UPPER TRIANGULAR TECHNOLOGY AND THE
ARFKERVAIRE INVARIANT
VICTOR P. SNAITH
Abstract. This paper introduces the upper triangular technology (UTT)
into classical homotopy theory. This is a new and easy to use method to
calculate the effect of the left unit map in 2adic connective Ktheory; *
*the
map which is the basis for operations in butheory. By way of application,
UTT is used to give a new, very simple proof of a conjecture of Barratt
JonesMahowald, which rephrases Ktheoretically the existence of framed
manifolds of ArfKervaire invariant one.
1.Introduction
1.1. The upper triangular technology (UTT) referred to in the title con
sists of the following two results. Let bu and bo denote the stable homotopy
spectra representing 2adically completed unitary and orthogonal connective
Ktheory respectively. Thus the smash product, bu ^ bo is a left bumodule
spectrum and so we may consider the ring of left bumodule endomorphisms
of degree zero in the stable homotopy category of spectra [2], which we shall
denote by Endleftbumod(bu ^ bo). The group of units in this ring will be
denoted by Autleftbumod(bu ^ bo), the group of homotopy classes of left
bumodule homotopy equvialences and let Aut0leftbumod(bu ^ bo) denote the
subgroup of left bumodule homotopy equivalences which induce the identity
map on H*(bu ^ bo; Z=2).
Let U1 Z2 denote the group of infinite, invertible upper triangular matrices
with entries in the 2adic integers. That is, X = (Xi,j) 2 U1 Z2 if Xi,j2 Z2
for each pair of integers 0 i, j and Xi,j= 0 if j < i and Xi,iis a 2adic uni*
*t.
Theorem 1.2. ([20] x2.1)
There is an isomorphism of the form
~=
: Aut0leftbumod(bu ^ bo) ! U1 Z2.
Let _3 : bo ! bo denote the Adams operation.
This isomorphism is defined up to inner automorphisms of U1 Z2. Given
an important automorphism in Aut0leftbumod(bu ^ bo) one is led to ask what
is its conjugacy class in U1 Z2. By far the most important such automorphism
is 1 ^ _3.
___________
Date: 26 September 2006.
1
Theorem 1.3. ([10] x1.1)
Under the isomorphism the automorphism 1 ^ _3 corresponds to an ele
ment in the conjugacy class of the matrix
0 1
1 1 0 0 0 . . .
BB C
BB CC
BB 0 9 1 0 0 . . .CC
BB CC
BB 0 0 92 1 0 . . .CC.
BB CC
BB 0 0 0 93 1 . . .CC
BB CCC
@ . A
.. ......... ......
To be precise, the proof in ([10] x1.1) shows that 1 ^ _3 can be conjugated
to this form in its first N columns for arbitarily large N. In Example 3.2 I
shall explain what this means in practice.
The main purpose of this paper is to illustrate UTT at work in an appli
cation. By way of illustration I shall use UTT to give an elementary proof of
the following result, whose terminology and proof will be given in x3.
Theorem 1.4.
Let m be a positive integer and let 8m2 : S8m2  ! RP 8m2 be a mor
phism in the 2local stable homotopy category. Then the bo einvariant of
8m2 is (34m  1)=4 (modulo 24m1) if and only if m = 2q and 8m2 is
q+2
detected by the Steenrod operation Sq2 .
Any 2adic stable homotopy class " 2 ss8m2(S0) Z2 lifts canonically (via
the KahnPriddy Theorem [3] [12]) to an element of the stable homotopy
of RP 1 and thence to 8m2. Detection by a primary mod 2 cohomology
operation can only occur if m is a power of 2 and is equivalent to " being
represented by a framed manifold of ArfKervaire invariant one ([8], [11], [18],
[21]). The existence or otherwise of framed manifolds of ArfKervaire invari
ant one is a classical unsolved problem in homotopy theory. As explained in
[13], the alternative formulation of Theorem 1.1 is equivalent to a conjecture
of [7] which was first proved in [14] by a very difficult study of Adtheory
and is proved in [21] by a straightforward but brutally longwinded use of
BPoperations. The attempted proof of [13] contains a gap caused by lack of
control of the filtration in an Adams spectral sequence. Intuitively, the UTT
proof of Theorem 1.1 is conceptually simple because it amounts to inflicting
the relevant mod 2 Adams spectral sequence with a "mixed Hodge structure";
that is, a direct sum decomposition (corresponding to the entries in U1 Z2)
compatible with the usual Adams filtration.
Incidentally Theorem 1.4 poses the existence of framed manifolds of Arf
Kervaire invariant one in a form which is similar to the formulation of the exi*
*s
tence of framed manifolds of Hopf invariant one in terms of
2
2m1 : S2m1 ! RP 2m1 (see [19]). These only exist for dimensions 1, 3, 7
and currently framed manifolds of ArfKervaire invariant one have been con
structed in dimensions 2, 6, 14, 30, 62 (see [18] and [15]  I believe that [16]
has a gap in its construction). Accordingly the following conjecture seems
reasonable:
Conjecture 1.5.
Framed manifolds of ArfKervaire invariant one exist at most in dimen
sions 2, 6, 14, 30 and 62.
The paper is organised in the following manner. The basis of Theorems 1.2
and 1.3 is the left bumodule splitting of bu ^ bo into a sum of spectra of the
form bu ^ (F4k=F4k1) for k 0. In x2 the mod 2 Adams spectral sequence
for the homotopy of bu ^ (F4k=F4k1) ^ RP 2mis described. The crucial result
is the multiplicative structure stated in Theorem 2.6 and proved in x2.8.
Combined with results from [10], Theorem 2.6 yields Proposition 2.10, which
evaluates the effect on homotopy of the maps corresponding to the super
diagonal entries of an uppertriangular matrix. These maps correspond to the
1's in the matrix for 1 ^ _3 in Theorem 1.3. In x3 Theorem 1.3 is combined
with Proposition 2.10 to transform expressions for the bo einvariant of 8m2
into a series of 2adic equations in Example 3.4 (and Propositions 3.5 and 3.6)
from which Theorem 1.4 is easily deduced in x3.8.
