ALGEBRAIC STRUCTURE IN LOOP SPACE HOMOLOGY

Jonathan A. Scott

A thesis submitted in conformity with the requirements

for the degree of Doctor of Philosophy

Graduate Department of Mathematics

Universi@ of Toronto

@ Copyright by Jonathan A. Scott (2000)

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ALGEBRAIC STRUCTURE IN LOOP SPACE HOMOLOGY

by Jonathan A. Scott

Degree of Ph-D., 2000

Graduate Depôrtment of Mathematics, University of Toronto

Abstract

Let X be a finite, n-dimensional, r-connected C W cornplex Suppose p 2 n/r is

an odd prime. We show that each term in the Ioop space homology Bockstein spectral

sequence modulo p is the universal enveloping algebra of a differential graded Lie dgebra-

In particular, as shown in [la], there &s a natural isomorphism of Hopf algebras

H@X; F,) UEx, where Ex is a naturally dehed graded Lie dgebra Let Lx be the

plocal Ani& differential graded Lie algebra model for X [3], which plays the rôle of the

Quillen model [23] in rational homotopy theory We prove that there is an isomorphism of

graded Lie dgebras H(Lx 8 Fp) Ex x K, where K is an abelian (rp + p - 3)-connected

ideal. Using Anick's mod p Hurewicz theorem [5], we then have E, Y x,(QX; Fp) for

m 5 min(r +2p - 3, rp - 1) , where z,(-; F,) is the mth homotopy group with coefficients

in Fp [22].

There are many people who have helped me through the writing of this thesis, to one

extent or another-

First and foremost, 1 thank my supervisor, Steve Halperin, for his profound insight,

patience and support,

To rny parents, Thomas and Evelyn Scott, 1 express my love and gratitude- They

instilled in me a deep thirst for lmowledge that has driven me through He- My brothers,

David and Mark, have been great fiends Without them 1 would not be the person 1

m-

I would Iike to thank my extemal examiner, Biü Dwyer, whose comments helped to

improve the presentation of the thesis and to put the work in greater context.

1 am indebted to Peter Bubenik for his meticulous proofreading, and to the senetarial

staff of the Department of Mathematics at the University of Toronto, particularly Ida

Bulat and Pat Broughton, for making the whole process so painless.

Findy, 1 would like to thank my friends, who helped me tkough the hard times

and celebrated with me the good tirna- 1 would hate to incriminate any of them by

mentioning them by name, but they know who they are!

Contents

1 Introduction 1

2 Preliminaries 7

2.1 Graded coalgebras and graded Hopf algebras . . . . . . . . . . . . . . . . 10

2.2 Graded Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Dividcd powers algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 .3.1 r-algebras and homology . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.2 The Cartan-Chedey-Eilenberg-Cartan complex . . . . . . . . . 18

2.4 Bockstein spectral sequences . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Semifree resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Acyclic closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Bockstein spectral sequences and universal enveloping algebras 24

3.1 The image of H(L) + H(UL) . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 The dual of a î-derivation . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 BoCkstein spectral sequence of a

universal enveloping algebra . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Sullivan Decomposition 32

. . . . . . 4.1 Proofofthedecomposition. .. .. . . . . . . . . . . . . . . . 34

4-2 Acyclic Sullivan algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Homology of a Merential graded Lie algebra 43

5.1 DGL's and their homotopy Lie algebras . . . . . . . . . . . . . . . . . . . 45

5.2 Lie algebras reduced modulo p . . . . . . . . . . . . . . . . . . . . . . . . 47

6 Some examples 49

6.1 Caiculations using acyclic closures . . . . . . . . . . . . . . . . . . . . . . 51

6.2 Counterexamples to rationai behaviour . . . . . . . . . . . . . . . . . . . 55

6.3 Behaviour of the Bockstein spectral sequence . . . . . . . . . . .. . . . . 57

Chapter 1

Introduction

Let O X be the Moore loop space on a pointed topological space X. If R Q is a principal

ideal domain, then H,(QX; R) has a natural Hopf algebra structure via composition of

loops, as long as there is no torsion. The submodule P c H,(QX; R) of primitive

elements is a graded Lie subalgebra; in [24, Milnor and Moore showed that if R = Q and

X is simply connected then H,(OX; Q) is the univasal enveloping algebra of P. In [181,

Halperin established the same conclusion for R C Q when X is a finite, simpiy-comected

CW cornplex, provided that H*(QX; R) is torsion-free and the least non-invertibIe prime

in R is d c i e n t l y Iarge-

In the presence of torsion, the loop space homology algebra does not have a natural

Hopf algebra structure. However, in [SI Browder showed that the BockStein spectral

sequence

H. (0% Fp) =+ (H. (RX; Z) /torsion) 631 Fp

is a spectral sequence of Hopf algebras. Halperin also proved in [18] that for large enough

1

primes, H*(OX; Fp) is the universal enveloping algebra of a graded Lie algebra. The fkst

part of the thesis establishes this for every term in the Bockstein spectral sequence.

Theorem A. Let X be a finite, n-dimensional, q-connected CW complez (q 2 1). I f p

is a n odd prime and p 2 n/q, then each t e m in the mod p homology Bockstein spectml

sequence for QX is the universal enveloping algebm of a diflemntial graded L e algebm

(Lr , a). fifihennore, the sequence conveqes to a universal envebping algebra U(LOD) .

It is important to stress at this point that the association X -r {Lr) is not natural;

see Example 3-7-

In [3], under the hypotheses of Theorem A, Ani& associates to X a diaerential graded

Lie algebra Lx over Zb) and a naturai quasi-isomorphism UAx -t C.(RX; Zb)) of Hopf

algebras up to homotopyi. It follows that there is an isomorphism of mod p Bockstein

spectral sequences F ( U L x ) C(S2X). The inclusion LX : Lx + ULx therefore induces

a transformation of Bockstein spectral sequences F ( L ~ ) : G ( L x ) + F ( 0 X ) .

Theorem B. Given the notation and hypotheses above, the image of each F ( L ~ ) is

contained in Lr .

Given the results of [3], Theorems A and B foIlow immediately hom the fouowing

purely algebraic result:

Theorem C. Let (L, a) be a diffemnticu gmded Lie algebm over Zbl which is connected,

free as a gmded module, and of finite type- Then the mod p honology Bochtein spectml

sequence of U(L, a) D a sequence of universal enveloping algebras, F ( U L ) = U(Lr, p).

f i~thennore, c : L + UL is the inclusion, then the image of 6(&) is contained in Lr.

The proof of Theorem C depends in an essential way on the work of An&& [2] and

Sjodïn [25], which characterizes the cocommutative Hopf algebras of finite type over a

field k which can be written as universal enveloping aIgebras- NameLy, such a Hopf

algebra A can be Wntten as UL if and only if the dual Al can be given the structue of

a Hop£ algebra with divideci powers (see Theorem 2.5).

In the remainder of the thesis we continue the process begun in [6] of adapting the

machhery of rationd homotopy theory to more general codcient rings.

It is a classical theorem of Sulli- [26, Theorem 2.21 that any Sullivan algebra (over

Q) is the tensor product of its unique minimal model and a contractible algebra- A

contractible algebnr over Q is a free graded commutative dgebra A(X @ dX). Over an

arbitrary commutative ring, such an algebra is not necessarily acyclic. Loosely speaking,

we will c d a commutative dinerential graded algebra (A, d) R-contmctible if it is a

Sullivan model of A(dX) @ r ( X ) , where the graded R-module X is determined by (A, d)

and r(X) denotes the fiee divided powers algebra on X . If R = Q, then we recover the

original d a t i o n . R e c d fiom [6] that a commutative DGA (A, 8) is admikszble if there

exists a DGA surjection a : (B, d ) ++ (A, a), where (B, d) is an R-fkee CDGA that satisfies

H(B, d) = Eo(B, d ) = R. We will call a an acyclic cover of (A, a). Let R be a local

principal ideal domain containhg 1/2- We prove

Theorem D. Let ( A v d) be an admissible Sulliuan algebra over R, such that V = VZ2

and each Vi is R-jke of finite type.

1. There twists a minimal mode1 m : (hW, d) 7 (A% d) which ii split injective.

2. (AK d ) (AW, d) 8 (AU, d) where (AU, d) is an R-contructible algebra

3. A quasi-.EsomorphZsrn between minimal Sullivan algebnrs over R is an 2somorphism-

4. Al1 of the minimal Sulliuun rnodels of an admissible commutatiue diflerential gnrded

algebra are isomorphz'c,

In [23], Quillen formulates rational homotopy theory in te- of differentid graded

Lie algebras. The Quillen mode1 Lx of a simply-connected space X is a ciifkentia1

graded Lie algebra over Q that satisfies H(U&) H.(QX; Q) U(,(nX) 631 Q), so

H(&) G(QX) @ Q-

Let (L, a) be a differentid gradecl Lie dgebra over Fp whose cochai. algebra C*(L)

(see Section 2.3.2) is admissible. Let L : L -t UL be the canonical inclusion. By [18],

there is a natural isomorphism of Hopf algebras H(UL) "- UE, where E is a naturally

defined graded Lie algebra. We prove

Theorem E. With the hypotheses and notation above,

1. H(L) E x K as Lie algebras, where K = kerH(~) is an abelian ideal, and

2. if L is (q - 1) -connected, then K is (pq + p - 3) -connected, so Hm(L) E, for

m 5 p q - f - p - 3 .

- - suppose now that (L, a) = (t, 8) @ Fp for some Z@)-free DGL (L, a) over Z6) such

that c*(E, a) is admissible. Then 8 induces Bockstein operators on H(L) and on UE.

Since UE is the first term of the mod p Bockstein spectral sequence for U& Theorem C

says that the Bockstein presmes E in UE.

