ContemporaryVMathematicsolume 00, 0000
SU (n) does not Split in 2 Suspensions, for n 3
MARK MAHOWALD AND WILLIAM RICHTER
0. Introduction
Solving a conjecture of Hopkins and Mahowald, the second author [Ri ] showed
that Mitchell's [Mi3 ] filtration {Fn;k}1k=1of SU(n) splits stably, analogous*
* to
the Snaith [Sn2 ] splitting of BU. Crabb and Mitchell [CM ] then gave similar
splittings of U(n)=O(n) and U(2n)=Sp(n). The first filtration Fn;1is the in
clusion CP n1 SU(n) , which was actually known to split off by the work of
James [Ja ], which was refined by Miller [Mi2 ]. James split CP n1 off SU(n),
with a map J: SU(n) ! N N CP n1 , whose loop (J):SU(n) !
N+1 N+1 CP n1splits CP n1 stably off SU(n) . Cohen and Peterson [CP ],
using DyerLashof operations, showed that N > 2 for any such map J. How
ever there is no DyerLashof obstruction to factoring (J)through 22CP n1.
That is, the image of (J)consists only of monomials in H* CP n1 ; Z=2. The
question arose: does there exist a map ae: SU(n) ! 22CP n1 which splits
off CP n1 stably? The existence of such a map ae would have implied the sta
ble splitting of`SU(n) , provided the MitchellSegal [Mi3 , Se1 ] group com
pletion model kFn;k U(n) has a C2structure (cf. May [Ma3 ]). Following
Snaith [Sn2 ], the kthsplitting map could have been constructed as the composite
+
SU(n) ae!22CP n1 k! Q F (R2; k) xk CP n1 k
Q(c2)!Q F +
n;k ! Q (Fn;k=Fn;k1):
Here F (R2; k) is the ordered configuration space of k points in R2, k is the k*
*th
groupcompleted Hopf invariant [Sn1 , Co , KP , Se2 ], popularized by Segal,
and c2 is one of the C2structure maps. It was furthermore speculated that ae
____________
1991 Mathematics Subject Classification. Primary 55P35, 55P40.
Research of the first author supported by a grant from the NSF.
Research of the second author supported by an NSF postdoctoral fellowship.
cO00000American0Mathematical0Societ*
*y00000/00 $1.00 + $.25 per page
1
2 MARK MAHOWALD AND WILLIAM RICHTER
could be constructed by embedding SU(n) in the configuration space model
C R2; CP n1 for 22CP n1. We prove here that no such map ae exists.
Theorem 0.1. For n 3, there is no homotopy retraction 2SU(n) !
2CP n1 of the inclusion 2CP n1 ! 2SU(n) .
We prove slightly more, that CP n1 does not split off the second filtration
Fn;2after 2 suspensions. Earlier [MR ] we proved Theorem 0.1 for the case
n = 3, so it is not surprising that Theorem 0.1 is true. We feel, however, that
our proof for the general case is sufficiently interesting to justify our effor*
*ts. Our
previous proof used the BarrattGaneaToda relative Hopf invariant to show that
the attaching map of 2F3;2(j2 on the bottom cell in 2CP 2) was essential.
We note in particular that Theorem 0.1 does not follow from our earlier proof,
as there exists a map 2F3;2! 2CP 4 extending the inclusion CP 2,! CP 4.
Our proof is based on a relation between unstable cohomology operations due
to the first author [Ma1 ]. Let be the unstable secondary operation studied by
Mahowald and Peterson [MP , Ma2 ], which arises from the relation Sq2Sq4m =
0 on Z=4 cohomology classes of dimension 4m + 1. For us n is either 2m + 1 or
2m + 2. Mahowald's work [Ma1 ] implies the unstable relation
Sq2 = Sq4m+1 Sq2 + cup product terms: (1)
We write 2Fn;2as the cofiber of a composite, and assume that a homotopy
retraction 2Fn;2! 2CP n1 exists, which implies that our composite is null
homotopic, which allows us to construct a "threecell complex" N. We obtain a
contradiction by applying (1)to a cohomology class in N. In order to construct
the cohomology class we need the fact, proved with the Chern character in x4,
that any stable map CP n1^ CP n1!`CP n1 is zero in Z=4 cohomology.
