On adic genus, Postnikov conjugates, and
lambda-rings
Donald Yau
April 25, 2002
Department of Mathematics, MIT Room 2-588, 77 Massachusetts Av-
enue, Cambridge, MA 02139, USA
Abstract
Sufficient conditions on a space are given which guarantee that the
K-theory ring and the ordinary cohomology ring with coefficients over
a principal ideal domain are invariants of, respectively, the adic genus
and the SNT set. An independent proof of Notbohm's theorem on the
classification of the adic genus of BS3 by KO-theory ~-rings is given.
An immediate consequence of these results about adic genus is that
for any positive integer n, the power series ring Z[[x1, . .,.xn]] admits
uncountably many pairwise non-isomorphic ~-ring structures.
MSC: 55P15; 55N15, 55P60, 55S25.
Keywords: Adic genus, Postnikov conjugates, SNT, lambda-rings.
1 Introduction and statement of results
In this paper we study some problems about the adic genus and the SNT
set of a space. For a nilpotent finite type space X, we denote by Genus (X),
call the adic genus of X, the set of homotopy types of nilpotent finite type
spaces Y such that the p-completions of X and Y are homotopy equivalent
for each prime p and also their rationalizations are homotopy equivalent.
The adic genus of an infinite dimensional space X is often a very big
set. For instance, Møller [17] proved that whenever G is a compact con-
nected non-abelian Lie group, the adic genus of its classifying space BG is
1
uncountably large. It is an important problem to find computable homotopy
invariants which can distinguish between spaces with the same adic genus.
Such a result was achieved by Notbohm and Smith [19] (see also [11]). Re-
call that a space X is said to be an adic fake Lie group of type G if (1)
X = BX is a finite loop space and (2) BX lies in the adic genus of BG.
Notbohm and Smith showed that if X is an adic fake Lie group of type G,
where G is a simply-connected compact Lie group, then BX is homotopy
equivalent to BG if and only if K(BX) ~=K(BG) as ~-rings. (Here K(Y )
denotes the complex K-theory of Y .) In fact, Notbohm [18] showed that
K-theory ~-rings classify the genus of BG, provided G is a simply-connected
compact Lie group. Note that what we call a ~-ring is what used to be called
a "special" ~-ring.
With this result in mind, a natural question is then the following:
Is it really necessary to take into account the ~-operations in
order to distinguish spaces in the adic genus of BG?
To describe our answer to this question, we need the following definition.
A filtered ring is a commutative ring R with unit together with a decreasing
filtration
R = I0 I1 I2 . . .
of ideals such that Im In Im+n . A map of filtered rings is required to
respect filtrations. The filtration in the K-theory of a space X is given by
a skeletal filtration of X; that is, K(X) is filtered by the kernels
In (X) = ker(K(X) ! K(Xn-1))
of the restriction maps, where Xn-1 denotes the (n - 1)-skeleton of X. By
using the Cellular Approximation Theorem, it is easy to see that the filtered
ring isomorphism type of a given space X is well-defined, though there are
many different filtrations on X. Clearly, a map between two spaces induces
a filtered ring map between the respective K-theories.
Our first main result is then the following, which shows that for a tor-
sion free classifying space, K-theory filtered ring cannot tell the difference
2
between spaces in its adic genus. Consequently, ~-operations are necessary
in order to distinguish these spaces.
Theorem 1.1. Let X be a simply-connected space of finite type whose in-
tegral homology is torsion free and is concentrated in even dimensions and
whose K-theory filtered ring is a finitely generated power series ring over Z.
If a space Y belongs to the adic genus of X, then there exists a filtered ring
isomorphism K(X) ~=K(Y ).
For example, in Theorem 1.1 the space X can be BSp(n) (n 1),
BSU(n) (n 2), or any finite product of copies of these spaces and of
infinite complex projective space.