Theorem F . The natuml map H(L) induces an isomorphisrn of gmded Lie algebm

H(H(L), P ) q m% Pl -

4

W e next apply Theorems E and F to the topological setting- Let X be a finite, q-

connected, n-dimensional C W cornplex, and suppose that p 2 n/q. The Anick mode1

of X, Lx, is a DGL over Zbl susuch that ULx 7 Ce(- Zb)) as Hopf algebras up to

homotopy [3]. Applying Halperin's resdt [Ml, we get a natural Hopf algebra isomorphism

K(U(Lx 8 F,)) = UEx. By analogy to the rational case, the graded Lie algebra Ex is

called the mod p homotopy Lie algebra of X-

Theorem G. With the notation and hypotheses above,

1. H(Lx @ F,) Ex x K, where K is an abelian ideal.

3. LX : LX + ULX induces on isomorph2sm H(H(Lx 8 Fp), ,û) H(Ex,P) .

Let T,(-; Fp) denote the mth homotopy group with coefficients in Fp [22]. In [4],

Ani& showed that the mod p Hurewicz homomorphism ~r,(aX; Fp) + ET,(& @ Fp)

(m s p q - l ) , is an isomorphismifrn 5 min(qf2p-3,pq-1) . Sinceq+2p-3 <pq+p-3

i f q ? 2 andpq- 1 S p q t p - 3 if q=1 , wehaveproved

Theorem H . (cf. [26, Theom 10.1]) (Ex), S rm(QX; Fp) for m 5 min(q + 2p -

3, w - 1).

The thesis is organized as follows. Chapter 1 is this introduction. Chapter 2 outlines

notation, terminology, and other required background material. In Chapter 3 we prove

Theorems A, B, and C. Chapter 4 is devoted to the proof of Theorem D, as well as

the carefur analysis of R-contractible algebras. In Chapter 5, we apply the results of the

previous chapter to prove Theorems E, F, G, and H- W e conclude by presenting some

relevent examples in Chapter 6-

Chapter 2

Preliminaries

Let R be a commutative ring with unit. The ring of integers localized at a prime p is

denoted Zb), while the field of integers modulo p is denoted by Fp-

A graded R-module M is a collection of R-modules {Mi) indexed by the integers.

We Say that x E M if x E Mi for some j E 2. In this case we say that x has degree j,

denoted 1x1 = j. We use the convention Mj = M-j to raise and lower degrees. The fkee

graded R-module on (graded) basis {x,) is denoted R{xm). A graded R-module M is

said to have finite type if each Mi is hitely generated as an R-module. The suspension

of a graded R-module M is the graded R-module SM with SM)^ = Mi-1. The element

of SM corresponding to x E M is denoted sx.

An R-linear map of graded modules cp : M + N of degree k E Z is a collection of

R-linecur transformations

{ ~ j : Mj Nj+k)jE~-

An R-linear rnorphisrn is an R-linear map of degree zero. Hom(M, N) is the graded

7

module where (Hom(M, N))i is the module of Iinear maps M + N of degree k.

A differentfal a on a graded module M is a linear map of degree -1 satisfying t3a = 0.

In this case the pair (M, a) is cailed a diflerential gmded modde, DGM for short. The

homology of (M, 8) is defined as the quotient module H(M, a) = ker a/ima. Elements of

ker a are c d e d cycles while dements of ima are c d e d boundaràes- The homology class of

a qc le z is denoted [%Ir]. A DGM morphism p r (M, a) -t (Mt, 8) is an R-linear morphisrn

satissinp <p8 = a(p. The induced b a r morphism H(9) : H(M, a) + H(Mr, a) is dehed by H ( 9 ) [z] = [ ~ ( z ) ] . A (co)chain cornplex is a DGM concentrated in non-negative

(non-p ositive) degrees .

If (M, a) and (Mt, û') are DGM'S, then Hom(M, Ml) is naturdy one too. Indeed,

let f : M + Mt be a linear map of degree k. We define df by (df )(s) = û'( f (x)) -

( - I )~ f (Bx), for x E Mt. In this formulation, a DGM homomorphism is just a O-cycle

in Hom(M, Mt). The dval of (M, a) is the DGM, (M, a)fl = Hom(M, R), where R is

considered to be a DGM concentrated in degree O, with trivial diEerentïaI.

A graded R-algebra A is a graded R-module together with R-lineas morphisms cp :

A @ A + A (the muZtiplàcation) and r ] : R -t A (the unit) satisfying the appropnate

diagrams for associativity, etc. [21]. A graded algebra is called connected if A. = R and

A, = O for either n < O or n > 0.

A graded algebra morphism is an R-linear morphism which respects the multiplicative

structures. A derivation 0 : A -+ A is a linear map of degree k such that 8(aut) =

8(a)ar + (-l)klala~(at). A diffeential gmded algebra, DGA for short, is a pair (A, a), where A is a graded algebra and t3 is a derivafion on A of degree -1 satisfying at3 = O. A

DGA (A, B) satisfying A = AZo (resp. A = AZo) is cded a chain aalgebm (resp. cochain

algelira).

An augmentation is a graded algebra motphism E : A -t R. A graded algebra A dong

with an augmentation E ïs caUed an avgmented graded algebra. We henceforth assume

that aI.L graded algebras are augmented. The module of indecomposable elements of an

augmented graded dgebra A is the graded module Q(A) = ker E/ ker E - km&. Let (A, a) be a DGA. A lefi (A, &nodule is a DGM (M, d) over R dong with a

structure morphism A @ M -t Ml a @ rn t+ a mm, that satlsfies d(a m) = (au) - m + (-i)laia - dm.

The tensor algebm on a graded R-module V is denoted TV. H V is the fiee R-

module on basis {xl, xz, . . . ), then we will sometimes denote TV by T(xl, xz, . . . ). Note

that TV = @k>oTkV - where T ~ V consists of a,Il elements of TV of wordength k. Set

TCV = @&T~v. TV is naturally augmentecl, where kerr = TCV and Q(TV) V.

The symmetric group Sk acts on TkV via

where the sign is determincd by the rule x @ y C, (-l)l=llvly @ x.

A graded algebra A is commutative if ab = (-l)[allblba for all a, b E A. The f i e

commutative algebra on a graded R-module V is denoted by AV. It is the quotient

AV = TV/I, where I is the ideal generated by elements u @ u - (-l)lullvlv 8 u for

u, v E V. Note that AV = @&AkV, where AoV = R, and Akv consists of elements

of wordlength k- Denote by A+V = @&AkV the kernel of the natural augmentation.

Note that Q(AV) V.

A commutative DGA ( s h o r t 4 to CDGA) of the fonp (AV, d) is caJled a Sullivan

algebm if V is R-hee, V = VzL, and there exkts an increasing sequence of submodules

such that d : V(k) + AV(k - 1). We can write the MerentiaI as the sum d = 4 of derivations 4 increasing wordlength by i-

Now let R be a principal ideal domain, and consider a Sullivan algebra of the form

(AW, d ) over R, where W = Wz2 is R-f ie of finite type.

Defmition 2.1. The Sullivan algebra (Aw d ) above is R-minimal if for i 2 2, do :

WC + riWi+' where E R is noninvertible.

Remark 2.2. If R = Zb), then : W + pW in a minimal Sullivan algebra; in an

F,-minimal Sullivan algebra, t& = O.

Remark 2.3. If R is, in addition, a local ring with residue field k, then a Sullivan algebra

(AW, d ) over R is R-minimal if and only if (AW, d ) 8 k is k-minimal.

Suppose (A, a) is a commutative cochain aigebra satisfying Ho(A) = R, K1(A) = 0,

R2(A) is R-fiee, and K*(A) is of f i t e type. Then by [18, Theorem 7-11, there d s

a quasi-isomorphism m : (AW, d) 7 (A, a) h m an R-minimal Sull ivan algebra. This

quasi-isomorphism is called a minimal mode1 for (A, a).

2.1 Graded coalgebras and graded Hopf algebras

A graded coalgebm is a graded module C dong with structure morpbisms A : C -t C @ C

(the comultiplication, coproduct, or diagonal) and E : C + R (the CO-unit) satisfying the

usual diagcams for coassociativi~ etc. [21]. A graded coalgebra C is cocommutatTve if

TA = A, where r : G 8 C + C 8 C is given by T(X 8 y) = (-l)lZllyly @ x for x, y E C.

A linear morphism cp : C -î Cf is a morphisrn of coalgebm if it cornmuttes with the

structure morphisms- A coderivation of degree k on C is a linear map 0 : C + C of

degree k making the following diagram cornmute.

A differential graded coalgebm (DGC for short) is a DGM (C,a) where C is a graded

coalgebra and i3 is a coderivation. If (C, a) is a DGC and H ( C ) is torsion-free, then H ( C )

is naturally a coalgebra. The dual of a (differential) graded coalgebra is a (differential)

graded algebra. The dual of an R-projective (differential) graded algebra of fmite type

is a (Merential) graded coalgebra.

A graded Hopf algebra is a graded module H equipped with structure morphisms

p : H @ H + H , A : E + H @ H , ~ ) : R + H , a n d ~ : H + R , s u c h t h a t K w i t h p a n d

7 ïs a graded algebra, H with A and E is a graded coalgebra, and A and E are algebra

morphisms (equivalently, p and q are codgebra morphisms).

A Hopf algebm morphisrn is a linear morphism respecting all of the various structure

morphisms. A Hopf derivation is a derivation which is also a coderivation. A diflerential

graded Hopf algebm (DGH for short) is a DGM (61, a) where H is a graded Hopf algebra

and 3 is a Hopf derivation. If (H, 3) is a DGH, then H(H) is a graded algebra; if, in

addition, H ( H ) is torsion-fiee, then it is a graded Hop£ algebra.

