MAPS TO SPACES IN THE GENUS OF INFINITE
QUATERNIONIC PROJECTIVE SPACE
DONALD YAU
Abstract. Spaces in the genus of infinite quaternionic projective space
which admit essential maps from infinite complex projective space are
classified. In these cases the sets of homotopy classes of maps are de-
scribed explicitly. These results strengthen the classical theorem of
McGibbon and Rector on maximal torus admissibility for spaces in the
genus of infinite quaternionic projective space. An interpretation of
these results in the context of Adams-Wilkerson embedding in integral
K-theory is also given.
1. Introduction and statement of results
In an attempt to understand Lie groups through their classifying spaces,
Rector [9] classified the genus of HP1 , the infinite projective space over
the quaternions, considered as a model for the classifying space BS3. The
homotopy type of a spaces X is said to be in the genus of HP1 , denoted
X 2 Genus (HP1 ), if the p-localizations of X and HP1 are homotopy
equivalent for each prime p. One often speaks of a space rather than its
homotopy type when considering genus. Rector's classification [9] of the
genus of HP1 is as follows.
Theorem 1.1 (Rector). Let X be a space in the genus of HP1 . Then for
each prime p there exists a homotopy invariant (X=p) 2 { 1} such that the
following statements hold.
(1) The (X=p) for p primes provide a complete list of homotopy classi-
fication invariants for the genus of HP1 .
(2) Any combination of values of the (X=p) can occur. In particular, the
genus of HP1 is uncountable.
(3) The invariant (HP1 =p) is 1 for all primes p.
(4) The space X has a maximal torus if and only if X is homotopy
equivalent to HP1 .
____________
Date: October 29, 2001.
2000 Mathematics Subject Classification. Primary 55S37; Secondary 55S25.
Key words and phrases. Genus, maximal torus, quaternionic projective space.
1
2 DONALD YAU
The invariant (X=p) is now known as the Rector invariant at the prime
p. Actually, for the last statement about the maximal torus, Rector only
proved it for the odd primes. That is, if X has a maximal torus, then (X=p)
is equal to 1 for all odd primes p. Then McGibbon [6] proved it for the prime
2 as well. Here X is said to have a maximal torus if there exists a map from
CP1 , the infinite complex projective space, to X whose homotopy theoretic
fiber has the homotopy type of a finite complex.
For a space X in the genus of HP1 which is not homotopy equivalent to
HP1 , the nonexistence of a maximal torus does not rule out the possibility
that there could be some essential (that is, non-nullhomotopic) maps from
CP1 to X. The main purposes of this paper are (1) to describe spaces
in the genus of HP1 for which this can happen (Theorem 1.2) and (2) to
compute the maps in these cases (Theorem 1.7).
The following is our first main result, which classifies spaces in the genus
of HP1 which admit essential maps from CP1 .
Theorem 1.2. Let X be a space in the genus of HP1 . Then the following
statements are equivalent.
(1) There exists an essential map from CP1 to X.
(2) There exists a nonzero integer k such that (X=p) = (k=p) for all but
finitely many primes p.
(3) There exists a cofinite set of primes L such that HP1 and X become
homotopy equivalent after localization at L.
Here (k=p) is the Legendre symbol of k, which is defined whenever p does
not divide k. If p is odd and if p does not divide k, then (k=p) = 1 (resp.
-1) if k is a quadratic residue (resp. non-residue) mod p. If p = 2 and if k is
odd, then (k=2) = 1 (resp. -1) if k is a quadratic residue (resp. non-residue)
mod 8.
Before discussing related issues, let us first record the following immediate
consequence of Theorem 1.2.
Corollary 1.3. There exist only countably many homotopically distinct
spaces in the genus of HP1 which admit essential maps from CP1 .
Indeed, each nonzero integer k can determine only countably many ho-
motopically distinct spaces X in the genus of HP1 satisfying the second
condition in Theorem 1.2.
The second condition of Theorem 1.2 gives an arithmetic description of
spaces in the genus of HP1 which occur as the target of essential maps
from CP1 . Since it involves Rector invariants, it is specific to the genus
GENUS OF INFINITE QUATERNIONIC PROJECTIVE SPACE 3
of HP1 and is not very convenient for generalizations. The last condition
of Theorem 1.2, on the other hand, is geometric and is more suitable for
possible generalizations of the theorem.
