An alternative approach to homotopy
operations
Marcel B"okstedt & Iver Ottosen
May 17, 2004
Abstract
We give a particular choice of the higher Eilenberg-MacLane maps
by a recursive formula. This choice leads to a simple description of
the homotopy operations for simplicial Z=2-algebras.
1 Introduction
This paper is about the ring of homotopy groups of a simplicial ring. This
ring of homotopy classes has a huge amount of additional structure. The
theory is best worked out for algebras over F2, and we will restrict ourselves
to this case. xx2-3 in [3] contain a good survey with references to the original
articles. We just recall the points which are most important to us.
The main observation is that the square of every element in positive
degree is zero. Analyzing this fact gives rise to a divided power structure on
the ring of homotopy groups. There is a refinement of this, which constructs
a sequence of homotopy operations ffii. These are defined by Dwyer in [2] and
also by Bousfield.
The homotopy of a simplicial algebra Ro is isomorphic to the homology of
the associated chain complex C*(R). We can represent an element in ssn(R)
by a cyclePin the the chain complex, that is by a class z 2 Rn = Cn(R), such
that 0 i n di(z) = 0.
The problem which this article wants to solve is the following: Suppose
that ffii is defined on ssn(Ro). Can we give an explicit formula for an element
in Rn+i that represents the element ffii(z)?
The reason that we care is that we are interested in explicit applications,
like in [1] where we compute these operations.
We will focus on Dwyer's approach. In order to define the operations, he
considers a sequence of natural transformations, defined for pairs of simplicial
1
F2-vector spaces Vo and Wo as follows:
Dk :[C(V ) C(W )]m ! [C(V W )]m-k .
The transformations satisfy recursive conditions, which we write down in the
beginning of section 2.
These maps are considered to be higher analogues of the Eilenberg-Mac-
Lane map. Dwyer proves an existence and uniqueness result for these maps,
but he does not give an explicit formula. He then uses them to define homo-
topy operations. We follow the reformulation of [3] at this point.
Let z 2 Rn and define i(z) (for 1 i n) as the element
i(z) = ~Dn-i(z z) + ~Dn-i-1(z @z).
Here ~ is multiplication in the simplicial algebra. If z is a cycle, and 2
i n, one sees that i(z) is also a cycle, and that the formula defines an
operation ffii: ssn(Ro) ! ssn+i(Ro).
We are going to give simpler formulas for Di which satisfy the defining
recursive relations. The derivation of these is where the hard work of this
article is done. When we have obtained the formulas, we can plug them into
the definition of i, and obtain an explicit formula for ffii(z). We now explain
the formula we obtain in this fashion.
Write N*R for the normalized chain complex with NqR = \1 i qker(di)
and differential d0. Let ZqR NqR denote the cycles i.e. the elements
z 2 Rq with diz = 0 for 0 i q.
To make the final formulas appear simpler, we assume that z is a normal-
ized chain. This is no restriction, since the associated chain complex con-
sidered above contains the normalized chain complex as a quasi-isomorphic
subcomplex.
Definition 1.1. For integers q, i with 1 i q we define U(q, i) to be the
set of pairs (~, ) of ordered sequences ~1 < . . .< ~i, 1 < . . .< i with
disjoint union
{~1, . .,.~i} t { 1, . .,. i} = {q - i, q - i + 1, . .,.q + i - 1}.
Let V (q, i) U(q, i) be the subset with ~1 = q - i.
Definition 1.2. For a cycle z 2 ZqR we define ffii(z) 2 Rq+i by
X
ffii(z) = s i. .s. 1(z)s~i. .s.~1(z).
(~, )2V (q,i)
2
Note the close relationship to the Eilenberg-MacLane map D:
X
~D(z z) = s i. .s. 1(z)s~i. .s.~1(z).
(~, )2U(q,q)
Theorem 1.3. When 2 i q the formula in Definition 1.2 defines a map
ffii : ZqR ! Zq+iR. The induced map on homotopy ffii : ssqR ! ssq+iR is the
Bousfield-Dwyer homotopy operation.
Proof. Let z 2 NqR be a cycle. It is shown in Lemma 3.1 that ffii(z) is a cycle
in Nq+iR.
We use Definition 2.6 as our choice of higher Eilenberg-MacLane maps.
The suspension operator S in the definition increases the simplicial degree by
one. It preserves composition of simplicial maps and S(dj) = dj+1, S(sj) =
sj+1, S(id) = id.