Remark 1.2. The proof of Theorem 1.1 is independent of Notbohm's result
mentioned above. It consists of first showing the weaker statement that the
filtered rings K(Y )=In (Y ) and K(X)=In (X) are isomorphic for all n suffi-
ciently large. Since K(Y ) can be recovered from the quotients K(Y )=In (Y )
by taking inverse limits, a filtered ring analog of a classification result of
Wilkerson [25] implies that to prove Theorem 1.1 it is sufficient to prove the
triviality of a lim-1set.
Remark 1.3. This theorem has a variant in which complex K-theory (resp.
Z) is replaced with KO*-theory (resp. KO* = KO*(pt)), provided the inte-
gral homology of X (which is simply-connected of finite type) is torsion free
and is concentrated in dimensions divisible by 4 (e.g. X = BSp (n)). This
variant admits a proof which is essentially identical with that of Theorem
1.1.
Using Theorem 1.1 we will show the following KO-theory analog of Not-
bohm's result (applied to BS3) mentioned above.
Theorem 1.4. Let X and Y be spaces in the adic genus of BS3. Then X
and Y are homotopy equivalent if and only if there exists a filtered ~-ring
isomorphism KO*(X) ~=KO*(Y ).
In Theorem 1.4, by a filtered ~-ring we mean a filtered ring R which is
also a ~-ring for which the filtration ideals are all closed under the operatio*
*ns
3
~i (i > 0). The proof of of this theorem uses directly Rector's classification
of the genus of BS3 [20] and the KO-theory analog of Theorem 1.1 when
X = BS3.
Our next result is a purely algebraic application of Theorem 1.1 to the
study of ~-rings.
Now since the adic genus of BSp(n), n any positive integer, is uncount-
able [17], Theorems 1.1 and Notbohm's result alluded to above imply the
following purely algebraic statement about ~-rings.
Corollary 1.5. Let n be any positive integer. There exist uncountably many
distinct isomorphism classes of ~-ring structures over the power series ring
Z[[x1, . .,.xn]] on n indeterminates.
Remark 1.6. As far as the author is aware, this result is new for every n.
It exhibits a huge diversity of ~-ring structures over some rings. The author
has recently obtained results about classifying filtered ~-ring structures over
a filtered ring, exhibiting this set as a moduli space, and also results about
how some filtered ~-rings give rise to unstable algebras over the mod p
Steenrod algebra.
Remark 1.7. Notice that the way we obtain Corollary 1.5 is by combining
three topological results about spaces in the genus of classifying spaces and
their K-theories, even though the statement of the result is purely algebraic.
It would be nice if this statement can be demonstrated in an algebraic way.
Remark 1.8. The analogous question of how many ~-ring structures the
polynomial ring Z[x] on one indeterminate supports has been studied by
Clauwens [7]. Employing the theory of commuting polynomials, he showed
that there are essentially only two non-isomorphic ~-ring structures on the
polynomial ring Z[x]. We, however, have not been able to establish any
connections between Clauwens' result and our Corollary 1.5. As pointed out
by Clauwens, the same question for the polynomial ring on n indeterminates,
n > 1, is still open.
4
The last main result of this paper concerns the SNT set of a space. For
a given nilpotent finite type space X, SNT (X) is by definition the set of
homotopy types of nilpotent finite type spaces Y such that for every n the
Postnikov approximations through dimension n of X and Y are homotopy
equivalent. Spaces in the same SNT set are also known as spaces of the same
n-type for all n (hence the notation SNT ). The SNT set of a space is closely
related to its adic genus. Indeed, according to a result of Wilkerson [25], if
X is a connected nilpotent finite type space, then there is an inclusion of
pointed sets
SNT (X) Genus (X).
The adic genus of a space, when it is nontrivial, is often strictly bigger than
its SNT set.