Having characterized spaces in the genus of HP1 which admit nontrivial
maps from CP1 , we proceed to compute the maps themselves. Now for any
space X in the genus of HP1 , the K-theory K(X) of X, as a filtered ring, is
a power series ring Z[[b2uX ]] (see Proposition 2.1), where uX is some element
in K4(X) and b is the Bott element in K-2 (pt). Here, and throughout the
rest of the paper, K(-) denotes complex K-theory with coefficients over
the integers Z. So if f :CP1 ! X is any map, then its induced map in
K-theory defines an integer deg(f), called the degree of f, by the equation
(1.4) f*(b2uX ) = deg(f)(b,)2 + higher terms inb,,
where b, is the ring generator in the power series ring K(CP1 ) = Z[[b,]].
Note that deg(f) is, up to a sign, simply the degree of the induced map of
f in integral homology in dimension 4. According to a result of Dehon and
Lannes [4], the homotopy class of such a map f is determined by its degree.
We will therefore identify such a map with its degree in the sequel. The
degrees of a self-map of X, a self-map of the p-localization HP1(p)of HP1 ,
or a map from CP1 to HP1(p)can be defined similarly.
To describe the maps from CP1 to X 2 Genus (HP1 ) up to homotopy,
we need only describe the possible degrees of such maps. Let's first consider
the classical case. There is a maximal torus inclusion i: CP1 ! HP1
of degree 1, and any other map f :CP1 ! HP1 factors through i up to
homotopy. A special case of a theorem of Ishiguro, Møller, and Notbohm
[5, Thm. 1] says that for any space X in the genus of HP1 , the degrees of
essential self-maps of X consist of precisely the squares of odd numbers. For
the classical case, X = HP1 , this result is due to Sullivan [10, p. 58-59].
Therefore, the degrees of essential maps from CP1 to HP1 also consist of
precisely the odd squares.
The situation in general is quite similar. Recall that any space X in the
genus of HP1 can be obtained as a homotopy inverse limit [3]
n rq nq o
(1.5) X = holimq HP1(q)-! HP1(0)-! HP1(0) .
Here q runs through all primes, rq is the natural map from the q-localization
to the rationalization of HP1 , and nq is an integer relatively prime to q,
satisfying (nq=q) = (X=q). The integer n2 also satisfies n2 1 (mod 4).
Now if X admits an essential map from CP1 , and thus satisfies the
condition in Theorem 1.2 for some nonzero integer k, then the integers nq
can be chosen so that the set {nq: q primes} contains only finitely many
distinct integers. So it makes sense to talk about the least common multiple
4 DONALD YAU
of the integers nq, denoted LCM (nq). Now we define an integer TX as
(1.6) TX = min{LCM (nq) :X = holimq(nq O rq)}.
That is, choose the integers nq as in (1.5) so as to minimize their least
common multiple, and TX is defined to be LCM (nq).
We are now ready for the second main result of this paper, which describes
the maps from CP1 to X 2 Genus (HP1 ).
Theorem 1.7. Let X be a space in the genus of HP1 which admits an
essential map from CP1 . Then the following statements hold.
(1) There exists a map iX :CP1 ! X of degree TX .
(2) The map iHP1 :CP1 ! HP1 is the maximal torus inclusion.
(3) Given any map f :CP1 ! X, there exists a self-map g of X such
that f is homotopic to g O iX .
(4) The degrees of essential maps from CP1 to X are precisely the odd
squares multiples of TX .
It should be noted that the integer TX does not determine the homotopy
type of X. For example, consider the spaces X and Y in the genus of HP1
with Rector invariants
( (
1 ifp 6= 3 1 ifp 6= 5
(1.8) (X=p) = , (Y=p) =
-1 ifp = 3 -1 ifp = 5.
Then, of course, X is not homotopy equivalent to Y because their Rector
invariants at the prime 3 are distinct. But it is easy to see that TX = 2 = TY .
Theorems 1.2 and 1.7 are closely related to the (non)existence of Adams-
Wilkerson type embeddings of finite H-spaces in integral K-theory. As
mentioned before, a map f :CP1 ! X 2 Genus (HP1 ) is essential if and
only if deg(f) is nonzero. Thus, if there exists an essential map from CP1
to X 2 Genus (HP1 ), then K(X) can be embedded into K(CP1 ) as a
sub-~-ring. The converse is also true. Indeed, a theorem of Notbohm and
Smith [8, Thm. 5.2] says that the function
ff: [CP1 , X] ! Hom ~(K(X), K(CP1 ))
which sends (the homotopy class of) a map to its induced map in K-theory,
is a bijection. (Here [-, -] and Hom ~(-, -) denote, respectively, sets of
homotopy classes of maps between spaces and of ~-ring homomorphisms.)