We have that i(z) = ~Dq-i(z z). The statement we must show is that
i(z) = ~Sq-i(D0)(z z). So it suffices to see that Dk(z z) = Sk(D0)(z z),
where q - i = k.
For 0 j k - 1 we have
(
Sj(Dk-j-1(d0 id)) , k - j even
Sj(Dk-j) = Sj+1(Dk-j-1) +
Sj(Dk-j-1(id d0)) , k - j odd
(
Sj(Dk-j-1)(dj id) , k - j even
= Sj+1(Dk-j-1) +
Sj(Dk-j-1)(id dj) , k - j odd.
Thus Sj(Dk-j)(z z) = Sj+1(Dk-j-1)(z z). We iterate this result and find
__
that Dk(z z) = S0(Dk)(z z) = Sk(D0)(z z). |__|
2 Higher Eilenberg-MacLane maps.
Let Vo, Wo be simplicial F2-vector spaces. The Eilenberg-MacLane map D
can be considered as a set of linear maps
Di,j:Vi Wj ! Vi+j Wi+j. (1)
There is an explicit formula for these maps
X
Di,j= s j. .s. 1 s~i. .s.~1,
where the sum is indexed by (i, j)-shuffles. An (i, j)-shuffle consists of two
increasing sequences ~ = (~1, ~2, . .~.i) and = ( 1, 2, . .,. j) such that
3
each integer from {0, 1, . .,.i + j - 1} occurs exactly once as either a ~r or a
s. There is an excellent discussion of this map in [4], chap. VIII x8.
We will also consider the map OEk with (OEk)i,j:Vi Wj ! Vi+j-k Wi+j-k,
defined to be the zero map, unless i = j = k. In this exceptional case, the
map is the identity.
Let Ti,j:Vi Wj ! Wj Vi be the map permuting the factors. According
to [2] there is a sequence of maps Dk, k = 0, 1, 2, . .w.ith
Dki,j:Vi Wj ! Vi+j-k Wi+j-k,
only defined under the conditions that
0 2k i + j, (2)
which satisfy that
D0 + T D0T = D + OE0, (3)
Dk + T DkT = Dk-1@ + @Dk-1 + OEk. (4)
There are many choices for the maps Dk, but Dwyer proves that the
choices are equivalent up to homotopy (in a strong sense).
Before we give our definition of the maps Dk, we introduce some language,
for the purpose of avoiding a nightmare of indices. We are going to write
down various formal sums of products of the simplicial generators dr, sr.
Such an expression sometimes but not always defines a map Vi ! Vj for
all simplicial vector spaces Vo. For instance, consider the simplicial relation
drsr = id. The right hand side of this relation defines a map (the identity)
Vi ! Vi for all i. The left hand side does not define a map on Vi if r > i.
But if r i, both sides of the equation defines maps, and in this case the
relation says that the two maps Vi ! Vi induced by the two sides agree.
We are going to consider tensor products of pairs of simplicial groups
Vo Wo. For pairs of integers (i, j), (k, l), we will write down natural trans-
formations Vi Wj ! Vk Wl and relations between such. We will specify
them as sums of formal sequences of generators dr, sr. Every time we do
this, we have to keep track of whether the formal sequences do indeed define
natural transformations of the groups we write down.
We start by introducing a "suspension operator" S.
Definition 2.1. Let denote the simplicial category. Define the functor
S0 : ! by S0([n]) = [n + 1] on objects, and
(
ff(i - 1) + 1 ifi 1
(S0ff)(i) =
0 ifi = 0.
4
on morphisms. Let S = (S0)op : op ! op be the corresponding functor on
the opposite category.
We can picture the suspension operator as follows:
3 == 3
==
==
OEOE=
2 == 2 2 ____//_2@@
==
==
=OEOE
1 ____//_1@@ 1 1
0 0 0 ____//_0
ff :[2] ! [2] S0ff :[3] ! [3]
Note that d0 defines a natural transformation d0 : Id ! S0. So we have a
natural transformation d0 : S ! Id.
For a simplicial vector space V : op ! {F2-vector spaces}, we define a
new simplicial vector space S(V )o as the functor V O S. There is a natural
transformation of simplicial vector spaces P :S(V )o ! Vo, given degreewise
as P :S(V )i = Vi+1 d0!Vi.