I am very grateful for help and advice to Huajian Yang, my postdoc at
McMaster University in 199698, to Jonathan Barker, my PhD student at
the University of Southampton in 20032006 and to Francis Clarke.
2. Ktheory examples
2.1. Let bu*(X) (resp. KU*(X)) denote the reduced, connective (resp. pe
riodic) complex Ktheory of a (based) CW complex X. When X equals the
zerodimensional sphere we have bu*(S0) ~=Z[u] and KU*(S0) ~=Z[u 1] where
deg(u) = 2. Let RP ndenote ndimensional real projective space. Let Z=t
denote a cyclic group of order t with generator w The following result is well
known.
Proposition 2.2. For 1 m 1
8
>>>0 ifj is even,
>><
buj(RP 2m) = > Z=2i if1 j = 2i  1 < 2m,
>>>
>:
Z=2m if2m < j = 2i  1.
In addition, the generators may be chosen to satisfy uv2i1= 2v2i+1for 1
i m  1.
3
Proof
The AtiyahHirzebruch spectral sequences for computing bu*(RP 2m) and
KU*(RP 2m) collapse for dimensional reasons. This implies that buj(RP 2m)
has the correct order. It also implies the injectivity of the canonical maps
hj(RP 2m) ! hj(RP 2m+2) (h = bu, KU), ~* : buj(RP 2m) ! KUj(RP 2m).
However, by the universal coefficient theorem for KU and the results of ([5]
p.107) we have KU2i1(RP 1 ) ~= Z=21 so each bu2i1(RP 2m) is cyclic. The
relation uv2i1= 2v2i+1follows from Bott periodicity and the fact that the
injection ~* commutes with multiplication by u. 2
2.3. Ext*,*B(H"*(RP 2m; Z=2), Z=2)
Let B = E(Sq1, Sq0,1) denote the exterior subalgebra of the mod 2 Steen
rod algebra A [22] generated by Sq1 and Sq0,1= [Sq1, Sq0,1]. There is an
isomorphism of bigraded algebras Ext *,*B(Z=2, Z=2) ~= Z=2[a, b], the poly
nomial algebra on a and b with bideg(a) = (1, 1), bideg(b) = (1, 3). Also
H"*(RP 2m; Z=2) = .=(x2m+1) with Sq1(xn) = nxn+1, Sq0,1(xn) =
nxn+3.
Consider the bigraded Z=2[a, b]module Ext*,*B(H"*(RP 2m; Z=2), Z=2). De
note the nonzero element of Ext0,2i1B(H"*(RP 2m; Z=2), Z=2) by "v2i1for 1
i m.
Proposition 2.4. For 1 m 1 the bigraded Ext *,*B(Z=2, Z=2)module
Ext*,*B(H"*(RP 2m; Z=2), Z=2) is equal to
Z=2[a,_b]<"v1,_"v3,_._.,."v2m1>_
.
{ai"v2i1, b"v2i1 a"v2i+1}
Proof
We prove this by induction on m. When m = 1 we have
Ext*,*B(H"*(RP 2; Z=2), Z=2) ~=Ext*,*B(E(Sq1)[1], Z=2) ~=Z=2[b]<"v1>
where X[n] denotes X with a dimension shift by n so that X[1] = X in the
notation of [2] and [20]. We have a short exact sequence of Bmodules
(x2.) * 2m * 2
0 ! "H*(RP 2m2; Z=2)[2]  ! "H (RP ; Z=2) ! "H (RP ; Z=2) ! 0.
By induction, for each nonnegative integer r the resulting long exact sequence
yields an upper bound for the sums of F2 dimensions
X1 X
dim F2(Ext s,tB(H"*(RP 2m; Z=2), Z=2)) dr
s=0 ts=r
where dr = 0 if r is even, d2i1= i for 1 i m and d2i1= m for m i.
On the other hand, if Z2 denotes the 2adic integers, the Adams spectral
sequence ([2]) for ss*(bu ^ RP 2m) Z2 = bu*(RP 2m) Z2 has the form ([10];
[20])
Es,t2= Exts,tB(H"*(RP 2m; Z=2), Z=2) =) buts(RP 2m) Z2
4
and collapses for dimensional reasons, being concentrated where t  s is odd.
Therefore Proposition 2.2 shows that dr is also a lower bound. The relations
folllow from the fact that a and b represent 2 and u respectively in the Adams
spectral sequence for bu*(S0) Z2. 2
2.5. Ext*,*B(H"*(X ^ (F4k=F4k1); Z=2), Z=2)
Consider the second loopspace of the 3sphere, 2S3. There exists a model
for 2S3 which is filtered by finite complexes ([9],[17])
[
S1 = F1 F2 F3 . . . 2S3 = Fk
k 1
and there is a stable homotopy equivalence, an example of the socalled Snaith
splitting, of the form 2S3 ' _k 1Fk=Fk1.
Consider the finite complexes F4k=F4k1with the convention that F0=F1 =
S0, the 0sphere. Let ff(n) denote the number of 1's in the dyadic expansion
of the positive integer n. The results of AdamsMargolis ([2], [4]; see also [1*
*0]
and [20]) yield Ext*,*B(Z=2, Z=2)module isomorphisms of the form
Exts,tB(H"*(X ^ (F4k=F4k1); Z=2), Z=2)
~=Ext s+2kff(k),t2kff(k)B(H"*(X; Z=2), Z=2)
for all s > 0. We shall need this isomorphism in the case where X is either a
real projective space or a sphere. The case when X is a sphere is described
extensively in [20] in connection with the left bumodule equivalence of 2local
spectra (see also ([10] x2)
^L: _k 0 bu ^ (F4k=F4k1) '! bu ^ bo.