2.2 Graded Lie algebras

Let R be a commutative ring containing 1/2- A grnded Lie algebra is a graded R-module

L = eLZOLk - dong with a degree-zero R-linear map [,] : L @ L + C, cded the Lie

bracket, satisfyuig

2. graded Jacobi identity:

for x , y , z E L;

3. [x, [x, XI] = O for z E Ldd-

A graded Lie algebra is cded connected if it is concentrated in strictly positive

degrees.

We may consider any non-negatively graded associative algebra A to be a graded Lie

algebra via the graded cornmutaior bracket [a, b] = ab - (-l)["llblbu, for a, b E A.

Associated to an F,-minimal mode1 m : (AW, d ) 7 (B, a) of a 1-connected CDGA

(B , a) of h i t e type is its homotop y Lie algebra, E. As a graded vector space, E = (s w ) I ;

the bracket is defined by the relation

for w E W , x, y E E. If there is a DGA surjection (C, d) + (B, a), where H(C,d) =

Ho(C, d ) = Fp, then it foilows fiom Theorem D(4), below, that the assignment (B, a) H

E is functorial. Eûrthermore, it follows fkom Proposition 6.2 and Theorem E that E is a

direct factor of a DGL consmicted from (B, a). The fiee Lie algebrn on a graded R-module V is denoted by L(V). The abelian Lie

algebm on a graded set {sa) is denoted by Lab(xQ); its uuderlying graded R-module is

R{xa), given the trivial bracket.

Let L be a graded Lie algebra over R, and denote by L~ the underlying graded module.

The universal enveloping algebm of L is the associative algebra UL = (TL~)/I, where 1

is the ideal generated by elements of the form x 8 y - (-l)l=lluiy @ x - [x, y], for x, y E L. If

L is R-fiee then the inclusion L~ + TL^ induces a Lie monomorphism L : L -t UL. The

dgebra UL is universal in the sense that if A is any associative dgebra, and if f : L + A

is a Lie morphism from L to the Lie algebra associated to A, then f extends to a unique

algebra morphism f : UL + A such that f = fi. UL has the natural structure of a

graded Hopf algebra with L primitive; indeed, the diagonal map L + L x L extends to

an algebra morphism A : UL + UL @ UL-

A Lie derivation on a graded Lie algebra L is a linear operator 8 on L of degree k

such that for x, y E L,

A diflerential gmded Lie algebm (DGL for short) is a pair (L, a), where L is a graded Lie

dgebra, and a is a Lie derivation on L of degree -1 sat ismg ûô = O, The homology of

a DGL is a graded Lie algebra. If (L, a) is a DGL, then a extends to a derivation on UL,

making U(L, a) into a DGA.

2.3 Divided powers algebras

Divided powers dgebras arise here as the duals of universal enveloping algebras, in a

sense to be made precise in Theorem 2.5.

Dennition 2.4. A divided powers algebm, or L'-dgebra, is a graded commutative algebra

A, such that A = Aro or A = Aa, equipped with set maps (k 2 O, n 2 1) yk : AZn -t

A~" s a t i m g the following Iist of conditions.

1- ro(a) = i;rl(a) = a for a a8A;

2. 9 (a + b) = x:=., rj(a)rk-j(b) for a, b E A*";

( j k ) ' 4. 7j(yk(a)) = _ j i + k ( a ) for a E Ah; 3 !kk!

akrk(b) if la[ and lbl even, Ibl # O , 5. yk(ab) -

if la1 and Ibl odd.

A r-morphism is an algebra morphism which respects the divided powers operations.

A I'-defivation on a L'-dgebra A is a derivation @ on A satisfying O(yk(a)) = @(a)+-'(a)

for a E A2*, k 2 1. A differntial gmded r-algebm, or r-DGA, is a DGM (A, a), where A

is a r-algebra and a is a r-derivation.

If A and B are r-algebras, then A @ B is given a divided powers structure via condi-

tions (2) and (5) of Definition 2.4, noting that a 8 b = (a 8 1) ( 1 8 b).

A Hopf algeàra with divided powers, or Hopf I?-algebm, is a Hop£ algebra whose

comultiplication respects the divided powers.

Theorem 2.5. (André-Sjadin) Let A be a connected Hopf algebm of finite type over a

field k , where char h # 2. Then its duaL Hopf algebm Ag is isomorphic €0 the universal

enveloping algebm of a gmded Lie algebm over k i f and only if A A a Hopf I'-algebm.

Remark 2.6. Theorem 2.5 was proved first by André in [2] in dud form; Sjodin proved

the result directly in [25], extending it to arbitrary characteristic using the notion of "ad-

justed" graded Lie algebras in characteristic 2. Sjodin &O proved for the "if9 direction

that A is fiee as a r-algebra (see below for the definition of a fiee I'-algebra).

A difleerenttiaal gmded Hopf ï-algebrq or CDGH, is a EDGA (A, a), where A is a Hopf

I'-aIgebra and a is a coderivation.

Recall the tensor wdgebm G V on a graded R-module V- As a graded R-module,

TcV = TV; elements of T$V are denoted by [vil - -- luk]- The diagonal is given by

where we use the convention [vi 1 - - luo] = [ ~ k + ~ l - - luk] = 1. The shutae produd [18,

Appendix] gives the tensor coalgebra a commutative (but not CO-commutative) Hopf

algebra structure.

Let rk(V) be the graded submodule of TZV of elements fixed by the action of the

symmetric group Sk. Then r(V) = @krk(V) is a Hopf subalgebra of Tc(V), called the

free r-algebra on V. Divided powers are defined on r(V) by

1. 7°(v) = 1, yl(v) = v for v E V7

2. 7k(v) = [ V I - - - IV] for v E - k timeS

and then extending via conditions (4) and (5) of Definltion 2.4. If f : V + A is any linear

rnap of degree zero from V into a r-dgebra A, then f extends to a unique I?-morphism

f : r(V) + A. If V is R-free on the countable, well-ordered bais {ut), then T'(V) is

R-fke, with basis consisting of elements yb(ul) - - - y4 (us) where 4 2 O and kj = O or 1

if lvjl is odd. For further properties of î(V) , see for example [18, Appendix].

We will abose notation, and c d a r-DGH (A, a) f i e if A is free as a r-dgebra-

where f: is the sign of the permutation

231, - - - ,%rwl,--- , W k e Vl>wl,--- xvktwk-

n i e pairing (2.1) in tum induces a pairing

If V is R-free, of finite type, and V = V& or V = V>a, am

(2.3) induce Hopf algebra isornorphisms

i that U

T~(V$) (TV)$ and I?(v~) (AV)~.

16

(2.3)

1) and

2.3.1 r-algebras and homology

In gened, a a-DGA does not pas its divided powers structure dong to its homology

dgebra.

Example 2. % Let our ground ring be Fp- Consider the CDGA @?(y, z) , a(z) = y) , where

1 y1 = 272. Then y is a bound- but yP(y) is not.

Example 2.8. Let our ground ring R be a proper subring of Q or a field of nonzero char-

acteristic. Let p E R be the least non-invertible prime or the characteristic, respectively,

of RI

Let ( A , a ) = ( r ( ~ , Y , z ) , & ) = Y ) with 1x1 = [ y [ = 2n, Ir[ = 2n - 1. We show that

the rule +[a] = [yk(a)] is not weU-dehed for cycles a E A. Specfically, [ X I = [x + y] but

[ y ( x ) ] # [ v ( x + y)]. Indeed, for O < j' < p - 1,

while [yP(y)] is a nontrivial homology class of order p. Thus

from which it follows that [.yP(x + y)] = [yP(x)] + [+yp ( y ) ] # [y (x) ] - There are r-morphisms p,+ : (r(u,v, w),O) + ( A , a ) (lu1 = 2n, I V [ = 2np, and

lwl = 2np-1) inducing isomorphisms in homology, yet H($)-'R(<p) is not a r-morphism.

Indeed, let p(u) = 2, p(v) = ~ ~ ( y ) , p(w) = T 1 ( y ) r, dli.Ie $(u) = x f y, é ( v ) = rPQ, and @(w) = y v 1 ( y ) - Z. Then û := H($)- lH(p) is not a r-morphism-- 8(u) = u, while

0(rP(u)) = yP(u) + v. Note that this is the dual phenornenon, à h André [2], Sjodin [25]

and Theorem 3.3, to that occuning in Example 3.7.

2.3.2 The Cartan-Chevalley-Eilenberg-Cartan complex

We r e c d the definition of the bar construction. Let (A, a ) be an augmented DGA, and

let A be the augmentation ideal. The bar c07tstruction on A is a DGC denoted B(A). The

underlying coalgebra is the tensor coalgebra rc(sÂ). The differential is the sum & f

of coderivations dehed as

and

where E j = ci<j Is&[ Note that do preserves wordlength while al decreases wordlength

by one. Furthermore, while & is both a derivation and a coderivation, is not a

derivation unless A is commutative-

Let (L, 8) be a DGL. Then r(sL) C r ( s m ) c B(UL) is a Hopf subalgebra, preserved

by the dinerential a. +Ol; indeed, (sz) = -sa(z) and & (sz -*sy) = (-l)l%[z, y]. Thus

(r(sL), & + al) is a sub-DGC of B(UL), denoted by C.(L, O) and cailed the chains on

(L, a). The inc1usion C,(L) ~t B(UL) is a DGC quasi-isomorphism [18, CoroUary 1-61.

The Cartan-Chevalley-Eilenberg-Cartan complex on (L, O) is the commutative cochain

algebra C*(L, a) = (AV, d), dual to C,(L, a), where V = (sL)@, and the Werential d is

the sum of derivations do and dl . The h e u r part 6 preserves word length and is dual

to d in that

(&v, sx) = (-I)'~' (v, Sax)

for v E V, x E L. The quadmtrc part di increases word-length by one and is dud to the

Lie bracket in L:

(div, sz - s y) = (-1) '4 (u, S[Z, y]) (2-4)

where the pairing is (2.3) above with W = sL = Vd. We will refer to the Cartan-

Chevalley-Eienberg-Cartm complex as the cochains on (L, a).

