THE HOMOLOGY OF CERTAIN SUBGROUPS OF THE
SYMMETRIC GROUP WITH COEFFICIENTS IN Lie(n)
GREG ARONE AND MARJA KANKAANRINTA
September 10, 1996
Abstract. The authors compute the mod p homology of groups of the form
n1x . .x.nk with coefficients in Lie(n), where n1+ . .+.nk = n.
0. introduction
For a commutative ring R and a positive integer n, let LieR (x1; : :;:xn) be *
*the
free Lie algebra over R generated by x1; : :;:xn. Let LieR(n) LieR(x1; : :;:xn*
*) be
the submodule spanned by all bracket monomials containing each xiexactly once. *
*In
what follows, R will almost always be Z or Z=p for some prime p, and we often s*
*upress
R from the notation. Whenever we perform explicit calculations, we assume R = Z*
*=p,
otherwise, R = Z. The submodule Lie(n) is invariant under the obvious action of
the symmetric group n on Lie(x1; : :;:xn). We consider Lie(n) as a representati*
*on
of n. It is well-known that as an R-module
M
Lie R(n) ~= R:
(n-1)!
As a n-representation, Lie(n) can not, in general, be described so trivially. *
*It is
known (see, for instance, [9]) that if R = C then
Lie(n) ~=hom (IndnZ=n; C[-1])
2ssi_
where is the representation of Z=n sending the generator to e n and C[-1] is
the sign representation. However, if the ground ring is Z or Z=p then it is ea*
*sy to
check directly that Lie(n) is not closely related to any representation induced*
* from
a one-dimensional representation of Z=n. Following a suggestion of the referee*
*, we
also remark that if p|n then Lie(n) can not be projective in characteristic p s*
*ince
___________
1991 Mathematics Subject Classification. 55P99 .
Key words and phrases. Goodwillie calculus, Hilton-Milnor theorem, group homo*
*logy.
The first author was partially supported by the Alexander von Humboldt founda*
*tion.
The second author was supported by the Emil Aaltonen foundation and the Acade*
*my of Finland.
1
2 GREG ARONE AND MARJA KANKAANRINTA
projectivity in characteristic p would imply freeness over the p-Sylow subgroup*
* of
n which can not happen since the order of the p-Sylow subgroup does not divide
(n - 1)! when p divides n.
Let n1; : :;:nk be positive integers such that n = n1+ . .+.nk. Consider the *
*group
n1x. .x.nk as a subgroup of n. The homology groups H*(n1x . .x.nk; Lie(n))
arise in various contexts in mathematics. In particular, they arise as basic bu*
*ilding
blocks for homology of spaces of unordered configurations of points in manifold*
*s and
for homology of Lie algebras (see, for instance, [2, 3, 4]). The homology grou*
*ps
H*(n; Lie(n)) (with the ground ring Z=p) were computed in [1] (in a certain gui*
*se,
as will be explained below). The main purpose of this paper is to use this comp*
*utation
to describe
H*(n1x . .x.nk; Lie(n))
(with the same ground ring).
Thus the main result of this paper is a recursive formula (theorem 0.2) which*
* re-
duces computation of H*(n1x . .x.nk; Lie(n)) to computation of H*(d; Lie(d))
for various values of d. Our methods involve some homotopy theory, namely the
Hilton-Milnor theorem and Goodwillie's calculus of functors ([5, 6, 7]). In fa*
*ct we
do a little more than prove theorem 0.2 in the sense that rather than prove tha*
*t two
graded vector spaces are isomorphic, we prove that two spectra are homotopy equ*
*iva-
lent and then obtain the result on vector spaces by passing to homology. We use*
* that
the n-th derivative (in the sense of Goodwillie) of the functor S (loop - suspe*
*nsion)
is a spectrum with an action of n, whose homology is concentrated in one degree,
and in this degree it is precisely Lie(n). In more detail, we consider the Good*
*willie
tower (the Taylor tower of [7]) of the multi-variable functor S(X1 _ X2 _ : :_:*
*Xk)
and we evaluate its differentials in two different ways. The first way is to co*
*nsider the
Goodwillie tower of the functor S and evaluate it at X1_X2_: :_:Xk. The n-th la*
*yer
is given, as a spectrum, by the following formula (we denote Whn = W ^n En +
for any space or spectrum W ):
i ^nj
Map * S2Kn; 1 (SX1 _ SX2 _ : :_:SXk) h
n
where Kn is a complex with an action of n, that is non-equivariantly homotopy
equivalent to a wedge of (n - 1)! copies of the (n - 2)-dimensional sphere, and*
* whose
only non-trivial reduced homology group satisfies
hom (Hn-2(Kn); Z[-1]) ~=Z[n]Lie(n):
This is proved as fact 2.3 below. Kn is essentially the geometric realization *
*of the
poset of partitions of the set with n elements (the precise definition is given*
* on
page 9). By using the binomial expansion, it should be intuitively clear (to an*
*yone
familiar with the Goodwille calculus) that the (n1; : :;:nk)-differential of th*
*e functor
H*(n1x . .x.nl; Lie(n)) 3
S(X1 _ X2 _ : :_:Xk), which is the counterpart of
____@nf_____ xn11. .x.nkk
(0; : :;:0)__________
@xn11. .@.xnkk n1! . .n.k!