So a ~-ring embedding K(X) ! K(CP1 ) must be induced by an essential
map from CP1 to X. Therefore, Theorem 1.2 and Corollary 1.3 can be
restated in this context as follows.
GENUS OF INFINITE QUATERNIONIC PROJECTIVE SPACE 5
Theorem 1.9. Let X be a space in the genus of HP1 . Then K(X) can
be embedded into K(CP1 ) as a sub-~-ring if, and only if, there exists a
nonzero integer k such that (X=p) = (k=p) for all but finitely many primes
p. This is true if, and only if, there exists a cofinite set of primes L such
that HP1 and X become homotopy equivalent after localization at L.
In particular, there exist only countably many homotopically distinct
spaces X in the genus of HP1 whose K-theory ~-rings can be embedded
into that of CP1 as a sub-~-ring.
Before Theorem 1.9, there is at least one space in the genus of HP1
whose K-theory ~-ring was known to be non-embedable into the K-theory
~-ring of CP1 . This example was due to Adams [1, p. 79].
Remark 1.10. Theorems 1.2 and 1.7 can also be regarded as an attempt
to understand the set of homotopy classes of maps from X to Y , where X 2
Genus (BG) and Y 2 Genus (BK) with G and K some connected compact
Lie groups. This problem, especially the case G = K = S3 x . .x.S3, was
studied extensively by Ishiguro, Møller, and Notbohm [5].
This finishes the presentation of our main results. The rest of this paper
is organized as follows. Theorems 1.2 and 1.7 are proved in x3 and x4,
respectively. For use in x3, we recall in x2 some results from [12], with
sketches of proofs, about K-theory filtered rings.
Acknowledgment
Thanks are due to Dietrich Notbohm for his interest and comments in
this work, and to the referee whose suggestions are very helpful. The author
would like to express his deepest gratitude to his thesis advisor Haynes Miller
for many hours of stimulating conversations and encouragement.
2.K-theory filtered ring
In preparation for the proof of Theorem 1.2 in x3, a result from [12] about
K-theory filtered ring is reviewed in this section.
To begin with, a filtered ring is a pair (R, {IRn}) consisting of:
(1) A commutative ring R with unit;
(2) A decreasing filtration R = IR0 IR1 . .o.f ideals of R such that
IRiIRj IRi+jfor all i, j 0.
A map between two filtered rings is a ring homomorphism which preserves
the filtrations. With these maps as morphisms, the filtered rings form a
category.
6 DONALD YAU
Every space Z of the homotopy type of a CW complex gives rise naturally
to an object (K(Z), {Kn(Z)}), which is usually abbreviated to K(Z), in this
category. Here K(Z) and Kn(Z) denote, respectively, the complex K-theory
ring of Z and the kernel of the restriction map K(Z) ! K(Zn-1), where
Zn-1 denotes the (n - 1)-skeleton of Z. Using a different CW structure
of Z will not change the filtered ring isomorphism type of K(Z), as can be
easily seen by using the cellular approximation theorem. The symbol Krs(Z)
denotes the subgroup of Kr(Z) consisting of elements whose restrictions to
Kr(Zs-1) are equal to 0.
The following result, which is proved in [12], will be needed in x3. For the
reader's convenience we include here a sketch of the proof. In what follows
b 2 K-2 (pt) will denote the Bott element.
Proposition 2.1. Let X be a space in the genus of HP1 . Then the fol-
lowing statements hold.
(1) There exists an element uX 2 K44(X) such that K(X) = Z[[b2uX ]]
as a filtered ring.
(2) For any odd prime p, the Adams operation _p satisfies
2 2 p 2 (p+1)=2 2
(2.2) _p b uX = b uX + 2 (X=p) p b uX + pw + p z,
where w and z are some elements in K02p+3(X) and K04(X), respec-
tively. In particular, we have
2 2 (p+1)=2 0 2
(2.3) _p b uX = 2 (X=p) p b uX mod K2p+3(X) and p .
Sketch of the proof of Proposition 2.1.For the first assertion, Wilkerson's
proof of the classification theorem [11, Thm. I] of spaces of the same n-type
for all n can be easily adapted to show the following. There is a bijection
between the following two pointed sets:
(1) The pointed set of isomorphism classes of filtered rings (R, {IRn})
with the properties:
(a)The natural map R ! lim-nR=IRnis an isomorphism, and
(b) R=IRnand K(HP1 )=Kn(HP1 ) are isomorphic as filtered rings
for all n > 0.