Consider any natural transformation defined on the category of simplicial
vector spaces of the form
`V :Vi ! Vj,
for instance, this could be a map induced by a morphism [j] ! [i] in , or
a linear combination of such maps.
Definition 2.2. The suspension of ` is the natural transformation
`SV
(S`)V :Vi+1 = S(V )i -! S(V )j = Vj+1.
We obtain a commutative diagram:
Vi+1 _______S(V )i -- P-! Vi
? ? ?
(S`)V?y `SV?y `V?y
Vj+1 _______S(V )j -- P-! Vj
The composite of both rows is d0, so it follows from this diagram and the
naturality of ` that we have a relation
d0(S`) = `d0: S(V )i ! Vj. (5)
5
We can also define a suspension on formal products of the simplicial gen-
erators di, siby S(di) = di+1 and S(si) = si+1. This suspension is compatible
with the (formal) simplicial relations. In this way, we can formally define S
on strings of simplicial generators. by adding 1 to the index of every occurring
generator si or dj.
Lemma 2.3. Let Vo be a simplicial F2-vector space.
o Let ff be a formal products of generators dr, sr. Then ff defines a natur*
*al
transformation ff*: Vi ! Vj if and only if the corresponding natural
transformation S(ff)* : Vi+1 ! Vj+1 is also defined.
o Suspension commutes with passage to natural transformation, if the
natural transformation is defined. That is
S(ff*) = (Sff)*.
P
o If a formalPsum ff*: Vi ! Vj is defined and equals the zero map for
i, then Sff*: Vi+1 ! Vj+1 is defined, and also equals the zero map.
Proof. By induction on the number of factors in a product of generators, it
is enough to check the first statement for the case of a generator sj or dj.
For example, (sj)* is defined on Vk if and only if k j. But it is also true
that (S(sj))* = (sj+1)* is defined on Vk+1 if and only if k + 1 j + 1. The
case dj is treated in exactly the same way.
To prove the second statement, by induction it is enough to consider the
case of a simplicial generator. That is, we have to check that S(si) = si+1
and that S(di) = di+1. But this follows directly from the definition of S.
The third statement follows from the second, since
X X
( Sff)* = S( ff*) = S0 = 0
__
|__|
All we have said can be generalized to tensor products of two simplicial
vector spaces.
Lemma 2.4. Let Vo, Wo be simplicial F2-vector spaces. We can define the
suspension of a natural transformation ` :Vi Wj ! Vk Wl as a natural
transformation S` :Vi+1 Wj+1 ! Vk+1 Wl+1.
o Let ff, fi be formal products of generators di, si. If ff* fi* is define*
*d for
the index (i, j) then S(ff* fi*) is defined for (i + 1, j + 1).
6
o Suspension commutes with passage to natural transformation.
P
o If a formalPsum ff* fi* is defined and equals the zero map for (i, j),
then S( ff* fi*) equals the zero map for (i + 1, j + 1).
We leave the proof to the dedicated reader.
Note that the map (OEk)i,jconsidered above is a natural transformation
which is not defined as a formal combination of the simplicial generators
di, si. But the suspension on it is still defined, and actually
(OEk+1)i+1,j+1= S((OEk)i,j) :Vi+1 Wj+1 ! Vi+j+1-k Wi+j+1-k.
However, OEk is an example of a map with the following property.
Definition 2.5. An EM type transformation F consists of the following
data.
1. An index function IF which to each pair of integers (i, j) associates a
pair of integers (k, l) = IF (i, j).
2. For each (i, j), we have a natural transformation on the category of
simplicial vector spaces Vi Wj ! Vk Wl. Here, we conventionally
define Vi Wj = 0 if either i < 0 or j < 0.
For instance, OEk defines such an EM type transformation. The index
function for OEk is IOEk(i, j) = (i + j - k, i + j - k).
The main example of such a map is the Eilenberg-MacLane map D. It
is a collection of maps Di,j:Vi Wj ! Vi+j Wi+j. The index function is
ID (i, j) = (i + j, i + j).
We can always compose two EM type transformations F, G. We can add
them if the index function of F agrees with the index function of G. If F
and G does not have the same index function, their sum is not defined. This
is the price we pay for keeping easy control of the indices involved.
We can suspend an EM type transformation. We define ISF (i+1, j +1) =
(1, 1) + IF (i, j) and (SF )i+1,j+1= S(Fi,j). This is to be interpreted so that
if either i < 0 or j < 0, then (SF )i+1,j+1= 0.