is given by
i j
Map * S2Kn; 1 ((SX1)^n1^ . .^.(SXk)^nk) h(
n1x...xnk)
where the action of n1 x . .x.nk on Kn is given by restriction from the action
of n and the action on (SX1)^n1^ . .^.(SXk)^nk is the obvious one (this analogy
with ordinary calculus is addressed in lemma 1.3 and the explanation following *
*it).
The second way is to use the Hilton-Milnor theorem. According to this theorem,Y
S(X1 _ X2 _ : :_:Xk) is homotopy equivalent to a weak infinite product SYi,
i
where Yi are smashYproducts of some of the Xjs. Evaluating the (n1; : :;:nk)-th
differential of SYiand comparing the two outcomes yields the following theore*
*m.
i
Theorem 0.1. For any connected based spaces X1; : :;:Xk, there is an equivalen*
*ce
of spectra, which is natural in X1; : :;:Xk
i j
Map * S2Kn; 1 ((SX1)^n1^ . .^.(SXk)^nk) h(
n1x...xnk)
0 1
_ B _ i n1_ nk_j^d C
' @ Map * S2Kd; 1 S(X1)^ d ^ . .^.(Xk)^ d A :
d|n0B(n1_d;:::;nk_d) hd
(Here n0 = gcd(n1; : :;:nk) and the definition of B(n1_d; : :;:nk_d) is given i*
*n formula 3
on page 12).
Next we take each one of the spaces Xj to be a wedge of ij copies of S2m , an
even-dimensional sphere (m is arbitrary, but it is the same for all j). Thus, *
*for
ij_
j = 1; : :;:k we take Xj = S2m where ij is some positive integer. Passing to
j=1
homology we obtain the following theorem as a corollary
Theorem 0.2. Let n0 and B(n1_d; : :;:nk_d) be as in the previous theorem. For*
* any
ground ring R, and any finitely generated free R-modules N1; : :;:Nk
i j
H* n1x0. .x.nk; LieR(n) R Nn11 R . .R.Nnkk ~=
! 1
M B M n1_ nk_ d C
@ H* d; LieR(d) R N1d R . .R.Nkd A :
d|n0 B(n1_d;:::;nk_d)
4 GREG ARONE AND MARJA KANKAANRINTA
Now take the ground ring to be Z=p for some prime p, and let N1 ~=. .~.=Nk ~=R
(which corresponds to taking all the spaces Xj in theorem 0.1 to be S2m ). The *
*groups
H*(d; Lie(d)) have been computed in [1]. In particular, if d is not a power of*
* p,
then H*(d; Lie(d))i~= 0. Ifjd = pi for some non-negative integer i, then let us
denote Mi = H* pi; Lie(pi). We will describe Mi explicitly in section 3. Rec*
*all
that n0 = gcd(n1; : :;:nk) and let us write n0 = mpj where p does not divide m.
Then the only values of d in theorem 0.2 that we need to consider are those tha*
*t are
powers of p. Therefore
_n_)!
pi_X (e)____(epi_____
Mj ne|mpj-i (_n1_epi)! . .(.nk_epi)!
H*(n1x . .x.nk; Lie(n)) ~= Mi :
i=0
Here denotes the number-theoretic M"obius function. Because of the properties *
*of
, the only values of e we need to consider in the exponent of the right hand si*
*de are
m0and (if j - i > 0) m0p where m0|m. We obtain the following theorem
Theorem 0.3. Let n = n1+ . .+.nk. Let n0 = gcd(n1; : :;:nk). Let the ground ri*
*ng
be Z=p. Write n0 = mpj where p does not divide m. Then
H*(n1x . .x.nk;0Lie(n)) ~= 1
__n_)! (__n__)!
pi_X (m0) @_____(m0pi_____ - ______m0pi+1_______A
j-1M nm0|m (_n1_m0pi)! . .(.nk_m0pi)!(__n1_m0pi+1)! . .(._nk_m0p*
*i+1)!