There are many interesting spaces whose SNT sets are nontrivial (in fact,
uncountable); see, for example, [9, 15, 22, 23]. Several homotopy invariants
which can distinguish between spaces of the same n-type for all n have been
found. These include [16] Aut(-) and WI (-), the group of homotopy classes
of homotopy self-equivalences and the group of weak identities, and [24]
End(-), the monoid of homotopy classes of self-maps. One might wonder
if there are more familiar and more computable homotopy invariants, such
as integral cohomology and K-theory, which can distinguish between spaces
in the same SNT set. Of course, our Theorem 1.1 together with the above
inclusion imply that if a space X is as in Theorem 1.1 and if Y lies in
SNT (X), then the integral cohomology (resp. K-theory) rings of X and Y
are isomorphic.
Does this hold for more general spaces?
The following result gives an answer to this question for the ordinary
cohomology case.
Theorem 1.9. Let be a principal ideal domain. Let X be a connected
space of finite type whose ordinary cohomology ring with coefficients over
is a finitely generated graded algebra over . If Y has the same n-type
5
as X for all n, then there exists an isomorphism H*(X; ) ~=H*(Y ; ) of
cohomology rings.
Remark 1.10. It should be emphasized that in Theorem 1.9 the hypothesis
Y 2 SNT (X) cannot be weakened to the condition Y 2 Genus (X). Indeed,
in [5] Bokor constructed two spaces, both two cell complexes, with the same
genus but whose integral cohomology rings are non-isomorphic.
This finishes the presentation of the results in this paper. The rest of the
paper is organized as follows. In x2 we recall the definitions of a (filtered)
~-ring and of Adams operations. The proofs of Theorems 1.1, 1.4, and 1.9
are given in this order, one in each section, in x3 - x5.
2 ~-rings and Adams operations
In this section, we recall the definitions of a (filtered) ~-ring and of Adams
operations. The reader is referred to Atiyah and Tall [3] or Knutson [13] for
more information about ~-rings.
2.1 ~-rings
A ~-ring is a commutative ring R with unit together with functions
~i:R ! R
for i = 0, 1, . .s.atisfying the following properties: For any elements r and s
in R one has
o ~0(r) = 1
o ~1(r) = r, ~n(1) = 0 for all n > 1
P n
o ~n(r + s) = i=0 ~i(r)~n-i(s)
o ~n(rs) = Pn(~1(r), . .,.~n(r); ~1(s), . .,.~n(s))
o ~n(~m (r)) = Pn,m(~1(r), . .,.~nm (r))
6
Here the Pn and Pn,m are certain universal polynomials with integer coef-
ficients. Note that in the literature (for example, Atiyah and Tall [3]) the
terminology "special" ~-ring is used.
A ~-ring map f :R ! S between two ~-rings is a ring map between the
underlying rings which respects the operations ~i: f~i= ~if (i 0).
2.2 Filtered ~-rings
A filtered ~-ring is a filtered ring R = (R, {In }) which is also a ~-ring such
that the ideals In are all closed under the operations ~i (i > 0).
A filtered ~-ring map is a ~-ring map which is also a filtered ring map.
2.3 Adams operations
Given a ~-ring R, the Adams operations
_k :R ! R
for k = 1, 2, . .a.re defined inductively by the Newton formula:
_k(a) - ~1(a)_k-1(a) + . .+.(-1)k-1~k-1(a)_1(a) = (-1)k-1k~k(a).
The Adams operations satisfy the following properties.
1. All the _k :R ! R are ~-ring maps.
2. _1 = Idand _k_l = _kl for any k, l 1.
3. _p(a) ap (mod pR) for each prime p and element a in R.
If R is a filtered ~-ring, then the Adams operations are all filtered ~-ring
maps.
3 Proof of Theorem 1.1
In this section we prove Theorem 1.1, which consists of a few lemmas.
Throughout this section we work in the category of filtered rings.
7
We will make use of the following observation, whose proof is a straight-
forward adaptation of Wilkerson's proof of the classification theorem [25,
Theorem I] of spaces of the same n-type for all n. Now let X be as in
Theorem 1.1.
Lemma 3.1 (Wilkerson). There is a bijection between the following two
pointed sets:
1. The pointed set of isomorphism classes of filtered rings (R, {In}) with
the properties:
(a) The natural map R ! lim-nR=In is an isomorphism, and
(b) R=In and K(X)=Kn(X) are isomorphic as filtered rings for all
n > 0.