The groups Ext s,tB(H"*(RP 2m ^ (F4k=F4k1); Z=2), Z=2), when depicted in
the traditional Adams spectral sequence manner with s along the vertical
and t  s along the horizontal axis, looks as in the figure below. The figure
is interpreted as follows: the groups are F2vector spaces which are possibly
nonzero only when s = 0 and t  s 4k + 1 or a copy of Z=2 at each
point with (s, t  s) = (v, 8k  2ff(k)  1 + 2w + 2v) with v = 1, 2, 3, . .a.nd
1 w m.
5
66


s 




















 . . .. . .


2 

 . . .. . .
1 

_______________o_____o_______________________o_____o_____o_________o
4k + 1 4k + 5 8k  2ff(k) + 1
4k + 3 _______________ 8k  2ff(k)  1 + 2m
t  s
We have the following result describes the important aspects of the mod
ule structure over the bigraded algebra Ext *,*B(Z=2, Z=2) ~= Z=2[a, b]. Let
"v2i12 Ext0,2i1B(H"*(RP 2m; Z=2), Z=2) be as in Proposition 2.4 and let ^z4k2
Ext0,4kB(H"*(F4k=F4k1; Z=2), Z=2) be element represented as a homomorphism
on mod 2 cohomology by the inclusion of the bottom cell of F4k=F4k1 ([10]
Theorem 2.12). Hence we have a (nonzero) external product "v2i1^z4k2
Ext0,4k+2i1B(H"*(RP 2m ^ (F4k=F4k1); Z=2), Z=2) for 1 i m.
6
Theorem 2.6.
In Exts,tB(H"*(RP 2m ^ (F4k=F4k1); Z=2), Z=2) we have:
(i) "vi^z4k6= 0 for each 1 i m,
(ii) b"vi^z4k= 0 = a"vi^z4kfor i = 1, . .,.2k  ff(k),
(iii) be"vi^z4k6= 0 for e 1 and i = 2k  ff(k) + 1, . .,.m.
The proof of Theorem 2.6 will be given in x2.8 after some preliminaries
concerning Bresolutions.
2.7. Resolutions
For 1 m 1, the Baction on H"*(RP 2m; Z=2) = is
given by Sq1(xi) = ixi+1 and Sq0,1(xi) = ixi+3. The beginning of a free
Bresolution
. ..d!P1 d! P0 ffl!"H*(RP 2m; Z=2) ! 0
may be given by P0 = B and P1 = B< 5, 4, 6, . .,. 2m>
where deg(oei) = i, ffl(oei) = xi and d( 2t) = Sq1oe2t1+ Sq0,1oe2t3and d( 5) =
Sq1Sq0,1oe1. By Proposition 2.4 b"v2i1is the only nonzero element in the
group Ext1,2i+2B(H"*(RP 2m; Z=2), Z=2) for 1 i m. Therefore it must be
represented by homomorphism hi 2 Hom B (P1, Z=2) given by hi( 2i+2) 1
(modulo 2) and hi( j) 0 otherwise.
Let H(k) be the graded F2vector space with basis
yk,2k, yk,2k+2, yk,2k+4, . .,.yk,2k+12, yk,2k+3, yk,2k+5, . .,.yk,2k+11
where deg(yi) = i, with the "lightning flash" Bmodule structure given by
Sq0,1yk,2k= yk,2k+3= Sq1yk,2k+2, . .,.Sq0,1yk,2k+14= yk,2k+11= Sq1yk,2k+12.
We define the start of a free Bresolution
d(k) ffl(k)
. ..! Rk,1! Rk,0! H(k) ! 0
by Rk,0 = B< k,0,2k, k,0,2k+2, . .,. k,0,2k+12> where deg ( k,0,i) = i and
Rk,1= B< k,1,2k+1, k,1,2k+3, . .,. k,1,2k+1+1> where deg( k,1,i) = i. Also ffl*
*(k)
and d(k) are given by ffl(k)( k,0,2i) = yk,2iand
d(k)( k,1,2k+1) = Sq1 k,0,2k,
d(k)( k,1,2k+3) = Sq1 k,0,2k+2+ Sq0,1 k,0,2k,
d(k)( k,1,2k+5) = Sq1 k,0,2k+4+ Sq0,1 k,0,2k+2,
.. . .
. .. ..
d(k)( k,1,2k+1+1) = Sq0,1 k,0,2k+12.
We are now ready to embark on the proof of Theorem 2.6.
7
2.8. Proof of Theorem 2.6
Part (i) follows since the exterior product of two nonzero Bhomomorphisms
to Z=2 is also nonzero and part (ii) follows because the elements in question
lie in groups which are zero, by Proposition 2.4 and the discussion of x2.5.
Part (iii) is more substantial. By naturality it suffices to work with m = 1.
Let k = 2ffl1+ 2ffl2+ . .+.2ffltwith 0 ffl1 < ffl2 < . .<.fflt so we are in*
*terested
in i 2ffl1+1+ 2ffl2+1+ . .+.2fflt+1 t + 1 and 4k = 2ffl1+2+ 2ffl2+2+ . .+.2f*
*flt+2.
From ([2] pp.3412) or ([20] p.1267) H"*(F4k=F4k1; Z=2) ~= tj=1H(fflj + 2).
We have a free Bresolution given by the tensor product
. .!. tj=1, P aj=1Rfflj+2,ajd! tj=1Rfflj+2,0! "H*(F4k=F4k1; Z=2) ! 0.
We introduce the convention that
fflj+2,1,2s+1= 0 = fflj+2,0,2sifs 2fflj+1or2fflj+2 s.
With this convention the differential has the form
d(fflj + 2)( fflj+2,1,2s+1) = Sq1 fflj+2,1,2s+ Sq0,1 fflj+2,1,2s2.
The element ^z4kis represented by the Bhomomorphism
gk 2 Hom B ( tj=1Rfflj+2,0, Z=2)
given by gk( tj=1 fflj+2,0,2fflj+2) 1 (modulo 2) and gk( tj=1 fflj+2,0,wj) 0
otherwise.