Mi
i=0 n
____)!
pj_X 0 _____(m0pj_____
n (m ) _n1_ _nk_
L m0|m (m0pj)! . .(.m0pj)!
Mj :
The paper is organized as follows: in section 1 we present a very brief surve*
*y of
some of Goodwillie's theory of Taylor towers and state it in the multi-variable*
* form
that we need. All the material in this section is either quoted from [7] or is *
*adapted
from there in a straightforward way. In section 2 we recall various facts from *
*algebraic
topology that we need. In section 3, we prove theorems 0.1 and 0.2 and also des*
*cribe
the Mis of theorem 0.3.
1.Multivariable calculus
Let U denote the category of spaces with a non-degenerate basepoint. We will
be concerned with calculus of homotopy functors F : Uxk ! U. We begin with an
overview of the case k = 1, which is the subject of [5, 6, 7]. According to Goo*
*dwillie's
calculus, certain functors are analogous to polynomial functions. Namely, a fun*
*ctor
is analogous to a polynomial of degree n if it is n-excisive in the sense of [6*
*, definition
3.1]. One of the main theorems of [7] is that if a functor F is ae-analytic in*
* the
H*(n1x . .x.nl; Lie(n)) 5
sense of [6, definition 4.2], then there exists a tower of functors {PnF }n0 w*
*ith
natural transformations . .!.PnF ! Pn-1F ! . . .such that PnF is of degree n
and there exist natural transformations pn : F ! PnF which are the unique best
approximations of F by n-excisive functors in some precise sense. Since [7] has*
* not
been published yet, we spell out the details of the construction, but not of th*
*e proofs.
Let C(n) denote the category whose objects are subsets of n_= {1; : :;:n} and
whose morphisms are inclusions. Let C0(n) denote the full subcategory of C(n) w*
*hose
objects are the non-empty subsets of n_. Let C be a category. An n-dimensional *
*cubical
diagram in C is a functor C(n) ! C. A punctured n-dimensional cubical diagram is
a functor C0(n) ! C. For our purposes, C will always be any well-behaved versio*
*n of
the category of either based spaces or spectra.
Pn involves the infinite iteration of another construction Tn. For X 2 U the
diagram U 7! X * U, U n_+_1_(here * denotes join) determines a strongly co-
cartesian n_+_1_-cube [6, definition 2.1] and a map, as in [6, definition 1.2],
F (X) ! (Tn1F )(X) def.=holim{F (X * U)|U 2 C0(n + 1)}:
For i > 1, define the functor TniF inductively by TniF = Tn1Tni-1F . It is easy*
* to see
that n o
TniF (X) ~=holim F (X * U1 * . .*.Ui)|U1 x . .x.Ui2 C0(n + 1)i
and we might as well have taken this to be the definition. We will do this when*
* we
define the multi-variable analogue of Tni. Clearly, there are natural transform*
*ations
TniF ! Tni+1F . PnF is defined to be the homotopy colimit of the diagram
F = Tn0F ! Tn1F ! Tn2F ! . .!.TniF ! . .:.
It is shown in [7] that PnF is n-excisive and the map pn : F ! PnF is character*
*ized,
up to weak homotopy equivalence, among natural transformations from F to n-
excisive functors by the property that pn(X) : F (X) ! PnF (X) is (k - ae)(n + *
*1) - c-
connected, where k is the connectivity of X and c is constant which does not de*
*pend
on X or n.
According to [7], this tower should be thought of as the Taylor expansion of *
*F at the
one-point space (but we will subsequently call it the Goodwillie tower of F ). *
*Thus,
PnF is the n-th "Taylor polynomial" of F and in particular P0F is a (homotopy)
constant functor with P0F (X) ' F (*) for all X. Note also that if F is reduced*
*, i.e
F (*) ' *, then T1F (X) ' F (SX) and P1F , the linearization of F , is, essenti*
*ally,
1 F (S1 X). Let DnF denote the homotopy fiber of the map PnF ! Pn-1F . This
is a well defined functor and we call it the n-th differential of F . It can b*
*e shown
that DnF is homogeneous of degree n in the sense that it is excisive of degree *
*n and
Pn-1DnF ' *. Homogeneous functors were classified in [7]: a homogeneous functor
of degree n is determined, at least up to homotopy, by a spectrum A endowed with
an action of the symmetric group n and (up to weak homotopy equivalence) it has
6 GREG ARONE AND MARJA KANKAANRINTA
i j n
the form 1 (A ^ X^n) hn . Note the visual resemblance to the formula ax_n!for*
* the
i j
n-th summand in a Taylor expansion. If DnF (X) ' 1 (A ^ X^n) hn , then we
say that the spectrum A, together with the action of n, is the n-th derivative *
*of F .