(2) The pointed set lim-1nAut(K(HP1 )=Kn(HP1 )).
Here Aut (-) denotes the group of filtered ring automorphisms, and the
lim-1of a tower of not-necessarily abelian groups is as defined in [3]. It is
not difficult to check that the hypothesis on X implies that the filtered ring
GENUS OF INFINITE QUATERNIONIC PROJECTIVE SPACE 7
K(X) has the above two properties, (a) and (b). Moreover, by analyzing
the subquotients Kn(HP1 )=Kn+1(HP1 ), one can show that the map
Aut (K(HP1 )=Kn+1(HP1 )) ! Aut(K(HP1 )=Kn(HP1 ))
is surjective for each integer n greater than 4. The point is that any au-
tomorphism of K(HP1 )=Kn(HP1 ) can be lifted to an endomorphism of
K(HP1 )=Kn+1(HP1 ) without any difficulty. Then, since the quotient
K(HP1 )=Kn+1(HP1 ) is a finitely generated abelian group, one only has
to observe that the chosen lift is surjective. Thus, the above lim-1is the one-
point set, and hence K(X) and K(HP1 ) are isomorphic as filtered rings.
This establishes the first assertion.
The second assertion concerning the Adams operations _p is an easy
consequence of the first assertion, Atiyah's theorem [2, Prop. 5.6], and the
definition of the Rector invariants (X=p) [9].
3. Proof of Theorem 1.2
In this section the proof of Theorem 1.2 is given.
Recall that the complex K-theory of CP1 as a filtered ~-ring is given by
K(CP1 ) = Z[[b,]] for some , 2 K22(CP1 ), where b 2 K-2 (pt) is the Bott
element. The Adams operations on the generator are given by
(3.1) _r(b,) = (1 + b,)r - 1 (r = 1, 2, . .)..
Fix a space X in the genus of HP1 and write Z[[b2uX ]] for its K-theory
filtered ring (cf. Proposition 2.1).
We will prove Theorem 1.2 by proving the implications (1) ) (2) ) (3)
) (1). Each implication is contained in one subsection below.
3.1. Proof of (1) implies (2). This part of Theorem 1.2 is contained in
the next Lemma.
Lemma 3.2. Let p be an odd prime and k be a nonzero integer relatively
prime to p. If there exists an essential map f :CP1 ! X of degree k, then
(X=p) = (k=p).
Proof.We will compare the coefficients of (b,)p+1 in the equation
2 p * 2 0 1 2
(3.3) f*_p b uX = _ f b uX mod K2p+3(CP ) and p .
Working modulo K02p+3(CP1 ) and p2, it follows from (1.4)and (2.3)that
2 2 2(p+1)=2
f*_p b uX = 2 (X=p) p kb ,
= 2 (X=p) p k(p+1)=2(b,)p+1.
8 DONALD YAU
Similarly, still working modulo K02p+3(CP1 ) and p2, it follows from (1.4)
and (3.1)that
2 p 2 2
_pf* b uX = k_ (b , )
= k_p(b,)2
= 2pk(b,)p+1.
Thus, we obtain the congruence relation
(3.4) 2 (X=p) p k(p+1)=2 2pk (mod p2).
Since (k=p) is congruent to k(p-1)=2(mod p) (see, for example, [7, Thm.
3.12]) and since p is odd and relatively prime to k, (3.4)is equivalent to the
congruence relation
(3.5) (X=p) (k=p) 1 (mod p).
Hence (X=p) = (k=p), as desired.
This finishes the proof of Lemma 3.2.
This shows that (1) implies (2) in Theorem 1.2.
3.2. Proof of (2) implies (3). Suppose that there exists a nonzero integer
k such that (X=p) = (k=p) for all primes p, except possibly p1, . .,.ps. The
prime factors of k are among the pi. Let L be the set consisting of all primes
except the pi, 1 i s. We will show that HP1 and X become homotopy
equivalent after localization at L.
First note that for any space Y in the genus of HP1 and for any subset
I of primes, the I-localization of Y can be obtained as
n nqO rq o
(3.6) Y(I) = holimq2I HP1(q)----! HP1(0) .
In particular, we have
æ oe
k O rq 1
(3.7) X(L) = holimq2L HP1(q)---! HP(0)
and
n rq o
(3.8) HP1(L)= holimq2L HP1(q)-!HP1(0) .