We must show that there does not exist a Bhomomorphism
f 2 Hom B (P0 ( tj=1Rfflj+2,0), Z=2)
such that f . d = (0, hi gk) in the group of Bhomomorphisms
Hom B(P0 ( tj=1, P aj=1Rfflj+2,aj) P1 ( tj=1Rfflj+2,0), Z=2)
when m = 1 and i lies in the range i 2k  ff(k). This will show that
b"vi^z4k6= 0 from which be"vi^z4k6= 0 for e 1 follows because the isomorphism
of x2.5 commutes with multiplication by b when s > 0.
I shall first give the argument to prove that f does not exist and finally I
shall explain where the i 2k  ff(k) is necessary.
In degree 4k + 2i + 2, suppose that we have the relation (0, hi gk) = f . d.
Then we shall apply f . d to all the Bbasis elements in P* ( tj=1R(fflj+2,*))
in resolution degree 1 and homological degree 4k + 2i + 2 and add the results
in two ways to get a contradiction. The basis elements in question are
X
{ 2q+2 ( j fflj+2,0,2sj)  2q + 2 + 2sj = 4k + 2i + 2 }
j
P
and (where 0 aj 1 and jaj = 1)
X
{oe2q+1 ( j fflj+2,aj,2sj+aj)  2q + 2 + 2sj + aj = 4k + 2i + 2 }
j
disregarding, of course, the ones of this list which are zero by the convention
introduced above.
8
We have
f(d( 2q+2 ( j fflj+2,0,2sj)))
= f(Sq1oe2q+1 ( j fflj+2,0,2sj)) + f(Sq0,1oe2q1 ( j fflj+2,0,2sj))
P
and (where 0 aj 1 and jaj = 1)
f(d(oe2q+1 ( j fflj+2,aj,2sj+aj)))
= f(oe2q+1 . . .Sq1 fflj+2,0,2sj . .).
+f(oe2q+1 . . .Sq0,1 fflj+2,0,2sj2 . .).
where in the last expression the Sq's appear precisely in the unique factor for
which aj was equal to one. P
Now fix a t + 1tuple (q, s1, . .,.st) such that 4k + 2i + 2 = 2q + 2 + j 2sj
and consider the sum
P t 1
f(Sq1oe2q+1 ( j fflj+2,0,2sj)) + j=1f(oe2q+1 . . .Sq fflj+2,0,2sj . .).
= Sq1(f(oe2q+1 ( j fflj+2,0,2sj)))
= 0
because Sq1 acts trivially on Z=2 for dimensional reasons. Similarly
f(Sq0,1oe2q1 ( j fflj+2,0,2sj))
Pt 0,1
+ j=1 f(oe2q+1 . . .Sq fflj+2,0,2sj . .).
= 0.
Therefore applying f . d to each of the basis elements listed above and adding
the results yields zero modulo 2.
Now consider what happens if we apply (0, hi gk) to each of the basis
elements listed above and add the results. The sum equals 1 because the map
is zero on P0* ( tj=1, P aj=1Rfflj+2,aj) and is also zero on 2q+2 ( j fflj+2*
*,0,2sj)
unless q = i and 2sj = 2fflj+2for j = 1, . .,.t.
This contradiction completes the proof of part (iii) except that it remains
to explain why we need the condition that
i 2ffl1+1+ 2ffl2+1+ . .+.2fflt+1 t + 1.
We require that i be large enough so that all the elements 2q+2 ( j fflj+2,0,*
*2sj)
and oe2q+1 ( j fflj+2,aj,2sj+aj) over which we want to sum are permissible
within homological degree 2i + 2 + 4k. However if i 2ffl1+1+ 2ffl2+1+ . .+.
2fflt+1 t + 1 then
Xt
2i + 2 + 4k 2ffl1+3 2 + 2ffl2+3 2 + . .+.2fflt+3 2 + 2 2sj
j=1
9
for all possible choices of the sj's involved in the sum. 2
2.9. The maps 'k,l
As in x2.5 let bu and bo denote the 2localised, connective unitary and or
thogonal Ktheory spectra, respectively. Consider a leftbumodule spectrum
map
' : bu ^ (F4k=F4k1) ! bu ^ (F4l=F4l1).
This map is determined up to homotopy by its restriction, via the unit of bu,
to (F4k=F4k1). By Sduality this restriction is equivalent to a map of the
form
S0 ! D(F4k=F4k1) ^ bu ^ (F4l=F4l1),
which DX denotes the Sdual of X. Maps of this form are studied by means
of the (collapsed) Adams spectral sequence (see [20] x3.1)
Es,t2= Exts,tB(H"*(D(F4k=F4k1); Z=2) "H*(F4l=F4l1; Z=2), Z=2)
=) ssts(D(F4k=F4k1) ^ (F4l=F4l1) ^ bu) Z2
where Z2 denotes the 2adic integers. It is shown in [20] that such maps ' are
trivial when l < k and form a copy of the Z2 when l k. Following [10] and
[20] we choose leftbumodule spectrum maps
'k,l: bu ^ (F4k=F4k1) ! bu ^ (F4l=F4l1)
to satisfy 'k,k= 1, 'k,l= 'l+1,l'l+2,l+1. .'.k,k1for all k  l 2 and each 't*
*+1,t
is a Z2module generator of the group of such leftbumodule maps.
Let "z4k2 ss4k(bu ^ (F4k=F4k1)) Z2 denote the element represented by
the smash product of the unit j of the buspectrum with the inclusion of the
bottom cell jk into F4k=F4k1 (see [10] x2.12)
j^jk
S0 ^ S4k ! bu ^ F4k=F4k1
and let v2i12 ss2i1(bu ^ RP1 ) Z2 = bu2i1(RP 1 ) be as in Proposition 2.2.
Then we have the exterior product
v2i1"z4k2 ss4k+2i1(bu ^ RP 1 ^ (F4k=F4k1) Z2
which is nonzero and is represented by "v2i1^z4kin the collapsed Adams spec
tral sequence whose E2term is described in x2.5.