Note also that for n = 1 one gets the classical description of a generalized ho*
*mology
theory as given by the Brown representability theorem.
For a general k, we will say that a functor F : Uxk ! U is analytic if it is *
*analytic
considered in each variable separately. For a multi-index n= (n1; : :;:nk), we*
* say
that F is n-excisive if it is ni-excisive considered as a functor of Xi for any*
* choice
of fixed X1; : :;:^Xi; : :;:Xk. We say that F is multilinear if it is reduced a*
*nd linear
(1-excisive) in each variable separately. Given a subgroup H of k we say that F*
* is
symmetric with respect to H if it extends to a functor F : H o Uxk ! U. If H = *
*k,
then we simply say that F is symmetric. It is shown in [7] that homogeneous one-
variable functors of degree k are equivalent to symmetric multilinear functors *
*of k
variables. In fact, up to weak homotopy equivalence, a symmetric multilinear fu*
*nctor
has the form 1 (A ^ X1 ^ . .^.Xk), where A is a spectrum with an action of k.
By a k-index, or simply a multi-index, when k is clear from the context, we m*
*ean
an ordered k-tuple of non-negative integers. We denote multi-indices with bold *
*letters
such as i, j, n. Given multi-indices i= (i1; : :;:ik) and j= (j1; : :;:jk), we *
*say that
ij if il jl for l = 1; : :;:k. For a functor F : Uxk ! U, and multi-indices
i= (i1; : :;:ik) and n= (n1; : :;:nk) we define
TniF (X1; : :;:Xk) = holim{F (X1 * U11* U12* . .*.U1i1; : :;:Xk * Uk1* . .*.U*
*kik)|
(U11; : :;:U1i1; : :;:Ukik) 2 C0(n1 + 1)i1x . .x.C0(nk + 1)ik}:
By analogy with the one-variable case, it is clear that if i1 i2 then there is*
* a
canonical natural transformation Tni1F ! Tni2F and if also i2 i3, then the map
Tni1F ! Tni3F is the composition Tni1F ! Tni2F ! Tni3F . We define Pn to *
*be
the homotopy direct limit of the thus obtained k-dimensional diagram of all Tni.
Intuitively,
P(n1;:::;nk)F = T((1;:::;1)n1;:::;nk)F:
Clearly, Pn F is n-excisive. Let 1j denote the multi-index (0; : :;:1; : :;:0)*
* having
a 1 at the j-th place and zeros elsewhere. Just as in the 1-variable case ther*
*e is a
map Pn ! Pn-1 induced by restriction of homotopy inverse limits, in the k-varia*
*bles
case there are maps Pn ! Pn -1j. Thus the functors Pn F fit into a "k-dimension*
*al"
inverse limit system, that converges to F in an appropriate sense. When we speak
of Pn F , we allow that some of the indices of n are 1, in which case we mean t*
*hat
we take the inverse limit with respect to these indices. For example, P(1;n2;::*
*:;nk)=
holimnP(n1;n2;:::;nk).
1!1
In the case of a one-variable functor F , there is an important relation betw*
*een the
n-th differential of F and the n-th cross-effect of F . The connection is expla*
*ined in
H*(n1x . .x.nl; Lie(n)) 7
[7]. We present a brief summary. Given a functor F : U ! U, the n-th cross-effe*
*ct
of F , denoted OnF , is a functor of n variables defined as follows: For based *
*spaces
X1; : :;:Xn, we define an n-dimensional cubical diagram S(X1; : :;:Xn) by
_
n_r T 7! Xs
s2T
with the maps in the cube being the obvious retractions. Then OnF (X1; : :;:Xn)*
* is
defined to be the iterated homotopy fiber (or the total fiber, as defined in [7*
*, defi-
nitions 1.1-1.2a]) of S(X1; : :;:Xn). It is easily seen from the definitions t*
*hat OnF
is symmetric and reduced. It follows that P(1;1;:::;1)OnF (X1; : :;:Xn), the m*
*ultilin-
earization of OnF , is given, up to homotopy, by
hocolimk1+...+knOnF (Sk1X1; : :;:SknXn):
k1;:::;kn!1
We denote this functor D(n)F . D(n)F is a symmetric multilinear functor, and as*
* such
is represented by a spectrum with an action of n; denote this spectrum by An. T*
*he
following result is proved in [7]
Lemma 1.1. If An is as above, then An is the n-th derivative of F .