2. The pointed set lim-1nAut(K(X)=Kn(X)).
Here Aut (-) denotes the group of filtered ring automorphisms, and the
lim-1of a tower of not-necessarily abelian groups is as defined in Bousfield-
Kan [6].
The two lemmas below will show that, for every space Y in the genus of
X, the object K(Y ) lies in the first pointed set in Lemma 3.1.
Lemma 3.2. For every space Y in the genus of X (as in Theorem 1.1), the
natural map
K(Y ) ! lim-K(Y )=Kj(Y )
j
is an isomorphism.
Proof.According to [4, 2.5 and 7.1] the natural map from K(Y ) to
lim-jK(Y )=Kj(Y ) is an isomorphism if the following condition holds:
lim-1rEp,rq= 0 for all pairs(p, q). (3.1)
Here E*,r*is the Er-term in the K*-Atiyah-Hirzebruch spectral sequence
(AHSS) for Y . This condition is satisfied, in particular, when the AHSS
degenerates at the E2-term. Thus, to prove (3.1)it suffices to show that
8
H*(Y ; Z) is concentrated in even dimensions, since in that case there is no
room for differentials in the AHSS. So pick an odd integer N. We must show
that
HN (Y ; Z) ~=0. (3.2)
By the Universal Coefficient Theorem it suffices to show that the integral
homology of Y is torsionfree and is concentrated in even dimensions, which
hold because Y lies in the genus of X.
__
This finishes the proof of Lemma 3.2. |__|
Lemma 3.3. For every space Y in the genus of X (as in Theorem 1.1),
the filtered rings K(Y )=Kn(Y ) and K(X)=Kn(X) are isomorphic for each
n > 0.
Proof.We have to show that for each j > 0 there is an isomorphism of
filtered rings
K(Y )=Kj(Y ) ~=K(X)=Kj(X). (3.3)
It follows from the hypothesis that for each j > 0, the filtered r*
*ing
K(Y )=Kj(Y ) belongs to Genus (K(X)=Kj(X)), where Genus (R) for a fil-
tered ring R is defined in terms of R Q and R bZin exactly the same
way the genus of a space is defined. To finish the proof we will adapt two
results of Wilkerson [26, 3.7 and 3.8], which we now recall.
For a nilpotent finite type space X, Wilkerson showed that there is a
surjection
oe :Caut (X^0) ! Genus (X),
Q
where X^0is the rationalization of the formal completion X^= pX^pof X.
Notice that each homotopy group ß*(X^0) is a Q bZ-module, and Caut(X^0)
is by definition the group of homotopy classes of self-homotopy equivalences
of X^0whose induced maps on homotopy groups are Q bZ-module maps.
Note that the definitions Genus (-) and Caut (-) also make sense in
both the categories of nilpotent groups and of filtered rings. For instance, if
R = (R, {In }) is a filtered ring, then Caut(R Q Zb) is the group of filtered
ring automorphisms of R Q bZwhich are also Q bZ-module maps. Now
9
if A is a finitely generated abelian group, then Wilkerson showed that for
any class ['] 2 Caut (A Q bZ), the image oe([']) is isomorphic to A as
groups; that is, the image of oe is constant at A.
It is straightforward to adapt Wilkerson's proofs of these results to show
that for each j the map
oe :Caut ((K(X)=Kj(X)) Q bZ) ! Genus (K(X)=Kj(X))
is surjective and that the image of oe is constant at K(X)=Kj(X). In
other words, the genus of K(X)=Kj(X) is the one-point set. Therefore,
K(Y )=Kj(Y ) is isomorphic to K(X)=Kj(X).
__
This finishes the proof of Lemma 3.3. |__|
In view of Lemma 3.1, to finish the proof of Theorem 1.1 we are only
left to show that the classifying object
lim-1nAut(K(X)=Kn(X))
is the one point set. To do this, it suffices to show that almost all the
structure maps in the tower are surjective. This is shown in the following
lemma.