The following formula is central to the proof in x3.8 of our main theorem.
Proposition 2.10.
For l < k, for some 2adic unit ~4k,4l,
('k,l)*(v2i1"z4k) = ~4k,4l24k4lff(k)+ff(l)v2i+4k4l1"z4l
10
Proof
Since 'k,lis a leftbmodule map we have ('k,l)*(v2i1"z4k) = v2i1('k,l)*("z*
*4k)
and, by ([10] Proposition 3.2) ('k,l)*("z4k) = ~4k,4l22k2lff(k)+ff(l)u2k2l"z*
*4lfor
some 2adic unit ~4k,4l. The result follows since, by Proposition 2.2,
v2i1~4k,4l22k2lff(k)+ff(l)u2k2l"z4l= v2i+4k4l1~4k,4l24k4lff(k)+ff(l*
*)"z4l.
2
3. Applications
3.1. The main diagram
In this section we are going to apply the results of the previous section
together with the upper triangular yoga of [20] and [10] to the following
partially commutative diagram to prove Theorem 1.4. In the diagram j is
the unit of bu, c is complexification, ~ is the bumultiplication and _3 is the
Adams operation. The homomorphism ~* is equal to (~ ^ 1)* . (1 ^ c ^ 1)*.
The diagram does not commute because the righthand oblique vertical
rectangle does not commute. However the upper and lower triangles, the
back rectangle and the front lefthand oblique vertical rectangle do commute.
11
ssj(bo ^ X) ________________ ssj(bu ^ bo ^ X)
(j ^ 1 ^ 1)*

  B
 @  B
 @  B
 @  B
 @  B
 3 @ 3  B
(_ ^ 1)* @ (1 ^ _ ^ 1)* B
 @  B
 @ (c ^ 1)*  B
 @  B
 @  B
 @  ~*B
? @  B
@ ? B
@ B
@ B
(j ^ 1 ^ 1)*
ssj(bo ^ X) ________________ @ssj(bu@^ bo ^ X) BB
@ B
Z @ B
Z Z @ B
Z @ B
Z @A B
Z Z A@ B
Z (c ^ 1) A @ B
Z * A @ B
Z Z A @ B
Z A @ BN
Z @R
Z A
Z ~*A
Z A ssj(bu ^ X)
Z
Z A
Z A 
Z A 
Z
Z A 3
Z A (_ ^ 1)*
Z A
Z 
Z A 
Z Z" A 
AAU?
ssj(bu ^ X)
Example 3.2. RP 8m1
Let K"U denote reduced 2local periodic unitary Ktheory and let bu, bo
denote the associated connective Ktheories. In this example, by way of
illustration, we shall show how to use the results of [20] and [10] to calculate
12
the map
(j ^ 1 ^ 1)* : bo8m1(RP 8m1) ! ss8m1(bu ^ bo ^ RP 8m1).
0 2t1 0 2t2 t1
From [5] we have K"U (RP ) ~= KU" (RP ) ~= Z=2 and
K"U 1(RP 2t1) ~= Z2. The KUtheory universal coefficient theorem (proved
by the method of [6]) shows that
K"U 1(RP 2t1) ~=Z2 Z=2t1, KU" 0(RP 2t1) = 0.
The Adams operation _3 gives a stable operation on 2local KUhomology
and in the book review [19] a (then) new, oneline proof of the nonexistence
of maps of Hopf invariant one based on the formula
_3(F2) = ((3t 1)=2)F1 + 3tF2, _3(F1) = F1.
The canonical map from bu2t1(RP 2t1) to K"U2t1(RP 2t1) is an isomorphism
commuting with _3 so that
bu2t1(RP 2t1) ~=Z2 Z=2t1
with the _3 acting on the generators by the formulae
_3(F2) = F2 + ut((3t 1)=2)F1, _3(F1) = F1,
where ut is an odd integer. When t = 4m the complexification map is an
isomorphism giving, in the notation of Proposition 2.2,
bo8m1(RP 8m1) ~=bu8m1(RP 8m1) ~=Z2<'8m1> Z=24m1
where the second summand is bo8m1(RP 8m2) ~=bu8m1(RP 8m2) and
_3('8m1) = '8m1 + u4m((34m  1)=2)v8m3u, _3(v8m3u) = v8m3u.
Proposition 2.4, the discussion of x2.5 and Theorem 2.6 easily imply (c.f.
[2] Lemma 17.12) that the Adams spectral sequence
Exts,tB(H"*(RP 8m2^(F4k=F4k1); Z=2), Z=2) =) buts(RP 8m2^(F4k=F4k1))
collapses and that for 1 k 2m  1 and 4m 4k  ff(k) + 1
bu8m1(RP 8m1 ^ (F4k=F4k1)) ~=bu8m1(RP 8m2 ^ (F4k=F4k1))
~=Vk Z=24m4k+ff(k)
where Vk is a finitedimensional F2vector space consisting of elements which
are detected in mod 2 cohomology (i.e. in Adams filtration zero) in the
spectral sequence. If 8m  1 8k  2ff(k) + 1 then the group is a Z=2vector
space of the form
bu8m1(RP 8m1 ^ (F4k=F4k1)) ~=Vk Z=2
entirely in Adams filtration zero and if k 2m the group is zero.
By means of the 2local equivalence ^Lof x2.5 we have a direct sum decom
position
^L*: k 0 bu*(RP 8m1 ^ (F4k=F4k1)) ~=!ss*(bu ^ bo ^ RP 8m1)
13
and by means of this identification we may write the element
(j ^ 1 ^ 1)*('8m1) 2 ss8m1(bu ^ bo ^ RP8m1 ) as a vector (w0, w1, . .,.w2m1)
with wk 2 bu8m1(RP 8m1 ^ (F4k=F4k1)). In the diagram of x3.1 with
X = RP 8m1 and j = 8m  1 the map (c ^ 1)* is an isomorphism which
sends '8m1 to itself. Hence w0 = '8m1.