Let F : Uk ! U be a functor of k variables. Let n= (n1; : :;:nk). We define D*
*n F
to be the iterated homotopy fiber of the k-dimensional cube given by U 7! Pn -1*
*U,
where 1U is defined to be the multi-index which has in its j-th place 1 or 0 de*
*pending
on whether j 2 U or j =2U respectively. Dn F is n-homogeneous in the appropriate
sense. There is a natural (weak) homotopy equivalence
i j
Dn F (X1; : :;:Xk) ' 1 A ^ X^n11^ . .^.X^nkkh(
n1xn2x...xnk)
where A is a spectrum with an action of n1 x n2 x . .x.nk. We may think of
A, together with the action, as the n-th derivative of F . Again, we would lik*
*e to
establish the relation between derivatives and cross-effects. So, given F as be*
*fore, we
may form its n1-th cross-effect with respect to the variable X1, n2-th cross-ef*
*fect in
the second variable, and so on. One can easily see that the outcome does not de*
*pend
on the order of variables. The outcome is a functor of n1 + n2 + . .+.nk variab*
*les,
which is reduced and symmetric with respect to an action of n1x n2x . .x.nk.
Upon multilinearizing, we obtain a functor which is multilinear and is symmetric
with respect to n1x n2x . .x.nk. As such, it is represented by a spectrum with
an action of n1x n2x . .x.nk. Denote it by An.
Lemma 1.2. If An is as above, then An is the (n1; : :;:nk)-th derivative of F*
* .
Proof.Very similar to the proof of our lemma 1.1 as given in [7]. __|_ |
8 GREG ARONE AND MARJA KANKAANRINTA
k !
_
Lemma 1.3. Let F (X1; : :;:Xk) = G Xi , where G is some functor of one va*
*ri-
i=1
able. Let n= (n1; : :;:nk) and let n = n1 + . .+.nk. Then the n-th derivative*
* of
F is equivalent to the n-th derivative of G with the action of n1x n2x . .x.nk
obtained by restriction from the action of n.
Explanation. Let f(x1; : :;:xk) be an analytic function, say, from Rk to R. Ass*
*ume,
furthermore, that f(x1; : :;:xk) = g(x1 + . .+.xk) for some g : R ! R. Then
____@nf_____ xn11. .x.nkkdng xn11. .x.nkk
(0; : :;:0)__________= ____(0)__________:
@xn11. .@.xnkk n1! . .n.k!dxn n1! . .n.k!
Proof of lemma 1.3.Immediate from lemmas 1.1 and 1.2. __|_ |
i j
Lemma 1.4. Let F (X1; : :;:Xk) = G X^m11^ X^m22^ . .^.X^mkk . Then Dn F is
trivial if ni_miis not the same for all i = 1. Otherwise,
i j
Dn F (X1; : :;:Xk) ' DlG X^m11^ X^m22^ . .^.X^mkk ;
where l = ni_mi:
Explanation. Let f(x1; : :;:xk) be an analytic function. Assume, furthermore,
that f(x1; : :;:xk) = g (xm11. xm22. .x.mkk)for some g. Then the Taylor series*
* for g
evaluated at xm11.xm22. .x.mkkis clearly the multi-variable Taylor expansion of*
* f and it
can be easily concluded that if ni_miis not a constant non-negative integer ind*
*ependent
nf xn1...xnk n
of i then ___@_____@xn1(0; :nk:;:0)_1_____k= 0 and if _i_= l for all i, then
1 ...@xk n1!...nk! mi
____@nf_____ xn11. .x.nkkdlg (xm11. .x.mkk)l
(0; : :;:0)__________= ___(0)_____________
@xn11. .@.xnkk n1! . .n.k!dxl l!
Proof of lemma 1.4.In fact, we want to show that for any multi-index n
i *
* j
Pn F (X1; : :;:Xk) ' P(lm1;:::;lmk)F (X1; : :;:Xk) ' PlG X^m11^ X^m22^ . .^.X^*
*mkk
where l = min {b_ni_mic}. Without loss of generality, we may assume that l = b*
*_n1_m1c.