We can also twist an EM type transformation by defining T (F )i,j=
T Fj,iT . Suspension and twisting preserve sum and composition, and they
commute.
Here are some examples of EM type transformations, and relations be-
tween them. We insist that the relations are valid as relations between natu-
ral transformations with source Vi Wj for all pairs of integers (i, j). When
checking the formulas below, the main worry is to keep track of cases like
V0 Wj and Vi W0
7
The sequences d0 id, s0 idare EM type transformations, with index
functions (i, j) 7! (i-1, j) respectively (i, j) 7! (i+1, j). The thing to noti*
*ce
is that d0 id:V0 Wj ! V-1 Wj, makes sense (and equals the zero map)
since we define Vi = 0 for i = -1, and Wj = 0 for j = -1.
The simplicial relation give relations between these functors.
(d0 id)(s0 id) = id id (6)
is true as a relation between EM type transformations.
Another type of example is @P id, with index function (i, j) 7! (i - 1, j).
It is defined by (@ id)i,j= 0 r idr id. Similarly, we can define id @,
with index function (i, j) 7! (i, j - 1).
When we apply suspension, we have to be careful. Here is an example of
this: unless i > 0, j = 0, we have that S(@ id)i,j+ (d0 id)i,j= (@ id)i,j.
If we post-compose with an EM type transformation which vanishes on all
groups Vi W0, we get a genuine relation. For instance we have for any EM
type transformation F that:
SF S(@ id) + SF (d0 id) = SF (@ id). (7)
We can also use simplicial operations simultaneously in both factors. Us-
ing the index function (i, j) 7! (i - 1, j - 1), we put
X
ffi = dr dr: Vi Wj ! Vi-1 Wj-1 .
0 r min(i,j)
d0 d0 is another EM type transformation with the same index function, and
S(ffi) = ffi + d0 d0. (8)
Similarly (using the simplicial relations d0s0 = id = d1s0, dis0 = s0di-1
for i 2) we get that
(id @)(id s0) = (id s0)(id @) + (id s0d0). (9)
In the same way
(id @)(id d0) = (id d0)(id @) + (id d0d0). (10)
For any EM type transformation F , we have (because of (5))
(d0 d0)(SF ) = F (d0 d0). (11)
We now give our definition of the higher Eilenberg-MacLane maps.
8
Definition 2.6. Dk is the EM type transformation with index function
IDk (i, j) = (i + j - k, i + j - k), and defined by D0 = S(D)(id s0) and
inductively for k 1 by the formula
(
Dk-1(d0 id) if k is even
Dk = S(Dk-1) + (12)
Dk-1(id d0) if k is odd.
The sum on the right hand side of (12) is defined because the EM type
transformations S(Dk-1) and Dk-1(id d0) have the same index function.
This equation (12) is an equation of EM type transformations. If we
write it out, it means that we have natural transformations Dki,j:Vi Wj !
Vi+j-k Wi+j-k given inductively as
8
>>S(Dk-1 ) + Dk-1 (d0 id) if k is even, i, j 1,
>> i-1,j-1 i-1,j
>> k-1 k-1
< S(Di-1,j-1) + Di,j-1(id d0) if k is odd, i, j 1,
Dki,j= Dk-1i-1,j(d0 id) if k is even, i 1, j = 0,
>>
>>Dk-1 (id d ) if k is odd, i = 0, j 1,
>> i,j-1 0
: 0 else.
It is obvious that our D0 satisfies (3). We have to prove that our choices
of Dk, k 1 satisfy (4). Our strategy for proving this is first to prove rela-
tions between EM type transformations, that is a relation between natural
transformations valid for all pair of integers (i, j).
Definition 2.7. Let Ak be the EM type transformation with index function
IAk(i, j) = (i + j - k, i + j - k), defined by
A0 = D0 + T D0 + D,
Ak = Dk + T Dk + ffiDk-1 + Dk-1(@ id) + Dk-1(id @).
We have now defined all EM type transformations that we will need. The
main work of this article is done in proving the following recursion relation
for Ak:
Lemma 2.8. For k 1 we have that
(
Ak-1(id d0) if k is even
Ak = S(Ak-1) +
Ak-1(d0 id) if k is odd.