According to the main theorem of [20] a leftbumodule selfequivalence of
bu ^ bo inducing the identity on mod 2 homology determines a unique conju
gacy class in the upper triangular group with entries in the 2adic integers.
According to the main theorem of [10] the conjugacy class associated to the
map 1 ^ _3 is equal to
0 1
1 1 0 0 0 . . .
BB C
BB CC
BB 0 9 1 0 0 . . .CC
BB CC
BB 0 0 92 1 0 . . .CC.
BB CC
BB 0 0 0 93 1 . . .CC
BB CCC
@ . A
.. ...... .........
In practical terms this means that, for any positive integer N, we may
choose ^Lin x2.5 so that, for all k N, 1 ^ _3 maps the wedge summand
bu ^ (F4k=F4k1) to itself by 9k times the identity map, to bu ^ (F4k4=F4k5)
by 'k,k1and to all other wedge summands bu ^ (F4t=F4t1) trivially if t N.
If we choose ^Lin this manner, taking N very much larger than 8m  1, we
have
(1 ^ _3 ^ 1)*((j ^ 1 ^ 1)*('8m1))
= (1 ^ _3 ^ 1)*('8m1, w1, w2, . .,.w2m1)
= ('8m1 + ('1,0)*(w1), 9w1 + ('2,1)*(w2), 92w2 + ('3,2)*(w3), . .,.92m1w2m1*
*).
On the other hand this element is equal to
(j ^ 1 ^ 1)*((_3 ^ 1)*('8m1))
= (j ^ 1 ^ 1)*('8m1 + u4m((34m  1)=2)v8m3u)
= (', w1, w2, . .,.w2m1) + u4m((34m  1)=2)(v8m3u, "w1, "w2, . .,."w2m1)
where (j ^ 1 ^ 1)*(v8m3u) = (v8m3u, "w1, "w2, . .,."w2m1).
14
Equating coordinates we obtain a string of equations
('1,0)*(w1) = u4m((34m  1)=2)v8m3u 2 bu8m1(RP 8m1),
(9  1)w1 + ('2,1)*(w2)
= u4m((34m  1)=2)w"12 bu8m1(bu ^ RP 8m2^ (F4=F3)),
(92  1)w2 + ('3,2)*(w3)
= u4m((34m  1)=2)w"22 bu8m1(bu ^ RP 8m2^ (F8=F7)),
.. . . .
. .. .. ..
(9k  1)wk + ('k+1,k)*(wk+1)
= u4m((34m  1)=2)w"k2 bu8m1(RP 8m2 ^ (F4k=F4k1)),
.. . . .
. .. .. ...
There is a relation between wi and "wiof the form 2wi = r8m1w"ifor all i
where r8m1 is an odd integer. For we have a cofibration
OE 8m
S8m1 ss!RP 8m1 ! RP
in which ss is the canonical projection. Also OE*('8m1) generates bu8m1(RP 8m)
so, by Proposition 2.2, OE*(2'8m1  r8m1v8m3u) = 0 for some odd integer
r8m1. Therefore 2'8m1  r8m1v8m3u originates in bu8m1(S8m1 ) and
(1 ^ j ^ 1)*(2'8m1) = (2'8m1  r8m1v8m3u, 0, 0, 0, . .)..
If we write u04mfor the 2adic unit u4mr18m1our string of equations simpli
fies to
('1,0)*(w1) = u4m((34m  1)=2)v8m3u 2 bu8m1(RP 8m1),
(9  1)w1 + ('2,1)*(w2) = 2u04m((34m  1)=2)w1,
(92  1)w2 + ('3,2)*(w3) = 2u04m((34m  1)=2)w2,
.. . . .
. .. .. ..
(9k  1)wk + ('k+1,k)*(wk+1) = 2u04m((34m  1)=2)wk,
.. . . .
. .. .. ...
15
We conclude the discussion of this example with the following result con
cerning the homomorphism ('k,k1)*
bu8m1(RP 8m2^(F4k=F4k1)) Z2 ! bu8m1(RP 8m2^(F4k4=F4k5)) Z2.
Proposition 3.3.
In the notation of Proposition 2.10 and Example 3.2
(i) ('1,0)*(v8m5z"4) = ~4,022v8m3u,
(ii) If 2 k 2m  1 and 4m 4k  ff(k) + 1 then ('k,k1)*
Vk Z=24m4k+ff(k) ! Vk1 Z=24m4k+4+ff(k1)
satisfies ('k,k1)*(v8m4k1"z4k) = ~4k,4k424ff(k)+ff(k1)v8m4k+3"z4k4whe*
*re
~4k,4k4is a 2adic unit.
In particular, ('k,k1)* is injective on Z=24m4k+ff(k) in cases
(i) and (ii).
Proof
These formulae follow from those of Proposition 2.10, concerning RP 1
together with the injectivity of the map from bu8m1(RP 8m2^(F4k=F4k1)) to
bu8m1(RP 1 ^ (F4k=F4k1)), which follows from the Adams spectral sequence
via Proposition 2.4 and x2.5. The formulae make sense because ff(k  1) =
ff(k)  1 + 2(k). 2
Example 3.4. The maps 2j
In this example we shall study stable homotopy classes of maps of the
form 2j : S2j ! RP 2j with mapping cone C( 2j) such that, on 2local
connective Ktheory
bu2j+1(C( 2j)) ~=Z2<'2j+1> Z=2j
for some 2adic unit uj
(_3 ^ 1)*('2j+1) = '2j+1+ uj((3j  1)=4)v2j1u.
In other words, the _3 einvariant (see [1]) of C( 2j) is half that of RP 2j+1in
Example 3.2, which is the mapping cone of ` in the canonical cofibre sequence
RP 2j! RP 2j+1! S2j+1 `! RP 2j.
For simplicity we shall restrict ourselves to the case when j = 4m  1. In
this case there are isomorphisms of 2local Kgroups
bo8m1(C( 8m2)) ~=bu8m1(C( 8m2)) ~=bu8m1(RP 8m1)
and for each k 1
bu8m1(C( 8m2) ^ (F4k=F4k1)) ~=bu8m1(RP 8m2 ^ (F4k=F4k1)).