2.4 Bockstein spectral sequences

Fix a prime p. Let C be a fke chah complex over Zb). Applying C 8 - to the short

exact sequence of coefficient modules

yields a long exact sequence in homology which may be wrapped into the exact couple

fkom which we get the homology Bockstein spectml sequence modulo p of C, denoted

(Er (C), p) [8]. The phrase 'Bockstein spectral sequence' WU often be shortened to BSS.

If C = G.(X) is the norrnahed singular chain complex of a space X, then we refer to the

homology BSS mod p of C,(X) as the homology BSS mod p of X, denoted (ET(X), ,F).

If C is of finite type, then for a given m 2 O there &ts an r 2 O such that whenever

s 2 T, p. vanishes on E;,,,. - Therefore Fm = b,fl = s S . . We define Eg = Fm. Then

Em (H(C)/torsion) @ Fp-

There is the corresponding notion of cohomology Bockstein spectral sequence defined

in the obvious manner, ushg the fundor Hom(C, -) rather than C @ -.

The BSS mod p of C measmes ptorsion in H*(C): if x, y E F, x # O, satisfy

,F(y) = x, then x represents a torsion element of order pr in H*(C).

Notation. I f c E C is such that [CI E E1 lives until the F term then we will denote

the corresponding element of G by [cl,.

2.5 Semifree resolutions

A good reference for semifree resolutions is [15]. Let (A, a) be a DGA over R, and let

( M , d) be a (left or right) DGM over (A, a). A quasi-isomorphism (P, d ) 7 (M, d) is a

(left or right, respectively) (A, a)-semifiee msolution of (M, d) if there exists an increasing

filtration P(0) c P ( l ) c - - - of P such that P(0) and P(k + l)/P(k) are (A, a)-& on

a basis of cycles.

I£ for j = 1,2 we have (A, B)-semifiee resolutions (Pi, d ) 7 (Mj, d) and an (A, a)- linea morphism f : (Ml, d) + (Ma, d), then f Iifts up to chah homotopy to an (A, a)- linear morphism f : (Pl, d) + (P2, d) .

If (M, d) is a right (A, a)-module, and (N, d) is a left (A, &module, (P, d) 1 (M, d)

is a right (A, a)-semiûee resolution of (M, d), and (Q, d ) 7 (N, d) is a left (A, 8)-semiEtee

resolution of (N, d), then the differential torsion Metor is defined by or(^') (M, N) =

H ( P @A N ) = H ( P @A Q) = H(M @A 0)-

If (K, d) is a left (A, a)-module, and (S, d) ? (K, d) is a lefi (A, a ) - s d e e rem-

Lution of (K, d) , then the difjerential extension fvnctor is defined by ExtqAS8) (K, N) =

H(Hom(S, N)) = H(Hom(K, 9)).

2.6 Acyclic closures

IL this section we recall needed results fiom [18, Section 21. Consider the gradeci algebra

AV 8 r(sV) over R, with V = V Z 2 R-fÏee Extend the divïded powers operations on

r(sV) to R e (AV@ rc(sV)) via rule 5 of D a t i o n 2.4. Filter AV 8 I'(sV) by the ide&

Fi = AV 8 I'ii(sV). An element of AiV @ rj(sV) has A-wordlength i, r-wordlength j,

and total wordlength i + j . A derivation t9 : AV 8 SV) -t AV O r(sV) will be callecl a

I'-derivation if 19 restricts to a Dderivation on R @ (AV @ rC(sV)); similady, an algebra

morphism f : AV 8 r(sV) + AW 8 r(sW) will be cded a r-morphism if f restricts to

a r-morphism on R @ (AV 8 I"(sV)).

Definition 2.9. An acyc~ic dosun? of the DGA (AV, d) is a DGA of the form C = (Av 8

I'(sV), D) in which D is a I'-derivation restricting to d in AV and H(C) = Eo(C) = R.

We now give Ealperin's recipe for a constructible acyclic closure. The DGA (AV @

r(sV), 6), 6(1@ su) = v 8 1 for u E V, has acyclic augmentation ideal G. Grade C by

ëk = AV @ rk(sV). Since (C, 6) is an R-fiee acyclic chah complex, and 6 presemes

total wordlength while raising superscript degree by 1, we may construct an operator h

on C of superscript degree -1 such that h : AiV @ I'j(sV) + Aà-IV @ I'j+l(sV) and

6h + hd = le- We c d h a homotopy operutor.

D e h e ï-derivations recuri'vely for n >_ O by

By [18, Lemma 2.31, for each n 2 O, when restrïcted to AV, D(n) = d; furthermore,

B(n) : F' + Fi*. Let D = 6 + d 8 1 f En,, - 0(n). Then D defines a difierentitial on

AV 8 r ( s V ) . B y [la, Proposition 2-41, (AV @ SV), D) is an acyclic clomire for ( A v d) ,

called a constructible acyclic closure.

Let C = (AV 8 I'(sV), D ) be a constructible acyclic closure. Let D SV), D) =

R BAV C. We can write b = 4 t D,, where Do preserves I'(sV) wordlength and n+ increases it. By [18, Proposition 2.51, su) = - s e for v E V, where c& is the linear

part of the differentid in (AV, d ) . Furthermore, by [18, Corollary 2.61, D = O if and only

ifdo=o-

suppose (hW, d ) is another DGA with W = WZ2, and that (o : (hW, d) + ( A v d )

is a DGA morphism. Define rpo : (W, 4) -t (q 4) by <p - <po : W -+ hZ2v. Let

(hW @ I'(sW), D ) and (AV 63 r ( s V ) , D) be constructible acyclic closmes corresponding

to homotopy operaton h. Define a sequence of r-morphisms +(n) : AW 8 l?(sW) +

AV @ SV) by 11 (O) = rp 8 r(scpo) and

Proposition 2.10. [18, Proposition 2-71 With the hypotheses and notation above, we

have:

1. Tlie morphism rp : (Aw d ) -t (AV, d) &en& to a rnorphian @ : (AW@I?(sW), D) + (AV @ l?(sV), D) defined by @ = +(n).

3- For w E W , @(1@ SW) - 1 @ S(PO(W) hm total wordlength ut Ieast two.

Proof. Using induction and the fact that neithr h nor D decreases total wordlength

establishes that D@(n) - @(n)D increases total wordlength, proving the first statement.

The fact that $ is a r-morphlsm is now clear-

Since D = 6 + d @ 1 f Ce, O(k) and B(k) : P -t P-+~, D decreases r-wordiength - by at most one. Induction and the fact that h increases r-wordlength by one proves the

second statement - The final statement follows fiom the proof of the first, since 1 0 s<po(w) = +(O)(l@ sw)

and i>(n + 1) - $(n) increases wordlength for n 2 O. O

If R is a principal ideal domain, then E : (AV @ r ( s V ) , D ) -t R is a (Av d)-semifree

resolution of R by [18, Lemma 3-31.

Let (L, a) be a connected DGL over R that is R-free of finite type. Then C*(L) =

(AK d ) where V = (sL)fl. Let C be an acyclic closure for C*(L), and set (I'(sV), 6 ) =

R oc-(^ C. By the work of Halperin in [18], we identify U(C, a) = (I?(sV), b)fl and

H(U(L, 8)) = H([r (sV) , filfi). A minimal model m : (AW, d ) ? C*(L) extends to a

morphism of constructible acyciic closures th, as in Proposition 2.10. Set f i = R @, m.

Then H ( a ) : H(r(sW), b) 5 H ( ~ ( s v ) , b). If H(UL) is torsion-fie, then H(a)

dualizes to give an isomorphism H(UL) UE, where E is the homotopy Lie algebra of

the minimal model m.

Chapter 3

Bockstein spectral sequences and

universal enveloping algebras

The a h of this chapter is to prove the following theorems fiom the introduction.

Theorem A. Let X be a finite, n-dimensional, q-connected C W comptez (q 2 1). If p

is an odd prime and p 2 n/q , then each term in the mod p homology Bockstein spectrd

sequence for QX is the unlversnl enveloping algebm of a diflerentid gmded Lie algebm

(Lr, p). hrthermom, the sequence converges to a universal enveloping algebrn U(Lw).

Let Lx be the plocal Ani& mode1 for X [3]. There is an isomorphism of mod p BSSeS

Er (ULx) E(RX). Let LX : Lx + ULx be the canonical inclusion.

Theorem B. The image of each C ( L ~ ) iS contained in Lr.

Theorem C. Let (L, 0) be a differential gmded Lie algebm over Z@) which is connectai,

free as a gmded module, and of finite type. The mod p homology Bockstein spedral

24

sequence of U(L, a) is a sequence of universal enuelopuig algebrus, E (UL) = U(Lr, F).

Fhthennore, if& : L + UL is the inclwa'on, then the image of E ( c ) is contazned in Gr.

Proof of Theurems A and B. Ani& in [3] proves that there is a DGL Lx and a qua&

isomorphism ULx + C,(CK) @ Zbl of Hopf algebras up to homotopy. Thus as Hopf

algebras, for r 2: 1, F ( U L x ) = C ( R X ) and E,(ULx) = E,(RX). The result follows

by applying Theorem C to the DGL Lx. a

3.1 The image of H(L) +H(UL)

Let (L, a) be a comected DGL over Fp of finite type. By [18], the choice of minimal modd

rn : (AW, d) 7 C*(L) determines an isomorphism of graded Hopf algebras, H(UL) Y

UE, where E is the homotopy Lie algebra of m. Let L : L + UA be the canonical

inclusion.

