TORSION IN LOOP SPACE HOMOLOGY OF RATIONALLY
CONTRACTIBLE SPACES
RAN LEVI
Abstract. Let R be a torsion free principal ideal domain. We study the gr*
*owth of torsion
in loop space homology of simply-connected DGR-coalgebras C, whose homolo*
*gy admits an
exponent r in R. Here by loop space homology we mean the homology of the *
*loop algebra
construction on C. We compute a bound on the growth of torsion in such ob*
*jects and show
that in general this bound is best possible. Our methods are applied to *
*certain simply-
connected spaces associated with classifying spaces of finite groups, whe*
*re we are able to
deduce the existence of global exponents in loop space homology.
1. Introduction and Statement of Results
The growth of torsion in loop space homology is known to be rather hard to co*
*ntrol in
general. A remarkable example of this was discovered by D. Anick [2], who intr*
*oduced a
finite dimensional CW complex X with the property that the integral loop space *
*homology
H*(X; Z) has torsion of all orders. A similar example was also constructed by L*
*. Avramov
[3]. On the other extreme, Y. Felix, S. Halperin and J-C. Thomas showed in [6] *
*that if Y is
a space of finite Lusternik-Schnirelmann category, with finite type Z(p)-homolo*
*gy and such
that the dimension of the graded vector space H*(Y ; Fp) grows at most polynomi*
*ally, then
the torsion in H*(Y ; Z(p)) has an exponent, namely there exists an integer r s*
*uch that pr
annihilates all the torsion in the p-local loop space homology of Y .
The spaces constructed by both Anick and Avramov have the additional feature *
*that their
integral homology is torsion free. The theorem in [6] on the other hand is conc*
*erned with
neither spaces of infinite category nor with such, where the mod-p loop space h*
*omology
growth rate is larger than polynomial.
The motivation for this paper comes from considering spaces of the form BG^p,*
* where G
is a finite p-perfect group and by (-)^pwe denote the Fp-completion functor of *
*Bousfield and
Kan [4]. In [10] we show that for such groups G, the integral loop space homolo*
*gy of BG^p
has an exponent, given by the order of the Sylow p-subgroup of G. It is always*
* the case
that BG^phas infinite category when G is finite and in [11, 12] we present exam*
*ples of finite
p-perfect groups G such that the mod-p loop space homology of BG^pgrows exponen*
*tially.
Thus, in general, spaces of this form do not fit in the context of the theorems*
* mentioned
above.
We specialize to rationally contractible spaces with a reduced integral homol*
*ogy exponent
and attempt to understand the growth of torsion in their loop space homology. *
*Clearly
___________
Date: March 31 1995.
1991 Mathematics Subject Classification. Primary 55R35, Secondary 55R40, 55Q5*
*2.
The author is supported by a DFG grant.
1
2 Ran Levi
one should not expect an exponent result in general, even when restricting to t*
*his family
of spaces. For example the integral homology of the double loop space on a mod-*
*pr Moore
space has p-torsion of arbitrarily high order, even though the homology of the *
*single loop
space has an exponent pr. Indeed, our study here may suggest that it is hardly*
* ever the
case that an infinite dimensional rationally contractible space X, which admits*
* a homology
exponent, still has one in its loop space homology. We refer the reader to Sect*
*ion 9 below
for further discussion.
In the preceding we will abbreviate the words "differential graded" by the le*
*tters DG. Let
R be a commutative ring with a unit. The major tool we use is the loop algebra *
*functor (-)
from the category of DGR-coalgebras to DGR-algebras [7] (this functor is more c*
*ommonly
known as the Adams' cobar construction [1]). We shall generally assume that the*
* ground
ring R is a torsion free principal ideal domain (by torsion free we mean that t*
*he natural
map from the integers to R is injective). For every DGR-module, the underlying*
* graded
R-module is assumed to be dimension-wise free and graded by the non-negative in*
*tegers. A
DGR-module (M; d) is said to have a homology exponent if there exists a non-zer*
*o element
r 2 R, such that r . Hi(M; d) = 0 for all i 0. All topological spaces consider*
*ed in this note
are assumed to be pointed, connected and to have the homotopy type of a CW comp*
*lex. For
precise definitions of the terminology used below, we refer the reader to secti*
*on 2.
For a supplemented DGR-coalgebra C, let JC denote the augmentation coideal of*
* C. For
a supplemented DGR-algebra A, let IA denote the augmentation ideal of A. A coal*
*gebra
C, as above, is said to have a homology exponent r 2 R if r is an exponent for *
*H*(JC).
Similarly one defines what it means for a supplemented DGR algebra to have a ho*
*mology
exponent.
Theorem 1.1. Let C be a c-connected DGR-coalgebra, c 1. Suppose that C admi*
*ts a
homology exponent r 2 R. Let d and e be positive integers such that e d-c+1_c*
*. Then re
annihilates Hi(C) for each 1 i d.
We construct an example over the integers, which shows that in general the ap*
*proximation
given by Theorem 1.1 is best possible. Indeed if C is precisely (2k - 1)-connec*
*ted, for some
positive integer k and has a homology exponent r 2 R, then the theorem claims t*
*hat re-1
annihilates Hi(C) for 1 i e(2k - 1) - 1. Thus the least d, for which re-1 mig*
*ht fail to
be an exponent for Hd(C), is d = e(2k - 1).
Theorem 1.2. For each pair of positive integers t and k, there exists a (2k -*
* 1)-connected
DGZ-coalgebra Ck;twith a homology exponent t, such that for every positive inte*
*ger e,
He(2k-1)(Ck;t) contains an element of order te.
An interesting fact about the DGZ-coalgebra Ck;tis that its dual algebra (Ck;*
*t)* is com-
mutative and nilpotent of degree 3, in the sense that the third power of its au*
*gmentation
ideal vanishes. Indeed, it is a triviality that if the dual DGR-algebra C* of a*
* DGR-coalgebra
C is nilpotent of degree 2 (i.e. C has a trivial diagonal) and C has a homology*
* exponent r
then so does C. Our example shows that this fails to hold once the nilpotency a*
*ssumption
is relaxed.