9
Proof. The proof is by direct computation. It is divided into three cases:
k = 1; k 2 and k even; k 3 and k odd. The method used in the three
cases is the same. We will write down about a dozen relations, and then add
all of them to give the desired recursion formula.
The case k = 1. We want to prove that
D1 + T D1 + D0(@ id) + D0(id @) + ffiD0
(*)
+ S(D0 + T D0 + D) + (D0 + T D0 + D)(d0 id) = 0.
We have that D0 + (SD)(id s0) = 0. From this follows immediately the
relations
ffiD0 + ffi(SD)(id s0) = 0, (13)
D0(id d0) + SD(id s0d0) = 0, (14)
D0(@ id) + SD(@ s0) = 0, (15)
D0(id @) + SD(id s0)(id @) = 0, (16)
and
D0(d0 id) + SD(d0 s0) = 0. (17)
There is also the defining relation for D1, and we can apply the twist to
it. This gives
D1 + SD0 + D0(id d0) = 0, (18)
and
T D1 + ST D0 + T D0(d0 id) = 0. (19)
A fundamental property of the Eilenberg-MacLane map is that it is a
chain map. In our notation, this says that ffiD + D(@ id) + D(id @) = 0.
Suspending this, using 2.4, right multiplying with id s0 we get
S(ffiD)(id s0) + S(D(@ id))(id s0) + S(D(id @))(id s0) = 0.(20)
Now use that S is compatible with composition so that S(D(@ id)) =
(SD)S(@ id). The identitity (7) provides
S(D(@ id))(id s0) + SD(@ s0) + SD(d0 s0) = 0. (21)
Similarly, using twisted versions of (7) and (6) gives
S(D(id @))(id s0) + SD(id @)(id s0) + SD = 0. (22)
The relation (9) gives
SD(id @)(id s0) + SD(id s0)(id @) + SD(id s0d0) = 0. (23)
10
The relation (8) also gives a relation. We apply (11) for F = D to it, and
obtain:
ffi(SD)(id s0) + S(ffiD)(id s0) + D(d0 id) = 0. (24)
Adding the numbered relations (13)-(24) gives (*), and finishes the proof of
case k = 1.
The case k 2 and k even. The relation we want to prove is the following:
Dk + T Dk + Dk-1(@ id) + Dk-1(id @) + ffiDk-1
+ S(Dk-1 + T Dk-1 + Dk-2(@ id) + Dk-2(id @) + ffiDk-2)
+ (Dk-1 + T Dk-1 + Dk-2(@ id) + Dk-2(id @) + ffiDk-2)(id d0) = 0.
(**)
The definition says that
Dk + SDk-1 + Dk-1(d0 id) = 0. (25)
Applying the twist to this, we get
T Dk + ST Dk-1 + T Dk-1(id d0) = 0. (26)
Since k - 1 is odd, the definition says that Dk-1 = SDk-2 + Dk-2(id d0).
This gives us a sequence of relations:
Dk-1(d0 id) + SDk-2(d0 id) + Dk-2(d0 d0) = 0, (27)
Dk-1(id d0) + SDk-2(id d0) + Dk-2(id d0d0) = 0, (28)
ffiDk-1 + ffiDk-2(id d0) + ffiSDk-2 = 0, (29)
Dk-1(@ id) + Dk-2(@ id)(id d0) + SDk-2(@ id) = 0, (30)
and
Dk-1(id @) + SDk-2(id @) + Dk-2(id d0@) = 0. (31)
(10) gives us
Dk-2(id @)(id d0) + Dk-2(id d0@) + Dk-2(id d0d0) = 0. (32)
Using (11) for F = Dk-2 we get
(d0 d0)SDk-2 + Dk-2(d0 d0) = 0. (33)
Finally, the following equations follow from (7) and its twisted version since
in this case k 3 and from (8), using the compatibility of suspension with
products.
S(Dk-2(@ id)) + SDk-2(@ id) + SDk-2(d0 id), (34)
11
S(Dk-2(id @)) + SDk-2(id @) + SDk-2(id d0), (35)
and
S(ffiDk-2) + ffiSDk-2 + (d0 d0)SDk-2. (36)
If we add all numbered equations (25)-(36), we get formula (**). This finishes
the case k 2, k even.
Case k 3 and k odd. In this case, we want to prove that
Dk + T Dk + Dk-1(@ id) + Dk-1(id @) + ffiDk-1
+ S(Dk-1 + T Dk-1 + Dk-2(@ id) + Dk-2(id @) + ffiDk-2)
+ (Dk-1 + T Dk-1 + Dk-2(@ id) + Dk-2(id @) + ffiDk-2)(d0 id) = 0.