Proposition 3.1. With the notation aboue, the image of K(L) : H(L) + K(UL) lies in

E.

Proof. It s a c e s to constxuct the fo110wing commutative diagram-

RecaU that C*(L, a) = (AV, d), where V = ( S L ) ~ and d = + dl. Recall further

that the minimality condition on (AW, d) implies that the linear part of its differential

vanishes. The ltnear part of m is the linear map mo : (W, O) -t (V,do) dehed by the

condition m - rno : W -t Az2V. Recall that E = (sw)~.

The mode1 m extends to a morphism of constructible acyclic closures f i : (AW @

r(sW), D) + (AVO l?(sV), D) by Proposition 2.10. Since (AW* d) is FP-minimal, do = 0,

so b = O in (r(sW), 6). Apply Fp @, - to fk to get a ï-morphism fi% : (r(s W) , O) +

( ' ( s ~ ) , 6)-

Let TL : (I'(sV), 6) -* s(V; 4) and : ( ~ ( s W ) , O) -» s(W, O) be the projections.

For w E W, Proposition 2.10 states that m ( l 8 sw) - 1 @ smow has total wordlength

at l e s t ho . It follows that n ~ ( s w ) - smov has I'(sV)-wordlength at least two, so

rL(f i (sw)) = smow = sm,-,(?rE(sw)). Hence Diagram (3.2) commutes; duaüze and p a s

to homology to get (3.1). O

3.2 The dual of a r-derivation

André [2] and Sjodin [25] proved that the functor L -r ( U L ) ~ is a natural equivaience

kom connected graded Lie algebras of finite type over a field k to Hopf r-algebras of finite

type over k. It follows ïmmediately that if F : UL + UL' is a Hop£ algebra morphism,

then F = U V : G + L') if and only if F I is a l?-morphism preswing the diagonals.

Let R be a commutative ring containing 1/2.

Proposition 3.2. Let (UL, a) be a DGH ouer R of finite type, so ( U L ) ~ = W , where

VI = L- Then is a I'-derivation if und only ifa(L) C L.

Proof. It mfEces to prove the dual statement, namely that d : r V -t I'V factors over

the surjection ir : I'V -, V to induce a difEerentia.1 in V- But ker(?r) is generated as a

module by products and elements of the form 7k(v) for v E V, k 2 2. Since 8 is a

r-derivation, afl(7k(v)) = afl(~)-(c-~(v) is a product. It follows that 6 (ker(?r)) c ker(r),

compIeting 'ody if' portion of the proofi

Conversely, the work of Halperi. in [18] allows us to identify [U(L, a)]' with (~(LI) , b)

as DG Hopf algebras. Since b is a r-derivation, so too is au. [3

We can thus extend Theorem 2-5 to the differential categories-

Theorem 3.3. The functor L -+ (UL)$ is a natuml equivalence, Rom the category of

DGL Is of finite type ouer k to the category of r-DGH 's of finite type over k . 17

3.3 Bockstein spectral sequence of a

universal enveloping algebra

In this section, we prove the main algebraic result of the first halfof the thesis, Theorem C.

Unless o t h e h e stated, our ground ring will be Zb), the integers localized at p.

Let (AW, d) be a minimal Sullivan algebra over Zb) ,). Let C = (AW O r(sW), D) be a

constructible acyclic closme foi (AW, d) . Let (I'(sW), 6) be the quotient Z6) @{Am C.

Note that C @ Fp is a constructible acyclic closure for (hW, d) 8 Fp. Since (AW, d) is

2,)-minimal, p divides do, so the 1inea.r part of the difkential vanishes in (AW, 4 @ Fp.

By [18, Corollary 2-61, it follows that the dlffixentid in (r(s W ) , D) 8 Fp is nd, so that

p divide b. S a G = G([r(sW),D]H) and Er = E,(r(sW),d). Let p : ~ ( s W ) -t

I'(sW) @ Fp = & be the reduetion homomorphism.

Proposition 3.4. With the hypotheses and notation above, for r 2 1, the folloving

statements hold-

2 n e r e is a r-mo~phbm gr : Er -t & such that if g&) = p(a) for some r E Er,

a E I'(sW), then z = [a],.

3. There îg o gmded Lie algebm Lr such that (6, ,F) = CT(Lr, ,F) as Hopf algebms.

Furthemore, E, is a Hopf I'-algebru, so E" is Me universal enveloping algebm of a

graded Lie algebm

Proof. We proceed by induction. For r = 1, let Wl = W 8 Fp- Since p divides b,

& = I'(sWl) and ,& = D l p (reduced modulo p). Because D is a r-derivation, so is &,

establishing the fmt statement. For the second statement, let gl be the identiiy map

on El. The third statement follows from the fbst and Theorem 3-3. Kn fact, from the

definitions it follows that L1 is the homotopy Lie algebra of the identie on (AW, d) 8 Fp-

Now suppose the three statements are established for r-1. We may write C*(Lr-', fl-') =

(AWr-, , 6 ) ; let C(r - 1) = (hWw1 0 r (s Wr-1), D) be a constructible acydic closure. By

Lemma 5.4 of [Ml, then is a chah isomorphism .y,-i : U(Lr-', ,@-') 5 (r(s Wr-'), D)d-

It is implicit that .y,-l is a coalgebra isomorphism, which then induces an algebra struc-

ture on (~(SW,-~), D)u which makes ( ~ ( s W ~ - ~ ) , 6) into a r-DGH. In particdar, under

r,-l we identify b with the Mixentid in Er+

Let m : (hWr, d) 4 C*(LT-17 P-l) be a minimal model. Let Cr(r) be a coflStfuctib1e

acyclic closure of (hW,, d) ([18], Section 2) - Since (AWr7 d) is FP-minimal, 4 = O, so by

CoroUary 2.6 of [l8], FP aAW Cr(r) = (I'(sW,), O). By [l8], Proposition 2.7, m, induces

a r-morphism % : (r(sW,), O) -t (r(sWA), 6). Since Fp is a field, by Lemma 3.3

of [18], we may identify H(%) with ToP(F,, Fp), where Tor is the differential torsion

fundor [15]. Since m, is a quasi-isomorphism, H(%) : r(sWr) K(I'(SW,-~), Pr-1) =

Er, so Er is a free r-algebra. Furth-ore, by [Ml, ~ ( f i t ) : G = H(UL,-') 7 ULr as

Hopf algebras, where Lr = (sw,)~ as graded vector spaces. Therefore Er = (Er)g is a

Hopf r-algebra.

By the inductive hypothesis, there e s a I'-morphism 9,-1 : Er-1 -t & such that

z = whenever a E Er-l, a E I?(sW) satisfy g(z) = p(a). Let gr = gr-,%-

For u E Er h o s e a E r(sW) so that g,-l(m(u)) = p(a). Then *(a) = [a],-1,

hence ,&-l[a]r-l = O and [a], E Er is dehed. Since El(&) [a], = [a],-1, m+([a],) =

[a],-l +h-l(v) for some v E ET-i- Thus -(u - [a],) = /3,1(v), so u - [a], is a boundary

in (Er, O), whence u = [a],. This establishes the second statement.

Let u E Er, and suppose for some a E I'(sW) that p(a) = g,(u). Then u = [a],,

so DU = prb for some b E I'(s W) . Thus &(u) = [b],. Since g, and p are r-morphisms,

p(y (a ) ) = g,(~j(u)) so ~ j ( u ) = [yj(u)],. E'urthermore, b (7k(a)) = prb ~ ~ - ' ( a ) so

Therefore P, is a Cderivation, hally establisbing the first statement.

By Proposition 3.2, we have estabIished the third statement, completing the inductive

step.

The Ew t e m . It d c e s to show that E, is a Hopf r-algebra- From the definitions

it follows that E, is a Hopf algebra- We have a sequence of r-morphisms

where g, = gr-1%- Thus we may identify E, with Er, which is the subset of Gz1Er

consisting of sequences (x,) that satisfy xi = gT(xr) for all r 2 1. Since each gr is a r- morphism, a well-defhed system of divided powers on E, is given by y((xr)) = (yk(x,)).

From the definitions, the coproduct is a r-morphism- 0

Proof of Theorem C. Let m : (AW, d) 7 C*(L, a) be a minimal model. Recall that the

underlying algebra of C*(L, â) is AV, where V = (sL)k Let (AW 8 r(sW), D ) and

(AV @ r(sV), D) b e constructible acyclic cIosures for (AW, d) and C*(L, a), respectively.

The model na determines a r-morphism fi% : (r(s W) , b) -t (r(sV), 6) where H ( d ) is

an isomorphism- The composition

induces an isomorphism of Bockstein spectral sequences, establishing the first statement.

The reduced minimal model m@ Fp : (AW, d) Fp ? C*(L, a) 8 Fp has homotopy Lie

algebra L', so by Proposition 3.1, imE'(~) c L1. Suppose that imF-'(L) c Lr-'. Let

&-l) : Lr-l ULr-' b e the inclusion. Then im F(L) c im H(L(~-')). The homotopy Lie

algebra of the minimal model m, : (AWr, d) ? C*(Lr-', p-') is Lr, so Proposition 3.1

states that im~(r(~- ' ) ) c Lr, completing the induction and the proof. O

We now state a proposition to be used in Example 3-7-

Proposition 3.5. Define a DGL over Fp by (L, a) = (&(e, f), a f = e), where 1 f =

2n. Then Ce(& B) = (A(%, y), d) with dx = y and 1x1 = Zn. A minimal mode1 m :

(A(x1, YI), 0) 7 C*(L, B), given by xi ct x p and y1 rt xply, induces isomorphàsrns

I'(sxl, SV=) 5 H([UL]v and H(UL) % ULah(ei, fi) roith [ell = [szl[ = 2np - 1, 1 fi[ =

Isuil = 2np-

Proof- S traightforward. a Remark 3.6. Proposition 3.5 will be restated in Chapter 6 as Proposition 6.1.