Now for each prime q 2 L, let fq be a self-map of HP1(q)of degree k-1 . Since
k is a q-local unit (because q does not divide k), it is easy to see that each
fq is a homotopy equivalence. Moreover, the two maps
(3.9) rq, k O rq O fq: HP1(q)! HP1(0)
coincide. Therefore, the maps fq (q 2 L) glue together to yield a map
(3.10) f :HP1(L)! X(L)
GENUS OF INFINITE QUATERNIONIC PROJECTIVE SPACE 9
which is a homotopy equivalence, since each fq is.
This shows that (2) implies (3) in Theorem 1.2.
3.3. Proof of (3) implies (1). Suppose that there exists a cofinite set
of primes L such that HP1(L)and X(L) are homotopy equivalent. Write
p1, . .,.ps for the primes not in L, and write rL for the natural map from
X(L)to HP1(0).
To construct an essential map from CP1 to X, first note that X can be
constructed as the homotopy inverse limit of the diagram
rpi 1 npi 1 rL
(3.11) HP1(pi)--!HP(0)--! HP(0)- X(L)
in which i runs from 1 to s.
For each i, 1 i s, let fpi be a map from CP1 to HP1(pi)of degree
Q s
M=npi, where M = i=1 npi. Also, let fL denote a map from CP1 to
X(L)of degree M, which exists because X(L)has the same homotopy type
as HP1(L). It is then easy to see that the two maps
(3.12) rL O fL, npiO rpiO fpi:CP1 ! HP1(0)
coincide for any 1 i s. Therefore, the maps fpi(1 i s) and fL glue
together to yield an essential map
(3.13) f :CP1 ! X
through which all the maps fpiand fL factor.
This shows that (3) implies (1) in Theorem 1.2.
The proof of Theorem 1.2 is complete.
4. Proof of Theorem 1.7
In this section we prove Theorem 1.7.
Fix a space X in the genus of HP1 which admits an essential map from
CP1 .
First we note that part (2) follows from the discussion preceding Theorem
1.7, since it is obvious that the integer THP1 is 1.
Since any essential self-map of X is a rational equivalence, part (4) is an
immediate consequence of parts (1) and (3) and a result of Ishiguro, Møller,
and Notbohm [5, Thm. 1] which says that the degrees of essential self-maps
of X are precisely the odd squares.
Now we consider part (1). Suppose that the integers nq as in (1.5)are
chosen so that there are only finitely many distinct integers in the set
{nq: q primes} and that TX is their least common multiple (see (1.6)for
the definition of TX ). Denote by lq: HP1 ! HP1(q)the q-localization map
10 DONALD YAU
and by i: CP1 ! HP1 the maximal torus inclusion. A self-map of CP1
of degree m on H2(CP1 ; Z) is simply denoted by m. Now for each prime q
define a map fq: CP1 ! HP1(q)to be the composition
M=nq 1 i 1 lq 1
(4.1) CP1 ---! CP ! HP -! HP(q).
It is then easy to see that the two maps
(4.2) nq O rq O fq, nq0O rq0O fq0:CP1 ! HP1(0)
coincide for any two primes q and q0. Therefore, the maps fq glue together
to yield an essential map
(4.3) f :CP1 ! X
through which every map fq factors. The map f has degree TX because its
induced map in rational cohomology in dimension 4 does.
Finally, for part (3), suppose that f :CP1 ! X is a map. Write
fp: CP1 ! HP1(p)for the component map of f corresponding to the prime
p. That is, fp is the composition
(4.4) CP1 f!X ! HP1(p)
where the second map is the natural map arising from the construction of
X. Then for any prime p we have the equality
(4.5) deg(f) = np deg(fp).
Since each np divides deg(f), so does their least common multiple TX . More-
over, by writing (iX )p for the component map of iX corresponding to the
prime p, (4.5)implies that for any prime p we have the equalities
deg (fp) deg(f)=np deg(f)
(4.6) _________= __________ = _______.
deg(iX )p TX =np TX
Since there are self-maps of HP1(q)(q any prime) and HP1(0)of degree
deg(f)=TX , one can construct a self-map g of X such that deg(g) is equal
to deg(f)=TX and that f is homotopic to g O iX . This proves part (3).
The proof of Theorem 1.7 is complete.
GENUS OF INFINITE QUATERNIONIC PROJECTIVE SPACE 11
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Department of Mathematics
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donald@math.mit.edu