We conjecture that the monoid kFn;khas a C2structure. We think that
with even homotopy commutativity we could extend our splitting of SU(n) [Ri ]
to Vn;k(C). In x2 we prove homotopy commutativity at the first level : Fn;1x
Fn;1! Fn;2, which we require to construct our three cell complex N.
The relation (1), Theorem 1.4 below, contains some undetermined cup prod
ucts that we have recently computed together with Peterson [MPR ]. Our
paper with Peterson gives another proof of Theorem 0.1, using the relation (1)
to construct an unstable tertiary operation which is nonzero in 2Fn;2.
We are grateful to Matthew Ando, Haynes Miller and Norihiko Minami for
teaching us about the Chern character. We would like to thank Fred Cohen,
Frank Peterson, George Whitehead and the referee made many helpful sugges
tions. Thanks to Paul Burchard for developing the latex commutative diagram
package diagram.sty, and Michael Spivak for his lamstex fonts.
1. Relations between unstable secondary operations
We briefly recall the MahowaldPeterson operations [MP , Ma2 , MR ], and
the unstable relation (1)of Mahowald [Ma1 ]. See [MPR ] for a more leisurely
SU(n) 3
account. Recall first the dual EHP sequence of Whitehead [Wh2 ] and Barcus
and Meyer [BM ]. For any nice space X, there is a homotopy fibration
X ^ X H!X oe!X; (2)
where H is the Hopf construction of the loop multiplication. See [Wh1 ,
Ch. VIII: Thm. 1.6; ex.12,6] for a proof of the following.
Theorem 1.1 (Whitehead, Barcus and Meyer). In the metastable range
i 3 conn(X), the Serre exact sequence of fibration (2)yields an exact sequence
of groups
* oe* H*
Hi+1(X ^ X) ! Hi+1(X) ! Hi(X) ! Hi(X ^ X) :
Let XP (2) be "Xprojective plane", which sits in the cofibration sequence
X ^ X H!X h!XP (2) @!2X ^ X:
Let __oe:XP (2) ! X be the canonical map extending the evaluation oe. The
following result [St, Mi1 ], which we will need in x3, gives an alternative pro*
*of
of the exact sequence of Theorem 1.1.
Theorem 1.2 (Stasheff, Milgram). The following diagram is homotopy
commutative.
XP (2) _______2Xw^@X '
shuffle
 ')
_oe X ^ X
 [
u [^ oe^oe
X ____________Xw^ X
We review the MahowaldPeterson unstable secondary operation [MP ,
Ma2 ]. Let E be the fiber of the map ff2: K (Z=4; 4m)! K (Z=4; 8m), giving us
2
a fibration sequence K (Z=4; 8m 1)! E ss!K (Z=4; 4m)ff!K (Z=4; 8m). A
nullhomotopy of the composite Sq2 . ff2: K (Z=4; 4m)! K (Z=2; 8m+ 2)defines
a secondary operation : E ! K (Z=2; 8m+ 1). The nullhomotopic composite
2
K (Z=4; 4m 1)oe!K (Z=4; 4m)ff!K (Z=4; 8m)
defines a lifting "oe:K (Z=4; 4m 1)! E. We deduce from [MP , Ma2 , MR ]
Theorem 1.3 (MahowaldPeterson). There is a choice of the unstable
operation so that the composite
K (Z=4; 4m 1)"oe!E ! K (Z=2; 8m+ 1)
is adjoint to the cohomology class ff ^ Sq2ff 2 H8m (K (Z=4; 4m 1); Z=2).
4 MARK MAHOWALD AND WILLIAM RICHTER
The cohomology class 2 H8m+1 (E; Z=2) defines an unstable cohomology
operation with indeterminacy Im Sq2. We now consider the unstable secondary
relation of the first author [Ma1 ], involving the composition of Sq2 with . We
note that since Sq2Sq2 = 0 on Z=4 classes, the composite Sq2 has no inde
terminacy. Since we are working with Z=4 cohomology, we need the Bockstein
element fi 2 H4m+1 (K (Z=4; 4m); Z=2).