Lemma 3.4. The maps
Aut(K(X)=Kj+1(X)) ! Aut(K(X)=Kj(X))
are surjective for all j sufficiently large.
Proof.First note that by hypothesis the K-theory filtered ring of X has the
form
K(X) = Z[[c1, . .,.cn]] (3.4)
in which the generators ci are algebraically independent over Z and Kj(X)
is the ideal generated by the monomials of filtrations at least j. Suppose
that di is the largest integer k for which ci lies in filtration k. Let N be the
integer
N = max {di:1 i n} + 1.
10
We will show that the structure maps
Aut(K(X)=Kj+1(X)) ! Aut(K(X)=Kj(X))
are surjective for all j > N.
So fix an integer j > N and pick a filtered ring automorphism oe of
K(X)=Kj(X). We must show that oe can be lifted to a filtered ring auto-
morphism of K(X)=Kj+1(X). For 1 i n pick any lift of the element
oe(ci) to K(X)=Kj+1(X) and call it ^oe(ci). Since there are no relations among
the ci in K(X), it is easy to see that ^oeextends to a well-defined filtered ri*
*ng
endomorphism of K(X)=Kj+1(X), and it will be a desired lift of oe once it
is shown to be bijective.
To show that
^oe:K(X)=Kj+1(X) ! K(X)=Kj+1(X)
is surjective, it suffices to show that the image of each ci in K(X)=Kj+1(X)
lies in the image of ^oe, since K(X)=Kj+1(X) is generated as a filtered ring
by the images of the ci. So fix an integer i with 1 i n. We know that
there exists an element gi2 K(X)=Kj(X) such that
oe(gi) = ci. (3.5)
Pick any lift of gi to K(X)=Kj+1(X), call it gi again, and observe that (3.5)
implies that
^oe(gi) = ci+ ffi (3.6)
in K(X)=Kj+1(X) for some element ffi 2 Kj(X)=Kj+1(X). We will alter
gi to obtain a ^oe-pre-image of ci as follows.
Observe that the ideal Kj(X)=Kj+1(X) is generated by certain mono-
mials in c1, . .,.cn. Namely, the monomials
ci = ci11. .c.inn, i = (i1, . .,.in) 2 Jj (3.7)
where Jj is the set of ordered n-tuples i = (i1, . .,.in) of nonnegative intege*
*rs
satisfying
Xn
d . i = dlil = j.
l=1
11
Thus, for every element i in the set Jj, there exists a corresponding integer
ai such that we can write the element ffi as the sum
X
ffi = aici. (3.8)
i 2 Jj
Now define the element ~giin K(X)=Kj+1(X) by the formula
X
~gidef=gi - aigi, where gi= gi11. .g.inn. (3.9)
i 2 Jj
We claim that ~giis a ^oe-pre-image of ci. That is, we claim that
^oe(~gi) = ci in K(X)=Kj+1(X). (3.10)
In view of (3.6), (3.8), and (3.9), it clearly suffices to prove the following
equality for each element i in Jj:
i j
^oegi = ci in K(X)=Kj+1(X). (3.11)
Now in the quotient K(X)=Kj+1(X), one computes
i j Yn
^oegi = ^oe(gj)ij
j=1
Yn
= (cj + ffj)ij by (3.6)
j=1
= ci + (terms of filtrations> j)
= ci.
This proves (3.11), and hence (3.10), and therefore ^oeis surjective.
It remains to show that o^eis injective. Since any surjective endo-
morphism of a finitely generated abelian group is also injective and since
K(X)=Kj+1(X) is a finitely generated abelian group, it follows that ^oeis in-
jective as well. Thus, ^oeis a filtered-ring automorphism of K(X)=Kj+1(X)
and is a lift of oe.
__
This finishes the proof of Lemma 3.4. |__|
This proof of Theorem 1.1 is complete.
