp, 1*
*)), we
have D(E2~oe2~S, 2Pm+1 (F ^ ~E1~fm+1))|~S1^~k(Zp,1)= 0.
Let D(E2~oe2~S, 2Pm+1 (F ^ ~E1~fm+1))|fi00^~upn= ` 2 2ßm+1 (W (m+ 1)) (I)*
*. We define
a new multiplication M~(m+ 1) from W (m+ 1) ^ W (m+ 1) to W (m+ 1) by M~(m+ 1) =
i j
M(m+ 1)+j -2(`)(ø*i^ø*0+ø*0^ø*i) , where M(m+ 1): W (m+ 1)^W (m+ 1) ! W (m+ 1)
denotes the multiplication of W (m+ 1) and j: K(A, m+ 1) ! W (m+ 1) denotes the*
* natural
injection. It can be easily proved that M~(m+ 1) is also homotopy commutative *
*and
homotopy associative. Let ~Pm+1be the N(Zp, 0) action defined by M~(m+ 1). It c*
*an be
proved from (I) that E2~oe2~S 2P~m+1(E1^E~1~fm), that is, Prposition 15.1 hol*
*ds for m+ 1
with respect to the multiplication M~(m+ 1).
17 The proof of (h)m+1 for m+ 1 = 2pi.
It can be easily seen that
E2*~oe2*~S*(~ø00^ ~upi)
2P~m+1(F ^ ~E1~fm)*(~ø0^ ~upi)
= 2Pm+1 (~ø0^ ~øi)
= 2(~ø0, ~øi)
where ~ø02 H2(n(Zp, 1), Zp) denotes the generater of H2(n(Zp, 1), Zp) which sen*
*ds ~ø0of
i *
* 2+2pi
K(Zp, 0). So, S*(ø00^ upi) 6= 0. Since H2+2p(B2(k(Zp, 1), Zp) is generated by *
*j*_*H
46
(E1(W (m)) ^ E1(W (m)) ^ E1(W (m)), Zp) and s*(ø00^ upi) and oe2*j*_* = 0, by t*
*he
construction of ff = (ff1, . .,.ffs) we have ffk(~ø0, ~øi) = 0, k = 1, . .,.s. *
* Thus, we have
ff(~ø0, ~øi) = 0. Therefore, E*2( 2ff)s(ø00^ upi) = 0. So we have oe*2E*2( 2f*
*f) =). So,
fm+1 : k(Zp, 1) ! E1(W (m+ 1)) is an H-map.
18 Appendix
In what follows, we use D to denote the statement ff1fip1= 0. There are two di*
*fferent
proofs of D. The first was given in [14]. The second was given in [15]. We will*
* indicate
that both are incorrect.
We follow the notation in [14]. In [14] (P.841), a (2m+ 1) sphere bundle Bm*
* (p)
over S2m+2p+1 was defined and it satisfies that H*( (Bm (p)), Zp) = p(ai, bi).*
* Notice
that Bm (p) and S2m+1 x S2m+2p+1 are of different homotopy type and so (Bm (p)*
*) and
(S2m+1 x S2m+2p+1) are of different homotopy type, too. We only know that the*
*re is
a spectral sequence Er converging to H*( (Bm (p)), Zp) with E2 = H*( (S2m+1, Zp)
H*( (S2m++2p+1, Zp)) = p(ai) p(bi) = p(aibi). But we do not know whether *
*the
equality H*( (Bm (p)), Zp) = p(ai, bi) holds. So the proof is incorrect.
Now we will show that the second proof of D in [15] (P.198-203) is also inco*
*rrect. In
what follows, we use the same notations as that in [15]. The proof of D depends*
* on the
following statement. For p> 3 and n> 2, we have P*1Hpn+2(p-1)(epp-2(M(Zp, n), Z*
*p)) 6= 0.
However, using some elementary method, we can show that this conclusion is wron*
*g. We
have the following propositions
Proposition 19.1 For p= 3 and l> 2, we have P*1H3l+4(epp-2(M(Zp, l), Zp)) = *
*0.