It is known and in fact admits a rather easy proof that if X is a rationally *
*contractible
finite dimensional CW complex, which admits an integral homology exponent (triv*
*ially it
Torsion in loop-space homology of rationally contractible spa*
*ces 3
does if it has only finitely many cells), then X admits an integral homology ex*
*ponent.
However the usual proof, using the Serre spectral sequence for the path-loop fi*
*bration over
X, fails to give the best approximation. The methods used in the proof of Theo*
*rem 1.1
enables us to improve the existing bound.
Proposition 1.3. Let C be a c-connected DGR-coalgebra of homological dimension *
*d with
a homology exponent r 2 R. Then re annihilates Hi(C) for every e d-c-1_cand i *
*> 0.
Our bound here should be compared with the corresponding result using the Ser*
*re spectral
sequence which gives rd-c as a bound, under the same hypotheses. At the end of *
*Section 5
below, we consider finite sub DGR-coalgebra of those constructed in Theorem 1.2*
*, for which
the bound given in Proposition 1.3 is best possible.
Next we study an application of the techniques developed here. Recall the con*
*struction
Pn(-) on DGR-coalgebras given in [10]. Given any DGR-coalgebra C, we produce a*
*n n-
connected quotient DGR-coalgebra Pn(C), such that the natural projection from C*
* to Pn(C)
induces an isomorphism in homology above dimension n. This algebraic constructi*
*on, being
motivated by the Quillen "plus" construction, admits a topological analogue, wh*
*ich in a
sense is an easy generalization of Quillen's idea.
Proposition 1.4. Let X be a connected CW-complex and let n be a positive intege*
*r. Then
there exists an n-connected CW-complex PnX together with a map n : X -! PnX, in*
*ducing
an isomorphism on integral homology above dimension n. Furthermore, if C = S*(X*
*), the
integral singular chain coalgebra on X, then Pn(C) and S*(PnX) are quasi isomor*
*phic as
DGZ-coalgebras.
With the terminology of Proposition 1.4 we now restrict attention to classify*
*ing spaces
BG, where G is a finite group.
Theorem 1.5. Let G be a finite group of order N. Then H"*(PnBG; Z) is annihil*
*ated by
Nq, where q = 3 if n = 1 and q = 2 otherwise.
If G is a finite p-perfect group then (P1BG)^pis homotopy equivalent to BG^p.*
* Thus
Theorem 1.5 is, in a sense, a generalization of [10, I, thm. 1], although in t*
*he last the
exponent is given by the order of the group rather than a power of it.
All the results above are motivated by topology. However the loop algebra fu*
*nctor is a
purely algebraic construction and so one could consider loop space homology tor*
*sion in a
context that might have nothing to do with topology. We look at the loop algebr*
*a functor
(-) together with its left adjoint, the classifying coalgebra functor B(-) [7].*
* The adjunct
morphisms here are equivalences. Thus one could consider a DGR-algebra A, for w*
*hich the
classifying construction B(A) admits a homology exponent and ask whether the sa*
*me holds
for A itself. An example of this is given by the following.
Theorem 1.6. Let A be a nilpotent DGR -algebra of nilpotency rank n. Suppose*
* that B(A)
admits a homology exponent r 2 R. Then rn-1 is a homology exponent for A.
In section 8 below we observe that the mod-p homology of a p-local loop space*
* X is a
nilpotent algebra if and only if it is finite dimensional, in which case it can*
*not be rationally
contractible. Thus Theorem 1.6 does not correspond to a topological situation.
4 Ran Levi
The paper is organized as follows. Sections 2 and 3 are preliminary in natur*
*e and the
techniques developed are applied in Section 4 for the proof of Theorem 1.1 and *
*Proposition
1.3. In Section 5 we prove Theorem 1.2. Section 6 is dedicated to the proof of *
*an algebraic
analogue of Theorem 1.5. The theorem follows using Proposition 1.4 which we pr*
*ove in
Section 7. In Section 8 we prove Theorem 1.6. We conclude the paper in Sectio*
*n 9 by
making some remarks and speculations on the subject.
The author is grateful to the Mathematisches Institut Der Universit"at Heidel*
*berg for its
kind hospitality and support and to Hans-Werner Henn for some useful discussion*
*s.
2. Extended Maps On DGR-algebras
Our terminology is mostly borrowed from [7]. We begin this section by recalli*
*ng the basic
definitions we need.
Let R be a commutative ring with a unit. Recall that a DGR-module M is a gra*
*ded
R-module, also denoted by M, together with a differential dM of degree -1. A m*
*orphism
of DGR-modules is a morphism of graded modules, which commutes with the differe*
*ntials.
Morphisms of graded R-modules (differential or not) will generally be assumed t*
*o have degree
0. However we shall frequently use morphisms of non-zero degree, in which case *
*the degree
will be explicitly spelled out.
A DGR-module M is said to be positive if Mn = 0 for n < 0. The suspension fu*
*nctor
(-), given by (M)n = Mn-1 and (dM ) = -(dM )n-1, is an automorphism on the cat*
*egory
of DGR-modules and preserves the subcategory of positive DGR-modules.
We shall generally assume that R is an integrally torsion free principal idea*
*l domain. Let
GR m denote the category of positive graded R-modules M, such that Mn is a free*
* R-module
for every n. Similarly let DGR m denote the category of DGR-modules such that *
*for every
object M the underlying graded R-module is an object of GR m. From this point a*
*nd on by
a GR -module (DGR -module) we shall always mean an object of GR m (DGR m)
A DGR-coalgebra C is a positive DGR-module, also denoted by C, together with *
*a diag-
onal C : C -! C C and a morphism fflC : C -! R, such that is associative and *
*ffl is a
counit for . A morphism f : C -! D of DGR-coalgebras is a morphism of DGR-modul*
*es,
which commutes with the diagonals and such that fflD f = fflC. Notice that the *
*ground ring R
admits a unique structure of a DGR-coalgebra.
A supplemented DGR-coalgebra is a DGR-coalgebra C, together with a map jC : R*
* -! C
of DGR-coalgebras called an augmentation. A morphism f : C -! D between two su*
*ch
objects is a map of DGR-coalgebras, which in addition satisfies fjC = jD . For *
*an integer
n 0, a DGR-coalgebra C is said to be n-connected if (fflC)q is an isomorphism *
*for all q n.