Example 33. D e h e a DGL (L,a) over Zbl by L = Lab(e, f , g ) , where [el = 2n - 1,

I f [ = 191 = 2n, and a(f) = pe. Then LL = Lab(e, f,g) (over Fp), with pl(f) = e, and

C*(L1, PL) = (A(x, y), & = y) 63 (A(z) , O). Recall the model m nom Proposition 3.5-

Define DGA morphisms i , j : (A(r),O) -t C*(L1,P1) by i(z) = r, j(z) = t +y- Then

p = m @ i and $ = m 8 j are minimal models, both with homotopy Lie algebra =

Lab (a, b, c) , Ial = 2np - 1,l b( = 2np, and [cl = 2n. The two models determine Hopf

dgebra isomorphisms p*, +* : H(UL1) + UL2, given by cp*[efP-'] = +*[efP-'1 = a,

p*[g] = $* [g] = c, <p*[fP] = b, and i>*[fPl = b + d. The algebra isomorphisrn $*(p*)" :

ULab (a, b, c) + ULab(a, b, c) is not of the form UB for any Lie algebra morphism 8 :

Lab(a, b, c) -t Lab(a, b, c). Therefore the constmction involved in Theorem C is not

natural-

Chapter 4

Sullivan Decomposit ion

The goal of this chapter is to prove a generalization of a theorem of Sullivan [26, Theorem

2-21. Let R be a local principal ideal domain containing 1/2, with unique maximd ideal

I and residue field k.

Let (AU, d) be a CDGA over R where U = Uz2 and U is R-free of finite m e . The

differential d is the sum d = '&, d j of derivations d j raising wordlength by j. Altering - C u -

U by decomposables i£ necessaq~, we may write U @ k = a @ I @ %, where d @ k : X + Y and do @ kIZ =O. AS a result, U =x@Y@ 2, w h e r e d : ~ Y and& : 2 + IU.

Let V = vZ* be an R-free graded module. Define a differential d on the CDGA

AV @ r(sV) by 6(1@ su) = v @ 1 for v E V. It is a gratifying exercise to verify that

H(AVB q s v ) , ~ ) = P(AV@ r ( ~ v ) , 6 ) = R.

Dennition 4.1. With the hypotheses and notation above, the CDGA (AU,d) is called

R-contmctibk if there is a DGA quasi-isomorphism (Au, d) ? (AY 8 r(sY), 6) restricting

to the identity on Y.

Note that an R-contractible algebra has an acyclïc augmentation ideal.

Recd kom [6] that a commutative DGA (A, a) is admksible if there exists a DGA

surjection P : (B, d ) -u (A, a), where (B, d ) is an R-free CDGA that satisfies H(B, d) =

HO(B, d ) = R W e will c d cr an acyclic cover of (A, 8).

We now recall

Theorem D. Let (AV, d) be an admsssible Sullivan dgebm over R, svch that V = Vz2

and each V' is R-jke of finite type.

1. There d t s a minimal mode1 m : (AW, d) 7 (AV, d ) which ii split injective-

2. ( A q d ) "- ( A w d ) @ (AU, d) where (AU, d) is an R-contractible algebra.

3. A quasi-isomorphism between minimal Sullivan algebnrs over R is an komorphism.

4. Al2 of the minimal Sullivan modek, of an adrnbsible CDGA are isomorphic.

We will prove Theorem D in Section 4-1 below-

Remurk 4.2. The requirement that R be a local ring is only used to show that a quasi-

isomorphism into a minimal Sullivan algebra is split surjective; our approach is to reduce

over the residue field 11: and use Nakayama's Lemma. Perhaps the proof can be done

directly without the local hypothesis.

In Section 4.2 we tum our attention to certain R-contractible algebras. Let R = Fp

or Z@). Suppose (AU, d ) is an R-contractible algebra that satisfies U1 = O and each Uk

is R-free. We give an explicit description of the generators and differential in (AU, d)

in the following manner. Recall that d = Cj2,, dj> where di is a derivation raising

wordlengthby j. WemaywnteU~F, =X@Y@Z, where6 : X ~ Y and&lz=O.

Let { ~ a ) c U represent a basis for X. W e wil I consmict Sullivan models (Ma,6,) 7

A(dxa) @I'(xa), then map each Sullivan algebra (AM=, 6,) into (ACT, d) . We then multiply

these morphisms together to define p : @,(Mar 6,) + (AU, d) . We prove Theorem 4.12:

cp is an isomorphism.

4.1 Proof of the decomposition

We will need the following lemmas. The proofs of the nrst two are the same as in the

rational setting and are omitted.

Lemma 4.3. Let R be a commutative ring containing 1/2. Let v : (A, d ) (B, d ) Be

a surjective quasGIsomorphlisrn of commutative DGAS over R. Suppose that (Av,d) is a

SdIZvan algebm and that rp : (Av d ) + (B, d) is a DGA rnorphism- Then the* is a D G A

rnorphism @ : (AK d) + (A, d) such that <p = u$. U

Lemma 4.4. Cet k be o field. Suppose that rp : (A, a) ? (AW, d ) is a quasi-isomorphism

of CDGA 's over k , where (hW, d) is a minimal Sullivan algebra. Then cp Zs split sujective.

0

Lemma 4.5. Let R be a local principal ideal domain, containing 112, with maximal ideal

I and residue field k . Let <p : M + N be an R-linear rnorphrPm of gmded R-modules of

finite type. If cp 631 k is surjective, then so Zs cp.

Proof. Since <p @ k is surjective, N = p(M) +IN. It follows that N/<p(M) = I(N/(o(M)).

By Nakyama's Lemma, N/p(M) = O, so N = p(M) .

34

Proof of Theorem D. Step 1. First we prove Statement 1- There is a minimd model

m' : (AW, d) ? (Aq d) and an acycllc cover a : (B, d) ++ (AV, d). Multiplication defines

a sqiinective quasi4sornorphism mr -a : (hW, d) 63 (B, d) Z (Av d). By Lemma 4.3, rnr - c splits via p : (AK d) 7 (AW, d) 8 (B, d) . Compose p with the projection (AW, d) 63

(B, d ) Z (AW, d) to get a quasi-isomorphism r : (AV, d) ? (AW, d ) . We will show that

r splits; the desired model m wilI be the spIitting for r. R e c d that the Sull ivan algebra

(AW, d) is R-minimal if and only if (AW, d) @ k is k-minimal Since R is a principal ideal

domain, applying -@k to r , weget aquan-iomorphïsmr@k : (AKd)@k 7 (AW,d)bk

of Sullivan algebras over the fieId k. Since (AW, d) @ k is minimal over k, T 60 k is split

surjective by Lemma 4.4. Lemma 4.5 then implies that r is surjective. Since (AW, d) is

a Sullivan algebra, r splits. C d the splitting rn : (AW, d) ? ( A x d). This establishes

Statement 1.

Step 2. Recall m : (AW, d) + (AV, d) has splitting r. We then have a split short exact

sequence of graded R-modules O -+ kerr + A+V + A+W + O and so m and the

inclusion of ker r induce a graded module isomorphism A+V A+ W@ker T. Mhermore

A+V A+ V r A+W - ACW + J where J = (A+V) 0-ker r c ker r. Since kerr/ J is R-hee,

ker r -+ kerr/J splits M a a splitting we wilI c d a. Set U = imu. T h d o r e m and

the inclusion of kerr induce the R-linear isomorphism Q(AV) Q(AW) @ (kerr/J). If

i : U v AV is the inclusion, then i extends to an algebra morphism i : AU + AV, and multiplication defines an algebra isomorphism rn - i : AW 69 AU 5 AV.

Step 3. We show that (AU, d) t= R @ c ~ ~ q (AV, d) is R-contractible. Write U = Y @ Z,

where Z = ker{& : U + U). Since R is a principal ideal domain, Y and Z are both

R-free. Since V = d : Zi _t AW 60 A(U<i-2 @ Zi-l) and d : Yi + AW Q

[z'+' 8 (AW @ A(u~~-~ ) ) ] . If we choose a basis Cui, uz, - - - ) of U compatible with the

sequence of submodules C, Z3, Y2, Z4, Y3, - . - , Zi,Y'-', ZàtL, Yi,. .. then duj E AW O

A(ul, . . . , u$-~)- (Note that this is exactly a relative version of the argument given in the

paragraph preceding [18, Proposition 7-51.) It follows that (AK d) is a (hW, d) - sedee

module. Applying R - to the minimal mode1 rn : (AW, d) 4 (AV, d), we get a

quasi-isomorphism R 7 (A& a). Thus (AU, 4 is acyciic.

Next, note that A(X@Y) is a sub-DGA of (AK a), and that (AK d) is a (A(X@Y), z)- semifcee resolution of R. Therefore we can extend the obvious map (A(X @ Y), d) + (AY @ r(sY), 6) to a quasi-isomorphism (Ac d) 7 (AY @ I?(sY), 6).

Step 4. Recd that U = Y @ 2, where Z = ken&. Filter U by a sequence of submodules

as foiiows. Set U(1) = Z2, and for k 2 2, U(k) = U(k - 1) @ zk+' 83 Yk = Uk @ Zk+'.

Since V = V Z 2 and U c kerr, d : U(k) + AW @ A+U(k). Suppose inductively that

d : ( k - 1) + A - 1) Then the inclusion ( A w d ) @ (AU(k - 1), d) ~t (Av d)

induces a homology isomorphism in degrees < k- By the same argument as in Step 3,

Hq(AU(k - 1), 4 = O for 2 5 q 5 kk. Furthennore, it is straightforward to verify that

H~+' (AU(k - 1) , a) is torsion-kee.