Consider the functor P(n1;1;:::;1)F , which amounts to considering F as a func*
*tor
of X1 only, and taking the n1-th Taylor polynomial in this variable. P(n1;1;::*
*:;1)F
is n1-excisive (in X1) and the map F (X1; : :;:Xk) ! P(n1;1;:::;1)F (X1; : :;:X*
*k) is
(n1 + 1)(d1 - ae) - c-connected, where d1 is the connectivity of X1 and ae and *
*c are
some numbers, which dependion X2; : :;:Xk, but notjon X1. On the other hand,
consider the functor PlG X^m11^ X^m22^ . .^.X^mkk as a functor of X1 only. It*
* is
obvious that it is lm1-excisive and therefore n1-excisive, since n1 lm1. The n*
*atural
map
i j i j
F (X1; : :;:Xk) = G X^m11^ . .^.X^mkk ! PlG X^m11^ . .^.X^mkk
H*(n1x . .x.nl; Lie(n)) 9
is
(l + 1)(m1d1+ . .+.mkdk - ae0) - c0 ((l + 1)m1)(d1- ae0) - c0 (n1+ 1)(d1- ae0) *
*- c0
connected, for some constants ae0 and c0. By universality properties of the Ta*
*ylor
approximations it follows that there is a weak homotopy equivalence
i j
(1) P(n1;1;:::;1)F (X1; : :;:Xk) ' PlG X^m11^ X^m22^ . .^.X^mkk :
Now consider P(n1;:::;nk)F . It is weakly equivalent to P(1;n2;:::;nk)P(n1;1;:*
*::;1)F . But
by (1), P(n1;1;:::;1)F is lmi-excisive in the variable Xi for i = 1; 2; : :;:k.*
* Since we
defined l to be min {b_ni_mic}, it follows that ni lmi, and therefore P(n1;1;:*
*::;1)F is
ni-excisive in the variable Xi for i = 2; : :;:k. Hence
i j
P(n1;:::;nk)F (X1; : :;:Xk) ' PlG X^m11^ X^m22^ . .^.X^mkk :
The lemma readily follows. __|_ |
2.Miscellaneous preliminary results
Let W be a based space or a spectrum with an action of n. Let R be a commu-
tative ring. Let fH*(-; R) denote reduced homology with coefficients in R.
Fact 2.1.Suppose that fH*(W ; R) is concentrated in degree i. Then
Hf*(Whn ; R) ~=H*+i(n; fHi(W ; R)):
We will use the following "geometric realization" of Lie(n). Let kn be the ca*
*tegory
of unordered partitions of n_. Thus, the objects of kn are unordered partition*
*s of
n_, and there is a morphism 1 ! 2 iff 2 is a refinement of 1. The category of
partitions that we define here is the opposite of the category of partitions as*
* defined
in [1], but it makes no difference since we only are interested in the simplici*
*al nerve of
kn. Obviously, kn has an initial and a final object. Denote these ^0and ^1respe*
*ctively.
Let ekn= kn r {^0; ^1}. Let fKn be the geometric realization of the simplicial *
*nerve of
ekn. Let Kn be the unreduced suspension of fKn. Obviously, Kn has an action of *
*n.
Let Sn denote the n-dimensional sphere.
Fact 2.2.Non-equivariantly
_
Kn ' Sn-2:
(n-1)!
In particular, the reduced homology of Kn is concentrated in degree n - 2.
i j
Fact 2.3.As a n-representation, Lie(n) ~=hom Hfn-2(Kn); Z[-1] :
10 GREG ARONE AND MARJA KANKAANRINTA
Proof.It is known ([2, theorem 12.3, page 302]) that Lie(n) is isomorphic, as a
n-representation, to the top homology of F (R2k+1; n), the space of ordered con*
*figu-
rations of n points in an odd-dimensional Euclidean space (k is arbitrary). Now*
* view
F (R2k+1; n) as a n-equivariant subspace of R(2k+1)nand also, by adding a point*
* at
infinity, of S(2k+1)n. The top homology group of S(2k+1)nis isomorphic to Z[-1]*
* as a
n-representation. Let nS2k+1 denote the complement of F (R2k+1; n) in S(2k+1)n.
Let Hb denote the bottom reduced homology of nS2k+1. It follows by Alexander
duality that Hb is in dimension n + 2k - 1 and that Lie(n) ~=Z[n]hom (Hb; Z[-1]*
*). It
remains to show that Hb ~=fHn-2(Kn) as a n-representation. It is enough to prove
the following proposition (the first author is grateful to J. Rognes for explai*
*ning it
to him)
Proposition 2.4. There is a n-equivariant (weak) map
nSl ! SlKn
which, for large enough l, induces an isomorphism on the bottom homology of nSl.
The rest of the proof of the fact is occupied by the proof of the proposition.