For a supplemented DGR-coalgebra C let the augmentation coideal JC denote the c*
*okernel
of jC.
The dual terminology applies to define the concept of a DGR-algebra. Thus the*
* structure
jA
maps on a supplemented DGR-algebra A are a multiplication A, a unit R -! A and*
* an
augmentation A -fflA!R. For a supplemented DGR-algebra A, let the augmentation *
*ideal IA
denote the kernel of fflA. We let DGR c and DGR a denote the categories of s*
*upplemented
DGR-coalgebras and algebras respectively, such that for each object the underly*
*ing graded
Torsion in loop-space homology of rationally contractible spa*
*ces 5
R-module is an object of GR m. As before, by DGR -coalgebras and algebras we s*
*hall always
mean, from now on, objects of DGR c and DGR a respectively.
For a GR -module M, let T (M) denote the tensor algebra on M. Let j : M -! T *
*(M)
denote the natural inclusion map into elements of tensor filtration 1. For an e*
*lement x 2 M
let j(x) be denoted by [x]. The module T (M) is naturally a bigraded object. Na*
*mely, let
bideg([x]) = (-1; deg(x)), and if bideg(yi) = (ni; ki), i = 1; 2, then bideg(y1*
*y2) = (n1 +
n2; k1 + k2).
Definition 2.1. Let M be a GR -module and let F : IT (M) -! IT (M) be a map of *
*GR -
modules of some non-negative degree. We say that F is right extended if it is a*
* map of right
T (M)-modules. The map F is said to be a right extension of f : M -! IT (M) giv*
*en by
f = F O j.
The following two lemmas are rather elementary. The first appears in [10, I,*
* chp. 4]
and the second is an observation made to the author by John C. Moore. Short pro*
*ofs are
included for the convenience of the reader.
Lemma 2.2. For M 2 GR m, let f : M - ! IT (M) be a map of GR -modules of some
non-negative degree. Then there exists a unique right extension F : IT (M) -! I*
*T (M) of
f.
Proof.For x 2 M, define F [x] = f(x) and inductively
F ([x1| . .|.xn]) = F ([x1])[x2| . .|.xn]:
Using the fact that Mi is a free R-module for each i, one observes immediately *
*that F is a
well defined right extension of f in the sense given above. Uniqueness is immed*
*iate_from the
definitions. *
* |__|
For any R-module M and a non-zero element r 2 R, let OEr denote the endomorph*
*ism of
M given by multiplication by r.
Lemma 2.3. Let M be a DGR -module, with a homology exponent r 2 R. Then OEr *
*is null-
homotopic on M. Moreover, there is a choice of a null-homotopy s for OEr, such *
*that s2 = 0.
Proof.Since M is a free R-module in each dimension and since R is assumed to be*
* a principal
ideal domain, the short exact sequence of DGR -modules
0 -! Z(M) -i! M -d! B(M) -! 0
is split in GR m. Here Z(M) denotes the submodule of cycles and B(M) means the
suspension of the submodule of boundaries, where in both Z(M) and B(M) the diff*
*erentials
are taken to be trivial.
Let oe denote any right inverse for d and let j denote the left inverse for i*
*, given by
j(m) = i-1(m - oed(m)). Notice that m - oed(m) is a cycle for every m 2 M, so i*
*-1 is defined
on it. Since ij(m) is a cycle for every m 2 M, one has by hypothesis that r . *
*ij(m) is a
boundary. Define
s(m) = oe(r . ij(m)):
*
* __
The reader can easily verify that s is a null-homotopy for OEr and that s2 = 0.*
* |__|
6 Ran Levi
Let C be a simply-connected DGR -coalgebra. Recall that the loop algebra fun*
*ctor C is
given as a GR -algebra by C = T (-1JC). As a differential graded module, C is b*
*igraded
and has two differentials; an internal differential dI of bidegree (0; -1), giv*
*en by dI[x] =
-[dx], where d isPthe differential of C, and an external differential dE of bid*
*egree (-1; 0),
0| 0 00 P 0 00
given by dE[x] = i(-1)|xi[xi|xi], where x = ixi xi is the reduced diagona*
*l. Both
differentials are required to be derivations of the algebra structure and the t*
*otal differential
dT = dI + dE gives C the structure of a DGR -algebra.
Let C be a simply-connected DGR -coalgebra with a homology exponent r 2 R. *
*Let
s : C -! C be a choice of a null-homotopy for OEr. Let s1 : IC -! IC denote the*
* right
extension of the composite
j
-1JC --s!-1JC -! IC
in the sense of Lemma 2.2. It is easy to see then that s1 is a null-homotopy f*
*or OEr with
respect to the internal differential dI [10, I, chp. 4].
3. The obstruction Map and its iterations
In some sense, could the map s1, constructed in the previous section, be a ma*
*p of DGR -
modules with respect to the external differential, it would then be a null-homo*
*topy for OEr
with respect to the total differential dT. However, this is hardly ever the cas*
*e, due to the
fact that in general the null-homotopy s on C, "inducing" s1, fails to be, in a*
* sense that can
be made precise, a map of comodules over C. We now discuss a map of DGR -modu*
*les on
C, which can be considered as the obstruction for s1 to be a null-homotopy of O*
*Er.
Given the null-homotopy s for OEr on C, construct s1 on IC, as in Section 2, *
*and define
= (s) : IC -! IC by
= dEs1 + s1dE:
The following are basic properties of . The proofs are straight forward from *
*the definition
and are left to the reader.
Proposition 3.1. For any null-homotopy s for OEr on C, let denote the correspo*
*nding map
on IC. For each k 1 let k denote the k-th iteration of . Then
(1) k is a map of DGR -modules with respect to both differentials on C.
(2) k is homotopic to (-1)kOErk with respect to the total differential on IC.
(3) k is right extended.
(4) k has bidegree (-k; k).
Corollary 3.2. Let k 1 and suppose that k vanishes on elements of tensor filtr*
*ation 1.
Then k vanishes on IC. Moreover the map OErk is null-homotopic on IC if and onl*
*y if
k is. Thus if k = 0, for some k 1, then IC has a homology exponent rk.