Now, suppose z E Zk+? We may wrïte dz = 8 + a, where 101 = = k + 2, @ E

A+W 8 AcU(k - l) , and @ E AcU(k - 1). Thus d a = -d@. Since d is a derivation that

preserves A+W, dQ E A+ W O AU(k - 1). By the inductive hypothesis, d e E AU(k - 1).

It follows that both da and d* vanish. Thus iP is a cocycle in Hk+2(A+W @ A+U(k - 1)) - Since EQ(AU(k - l),Q) = O for 2 5 q 5 k, and H2(AW) and Hkf'(AU(k - l), d) are

torsion-kee, the Künneth formula irnplies that (A+ W @ A+U(k - 1)) = O. It follows

that O = df l for some R E A+W 8 A+U(k - 1) c kerr. Therefore d(r - 0) = B E

AU(k - 1) C AU(k) and we may replace z by r - C2 as a generator for AU-

Essentially the same argument estabkkhes that we mayreplace a basis element y E Yk

by y - 0, where d(y - 8) E AU(k).). The proof of Statement 2 is now complete.

step 5- Let cp : (hW, d ) ? (AX, d) be a quasi-isomorphism between R-minimal Sullivan

algebras. Since R is a principal ideal domain, - 8 k preserves quasi-isomorphisms of

R-fiee complexes. Thus cp 63 k : (AW, d ) 8 k ? (AX, d ) 8 k. By Lemma 4.4, cp 63 k is

surjective, so cp itselfis surjective by Lemma 4.5. Lemma 4.3 then implies that cp splits via

c : (AX, d) + (AW, d) . As CT too is a quasi-isomorphism, repeating the above argument

shows that n is a surjection and hence an isomorphism, This estabiishes Statement 3

step 6- Suppose that (A, d) is an admissible CDGA, and that rn : (AW, d ) 7 (A, d ) and

p : (a, d ) 7 (A, d ) are minimal models of (A, d ) . Use the argument of Step 1 to lift m

through p to get a quasi-isomorphism cp : (AW, d ) ? (LX, d) . Now apply Statement 3.

ul

4.2 Acyclic Sullivan algebras

The purpose of this section is to prove Theorem 4.12 below, which makes explicit the

generators and difkentials in any kee acyclic CDGA over Zb1. First, we give some

technical propositions.

Proposition 4.6. Let A be a r-algebm over Z@), x E A, lzl = 2n, n 2 1. The folLouring

relations hold:

1. For k >O,

where

2, for k _> 1,

where

) 1 (modp); ak+l -1 p k - 1

Proof. 1. Observe that pk+' - 1 = (pk+' - d ) + @C - 1)- Thus

But (cf- [9, 241)

and for n 2 1,

1 (mod p).

2. This is just Rule 4 of Definition 2-4.

CorolIary 4.7. Let (A, 0) be a r-DGA, x E Ah, n 2 1. The following identity holdsr

Let N = Zb1{ui, wjt i , - , where lvil = 2npi, lwjl = 2np"' - 1 (ia 2 1)- Define a

differential 6 on N V by making aH the vc coqdes, and setting b(wj) = - p ! c j ' ~ ~ + ~ + t$, where q- E Zb) is as in Proposition 4.6. Define a DGA homomorphism

where 1x1 = Pn, by u(wi) = r ~ ' ( ~ ) , ~ (q ) = O- Since F(z) is generated as an algebra by

the elements yP" (z) fm k 2 0, Y is surjective; by construction, H(ker V) = O, so we have

Proposition 4.8. The homomorphism v is a quasi-isomorphism.

Remark 4.9. See Example 6.4 for an interesting computation using the mode1 v : (AN, 6) ?

(W 7 0)-

Now let M = A(%) 8 NV, Ir1 = 2n + 1. Define a difïerential 6 on LW by

It is easily verified that a2 = 0.

Define p : (hP, 6) + r ( z , y) by z rt y, vk H ?PL(%), wj ++ 0.

Proposition 4d0. The homomorphàsm p is a quasi-iiomorphism.

Proof. Let ?r : (AP, 6) + (AN, ,6), z r+ 0, and let p : F(x, y) ++ r ( x ) , y ct O. Then

kera = ( M ) z and kerp = (r(x))y. Define DGM morphisms p : kerir + and

tr, : kerp -+ r ( ~ ) by rp(@ Z) = (a E AN) and +(@ -y ) = P (@ E r(~)). It is

apparent that p and $J are Zbl-Iinear isomorphisms, and it follows that p restricts to a

quasi-isomorphism of graded Z6)-modules ker lr 7 ker p. Form the commutative diagram

O - kerp d r(z,y) _O_* r(z) - O

where the horizontal rows are exact- Pass to homology and use the Five Lemma to

complete the proof, CI

Lemma 4.11. Let (M,d) be a Z@)-fize DGM over Zbl of finite type. Then there is a

direct s m decomposition

vhere d : Xo 5 Y', d(XC) C Y+, YC/d(XC) is a torsion module, and d vanishes on 2.

Let (AU, d ) be an acyclîc cochain algebra over Z@), such that U = Uz2 is Zb)-fiee of

h i t e type. By Lemma 4.11, we may assume that 17 = Xo @ Y. @ X+ @ Y+ @ 2, where

d : Xo - Yo, 4 : X+ -t Y,, Y+/&X+ is torsion, and hlz = O. Let {x,) be a basis of

Xo. This determines a basis {dz,) of Yo.

Theorem 4.12. The DGA morphisn

is an isomorphism.

Lemma 4.13. The d u c e d morphisrn

Proof. We consider everything to be reduced modulo p for the duration of the proof, and

omit the ubiquitous - 8 Fp. Fust we construct a splitting a for 9, then we show that cr

is in fact an inverse.

We may write U = X @ Y @ 2, where d : X 2 Y and hlz = O. Let {sa) be a basis

for X. Define the splitting CT on X @ Y by u(x,) = v,p, u ( h a ) = Za- Assume for k 2 2

that the sp1itting has been comtmcted on Zck. Let a E Zk be a basis element. Since

da E A(UCk), a(da) is d&ed and is a cocycle. Since (AM, 6) is acyclic, there &s

E 11M with bQ = o(&); hrthermore, d(rp(iP) - a) = O. Therefore there is an 7 E AU

that saWes dq = p(B) - a. Since [q[ = k - 1, a(7) is defined, so a = rp(Q + bu(q)).

Extend u by defining o(a) = O bu(^). This completes the construction of the spM5.ngg

Assume that UV = 1 on AMck. Let y E Mk be a basis element fiom the bask given

in the construction of ( A M , 6).

Case 1: y = Z, or y = va,o for some a. By construction, up(y) = y.

Case 2: y = va+ or y = Wa,j for some a and i 2 1 or j 2 O. Then by E (AbM)*+' C

AMck, so by the inductive hypothesis, o ~ ( 6 y ) = by. It follows that b(uv(y) - y) = O

so there is some element w E AM that satisnes 6w = urp(y) - y; that is, u v ( y ) =

y +- &W. Applying cp, we get cpo(p(y) = rp(y + bw ) ; since u splits (o, we get dv(w) = 0.

Since (AU, d ) is acyclic, there exists an element t7 E AU such that dB = rp(w). Since

Iw [ = k - 1, the inductive hypothesis says that ap(w) = w, so w = bu(0). Therefore

4~) = Y + 6 ( W @ ) ) = Y- O

Proof of Theorem 4.12. By Lemma 4.13, (o @ Fp is an isomorphism. Lemma 4.5 implies

t hat p is surjective. Since r p Q Fp îs injective and AM is Z@)-fiee, rp itself is injective. O

Chapter 5

Homology of a different ial graded

Lie algebra

The purpose of this chapter is to prove Theorerns E, F, G, and H, which we recall below.

Let p be an odd prime. Let (L, 3) be a connectecl DGL of f i t e type over Fp such that

C* (L) is admissible. Recall fiom [1&] that there is a natural isomorphism of Hop£ algebras

H(UL) UE; where E is a naturally denaed graded Lie algebra. Let L : L + UL be

the natural map.

Theorem E. With the hypotheses and notation above,

1. H(L) "- E x K as Lie algebras, where K = ker H(b) is an abelian ideal, and

2. if L is (q - 1)-connected, then K is @q + p - 3)-connected, so Hm(L) E E, for

m _ < p q + p - 3 -

Theorem E is proved in Section 5.1.

43

- - Suppose now that (L, 8) = (L, a) B Fp for some Zb)-kee DGL CL, a) over z ~ ) such

- - that C*(L, a) is admissible. Then a induces Bodrstein operators on K(L) and on UE.

Since UE is the h t term in the mod p BSS for UZ, the Bockstein preserves E in UE.

Let 6 : (AW, d) ? e ( L , a) be a minimal model. By Theorem D, fk is unique up

to isomorphism. The lin- part of 2, denoted CL, is divisible by p, and so induces a

Bockstein operator on w @ Fp dud to that on E; see [18, Section 91.

Theorem F. The natuml map H(c) induces an isornorphisrn of gmded Lfe aalgebsas

H(H(L) , B) q H(E, Pl-

Theorem F is proved in Section 5.2.

We now appLy Theorems E and F to the topological case. Let X be a finite, q-

connected, n-dimensional C W cornplex, and suppose that p 2 n/q. The Ani& mode1

of X , Lx, is a DGL over Zb) such that ULx 7 C@x; Z@)) as Hopf algebras up to

homotopy [3]. Applying Halperin's result [18], H(U(Lx @ Fp)) = UEx, where graded

Lie algebra Ex is cded the mod p homotopy Lie algebra of X .

Theorem G. With the notation and hypotheses above,

1. H(Lx @ Fp) s Ex x K, where K is an abelian ideal.

2. &(Lx 8 Fp) = (Ex) , for m 5 pq + p - 3.