(***)
The definition and its twist give
Dk + S(Dk-1) + Dk-1(id d0) = 0, (37)
T Dk + ST (Dk-1) + T Dk-1(d0 id) = 0. (38)
Since Dk-1 = S(Dk-2) + Dk-2(d0 id), we have relations:
Dk-1(id d0) + SDk-2(id d0) + Dk-2(d0 d0) = 0, (39)
Dk-1(d0 id) + SDk-2(d0 id) + Dk-2(d0d0 id) = 0, (40)
ffiDk-1 + ffiDk-2(d0 id) + ffiSDk-2 = 0, (41)
Dk-1(@ id) + Dk-2(d0@ id) + SDk-2(@ id) = 0, (42)
and
Dk-1(id @) + Dk-2(id @)(d0 id) + SDk-2(id @) = 0. (43)
The twisted version of (10) gives us
Dk-2(@ id)(d0 id) + Dk-2(@d0 id) + Dk-2(d0d0 id) = 0. (44)
Adding the numbered equations (37)-(44) and the numbered equations (33)-
__
(36), we get (***) |__|
Theorem 2.9. Assume that i + j 2k. Then
D0 + T D0T = D + OE0,
Dk + T DkT = Dk-1@ + @Dk-1 + OEk
as natural transformations Vi Wj ! Vi+j-k Wi+j-k.
12
Proof. The statement we need to prove is that Ak and OEk induce the same
natural transformation if i + j 2k. If k = 0, this is (3), which we have
verified already. Now we assume inductively that the statement is true for
k - 1, that is that Ak-1 and OEk-1 induce the same natural transformation if
i + j 2k - 2.
It follows that Ak-1(d0 id) and OEk-1(d0 id) induce the same transfor-
mation on Vi Wj if i + j 2k - 1. But OEk-1(d0 id) is only non-trivial if
i = k and j = k - 1, so Ak-1(d0 id) induces the trivial natural transforma-
tion on Vi Wj if i + j 2k. The same argument shows that Ak-1(id d0)
is trivial on Vi Wj for i + j 2k.
Because of of this and Lemma 2.8, we get that Ak induces the same
transformation as SAk-1 on Vi Wj for i+j 2k. Now we use the induction
assumption again, to see that this transformation agrees with SOEk-1 = OEk.
__
|__|
3 Appendix
Lemma 3.1. The formula in Definition 1.2 defines a map ffii : ZqR ! Zq+iR
when 2 i q.
Proof. Let z 2 ZqR. We must show that ffii(z) is a cycle ie.
X
djffii(z) = djs i. .s. 1(z)djs~i. .s.~1(z) = 0, 0 j q + i.
(~, )2V (q,i)
By one of the simplicial identities we have
8
>:
sjdi-1 , i > j + 1.
If j q-i we can commute the left dj all the way through the degeneracy
maps to z such that djffii(z) = 0.
If j = q - i + 1 we find that all terms vanish except for those with
1 = q - i + 1, ~1 = q - i. For such a term we have
djs i. .s. 1(z)djs~i. .s.~1(z) = s i-1. .s. 2-1(z)s~i-1. .s.~2-1(z)
so these cancels out in pairs.
If j q - i + 2 the non zero terms are those with j 2 , j - 1 2 ~ or
j - 1 2 , j 2 ~. For such a term we can interchange the corresponding ~r
with the corresponding s and get a new element in V (q, i) since j - 1
__
q - i + 1. So these terms cancels out in pairs. |__|
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Remark 3.2. For i = 1 we have that ffi1(z) = sq(z)sq-1(z). Thus djffi1(z) = 0
for j 6= q but dqffi1(z) = z2.
References
[1] M. B"okstedt & I. Ottosen, A splitting result for the free loop spaces of
spheres and projective spaces, preprint, Arhus 2004.
[2] W. G. Dwyer, Homotopy Operations for Simplicial Commutative Alge-
bras, Trans. Amer. Math. Soc. 260 (1980), 421-435.
[3] P. G. Goerss, On the Andr'e-Quillen Cohomology of Commutative F2-
algebras, Ast'erisque 186 (1990).
[4] S. MacLane, Homology, Springer Verlag, New York, 1967.
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