Define sk : IC -! IC by sk = k-1 O s1. Define oek : IC -! IC inductively by
oe1 = s1 and for every k 2, oek = rk-1s1- rk-2s2+ . .+.(-1)k-1sk: The followin*
*g equalities
provide some more insight into the structure of k and can be proven by an easy *
*induction.
Proposition 3.3. With the notation above, the following relations hold.
(1) skdIP+ dIsk = rk-1.
(2) k = ki=0(dEs1)i(s1dE)k-i.
Torsion in loop-space homology of rationally contractible spa*
*ces 7
(3) If s2 = 0 then k = (dEs1)k + (s1dE)k.
(4) k = dEsk + skdE:
(5) OErk= doek + oekd + (-1)kk:
4.Proof of Theorem 1.1 and Proposition 1.3
Let C be a c-connected DGR -coalgebra, c 1, such that H*(JC) has an exponen*
*t r 2 R.
Let s be a null-homotopy for OEr on C, such that s2 = 0. Let T denote the loop*
* algebra
construction C and let s1 and be the maps constructed in the previous sections*
* with
respect to s.
Since C is c-connected, T is (c - 1)-connected and so Tp;q= 0 for q -p(c + 1*
*) - 1. In
particular T-1;q= 0 if q c. For each positive integer i let T (i) denote the i*
*-th skeleton of
IT .
Lemma 4.1. Suppose e d-c+1_c. Then e vanishes on T (i), for i d.
Proof.By Corollary 3.2 it suffices to check that e vanishes on elements of tens*
*or filtration -1,
namely on T-1;i; i d+1. Since bideg(e) = (-e; e), it suffices to verify that T*
*-1-e;d+e+1= 0,
for each c d c(e + 1) - 1. Indeed, by the connectivity argument above T-1-e;q*
*= 0 if __
q ec + e + c, which is satisfied for q = d + e + 1. *
* |__|
Now suppose that C is d-dimensional for some d c. Then we have T-1;q= 0 if q*
* > d.
By the same argument as in Lemma 4.1 it follows that the map e vanishes on IT i*
*f e d-c_c.
However this can be slightly improved.
Lemma 4.2. Suppose that C is c-connected and d-dimensional for some d > c. T*
*hen e
vanishes on IT for e d-c-1_c.
Proof.It suffices to show that e vanishes on T-1;d. Indeed, since E0-1;d+1= 0 a*
*nd s2 = 0,
it follows from Lemma 3.3 that e = (s1dE)e. But (s1dE)e = s1((dEs1)e-1dE): Hen*
*ce the
restriction of e on T-1;dfactors through T-1-e;d+e-1, which vanishes since d+e-*
*1 ec+e+c._
Thus e = 0 on T-1;*and the lemma follows. *
* |__|
We are now ready to prove Theorem 1.1 and Proposition 1.3. Let C be an c-conn*
*ected
DGR -coalgebra with a homology exponent r 2 R. Let T denote C. By Lemma 4.1,*
* e
vanishes on T (i) for i d, if e d+c+1_c. Hence for any x 2 T (i); i d, we ha*
*ve by (5) of
Proposition 3.3
OEre(x) = (doee+ oeed)(x):
Thus re . Hi(C) = 0 for 0 < i d. This proves Theorem 1.1.
Next suppose that Cq = 0 for q > d. By Lemma 4.2, e vanishes on IT , if e d*
*-c-1_c.
Thus OEreis null-homotopic on IT and Proposition 1.3 is proved.
5. Examples
For each pair of positive integers (k; t) define Ck;tto be the graded free ab*
*elian group,
generated by elements {xn}n2 and {yn}n3 , where dimensions of the generators a*
*re given
below.
|x2n| = (2k - 1)(2n - 1) + 1;
8 Ran Levi
|x2n+1| = (2k - 1)(2n - 1) + 2;
|y2n+1| = (2k - 1)2n + 1;
|y2n+2| = (2k - 1)2n + 2:
Define a diagonal map : Ck;t-! Ck;t Ck;tby
x2n = y2n+1= 0
x2n+1= P n-1i=1x2i y2(n-i)+1+ y2(n-i)+1 x2i
y2n+2= P ni=1x2i x2(n-i+1)
where denotes the reduced diagonal and extends to in the obvious way.
Define a differential d on Ck;tby
dx2n = dy2n+1= 0; dx2n+1= tx2n; and dy2n+2= ty2n+1
for each i 1.
Define a map s of degree +1 on Ck;tby
sx2n+1= sy2n+2= 0; sx2n = x2n+1and sy2n+1= y2n+2:
Notice that s2 = 0.
One verifies by inspection that the triple (Ck;t; d; ) is a (2k-1)-connected *
*DGZ-coalgebra
with a homology exponent t and a null-homotopy for OEt given by s. Construct s1*
* and on
ICk;twith respect to the null-homotopy s.
Proposition 5.1. For each positive integer n the following equalities hold in C*
*k;t.
(1n) 2n-1[y2n+1] = (-1)n[x2]2n.
(2n) 2n[x2n+2] = (-1)n[x2]2n+1.
Proof.Since x2n and y2n+1 are primitive for all n 1 it follows from Propositio*
*n 3.3 that
k[x2n] = (dEs1)k[x2n] and k[y2n+1] = (dEs1)k[y2n+1].
One verifies directly that [y3] = -[x2]2 and that 2[x4] = -[x2]3. Thus the s*
*tate-
ment is true for n = 1. Assume that 2n-3[y2n-1] = (-1)n-1[x2]2n-2 and 2n-2[x2n*
*] =
(-1)n-1[x2]2n-1, namely that (1n-1) and (2n-1) hold. We prove (1n) and (2n).