12
4 Proof of Theorem 1.4
In this section we prove Theorem 1.4. The arguments in this section, espe-
cially Lemma 4.2 below, are inspired by Rector's [21, x4].
We begin by noting that an argument entirely similar to the proof of
Theorem 1.1 implies that whenever X belongs to Genus (BS3), one has
KO*(X) ~=KO*[[x]]
as filtered rings, where x 2 KO44(X) is a representative of an integral genera-
tor x4 2 H4(X; Z) = E4,20in the KO*-Atiyah-Hirzebruch spectral sequence
for X. Here KOab(X) denotes the subgroup of KOa(X) consisting of ele-
ments u which restrict to 0 under the natural map
KOa(X) ! KOa(Xb-1).
Such an element u is said to be in degree a and filtration b.
Now we recall the relevant notations, definitions, and results regarding
Rector's classification of the genus of BS3 [20]. Let , 2 ß-4KO and bR 2
ß-8KO be the generators so that ,2 = 4bR . As usual, denote by _k (k =
1, 2, . .).the Adams operations. Since X is homotopy equivalent to S3, it
follows as in [21, x4] that there exists an integer a, depending on the choice
of the representative x, such that the following statements hold.
1. _2(,x) = 4,x + 2abR x2 (mod KO09(X)).
2. The integer a is well-defined (mod 24). This means that if x0is another
representative of x4 with corresponding integer a0, then a a0 (mod
24), and if x4 is replaced with -x4, then a will be replaced with -a.
We can therefore write a(X) for a.
3. a(X) 1, 5, 7, or 11 (mod 24).
The last condition above follows from the examples constructed by Rector
in [21, x5] and James' result [10] which says that there are precisely eight ho-
motopy classes of homotopy-associative multiplications on S3. These eight
classes can be divided into four pairs with each pair consisting of a homotopy
class of multiplication and its inverse.
13
Rector's invariant (X=p) for p an odd prime is defined as follows [20].
The Adem relation P 1P 1= 2P 2implies that
P 1__x4= 2__x(p+1)=24
in H*(X; Z=p), where __x4is the mod p reduction of the integral generator
x4. Then (X=p) 2 { 1} is defined as the sign on the right-hand side of this
equation.
The invariant (X=2) and a canonical choice of orientation of the integral
generator are given as follows. Using the mod 24 integer a(X), define
8
>>>(1, 1) ifa(X) 1 mod 24;
>>>
< (1, -1) ifa(X) 5 mod 24;
((X=2), (X=3)) = (4.1)
>>>(-1, 1) ifa(X) 7 mod 24;
>>>
: (-1, -1) ifa(X) 11 mod 24.
The orientation is then chosen so that (X=3) is as given in (4.1). This
definition of Rector's invariants coincides with the original one (cf. [14, x9]*
*).
Now we can recall the classification theorem of the genus of BS3 [20].
Theorem 4.1 (Rector). The (X=p) for p primes provide a complete list
of classification invariants for the (adic) genus of BS3. Any combination of
values of the (X=p) can occur. If X is BS3 then (X=p) = 1 for any prime
p.
From now on in this section, X and Y are as in Theorem 1.4. Now we
can prove Theorem 1.4.
Proof of Theorem 1.4.In view of Theorem 4.1, to prove Theorem 1.4 it
suffices to show that if there exists a filtered ~-ring isomorphism
oe :KO*(X) ~=KO*(Y ),
then
a(X) a(Y ) (mod 24)
14
and
(X=p) = (Y=p)
for all odd primes p. We will prove these in Lemmas 4.2 and 4.3 below,
__
thereby proving Theorem 1.4. |__|
As explained above, KO*(X) = KO*[[x]] and KO*(Y ) = KO*[[y]] with
x 2 KO44(X) and y 2 KO44(Y ) representing, respectively, the integral gen-
erators x4 2 H4(X; Z) and y4 2 H4(Y ; Z).