Theorem 1.4. The composite of Sq2 with 2 H8m+1 (E; Z=2)factors through
the base K (Z=4; 4m), so that the following diagram
2
E __________Kw(Z=2; 8m+ 1)____Kw(Z=2;S8m+q3);
 ' '')
 ' ' '
ss ' '
 ' ' Sq4m+1Sq2ff+ffl1.Sq3ff ^ff+ffl2.fiff ^Sq2ff+ffl3.Sq2*
*fiff ^ff
 ' '
u ' '
K (Z=4; 4m)
commutes up to homotopy, where ffl1; ffl2; ffl3 are 0 or 1. Thus for any space *
*N and
any class ff 2 H4m (N; Z=4)such that ff2 = 0, we have
Sq2(ff) = Sq4m+1 Sq2ff + ffl1Sq3ff ^ ff + ffl2fiff ^ Sq2ff + ffl3Sq2fiff(^3*
*ff:)
Proof. By the theory of secondary operations, there exists a homotopy class
fl :K (Z=4; 4m)! K (Z=2; 8m+ 3)
such that Sq2 = ss*fl. In fact fl 2 1;1 ff2, where 1;1 is the secondary
operation defined by the relation Sq2 . Sq2 = 0 on Z=4 classes.
By Theorem 1.1 and Theorem 1.3, the homology suspension of fl is
oe*fl = Sq2("oe) = Sq2ff ^ Sq2ff = Sq4m+1 Sq2ff 2 H8m+2 (K (Z=4; 4m 1); Z=2):
By Theorem 1.1, for X = K (Z=4; 4m), fl then equals Sq4m+1 Sq2ff, plus possibly
some cup product terms. 
a
2. The MitchellSegal group completion model Fn;k U(n)
k
Mitchell [Mi3 ] defined a nonmultiplicative filtration {Fn;k} of SU(n) , w*
*here
Fn;1is the inclusion : CP n1 ! SU(n) . Letting Xn;k = Fn;k=Fn;k1,Wthe
splitting referred`to in the introduction is 1 SU(n) ' 1 1k=1Xn;k. The
disjoint union kFn;k has an interpretation as a group completion model of
U(n), due to G. Segal [Se1 ]. That is, there is a strictly associative monoid m*
*ul
tiplication k;l:Fn;kx Fn;l! Fn;k+.lWe show, following the work of Pressley,
Mitchell and Segal, that the groupcompletion model is homotopy commutative
at the first level. See [Pr , Mi3 , Se1, CM , Ri, MR ] for background materi*
*al.
Mitchell [Mi3 , Cor. 2.12] proves that the multiplication map : CP n1x
CP n1 ! Fn;2is a desingularization of projective algebraic varieties, that is
SU(n) 5
surjective, and injective on the complement of the subspace Gn;2;1 CP n1x
CP n1 of orthogonal lines (l; m) in Cn. Furthermore the restriction of to
Gn;2;1factors as the fiber bundle projection ae: Gn;2;1! Gn;2followed by the
natural inclusion Gn;2 Fn;2. One easily deduces from Mitchell's description
the following.
Corollary 2.1. Fn;2is homeomorphic to the identification space CP n1 x
CP n1 [aeGn;2; the multiplication map CP n1 x CP n1 ! Fn;2becomes the
quotient space projection.
Let T :CP n1 x CP n1 ! CP n1x CP n1 be the twist map sending the
pair of lines (k; l) to (l; k). We now prove the following result, whose proof *
*is is
suggested by result of Mitchell [Mi3 , Prop. 2.9] that the top Bruhat stratum of
Fn;2is the total space of the tangent bundle of CP n1, and that the composite
n1
CP n1x CP n1! Fn;2collapse!Fn;2=Gn;2~=T o # CP
is the ThomPontryagin collapse of the diagonal CP n1 CP n1x CP n1.
Theorem 2.2. The following diagram commutes up to homotopy:
CP n1 xuCP n1 ____Fn;2w
 []
T [ [
 [ [
CP n1x CP n1:
Proof. For perpendicular unit vectors u; v 2 Cn, and 2 [0; ss=2], let
[u; ; v] = (C{u}; C{cos()u + sin()v})2 CP n1x CP n1:
A tubular neighborhood of the diagonal CP n1 CP n1x CP n1can be given
by {(l; k) : l 6? k}, and the complement of this tubular neighborhood is the fl*
*ag
manifold Gn;2;1 CP n1 x CP n1. Note that [u; ; v] belongs to the tubular
neighborhood when 2 [0; ss=2) and to the complement when = ss=2. A
homotopy Ht: ' . T of the diagram is given by, for t 2 [0; 1],
ssit
Ht([u; ; v])= cos(t)u + sin(t)v; ; e ( sin(t)u + cos(t)v):
One sees easily that Ht is well defined and continuous. Note that if l ? k, then
Ht(l; k) = l k 2 Gn;2 Fn;2for all t 2 [0; 1]. 