2n-1[y2n+1] = -2n-2dE[y2n+2] =
P n
-2n-2 i=1[x2i|x2(n-i+1)] =
Pn 2n-2
- i=1( [x2i])[x2(n-i+1)]
By Lemma 4.1 all but the last summand vanish and hence by the inductive hypothe*
*sis
2n-1[y2n+1] = -(2n-2[x2n])[x2] = (-1)n[x2]2n
and (1n) is proved. Next
2n[x2n+2] = -2n-1dE[x2n+3] =
P n
2n-1 i=1[y2(n-i+1)+1|x2i] - [x2i|y2(n-i+1)+1] =
P n 2n-1 2n-1
i=1( [y2(n-i+1)+1])[x2i] - ( [x2i])[y2(n-i+1)+1]:
Torsion in loop-space homology of rationally contractible spa*
*ces 9
By Lemma 4.1 all summands vanish except for possibly
(2n-1[y2n+1])[x2] and (2n-1[x2n])[y2(n-i+1)+1]:
But
2n-1[x2n] = 2n-2[x2n] = (-1)n-1[x2]2n-1= 0:
Hence
2n[x2n+2] = (2n-1[y2n+1])[x2] = (-1)n[x2]2n[x2] = (-1)n[x2]2n+1:
*
* __
This completes the proof. *
* |__|
Proposition 5.2. For any positive integer m, the element [x2]m represents a non*
*-zero ho-
mology class in Hm(2k-1)(Ck;t).
Theorem 1.2 follows at once from Proposition 5.2. Indeed, set c = 2k - 1. W*
*e must
show that Hec(Ck;t) contains en element of order te. For e = 2n, consider the c*
*lass ffe 2
Hec(Ck;t), represented by the cycle [ye+1]. Since [ye+1] cannot possibly be a *
*boundary,
ffe 6= 0. By propositions 5.1 and 5.2, e-1*ffe 6= 0. Hence by Proposition 3.1, *
*te-1ffe 6= 0. On
the other hand by Theorem 1.1, teffe = 0. Hence ffe is an element of order te.
For e = 2n + 1, consider the class fie 2 Hec(Ck;t), represented by the cycle *
*[xe+1]. Again
fie is evidently non-zero. Propositions 5.1 and 5.2 imply that e-1*fie 6= 0. He*
*nce te-1fie 6= 0.
On the other hand, by Theorem 1.1 tefie = 0. Hence fie is an element of order *
*te. This
completes the proof of Theorem 1.2.
We proceed by proving Proposition 5.2. Let k and t be fixed and let C denote *
*Ck;t. Let
C* = Hom(C; Z) denote the dual of C. Thus C* is a DGZ-algebra with product giv*
*en
by dualizing the diagonal map on C. By abuse of notation, we denote the duals *
*of xi, yj
in C* again by xi and yj respectively. Let C*(j) denote the sub-algebra of C* *
*generated
by x2; x3; . .;.xj+2; y3; y4; . .;.yj+1 if j is odd and by x2; . .;.xj+1; y3; .*
* .;.yj+2 if j is even.
Define C(j) to be (C*(j))*. Thus C(j) is a quotient DGZ-coalgebra of C and for *
*every j 1
there are natural projections
ssj+1 'j+1
C -! C(j + 1) -! C(j):
The following lemma is immediate from the definitions.
Lemma 5.3. For each n 2, the submodule Ker('n) of C(n) is n(2k - 1)-connecte*
*d. Thus
so is the submodule Ker(ssn) of C.
Since the element [x2] 2 C is preserved under each of the maps ssn, it suffic*
*es to show
that [x2]n is not a boundary in C(n) to complete the proof of Proposition 5.2.
Proposition 5.4. The element [x2]n is not a boundary in C(n) but [x2]n+1 is. In*
* fact for
each m 1 there exist decomposable elements Am 2 C(2m) and Bm 2 C(2m - 1) such
that
(1m ) d(t2m-2[y2m+2] + Bm ) = [x2]2m in C(2m - 1),
(2m ) d(t2m-1[x2m+3] + Am ) = [x2]2m+1 in C(2m).
Furthermore, if z0n2 C(n) is such that dz0n= [x2]n+1, then z0nis indecomposable.
10 Ran Levi
Proof.We first show that [x2]n is not a boundary in C(n). This certainly holds *
*for n = 1.
Thus assume [x2]j is not a boundary in C(j) for j n - 1 and suppose n = 2m. T*
*he
kernel of '2m through dimension (2k - 1)2m + 2 = |y2m+2| is given by the elemen*
*t y2m+1.
Thus the kernel of '2m through dimension (2k - 1)2m + 1 is generated by the ele*
*ment
[y2m+1] if k > 1 and by [y2m+1], [y2m+1|x2] and [x2|y2m+1] if k = 1. Notice tha*
*t in either case
Ker('2m) through dimension (2k - 1)2m + 1 consists of cycles.
Suppose z 2 C(2m) is such that dz = [x2]2m. Then by naturality d'2mz = [x2]2m*
* in
C(2m - 1). But by our induction hypothesis it follows that '2mz is indecomposab*
*le in
C(2m - 1). Hence it contains an essential summand of tensor filtration 1. Sin*
*ce '2m
maps the single element of tensor filtration 1 and total dimension (2k - 1)2m +*
* 1, [y2m+2],
non-trivially, it follows that z itself is indecomposable. But in C(2m), [y2m+2*
*] is not a cycle
with respect to the internal differential. This yields a contradiction. The cas*
*e n = 2m + 1
follows by analogy.
Next we must show that [x2]n+1 is a boundary in C(n). Notice that d[y4] = [x*
*2]2 in
C(1) and that d(t[x5] - [y3|x3] - [x2|y4]) = [x2]3 in C(2). Thus (1m ) and (2m *
*) hold for
m = 1. Assume that for each j m - 1 there are decomposable elements A0jand B0j*
*in C
such that
(10j) d(t2j-2[y2j+2] + B0j) = [x2]2j- t2j-1[y2j+1],
(20j) d(t2j-1[x2j+3] + A0j) = [x2]2j+1- t2j[x2j+2].
Notice that (10j) and (20j) are satisfied for j = 1 with respect to A01= -([y*
*3|x3]+[x2|y4]) and
B01= 0. Proving (1m ) and (2m ) amounts to showing that there are decomposable *
*elements
A0mand B0min C, such that (10m) and (20m) hold. Indeed, reducing (10m) to C(2m *
*- 1)
and (20m) to C(2m) yields (1m ) and (2m ). For a graded algebra A and x; y 2 A,*
* write [x; y]
for the difference xy - (-1)|x||y|yx. We write down the formulas and leave it f*
*or the reader
to verify that the required equations hold.