Recall the definition of ep1(X). It is the identification space X3^ I+ = ~,*
* where the
equivalence relation ~ is defined by x1^x2^x3^0 ~ x2^x3^x1^1 and e^ e^ e^ t ~ e*
*0 (e
and e0 are respectively the base points of X and ep1(X)). Let P : X3 ! X3 be de*
*fined
by P (x1^x2^x3) = x2^x3^x1. It is obvious that if X is a cogroup, then ep1(X) i*
*s of the
same homotopy type as that of the mapping cone C(P - id) where id is the identi*
*ty map
of X3. So when X is taken to be M(Zp, n) for n> 2, ep1(M(Zp, n)) and C(P - id) *
*are of
the same homotopy type.
47
Before proving Proposition 19.1, we state some properties of Moore space. Fo*
*r sim-
plicity, we abbreviate M(Zp, n) to M(n) in what follows. M(n) = an [ bn, where *
*an is the
n-sphere and bn is the cone C(p(id)). Let l, m> 2 and !, ` are homotopy classes*
* of map
from M(l+ m) and M(l+ m+ 1) to M(l)^ M(m) determined by the following relations
!*(al+m) = al^ am !*(bl+m) = bl^ am
`*(al+m+1) = al^ bm + (-1)l+1bl^ am `*(bl+m) = bl^ bm
It is easy to check that the map ! _ `: M(l+ m) _ M(l+ m+ 1) ! M(l)^ M(m) is
a homotopy equivalence. Let X, Y be two spaces and ff, fi be respectively hom*
*otopy
J
classes of maps from M(l) and M(m) to X and Y . We use ff fi and ff fi to d*
*e-
note respectively the homotopy classes (ff^ fi)*(!) and (ff^ fi)*(`). It is ea*
*sy to check
that for any maps f: X ! X~ and g: y ! Y~, we have (f^ g)*(ff fi) = f*(ff) g**
*(fi),
J J
(f^ g)*(ff fi) = f*(ff) g*(fi).
Let l, m, n > 2, ff, fi, fl be respectively the identity maps of M(l), M(m),*
* M(n). We
*
* J
use q1, q2, q3, q4 to denote respectively the homotopy class of maps (ff fi) *
*fl, (ff fi) fl,
J J J
(ff fi) fl, (ff fi) fl. Then, it is easy to check that q1 _ q2 _ q3 _ q4 fr*
*om M(l+ m+ n) _
M(l+ m+ n+ 1) _ M(l+ m+ n+ 1) _ M(l+ m+ n+ 2) to M(l) ^ M(m) ^ M(n) is a homoto*
*py
equivalence. In what follows, we always assume that l = m = n and so ff = fi = *
*fl. Let
J *
* J
P from M(l)3 to itself be defined by P (x ^ y ^ z) = y ^ z ^ x, then P*((ff fi*
*) fl) =
J J J J J J J J *
* J J
(fi fl) ff = (ff ff) ff = (ff fi) fl = (id)*((ff fi) fl). So we have ((*
*ff fi) fl)P =
J J 3
((ff fi) fl)id, where id is the identity map of M(l) .
Now we prove Proposition 19.1. Let T1 = M(3l) _ M(3l + 1) _ M(3l + 2) and T2*
* =
M(3l + 2) and W = M(l) ^ M(l) ^ M(l) and q = q1 _ q2 _ q3 _ q4. Since q: T ! W
is a homotopy equivalence, we have that ep1(M(l)) = C(P - id) and C((P - id)q) *
*are
of the same homotopy type and therefore the two spaces may be considered the sa*
*me
space in homotopy theory. Let ~q= q1 _ q2 _ q3, then q = ~q_ q4 and (P - id)q =
(P - id)~q_ (P - id)q4. It follows from the previous argument that (P - id)q4 *
* 0. So
P 1 P
C((P -id)q4) = M(3l+2)_W and we have that ep (M(l)) = C((P -id)~q)_ M(3l+2).
Therefore, H3l+4(ep1(M(l)), Z3) = H3l+4(C((P - id)~q), Z3) H3l+4((M(3l + 2)),*
* Z3) =
P
H3l+4((M(3l + 2)), Z3) = Z3, so we have P*1H3l+4(ep1(M(l)), Z3) = P*1H3l+4( M(*
*3l +
48
2), Z3) = P*1H3l+4(M(3l + 2), Z3) = 0.
This proposition shows that the statement P*1H3l+2(p-1)(epp-2(M(l)), Zp) 6= *
*0 is false.
So, the second proof of D is also incorrect.
Since V (0) = {M(1), M(2), . .,.M(n), . .}., the above proposition implies t*
*he follow-
ing proposition
Proposition 19.2. P*1H4(ep1(V (0)), Z3) = 0.
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50