Mitchell's inclusion : CP n1! Fn;2is given by left multiplication with the
basepoint 0 x C Cn. By Theorem 2.2 this is homotopic to right multiplication
6 MARK MAHOWALD AND WILLIAM RICHTER
with the basepoint. By the homotopy extension property we have the homotopy
commutative diagram
CP n1 =(0xC;.)!Fn;2
x x
Fold?? ??
CP n1 _ CP n1 ! CP n1 x CP n1:
By taking cofibers we have the following result, which we will need in x3.
Corollary 2.3. There is a map :CP n1 ^ CP n1 ! Xn;2 making the
diagram homotopy commutative
CP n1 ! Fn;2 ss! Xn;2
x x x
Fold?? ?? ??
CP n1_ CP n1 ! CP n1 x CP n1 ss!CP n1^ CP n1:
3. The proof of the main theorem
We prove that 2CP n1 is not a retract of 2Fn;2, for n at least 3. We
suppose that a retraction exists, and derive a contradiction from our unstable
cohomology relation applied to a 4mdimensional class ff in a "3cell complex"
containing 2CP n1, where n = 2m + 1 if n is odd, and n = 2m + 2 if n is even.
Theorem 0.1 follows immediately from
Theorem 3.1. For n 3, there is no left homotopy inverse r :2Fn;2!
2CP n1 of the inclusion 2CP n1 ! 2Fn;2.
Proof. Suppose that for n 3 we have a retraction map r :2Fn;2 !
2CP n1. Consider the derived BarrattPuppe homotopy cofibration sequence
CPn1 !Fn;2ss!Xn;2@!CP n1!Fn;2ss!Xn;2@!2CP n1!2Fn;2
The existence of the retraction r :2Fn;2! 2CP n1 implies that the bound
ary map @ :Xn;2! 2CP n1 is nullhomotopic. We will construct our "3cell
complex" by factorizing the nullhomotopic map @ through a space Y .
As is wellknown the homology of SU(n) is a polynomial algebra on the
homology classes of CP n1; this is given by the generating map : CP n1 !
SU(n) . Recall also that the homology of SU(n) is an exterior algebra, with
a generating map given by the adjoint of the map , which we will call by the
same name : CP n1 ! X. A result of Mitchell [Mi3 , Thm. 2.3] calculates
the homology of Fn;2.
Theorem 3.2 (Mitchell). The inclusion Fn;2 ! SU(n) induces in ho
mology an isomorphism from H*(Fn;2)to the monomials of weight 2; H*(Xn;2)
is the monomials of weight = 2. In homology the multiplication map : CP n1x
CP n1 ! Fn;2is the map from the weight 2 tensors to the symmetric square of
H* CP n1 .
SU(n) 7
Let Y be the homotopy cofiber of the map of x2; we have a cofibration with
boundary
CP n1^ CP n1! Xn;2h!Y @Y!CP n1 ^ CP n1:
Define M to be the cofiber of the Hopf construction of the multiplication map
: CP n1x CP n1! Fn;2, giving us a cofibration sequence
CP n1 ^ CP n1H!Fn;2! M @M!2CP n1 ^ CP n1:
Using Corollary 2.3, we have an induced map of cofibers g :M ! Y , sitting in
the following homotopy commutative braid
2CP n1^uCPn1 _____2Fn;2wH________wMu
@M AAC   A AAC
A A @Y  2 A A
A A g   A
Mu____________wY u __________2CPwn1
AAC   AAC
A A  h  A A@ (4)
A A  ss  A
CP n1 _________Fn;2wu__________Xn;2w
 AAC
H  A A
 A
CP n1^ CPn1
of cofibration sequences. Our cofiber M is defined similarly to XP (2), for
X = SU(n). Therefore we have an induced map of homotopy cofibers h: M !