Define, for m 2
B0m;1= -([x2]A0m-1+ t2m-3([x2|x2m+1] + [x2m|x3])):
X
B0m;2= t2(m-j)-1[t2j-2[x2j+1] + tA0j-1+ [x3][x2]2j-2; [x2(m-j+1*
*)]]:
2j 0 consider the differential dn : C*
*n+1 -! Cn.
Since R is assumed a principal ideal domain and Cn is free, the image Bn(C) of *
*dn is a free
submodule of Cn. Thus there exists a right inverse ffln : Bn(C) -! Cn+1 for dn.*
* Let C0n+1
denote the image of ffln in Cn+1. Let D denote the sub-DGR -module of C given *
*by Di= Ci
for 0 < i n, Dn+1 = C0n+1and Di = 0 otherwise. Define PnC to be the quotient D*
*GR -
module C=D. Notice that PnC is a DGR -coalgebra by construction and that the p*
*rojection
n : C -! PnC is a map of DGR -coalgebras. The following are elementary proper*
*ties of
PnC and are proven in [10].
Proposition 6.1. Let C be a DGR -coalgebra. Then for every n > 0
(1) the projection n is a split epimorphism in DGR m.
(2) n induces a homology isomorphism above dimension n.
(3) PnC is n-connected.
12 Ran Levi
Let C be a DGR -coalgebra and let s denote a self map of C of degree +1. Let*
* n be a fixed
positive integer. Consider the projection = n : C - ! PnC, where PnC is const*
*ructed
with respect to some right inverse ffl for the n-th differential d as above. De*
*note elements of
PnC by x, where x 2 C is some pre-image. A right inverse j of in DGR m is gi*
*ven by
the identity in dimensions higher than n + 1 and for x 2 (PnC)n+1, define j(x) *
*= x - ffldx:
Then "s= sj can be identified with s in dimensions higher than n + 1 and for x *
*2 (PnC)n+1
one has "s(x) = s(x) - sffld(x):
The self map of C given by fs = ds + sd is a map of DGR -modules, for which *
*s is a
null-homotopy. Suppose that fs = OEr for some r 2 R. Then it is easy to verify *
*that "sis a
null-homotopy for the self map of PnC, given by "fs= fsj.
We now specialize to the case mentioned above. Let A be a DGR -algebra conce*
*ntrated in
degree 0. Let B(A) denote the classifying construction [7] for A. A self map s *
*of degree +1
on B(A) is said to be a transfer type null-homotopy, if for each n > 1
X
s([x1|x2| . .|.xn]) = ai[x0i|x00i|x2| . .|.xn];
i
P 0 00
whenever s([x1]) = iai[xi|xi]. Notice that transfer type null-homotopies s a*
*re a priori
unlikely to satisfy s2 = 0.
Proposition 6.2. Let A be a DGR -algebra concentrated in degree 0, which is fr*
*ee as an
R-module. For some r 2 R, let s be a transfer type null-homotopy for OEr on B(*
*A). For
some n 1 let be the corresponding map on PnB(A). Then for every k > n
([[x1| . .|.xk]]) =
(-1)n([s[x1| . .|.xn]|[xn+1| . .|.xk]]) - [sffld[x1| . .|.xn+1]|[xn+2| *
*. .|.xk]]):
Thus 2 = 0 if n > 1 and 3 = 0 if n = 1.
Proof.The proof amounts to calculating on the algebra generators for PnB(A) gi*
*ven by
[[x1|x2| . .|.xk]], xj 2 A, observing the fact that is right extended. In com*
*puting dEs1
on a typical generator [x] of homological degree k, we may assume that k 2n + *
*1, since
otherwise the element "s(x) is primitive in the coalgebra structure of PnB(A) a*
*nd thus dE
vanishes.
X
dEs1[[x1|x2| . .|.xk]] = -dE ai[[x0i|x00i|x2| . .|.xk]] =
P i
iai(-1)n[[x0i|x00i|x2| . .|.xn]|[xn+1| . .|.xk]] +
P P k-2n-1 n+j 0 00
iai j=1 (-1) [[xi|xi|x2| . .|.xn+j]|[xn+j+1| . .|.xk]] =
(-1)n[s[x1|x2| . .|.xn]|[xn+1| . .|.xk]] +
(-1)n+1[s[x1|x2| . .|.xn+1]|[xn+2| . .|.xk]] +
P k-2n-1 n+j+1
j=2 (-1) s1[[x1|x2| . .|.xn+j]|[xn+j+1| . .|.xk]]:
On the other hand
Torsion in loop-space homology of rationally contractible spa*
*ces 13
s1dE[[x1|x2| . .|.xk]] =
(-1)n+1s1[[x1|x2| . .|.xn+1]|[xn+2| . .|.xk]] +
P k-2n-1 n+j
j=2 (-1) s1[[x1|x2| . .|.xn+j]|[xn+j+1| . .|.xk]]:
Thus adding the two equations we get
[[x1|x2| . .|.xk]] =
(-1)n[s[x1|x2| . .|.xn]|[xn+1| . .|.xk]] +
(-1)n+1[s[x1|x2| . .|.xn+1]|[xn+2| . .|.xk]] +
(-1)n+1s1[[x1|x2| . .|.xn+1]|[xn+2| . .|.xk]] =
(-1)n[s[x1|x2| . .|.xn]|[xn+1| . .|.xk]] +
(-1)n+1[s[x1|x2| . .|.xn+1]|[xn+2| . .|.xk]] +
(-1)n[s[x1|x2| . .|.xn+1]|[xn+2| . .|.xk]] +
(-1)n+1[sffld[x1|x2| . .|.xn+1]|[xn+2| . .|.xk]] =
(-1)n([s[x1| . .|.xn]|[xn+1| . .|.xk]]) - [sffld[x1| . .|.xn+1]|[xn+2| *
*. .|.xk]]);
as claimed.