16
Furthermore if (j ^1^1)*('8m1) = ('8m1, w01, w02, . .,.w02m1) the einvariant
condition yields, as in Example 3.2, a string of equations
('1,0)*(w01) = u004m((34m  1)=4)v8m3u 2 bu8m1(RP 8m1),
(9  1)w01+ ('2,1)*(w02)
= u004m((34m  1)=4)w"12 bu8m1(bu ^ RP 8m2^ (F4=F3)),
(92  1)w02+ ('3,2)*(w03)
= u004m((34m  1)=4)w"22 bu8m1(bu ^ RP 8m2^ (F8=F7)),
.. . . .
. .. .. ..
(9k  1)w0k+ ('k+1,k)*(w0k+1)
= u004m((34m  1)=4)w"k2 bu8m1(RP 8m2 ^ (F4k=F4k1)),
.. . . .
. .. .. ...
where u004mis a 2adic unit and the "wk's are the same as in Example 3.2.
Suppose that we have an element
w 2 Vi Z=24m4i+ff(i)
we shall write w ' 2N if w = (x, 2N (2t + 1)v8m4i1"z4i) for some integers t
and N < 4m  4i + ff(i).
Also we observe that in Proposition 3.3(i) or (ii) we may choose Vk so that
('k,k1)*(Vk) = 0. Choosing Vk in this manner will simplify our subsequent
calculations.
Proposition 3.5.
Let m = (2p + 1)2q with p 1. In the notation of Examples 3.2 and 3.4
(i) For 2 k 2q+1, 'k,k1(wk) ' 24+q and 'k,k1(w0k) ' 23+q.
(ii) w2q+1' 1.
(iii) Under these hypotheses 8m2 does not exist.
Proof
First we observe that k 2q+1 implies that 4m = p2q+3+2q+2 4kff(k)+
1 so that we may apply Proposition 3.3. Therefore the relations ('1,0)*(w01) =
u004m((34m  1)=4)v8m3u and 2((92m  1)=4) = 3 + 2(2m)  2 = 2 + q
implies w01' 2q and similarly w1 ' 21+q (once we observe that the condition
1 + q < 4m  4 + ff(1) is fulfilled). The relation
(9  1)w1 + ('2,1)*(w2) = u4m((34m  1)=2)w"12 V1 Z=24m3
17
implies that ('2,1)*(w2) ' 24+q and similarly ('2,1)*(w02) ' 23+q, which starts
an induction on k.
Suppose 2 k < 2q+1 that
'k,k1(wk) ' 24+q 2 Z=24m4k+4+ff(k1).
Therefore wk 2 Z=24m4k+ff(k)satisfies wk ' 2ff(k)ff(k1)+q= 2q+1 2(k).
Then, since 2(9k  1) = 3 + 2(k),
(9k  1)wk + 'k+1,k(wk+1) = (92m  1)wk
implies that 'k+1,k(wk+1) ' 24+q, as required. Similarly 'k+1,k(w0k+1) ' 23+q.
Since ('2q+1,2q+11)*(v8m2q+31"z2q+3) = ~2q+3,2q+342q+4v8m2q+3+3"z2q+34we
see that w2q+1' 1 and that w02q+1cannot exist. 2
Proposition 3.6.
Let m = 2q. In the notation of Examples 3.2 and 3.4
(i) For 2 k 2q, ('k,k1)*(wk) ' 24+q and ('k,k1)*(w0k) ' 23+q.
(ii) In Z=2 V2q, w02q' 1.
Proof
This time we observe that k 2q implies 4m = 2q+2 4k  ff(k) + 1 so
that Proposition 3.3 applies and therefore part (i) follows as in Proposition
3.5. For part (ii) we have
('2q,2q1)*(v8m2q+21"z2q+2) = ~2q+2,2q+2423+qv8m2q+2+3"z2q+24
and the result follows. 2
Remark 3.7. (i) In Proposition 3.6 ('2q,2q1)* has the form
Z=2 V2q ! Z=2q+4 V2q1
so that the first component of ('2q,2q1)*(w2q) is zero and so is that of w2q.
(ii) When m = 2q we have shown that w0k' 2q 2(k)for 1 k 2q.
3.8. Proof of Theorem 1.4
Let m = 2q then we have '8m1 2 bo8m1(C( 8m2)) giving a stable homo
q+31
topy class '8m1 : S2  ! bo ^ C( 8m2). Let ' : bu ! HZ=2 be the
canonical cohomology class. Then, if h8m1 2 H2q+31(bo ^ C( 8m2); Z=2)
is the mod 2 Hurewicz image of '8m1, it is represented by either of the com
positions
q+31'8m1 (j^1^1) ('^1^1)
S2 ! bo^C( 8m2) ! bu^bo^C( 8m2) ! HZ=2^bo^C( 8m2)
or
q+31'8m1 0 ("j^1^1)
S2 ! S ^ bo ^ C( 8m2) ! HZ=2 ^ bo ^ C( 8m2)
where "jis the unit for HZ=2.
We have an isomorphism of Z=2vector spaces
H*(bo; Z=2) ~= k 0 H*(F4k=F4k1; Z=2)
18
and 8
>>>Z=2<'8m1> if j = 8m  1,
>><
Hj(C( 8m2); Z=2) ~=> Z=2 if 1 j = 8m  2,
>>>
>:
0 otherwise.