XP (2), giving us the homotopy commutative diagram
CP n1 ^ CP n1 H! Fn;2 ! M @M!2CP n1 ^ CP n1
? ? ? ?
^ ?y ?y h?y ?y2^
X ^ X H! X ! XP (2) @! 2X ^ X:
_oe
Now we define 1: M ! X = SU(n) to be the composite M h!XP (2) ! X.
The last square of this diagram combined with Theorem 1.2 and the naturality
of the diagonal map allows us to calculate the diagonal map of M. The following
diagram
M _______2CPwn1@^ CP n1
 '')shuffle

 CP n1 ^ CP n1 (5)
 [
u ^ [^ ^
M ^ M __________Xw^1X1
8 MARK MAHOWALD AND WILLIAM RICHTER
is homotopy commutative. By assumption (that the retraction r exists), and
diagram (4), we have a nullhomotopy of the composite
@ :Xn;2h!Y ! 2CP n1:
Let fl :2Xn;2! M be the resulting coextension of h. Let N be the homo
topy cofiber of map fl. We then have the the following homotopy commutative
diagram.
2CP n1 ^uCP n1 ___2CPwn1u __________Nwu______3CPwn1 ^uCP n1
   [ [] 
   @M [  2
 id  [  @Y
   [ 
   [ g 
Y u _________w2CP n1u _________Mwu ___________w2Y
   (6)
  fl
h  * 
  
  
Xn;2 ____________w* ____________2Xn;2w
The top row gives us a different cofibration for our "3cell complex" N.
Now we apply the secondary relation. Let m be (n  1)=2 if n is odd, and
n=2  1 if n is even. Thus n is 2m + 1 or 2m + 2 respectively. By Theorem 4.1
of the next section, there exists a (unique) cohomology class ff 2 H4m (N; Z=4)
extending x2m1 2 H4m2 (CP n1). We call ff0 2 H4m1 (M) the restriction
of ff under the inclusion M ! N. By Theorem 3.2 and diagram (4) the
inclusion CP n1 ! M induces an isomorphism in odddimensional homology.
Therefore the class ff0 is the pullback of a class ff1 2 H4m1 (X) under the map
1: M ! X = SU(n). The MahowaldPeterson operation is defined on the
class ff, that is, ff2 = 0, since the the evendimensional cohomology of N maps
injectively to the subspace 2CP n1. We apply equation (3)of Theorem 1.4 to
the class ff 2 H4m (N; Z=4). We first show that all the cup product terms vanis*
*h.
Recall that for any cofiber B = X[fCA the cup product pairing in H*(B) factors
through a relative cup product pairing ^ : H*(X) H*(B) ! H*(B). Using
the cofibration of the top row of diagram (6), it is then enough to show that t*
*he
restrictions to 2CP n1 of the classes fiff, Sq2fiff and Sq3ff vanish. But this*
* is
clear by dimensional reasons, so the three cup products in equation (3)vanish.
By Theorem 1.3, ff0 contains Sq2ff0^ ff0. By diagram (5),
2m1 2m
Sq2ff0^ ff0 = @*Moe2 x x :
Thus Sq2(ff), which has no indeterminacy, equals the class of the composite
oe3(x2mx2m)
N ! 3CP n1 ^ CP n1 ! K (Z=2; 8m+ 3);
which is the generator of H8m+3 (N; Z=2).
SU(n) 9
But Sq4m+1 Sq2ff = Sq1Sq4m Sq2ff = 0, since H8m+2 (N; Z=2)= 0. This gives
us our contradiction, and completes the proof of Theorem 3.1. 
4. Ktheory
We prove the following result, which we used in the proof of Theorem 0.1.
Theorem 4.1. Any stable map CP n^ CP n! CP n induces the zero map in
Z=4 cohomology, for any positive integer n.
The proof uses complex Ktheory and the Chern character. See the books of
Adams [Ad ] or Dyer [Dy ] for a treatment of the Chern character ch: K0(.)!
H* (.; Q).