Consider the covering of nSl by the subspaces Ui;j, 1 i < j n, where
n o
Ui;j= (s1 ^ . .^.sn) 2 nSl | si= sj :
The point is that the intersection poset of this covering is isomorphic to the *
*poset
of unordered partitions of n_and the intersections in this covering are highly *
*enough
connected. In more detail, let AT= {(i1; j1); (i2; j2); : :;:(iM ; jM )} be a c*
*ollection of
pairs 1 im < jm n. Let UA = (im ;jmU)2Aim.;jmWe associate with A a graph
on n vertices, labeled 1; : :;:n, as follows: There is an edge (i; j) iff (i; j*
*) 2 A. The
connected components of this graph determine a partition of n_. Clearly, UA dep*
*ends
only on the partition associated with A. Thus, the poset associated with the co*
*vering
of nSl by Ui;jis isomorphic to kn r ^1. Let J be the intersection diagram of t*
*his
covering. Thus, J is a functor
kn r ^1! Spaces
given by
n o
! (s1 ^ . .^.sn) 2 nSl | si= sjifi and j are in the same component of :
Thus nSl is the strict direct limit of J. Let nhSl be the homotopy direct limit*
* of
J. There is a canonical n-equivariant map
nhSl ! nSl
which is a homotopy equivalence.
H*(n1x . .x.nl; Lie(n)) 11
We now define another functor eJ: kn r {^1} ! Spaces as follows:
! {(s1 ^ . .^.sn) 2 nSl | Psi= sjifi and j are in the same component of;
n P n 2
i=1si= 0 and i=1ksik = 1}:
Here we think of nSl as a subspace of Snl, which is the one-point compactificat*
*ion
of Rnl. The functor Je takes values in subspaces of Rnl. Notice that Je(^0) =*
* ;
and therefore hocolim eJ~= hocolimJe|ekn. In general, if has k components, th*
*en
eJ() ~=Slk-l-1. Let enhSl denote the homotopy direct limit of Je. It is easy *
*to see
that
nhSl ~=Sl* enhSl
where * denotes join. Next we note that since enhSlis the homotopy direct limit*
* of
a functor fkn! Spaces and gKnis homeomorphic to the homotopy direct limit of the
constant functor fkn! *, there is a map
enhSl! fKn
which is ~ l-connected, since the values of eJare all ~ l-connected. All of the*
* above
assembles into the following chain of equivariant maps:
~= l n l l ' l
nSl ' nhSl ! S * ehS ! S * fKn! S ^ Kn:
The composed map is ~ 2l-connected since the map Sl*nehSl ! Sl*Kfn is. It follo*
*ws
that the map induces an isomorphism on the bottom homology for l large enough.
This completes the proof. __|_ |
The space Kn plays an important role in calculus of functors. Consider the id*
*entity
functor from based spaces to based spaces. The following is proved in [8] in a *
*slightly
different form:
Fact 2.5. i j
DnId(X) ' 1 Map * (SKn; 1 X^n) hn :
Proof.As mentioned above, this is a reformulation of the main theorem of [8]. F*
*or
proof of the equivalence with the result in [8] see [1, section 2]. __|_ |
Corollary 2.6.
i j
Dn(S)(X) ' 1 Map * S2Kn; 1 (SX)^n h :
n
Proof.For any functor F : U ! U, consider the functor F 0: X 7! F (SX). It foll*
*ows
immediately from the universal properties of Pn that (PnF 0)(X) ' (PnF )(SX) and
(DnF 0)(X) ' (DnF )(SX). The corollary follows by letting F to be the identity
functor. __|_ |
12 GREG ARONE AND MARJA KANKAANRINTA
Remark 2.7. It may seem that there is a problem with the definition of K1. In f*
*act,
there is not. Either by careful check of the definitions, or by convention, def*
*ine K1
to be the empty set. Then SK1 should be interpreted as S0, provided the suspens*
*ion
is unreduced. With these conventions fact 2.5 and corollary 2.6 hold for n = 1.
3. Proof of the main theorems
Let X1; : :;:Xk be sufficiently nice topological spaces, for example connecte*
*d CW -
complexes with non-degenerate basepoints. By the Hilton-Milnor theorem [10] the*
*re
is a natural homotopy equivalence
Y _k
(2) h: SYi! S Xj;
i2I j=1
where I denotes the set of basic products on k letters. Essentially, each Yiis *
*a smash
product of some of the spaces X1; : :;:Xk corresponding to the basic product i.*
* By
the formula due to Witt [11] the number of basic products involving Xj exactly *
*qj
times is
1 X (q_d)!
(3) B(q1; : :;:qk) = __ (d)____________qq;
q d|q0 (__d)! . .(._1kd)!
where q0 is the greatest common divisor of the numbers q1; : :;:qk, q = q1 + . *
*.+.qk
and is the number theoretic M"obius function.
k_
Let us denote Fi(X1; : :;:Xk) = SYi and G(X1; : :;:Xk) = S Xj. Let F be
Y j=1
Fi. Let n= (n1; : :;:nk), where n1; : :;:nk are positive integers and let n1 *
*+ . .+.
i2I
nk = n. The natural transformation h of (2) induces an equivalence
Dn h : Dn F ! Dn G:
Let I0 denote the set of basic products which involve Xj at mostYnj timesYfor j*
* =
1; : :;:k. Clearly, I0 is finite. Let I00= I \ I0. Then F = Fi0x Fi00.
i02I0 i002I00
The operator Dn commutes, up to homotopy, with finite products of functors.