If the einvariant is correct then we have shown that
h8m1 2 H*(bo; Z=2) H*(C( 8m2); Z=2)
has the form
2q+22X
h8m1 = "z0 '2q+31+ "z2q+2 v2q+21+ x8m1j vj
j=1
with x8m1j 2 H*(F2q+2+t=F2q+21+t; Z=2) with t 0.
q+2 q+2
If X 2 A2 is an element of the mod 2 Steenrod algebra of degree 2 we
write X* for the dual homomorphism on mod 2 homology, which decreases
dimensions by 2q+2. Since H*(bo; Z=2) is a cyclic Amodule generated by 1
in dimension zero there exists an X such that X*("z2q+2) = "z0. In fact, since
the Hurewicz image of the bottom cell in H4k(F4k=F4k1; Z=2) equals ,4k1,
unravelling the homology isomorphism between H*(bu; Z=2) H*(bo; Z=2)
and H*(bu; Z=2) H*(^l 0 F4l=F4l1; Z=2) described in [20] one can show
that Sq4k*("z4k) = "z0.
q+2
Since h8m1 is stably spherical we have 0 = Sq2* (h8m1) so that
q+2
"z0 Sq2* ('2q+31)
q+2
= "z0 v2q+21+ "z2q+2 Sq2* (v2q+21)
P2q+21 j 2q+2j P 2q+22 2q+2
+ j=1 Sq*("z2q+2) Sq* (v2q+21) + j=1 x8m1j Sq* (vj)
P2q+21 P 2q+22 a 2q+2a
+ a=1 j=1 Sq*(x8m1j) Sq* (vj)
q+2
which implies, comparing coefficients of "z0, that Sq2* ('2q+31) = v2q+21so
q+2
that '8m1 is detected by Sq2 on its mapping cone, as required. That is:
~= 2q+31
H2q+31(C( 2q+32); Z=2) ! H2q+31(S ; Z=2),
q+2 ~=
Sq2* : H2q+31(C( 2q+32); Z=2) ! H2q+21(C(`2q+32); Z=2),
q+32 ~=
H2q+21( RP 2 ; Z=2) ! H2q+21(C( 2q+32); Z=2).
q+2 0
Conversely, if this detection by Sq2 is correct then w2q' 1 and therefore
the einvariant is right.
When m is not a power of 2, Proposition 3.5(iii) shows that 8m2 with this
einvariant cannot exist. For mod 2 cohomology one can appeal to a theorem
19
of Browder [8] or alternatively wellknown formulae for the Amodule action
on the mod 2 cohomology of real projective space show that 8m2 cannot
be detected by a primary operation on H*(C( 8m2); Z=2). 2
Department of Pure Mathematics, University of Sheffield,
Sheffield S3 7RH, England.
v.snaith@sheffield.ac.uk
References
[1]J.F. Adams: J(X) IIV; Topology 2 (1963) 181195, Topology 3 (1964) 1371*
*71,
Topology 3 (1964) 193222, Topology 5 (1966) 2171.
[2]J.F. Adams: Stable Homotopy and Generalised Homology; University of Chica*
*go
Press (1974).
[3]J.F. Adams: The KahnPriddy Theorem; Proc. Camb. Phil. Soc. (1973) 4555.
[4]J.F. Adams and H.R. Margolis: Modules over the Steenrod algebra; Topology *
*10
(1971) 271282.
[5]M.F. Atiyah: KTheory; Benjamin, New York (1967).
[6]M.F. Atiyah: Vector bundles and the K"unneth formula; Topology 1 (1962) 24*
*5248.
[7]M.G. Barratt, J.D.S. Jones and M. Mahowald: The Kervaire invariant and the*
* Hopf
invariant; Proc. Conf. Algebraic Topology Seattle Lecture Notes in Math. #1*
*286
Springer Verlag (1987).
[8]W. Browder: The Kervaire invariant of framed manifolds and its generalizat*
*ions;
Annals of Math. (2) 90 (1969) 157186.
[9]E.H. Brown and F.P. Peterson: On the stable decomposition of 2Sr+2; Trans*
*. Amer.
Math. Soc. 243 (1978) 287298.
[10]J. Barker and V.P. Snaith: _3 as an upper triangular matrix; Ktheory 36 (*
*2005)
91114.
[11]J.D.S. Jones and E.G. Rees: Kervaire's invariant for framed manifolds; Alg*
*ebraic and
geometric topology, Proc. Symp. Pure Math XXXII (Part 2) Amer. Math. Soc. (*
*1978)
111117.
[12]D. S. Kahn and S. B. Priddy: The transfer and stable homotopy theory; Proc*
*. Camb.
Phil. Soc. 83 (1978) 103112 (See also: Applications of the transfer to sta*
*ble homotopy
theory; Bull. A.M.Soc. 741 (1972) 981987).
[13]J. Klippenstein and V.P. Snaith: A conjecture of BarrattJonesMahowald co*
*ncerning
framed manifolds having Kervaire invariant one; Topology (4) 27 (1988) 387*
*392.
[14]K. Knapp: Im(J)theory and the Kervaire invariant; Math. Zeit. 226 (1997) *
*103125.
[15]S.O. Kochman: Stable Homotopy Groups of Spheres; Lecture Notes in Math. #1*
*423,
SpringerVerlag (1990).
[16]R.J. Milgram: Symmetries and operations in homotopy theory; Proc. Symp. Pu*
*re
Math XXII Amer. Math. Soc. (1971) 203210.
[17]V.P. Snaith: A stable decomposition of nSnX; J. London Math. Soc. 2 (197*
*4)
577583.
[18]V.P. Snaith and J. Tornehave, On ss*(BO) and the Arf invariant of framed m*
*anifolds;
Contemporary Mathematics 12 (1989) 299313.
[19]V.P. Snaith: Review of Some applications of topological Ktheory by N. Mah*
*ammed,
R. Piccinini and U. Suter; Bull.A.M.Soc. v.8 (1) 117120 (1983).
[20]V.P. Snaith: The upper triangular group and operations in algebraic Ktheo*
*ry; Topol
ogy 41 (2002) 12591275.
[21]V.P. Snaith: Hurewicz images in BP and the ArfKervaire invariant; Glasgow*
* J.Math.
44 (2002) 927.
20
[22]N.E. Steenrod: Cohomology operations; Annals Math. Studies #50 (written and
revised by D.B.A. Epstein) Princeton Univ. Press (1962).
21