The Chern character is natural with respect to stable maps. Given a stable
map f 2 {CP n ^ CP n; CP n} we have the commutative diagram
*
K 0(CP n) f! K0(CP n ^ CP n)
? ?
ch?y ch?y (7)
*
H* (CPn; Q) f! H* (CPn ^ CP n; Q):
0 n n n+1 n+1
K 0(CP n)= Z[x]= xn+1 and K (CP ^ CP )= Z [x1; x2]= x1 ; x2 , where
the generator x = L  1 is the virtual bundle of degree zero, L being the Hopf
line bundle. Let t 2 H2 (CP n; Z)be the preferred generator. By definition of
the Chern character we have
Xn 1 dk fifi
ch(x) = et 1 = __ ____k(eu  1)ififitk 2 H* (CPn; Q); (8)
k=1k! du u=0
using differentiation with respect to the dummy real variable u. Write
X
f*(x) = ai;jxi^ xj 2 K0 (CPn ^ CP n); for some ai;j2:Z
1i;jn
Since the Chern character is a ring homomorphism, diagram (7)gives
t X t i t j * n n
f* e  1 = ai;je  1 e  1 2 H (CP ^ CP ; Q): (9)
1i;jn
By clearing out the denominators, we obtain the following result.
Lemma 4.2. The induced map f* :H* (CPn; Z) ! H* (CPn ^ CP n; Z)in
integral cohomology of our stable map f 2 {CP n^CP n; CP n} has the expression
X k + l
f* (tff)= Ck;l tk tl2 H* (CPn ^ CP n; Z);
k+l=ff; 1k;ln k
10 MARK MAHOWALD AND WILLIAM RICHTER
where
X dk ififi dl jfifi
Ck;l= ai;j____k(eu  1)fifi__l(eu  1)fifi: (10)
1i;jn du u=0 du u=0
We now express f* in Z=4 cohomology. Let ffi(k>1)= 1 when k > 1 and 0
when k = 0; 1. Similarly define the delta function ffi(k odd).
Lemma 4.3. The coefficient Ck;lof Lemma 4.2 satisfies the equation
Ck;l a1;1+ 2a2;1ffi(k>1)+ 2a1;2ffi(l>1)
+ 2a3;1ffi(k>1)ffi(k odd)+ 2a1;3ffi(l>1)ffi(l odd)(mod 4): *
* (11)
k u 2fifi dk u 4fifi
Proof. By the chain rule, _d_duk(e f1)iis even, and ___k(e  1)fi
k u=0 fi du u=0
0 (mod 4). By the product rule, d__duk(eu i1)fifiis then even if i 2 and
u=0
divisible by 4 if i 4. By the binomial theorem,
fi
_dk_(eu  1)2fifi= 2k  2 + ffi 2ffi (mod 4);
duk fiu=0 k;0 (k>1)
fi
_dk_(eu  1)3fifi= 3k  3 . 2k + 3  ffi 2ffi ffi (mod 4):
duk fiu=0 k;0 (k>1) (k odd)
Substituting these equations into (10)yields Lemma 4.3. 
Proof of Theorem 4.1. If ff > n, we have f* (tff)= 0, since tff= 0 2
H2ff(CPn ; Z)for ff > n. By Lemma 4.2, Ck;l= 0 if k + l > n and 1 k; l n.
Thus the right hand side of equation (11)is 0 (mod 4), for each pair (k; l)
with k + l > n and 1 k; l n. Theorem 4.1 now follows from "rowreducing"
a number of these mod 4 equations. The argument depends on the value of n.
For n = 2, consider the three equations arising from (k; l) equal to (2; 1), (1*
*; 2)
and (2; 2). If n 4 is even, let (k; l) equal (n; 1), (1; n), (n; 2), (n; 3) an*
*d (3; n).
If n 3 is odd, let (k; l) equal (n; 1), (1; n), (n; 2), (2; n) and (n; 3). One*
* shows
that
a1;1 2a1;2 2a2;1 2a1;3 2a3;1 0 (mod 4):
Substituting into equation (11)yields Ck;l 0 (mod 4) for all k; l. By Lemma 4.2
and Lemma 4.3, f* :H* (CPn; Z=4)! H* (CPn ^ CP n; Z=4)is the zero map.

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Mark Mahowald, Mathematics Department, Northwestern University, Evanston
IL 60208
Email address: mark@math.nwu.edu
William Richter, Mathematics Dept, MIT, Cambridge MA 021394307
Current address: Mathematics Department, Northwestern University, Evanston I*
*L 60208
Email address: richter@math.nwu.edu