Notice that [[x1| . .|.xk]] = 0 if k 2n. Since is right extended, it follow*
*s that 2 = 0
if n > 1. If n = 1 then 2 might not vanish but then 3 certainly does. This comp*
*letes_the
proof. *
* |__|
Corollary 6.3. Let G be a finite group of order prm with (p; m) = 1. Let B[G] d*
*enote the
classifying construction for the group ring Z(p)[G]. Then p2r annihilates H"*(*
*PnB[G]) if
n 2 and p3rannihilates "H*(P1B[G]).
Proof.For a finite group G of order N there is a canonical null-homotopy s for *
*the map OEN
on B[G] given by X
s([x1|x2| . .|.xn]) = [x|x1|x2| . .|.xn]:
x2G
The map s is easily seen to satisfy the hypotheses of Proposition 6.2. The resu*
*lt follows_from
Corollary 3.2. *
* |__|
Theorem 1.5 now follows by combining Corollary 6.3 and Proposition 1.4, which*
* we prove
in the next section.
7. Geometric Realization of the Construction Pn
Throughout this section whenever we say a space, we mean a pointed connected *
*CW
complex for which the cellular chain complex admits a natural strictly coassoci*
*ative diagonal.
This can of course always be obtained by geometrically realizing the associated*
* singular
simplicial set. This restriction is imposed here for simplicity and has no furt*
*her significance.
Let X(n)denote the n-skeleton of a given space X. Consider the space Pn0X obt*
*ained from
X by collapsing X(n)to the base point. Obviously the natural map 0n: X -! Pn0X *
*induces
14 Ran Levi
an isomorphism on homology groups Hi(-; Z) for i n + 2. Furthermore, Hn(X(n); *
*Z) is
free abelian and thus it follows by inspection of the associated long exact hom*
*ology sequence
that Hn+1(Pn0X; Z) ~=Hn+1(X; Z) F , where F is a free abelian group and that 0*
*ninduces
the obvious split monomorphism in this dimension. Notice that the free abelian*
* group F
is no other but the subgroup of boundaries Bn(S*(X; Z)), where by S*(-; Z) we d*
*enote the
cellular chain complex on X.
Next the Hurewicz isomorphism theorem implies that ssn+1Pn0X ~=Hn+1(X; Z)F . *
*Choose
fi 0
generators {fi}i2Ifor F and realize each fias a map Sn+1 -! PnX. Define PnX to*
* be the
cofibre in the sequence
_ W fi
Sn+1i-! Pn0X -! PnX:
i2I
It is immediate that the natural map n : X -! PnX induces an isomorphism on hom*
*ology
groups in dimensions larger than or equal to n + 1. This completes the proof of*
* the first part
of Proposition 1.4.
To prove the second part an intermediate step is needed. For a DGR -coalgebr*
*a C, define
Pn S as follows. Let F denote a free R-module on as many generators as the sub*
*module
of boundaries Bn(C). Without loss of generality we may identify F with Bn(C), t*
*hus the
differential d : Cn+1 -! F is an epimorphism and one can choose a right inverse*
* ffl for it.
Let PnC be the DGR -coalgebra given by dividing C by its n-skeleton with the a*
*dditional
modification that (Pn C)n+2 = Cn+2 F . Define d(x; y) = dx + ffly to be the di*
*fferential on
(Pn C)n+2. Require that every element in (Pn C)n+2 is primitive. It is thus eas*
*y to verify that
the obvious map : C -! PnC is a map of DGR -coalgebras inducing an isomorphis*
*m on
homology in dimensions larger than n. If C = S*X, then making compatible choice*
*s for the
attaching maps defining PnX out of Pn0X and the right inverse ffl, we immediate*
*ly get that
there is an isomorphism of DGR -coalgebras PnS*X ~=S*PnX.
Next for any DGR -coalgebra C, one can define PnC and PnC with respect to th*
*e same
right inverse ffl for the differential Cn+1 -d!Bn(C). Thus it is immediate that*
* there is a quasi
isomorphism of DGR -coalgebras PnC -! PnC. Applying this for C = S*X, complete*
*s the
proof of Proposition 1.4.
We end this section by pointing out that although Pn(-) and Pn(-) are equival*
*ent con-
structions on DGR -coalgebras, Pn(-) behaves nicer with respect to algebraic ma*
*ps due to the
fact that the projection C -n! PnC is a split epimorphism in the category of DG*
*R -modules.
On the other hand Pn(-) corresponds more naturally to the geometric analogue.
8. Nilpotent DGR -algebras
In this section we prove Theorem 1.6 and observe that it does not correspond *
*to a geometric
situation. We begin with a proof of the theorem.
Let A be a DGR -algebra and assume that the classifying construction B(A) ad*
*mits a
homology exponent r 2 R. Assume further that A is nilpotent as an algebra, name*
*ly that
some power of its augmentation ideal IA vanishes.
There is a quasi isomorphism of DGR -algebras [7]
ff(A) : B(A) -! A:
Torsion in loop-space homology of rationally contractible spa*
*ces 15
Since ff(A) is multiplicative, ff(A)(IB(A))n (IA)n, for every n 1.
Next, by hypothesis B(A) admits a homology exponent r 2 R. Thus we have a nu*
*ll-
homotopy s for OEr on B(A). Let denote the map constructed on B(A) with respec*
*t to s.
Then the k-th iteration k of increases tensor filtration by k and is homotopic*
* to OErk up
to a sign. Thus for every k 1 we get a commutative diagram
k ff(A)
IB(A) _______-(IB(A))k+1 _______-(IA)k+1
| | |
| | |
| | |
| | |
| | |
|= |inc |inc
| | |
| | |
| | |
| | |
|? |?k ff(A) |?
IB(A) _________-IB(A) ___________-IA
By hypothesis (IA)n = 0, which implies Theorem 1.6.
Next we observe that Theorem 1.6 does not refer to a topological situation, b*
*ut rather to
a purely algebraic one. The content of this remark is contained in the following
Proposition 8.1. Let X be a simply-connected, rationally contractible space of *
*finite type.
Then H*(X; Fp) is not a nilpotent algebra.