Therefore,
Y Y
Dn F ' Dn Fi0x Dn Fi00:
i02I0 i002I00
Proposition 3.1.
Y
Dn Fi00' *:
i002I00
H*(n1x . .x.nl; Lie(n)) 13
Proof.It follows immediately from the definition of I00and lemma 1.4 that for a*
*ny
i002 I00, Dn Fi00' *. The proposition does not follow automatically because in *
*general
Dn commutes only with finite inverse homotopy limits. However, Dn commutes with
arbitrary filtered homotopy direct limits, and since the product is a weak prod*
*uct, it
can be written as a homotopy direct limit of finite products. __|_ |
On the other hand, I0 is finite, and therefore there is a weak equivalence
Y
(4) D(n1;:::;nk)F ' D(n1;:::;nk)Fi' D(n1;:::;nk)G:
i2I0
Evaluating the right hand side of (4) at (X1; : :;:Xk) and using lemma 1.3, we *
*obtain
(5) D(n1;:::;nk)G(X1; : :;:Xk)
0 1
_k
= D(n1;:::;nk)@S XjA
j=1
i j
= 1 Map * S2Kn; 1 ((SX1)^n1^ . .^.(SXk)^nk) h( :
n1x...xnk)
Similarly, evaluating the middle part of (4) at (X1; : :;:Xk) and using lemma 1*
*.4, we
obtain
Y
(6) D(n1;:::;nk)Fi(X1; : :;:Xk)
i2I0
Y
= D(n1;:::;nk)SYi
i2I0
! B(n1_;:::;nk_)
Y i n1_ nk_j^d d d
= 1 Map * S2Kd; 1 S(X1)^ d ^ . .^.(Xk)^ d ;
d|n0 hd
where n0 is the greatest common divisor of the numbers n1; : :;:nk. Comparing (*
*5)
and (6), and passing from infinite loop spaces to spectra, we obtain theorem 0.*
*1.
Let us now take Xj to be a wedge of copies of an even-dimensional sphere S2m *
*for
every j. Then, using theorem 0.1, taking homology groups, and applying facts 2*
*.1
and 2.3, we obtain theorem 0.2.
From here on, we assume that the ground ring is Z=p. It remains to describe
the Mis of theorem 0.3. We have indicated in the introduction that the Mis were
computed in [1]. To be more precise, the homology groups of the spectrum
i 2mn_ j
Map * S2Kd; 1 (S d+1)^d h
d
with Z=p coefficients have been computed in [1]. In particular, if d is not a p*
*ower of
p, then
i 2mn_ j
H* Map * S2Kd; 1 (S d+1)^d h ; Z=p ~=0:
d
14 GREG ARONE AND MARJA KANKAANRINTA
Assumeitherefore d = pi forjsome non-negative integer i. The homology groups of
2mn_+1^d
Map * S2Kd; 1 (S d ) h are independent of m, up to a dimension shift, and
d
we may as well take m = 0. Using facts 2.1 and 2.3 once more, it is easy to see*
* that
i j
Mi~= H* Map * S2Kd; 1 Sd h ; Z=p :
d
Roughly speaking, up to a suitable suspension, Mi has a basis consisting of the
"completely inadmissible" Dyer-Lashof words of length i (which is almost the sa*
*me
as the set of admissible Steenrod words of length i).
Theorem 3.2. If d = pi, then the following constitutes a basis for 1+iMi:
if p > 2
n o
fiffl1Qs1. .f.iffliQsiu | si 1; sj > psj+1- fflj+1 81; j < i
if p = 2
n o
Qs1. .Q.siu | si 1; sj > 2sj+1 81 j < i:
Here u is of dimension 1, Qsjs are the Dyer-Lashof operations and fis are the h*
*omology
Bocksteins [2]. Thus Qs increases dimension by s if p = 2 and by 2s(p - 1) if p*
* > 2
and fi decreases dimension by one.
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H*(n1x . .x.nl; Lie(n)) 15
(G. Arone) The University of Chicago, Department of Mathematics, Chicago, IL
60637
E-mail address: arone@math.uchicago.edu
(M. Kankaanrinta) Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), S*
*F-
00014 University of Helsinki, Finland