Lemma 4.2. If there exists a filtered ~-ring isomorphism
~= *
oe :KO*(X) -! KO (Y ),
then
a(X) a(Y ) (mod 24).
Proof.Since oe is a ring isomorphism, we have
oe(,x) = ",y + oe2bR y2 (mod KO09(Y ))
for some integer oe2 and " 2 { 1}. Computing modulo KO09(X) we have
4oe(bR x2)= oe(,x)2
= ,2y2
= 4bR y2.
Therefore, one has
oe(bR x2) = bR y2 (mod KO09(Y )).
First we claim that there is an equality
a(X) = 6oe2 + ä (Y ). (4.2)
To prove (4.2)we will compute both sides of the equality
oe_2(,x) = _2oe(,x) (mod KO09(Y )).
15
Working modulo KO09(Y ) we have, on the one hand,
oe_2(,x) = oe(4,x + 2a(X)bR x2)
= 4(",y + oe2bR y2) + 2a(X)bR y2
= 4",y + (4oe2 + 2a(X))bR y2.
On the other hand, still working modulo KO09(Y ), we have
_2oe(,x) = "_2(,y) + oe2_2(bR y2)
= "(4,y + 2a(Y )bR y2) + oe2(24bR y2)
= 4",y + (16oe2 + 2ä (Y ))bR y2.
Equation (4.2)now follows by equating the coefficients of bR y2.
In view of (4.2), to finish the proof of Lemma 4.2 it is enough to establish
oe2 0 (mod 4). (4.3)
To prove (4.3), note that since oe is a KO*-module map, we have
,oe(x) = oe(,x).
Since oe is a ring isomorphism, we also have
oe(x) = "0y + oe02,y2 (mod KO49(Y ))
for some integer oe02and "02 { 1}. Therefore, working modulo KO09(Y ) we
have
,oe(x) = "0,y + oe02,2y2
= "0,y + 4oe02bR y2
= ",y + oe2bR y2.
In particular, by equating the coefficients of bR y2 we obtain
oe2 = 4oe02,
thereby proving (4.3).
__
This completes the proof of Lemma 4.2. |__|
16
Lemma 4.3. If there exists a filtered ~-ring isomorphism
~= *
oe :KO*(X) -! KO (Y ),
then
(X=p) = (Y=p)
for each odd prime p.
Proof.It follows from Theorem 1.1 that
K*(X) ~=K*[[ux]]
with ux 2 K44(X) a representative of the integral generator x4 2 H4(X; Z) =
E4,02in the K*-Atiyah-Hirzebruch spectral sequence. Moreover, we may
choose ux so that
c(x) = ux,
where
c: KO*(X) ! K*(X)
is the complexification map. Similar remarks apply to Y so that
K*(Y ) ~=K*[[uy]].
Now denote by b 2 ß-2K the Bott element and let p be a fixed odd
prime. We first claim that
_p(b2ux) = (b2ux)p + 2(X=p) p (b2ux)(p+1)=2+ p wx + p2x0 (4.4)
for some wx 2 K02p+3(X) and some x0 2 K04(X). To see this, note that since
b2ux 2 K04(X), it follows from Atiyah's theorem [2, 5.6] that
_p(b2ux) = (b2ux)p + p x1 + p2x0
for some xi2 K04+2i(p-1)(X) (i = 0, 1). Moreover, one has
__x 1_____2
1 = P b ux,
17
where __zis the mod p reduction of z and P 1is the Steenrod operation of
degree 2(p - 1) in mod p cohomology. Thus, to prove (4.4)it is enough to
show that
x1 = 2(X=p)(b2ux)(p+1)=2+ wx + p zx (4.5)
for some wx 2 K02p+3(X) and some zx 2 K02p+2(X). Now in H*(X; Z) Z=p
we have
__x 1_____2
1 = P b ux
= P 1__x4
= 2(X=p) __x(p+1)=24
_____(p+1)=2
= 2(X=p) b2ux .
Now (4.5) follows immediately. As remarked above, this also establishes
(4.4).