Proof.First observe that a finite loop space cannot possibly have a rationally *
*contractible
classifying space. Indeed R. Kane showed [8, pages 256 and 300] that the loop s*
*pace homology
of simply-connected finite H-spaces is torsion free and hence rationally non-tr*
*ivial. Thus by
C-class theory of Serre, a finite H-space cannot have a rationally contractible*
* classifying
space. Hence we may assume that X is infinite dimensional.
Consider the Hopf algebra H*(X; Fp). By assumption it is an infinite dimensio*
*nal, co-
commutative Hopf algebra, which is nilpotent as an algebra. But such Hopf algeb*
*ras do_not
exist by Proposition 8.2 below. *
* |__|
Proposition 8.2. Let A be a cocommutative Hopf algebra of finite type and infin*
*ite dimen-
sion over a field k of characteristic p > 0. Then A is not a nilpotent algebra.
The rest of this section is devoted to the proof of Proposition 8.2. The prop*
*osition is rather
elementary and should be well known to the expert. We include a proof for the c*
*onvenience
of the reader. Graded vector spaces are always assumed to be of finite type.
Lemma 8.3. Let A be a cocommutative Hopf algebra of finite type over a field *
*k of char-
acteristic p > 0, which is nilpotent as an algebra. Then the underlying algebra*
* of the dual
Hopf algebra A* is locally nilpotent, namely there exists no element x 2 A of i*
*nfinite hight.
Proof.Let A be a cocommutative Hopf algebra over a field k of characteristic p *
*> 0, which
is nilpotent as an algebra. Assume that the dual commutative Hopf algebra A* is*
* not locally
nilpotent as an algebra. Then we claim that A* contains a primitive element y o*
*f infinite
hight. Indeed choose an element x of infinite hight and minimal degree. Then, s*
*ince every
k
element of lower degree is nilpotent, some power y = xp must be primitive.
16 Ran Levi
Let y 2 A* be a primitive element of infinite hight. Then there is a monomor*
*phism of
Hopf algebras
OE *
P [y] -! A :
Dualizing we get an epimorphism of Hopf algebras
OE*
A -! [y];
where [y] denotes the divided polynomial algebra over k on one generator. In pa*
*rticular [y]__
contains products of arbitrary length and thus so does A contradicting its nilp*
*otency. |__|
Lemma 8.4. Let A be a commutative Hopf algebra of infinite dimension over a f*
*ield k of
characteristic p > 0, which as an algebra is locally nilpotent. Then the dual H*
*opf algebra A*
contains non-vanishing products of arbitrary length.
Proof.Since A is commutative the Borel structure theorem applies and A is isomo*
*rphic as
an algebra to a tensor product of monogenic algebras. Assuming that A is locall*
*y nilpotent
implies that the Borel decomposition for A does not contain polynomial factors.*
* Hence it
must be infinitely generated as an algebra. Consequently the module of primiti*
*ves in the
dual Hopf algebra P (A*) is infinite dimensional. But a product of an arbitrary*
* number of __
distinct primitives in a Hopf algebra is never zero. The result follows. *
* |__|
Lemmas 8.3 and 8.4 imply that a cocommutative Hopf algebra over k, for which *
*the
underlying algebra is nilpotent must be finite dimensional, which completes the*
* proof of
Proposition 8.2.
9.speculations, finiteness properties
In the study of loop space homology torsion, it seems reasonable to try and r*
*elate certain
finiteness conditions on the objects under consideration to the existence of ex*
*ponents. One
may take two different approaches here. Fix a DGR -coalgebra C with a reduced *
*homology
exponent r 2 R.
The first approach is to assume certain finiteness conditions on C and try to*
* gain control
on the growth of torsion in loop space homology. In particular one might be int*
*erested in
finding a loop space homology exponent. Thus one may assume for instance that C*
* is of finite
rank over R, in which case we have shown that C admits a homology exponent. If *
*C has a
trivial coproduct then it is immediate that C and C have the same homology expo*
*nent as
is the case when C is the chain coalgebra for a rationally contractible suspens*
*ion space. One
might thus expect that nilpotency of the Hom-dual of a DGR -coalgebra C should*
* imply the
existence of a loop space homology exponent. However by Theorem 1.2 this is not*
* nearly the
case. Indeed the DGZ-coalgebras Ck;tconstructed in the theorem have the proper*
*ty that
their duals are nilpotent of degree 3.
A finiteness condition on C, which we did not consider here is the existence *
*of only finitely
many primitives in C. Namely one may conjecture that if C contains only finite*
*ly many
primitives, then C admits a homology exponent. In fact we are not aware of a c*
*ounter
example to this.
The second approach to the problem is taken in Section 8 above and arises by *
*recalling the
fact that for every DGR -algebra A, the natural map B(A) -! A is a homotopy equ*
*ivalence.
Torsion in loop-space homology of rationally contractible spa*
*ces 17
Thus one may like to consider DGR -coalgebras of the form B(A), which satisfy *
*our basic
hypothesis of having a homology exponent and impose additional finiteness condi*
*tions on A.
If A is concentrated in degree 0, then the study of its homology doesn't make*
* too much
sense. But in this case one might like to study the associated DGR -algebras P*
*nB(A). If
A is of finite rank over R, as is the case if A = R[G] for a finite group G, th*
*en it is not hard
to construct a condition which imply that B(A) admits a transfer type null-homo*
*topy for
OEr, where r is the image of the rank of A in R under the canonical map from th*
*e integers.
However the existence of such a null-homotopy appears to impose severe restrict*
*ions on the
algebra structure of A. We thus find it reasonable to conjecture that there exi*
*st algebras A,
concentrated in degree 0, which are finitely generated as algebras or even of f*
*inite rank over
R, such that B(A) has a homology exponent but PnB(A) does not have one.
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[11]R. Levi; A Counter Example to a Conjecture of Cohen; to appear in the 1994 *
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[12]R. Levi; On Homological Rate of Growth and the Homotopy Type of BG^p; prepr*
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[13]J. W. Milnor and J. C. Moore; On the Structure of Hopf Algebras; Annals of *
*Math. 81 (1965), 211-264.
Mathematisches Institut, Universit"at Heidelberg, Im Neuenheimer Feld 288, 69*
*120 Hei-
delberg, Germany
E-mail address: rlvi@vogon.mathi.uni-heidelberg.de