Now the ~-ring isomorphism oe induces via c a ~-ring isomorphism
~= *
oec: K*(X) -! K (Y ).
By composing oec with a suitable ~-ring automorphism of K*(Y ) if necessary,
we obtain a ~-ring isomorphism
~= *
ff: K*(X) -! K (Y )
with the property that
ff(b2ux) = b2uy + higher terms inb2uy. (4.6)
Using (4.4)and (4.6)it is then easy to check that
ff_p(b2ux) = 2(X=p) p (b2uy)(p+1)=2 (mod K02p+3(Y ) and p2)(4.7)
and
_pff(b2ux) = 2(Y=p) p (b2uy)(p+1)=2 (mod K02p+3(Y ) and p2).(4.8)
Since ff_p = _pff it follows from (4.7)and (4.8)that
2(X=p) p 2(Y=p) p (mod p2),
18
or, equivalently,
2(X=p) 2(Y=p) (mod p).
But p is assumed odd, and so
(X=p) (Y=p) (mod p).
Hence (X=p) = (Y=p), as desired.
__
This finishes the proof of Lemma 4.3. |__|
The proof of Theorem 1.4 is complete.
5 Proof of Theorem 1.9
In this final section we give the proof of Theorem 1.9. The argument is
somewhat similar to the proof of Theorem 1.1. All cohomology groups and
rings have coefficients over a fixed principal ideal domain . Throughout this
section we are working in the category of (non-negatively) graded algebras
over .
We need the following two lemmas.
Lemma 5.1. For each n > 0, there is an isomorphism
H n (X) ~= H n (Y )
of graded algebras over .
Proof.This follows immediately from the assumption that X and Y have
__
the same n-type for all n. |__|
By hypothesis the cohomology ring of X has the form
H*(X) = [x1, . .,.xs]=J
for some homogeneous generators xi (1 i s) and some homogeneous
ideal J. Suppose that |xi| = di. Note that since the polynomial ring
19
[x1, . .,.xs] is Noetherian, the ideal J is generated by finitely many poly-
nomials, say, fm (1 m k). We can thus choose an integer N which
is strictly greater than the di (1 i s) and the degree of any nontrivial
monomial in any fm (1 m k).
Lemma 5.2. For each j > N, the natural map
j+1 j
Aut H (X) -! Aut H (X)
is surjective, where Aut (-) denotes the group of graded -algebra automor-
phisms.
Proof.Let j be any integer strictly greater than N and let oe be an au-
tomorphism of the graded algebra H j (X). We want to show that oe can
be lifted to an automorphism of H j+1 (X). Now oe(fm ) is 0 in H j (X)
for m = 1, . .,.k. But since each oe(xi) is homogeneous of degree di, each
monomial of
oe(fm ) = fm (oe(x1), . .,.oe(xs))
still has degree less than N, and in this degree the natural map
H j+1 (X) ! H j (X)
is an isomorphism. Therefore, there is a well-defined graded algebra map
^oe:H j+1 (X) ! H j+1 (X)
satisfying
^oe|H j (X) = oe :H j (X) ! H j (X).
We will be done once we show that ^oeis bijective.
It is clear that ^oeis surjective because H j+1 (X) is generated as an
algebra by the xi (1 i s) and each xi is in the image of ^oe, since this is
true for oe.
Finally, since H j+1 (X) is a finitely generated -module, the injectivity
of ^oenow follows from its surjectivity. Therefore, ^oeis an automorphism of
H j+1 (X) and is a lift of oe.
__
This finishes the proof of Lemma 5.2. |__|
20
Proof of Theorem 1.9.This is now an immediate consequence of Lemma
5.1, Lemma 5.2, and the analog of Wilkerson's Theorem 3.1 in the context
__
of (non-negatively) graded -algebras. |__|
Acknowledgements
Most of the results in this paper come from the author's Ph.D. thesis at
MIT. The author would like to express his sincerest gratitude to his advisor,
Professor Haynes Miller, for superb guidance and encouragement.
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