3. LX : LX -+ ULX induces an isomorphikm H(H(Lx @ Fp), P ) H(Ex,P).

Proof.. Since p 2 nlq, C'(Lx) is admissible 118, Lemma 7-61. Apply Theorems E and F.

O

Let T,(-; Fp) denote the mth homotopy group with coefficients in Fp [22). In [4],

Ani& showed that the mod p Hurewicz homomorphism -(SWT; F,) -t Hm(& 8 F,)

(m 5 pq-l), is an isomorphism i fm 5 min(qf2p-3 ,~-1) . Since q+2p-3 5 pq+p-3

i f q > 2 a n d p q - l ~ p q + p - 3 i f q = l , wehave

Theorem H. (Ex), rm(QX; F,) for m 5 min(q + 2p - 3, pq - 1).

5.1 DGL9s and their homotopy Lie algebras

Let (L, 8) be a connected DGL over Fp of finite type such that C*(L) is admissible. Then

H(UL) = UE for a naturally defined graded Lie dgebra E [18].

Since C*(L) is admissible and L is comected of finite type, C*(L) satisfis the hy-

potheses of Theorem D. Therefore there exïsts a split injective minimal model m :

(AW, d) 1 C*(L) and an F,-contractible S U ~ D G A i : (AU, d ) L) C*(L) such that

Since m is split injective, so is m, and so H(L) + E is split surjective. Let K = ker H(L);

then H(L) "- E @ K as graded vector spaces.

Linearize (5.1), then suspend and dualize to get the isomorphism of chah complexes

E : (L, a) Z (E, O) Q ( I , a), where 1 = ( S U ) ~ . Note that 1 = ker{L -+ E ) is a subchain

complex of L, and that H(1) = K. Recail from Theorem 4.12 the isomorphism rp :

(AM, 6) -% (AU, d ) , where (AM, 6) = @&(AMa, 6,). For each a, either Ma = FP{zo7

with I v ~ , ~ ~ odd, or Ma = F,{Z~, v , i , W, ,j) with I v ~ , ~ ~ even. The differentials are given by

J(vm,o) = &, b(va,k+i) = -&,,k)vg;:, and &&J-) = Ga.

Lemma 5.1. An element x of L is a h n d a r y if and only if c(x) uanîshes on s(W @

rpo(Mf)), where M' is the v a n of al2 the generators of M ezcept

Proof- Since E is an isomorphism of chah compIexes, x E L is a boundary if and

oxdy if E(x) is a bound- If E(x) is a boundary then it vanisbes on s(W @ rpo(Mf)),

which is generated by cocycles. On the 0th- hand, if c(x) Mnishes on s(W @ po(Mf))

then E(z) = &) , where g is defined by (s(span{v,il w a ) g) = 0 and (z,, sg) =

-(-l)'z' (va.0 4 ~ ) ) - 17

Pmof of Theorem E. To prove Statement 1, it suffices to show that [H(L), H(I)] = 0.

Suppose x E L and y E I are cycles and w E W . Since K = H(I) is a Lie ideal, we may

write [x, y] = z + Ba, where z E I is a cycle and a E L. Thus (w, sc[x, y]) = 0, since z

vanishes on sW and do = O in W-

Suppose now that u E M'. On Mf, bo = O, so cpo(u) E ker & C U. Mhermore,

Ji = O in AM, so a wordlength argument establishes that d l ~ ( u ) = 40, where R E A2U-

Since x and y are cycles, it follows that (<po (a), sc[x, y]) = (-1) lwi (dlpo (u) , sx sy) = 0.

Therefore [x, y] is a boundary, so [H(L) , K(I)l = 0.

For Statement 2, recall that K = H ( I ) = K([s(U, do)]g) = H([s(M, bo)]fl). Since

L = Lzq, M = M'nt'. The lowest degree basis element of M will be vafi for some a.

This means that the first nonvanishing cohomology class of H(s[M, bol) is that of sw,,~,

whose degree is at least pq + p - 2. O

5.2 Lie algebras reduced modulo p

Let (El 8) be a DGL over Zb) such that cyE, a) is admissible- Let (L, a) = ( E , a) @ Fp

be the reductîon modulo p; it is clear that C*(L, a) is admissible. By the uniqueness of

minimal models (Theorem D), we may suppose that any minimal model m : (AWz d ) 7 - -

C*(L, a) is of the form m = 6i @ Fp, where fii : (AW, d) 7 C*(L, 8) is a Zb)-minimal

model-

Since C*(L, 8) is admissible, Theorem E asserts that H(L) E x K as graded Lie

algebras, where K is a central ided Since (L, a) is reduced modulo p, H(L, ô) and

H(UL) corne equipped with Bockstein operators P. But H(UL) = UE, and P preserves

E. Therefore H(L) : H(L) -t H(UL) induces a morphism of DGL'S, (H(L), /3) + (E, P) ,

whose kernel is K-

Lemma 5.2. The difemntial ided (K, /3) is acyclic

Pmof. Recall that K = H([s(M, bo)]fl). We may assume that (AM, 6) = (a, 8) @ Fp

for a Z@)-contractible algebra (a, 8). Thus H ( K , P) = O if and only if (H(M, JO), P )

is acyclic, where ,û is the Bockstein operator associated to z0. By construction, M =

@,Ma = @,(a 8 F,). For a gi- a, if the lowest nonvanishing degree of Ma is odd

then Ma = Zbl{z, v) with bo(v) = z; so H(M,, &) = O. On the other hand, if the Iowest

nonvanishing degree in M, is even, then M= = Zb){z, vil wj)i,jz0, where &vO = Z, and

J0wj = -p!vj+=. It follows that H(Ma) = FP{vi, w ~ } ~ ~ ~ ~ ~ ~ with P(wj) = vj+l- Hence

H(H(Ma) , B) = 0- O

Proof of Theorem P. Since H(L) maps surjectively onto E, use Lemma 5.2 and pass to

the associated long exact homology sequence- O

The following coroUary m e r s a question raised in [18, Remark 9-31.

- - Coroiiary 5.3. The mod p Bocksteirr spectnrl sequenees for H(L, a) and for [s(w, &)Ir coincide after the first term. In par tic da^, the sequence E + EQ k a sequence of gmded

Lie algebras. O

Chapter 6

Some examples

We begin by stating propositions to be used in the Examples.

Proposition 6.1. Define a DGL over Fp by (L, a) = (Lab(e, f), a f = e), whez 1 f 1 =

272. Then C*(L, 3) = (A(x,y), d ) with dx = y and 1x1 = 2n. A minimal mode1 r n :

(A(xl, yl), O) 7 C*(L, O), @en by xi I+ XP and y1 c+ xP1y, induces iwmorphims

r(szl, syl) % H([UL]t) and H(UL) -% ULa6(el, fi) with lel 1 = Isz11 = 2np - 1, 1 fil =

Isyd = 2np.

Before stating the next Proposition, recall the cobar construction Cl(-) of Adams [l,

14, 191. Let (Cl B) be a connected chain coalgebra, and let I = Cc. The cobar con-

struction on (Cl a) is the DGA n(C, a) with underlying algebra T(KLI) (where s-' is

the desuspension, lowering degree by 1). The ditferentïd is the sum of derivations

d = + &, with preserving wordlength and & increasing wordength by 1- Specif-

ically, for x E 1, suppose the reduced coproduct of x is given by O(x) = x j x ; @ x;.

Then i30(s-'s) = -s-'a(x), and &(s-%) = ~j(-l)l~~.ls-'z~ 8 s%$.

The foilowing weU-known resdt is an excellent exercise for the budding doctoral

student in homotopy theory.

Proposition 6.2. Let R be a commutative ring conta2ning 1/2. Cet (C, a) be an R-fie,

cocomrnutatzue, connected chah coalgebra satisfing C = R @ C>2 - - Set 1 = - - Then

n(c, a) = u(L, a ) , where L = L(s-'~), and th ere ezists a naturaz injective DGC quasi-

isomorphLPm (C, a) ? C.(L), where C,(L) is the chain coalgebm on (L, a) dejhed in

Section 2.3.2. !a

For a more general version of the next proposition, see [16, Proposition If.2.61.

Proposition 6.3. Suppose (o : (A, d ) 7 (B, d ) i s a CDGA quasi-isomorphism. If (B, d)

is admissible, then so is (A, d ) .

Proof. Choose an acyclic cover (C, d ) = ( A q d ) obt (B, d ) , where H ( C ) = Eo(C) = R.

Let a E A. Since (B, d) is admissible, the morphism A(%, dx) + (B, d ) , x * -,(a), extends to a morphism P : (A(x) @ U, d ) -+ (B, d) , where (A(x) 8 M, d ) is a Sullivan

algebra satisfying H(A(x)@AX) = H O ( ~ ( x ) @ A X ) = R. Define cr : A(x, dx) -t (A, d ) by

a ( x ) = a, a (dx ) = da. It s a c e s to show that a extends to d : (A(x) @ AX, d ) + (A, d).

Multiplication in B defines a surjective quasi-isomorphism <p : (A, d ) 8 (C, d ) ? (B, d).

By Lemma 4.3, ,O Lifts through <p - +- Compose with lA @ EC : (A, d ) @ (G, d ) + (A, d ) to

get a' : (A(x) @ AX, d ) + (A, d ) extendhg a. Cl

6.1 Calculations using acyclic closures

In this section we work over the ground ring Zbl unless othemke stated. Exam-

ples 6.4 and 6.6 each compute the BSS of a universal enveloping. Each exampIe ïs

'almost topological'. That is, in each case we have a Chain dgebra quasi-isomorphism

U(L, a) 7 C.(QX; Z,,), but 1 have not been able to determuie whether the map com-

mutes with the coproducts up to DGA homotopy [7]. If so, then by [18], the examples

wodd compute the mod p loop space homology and cohomology BSS of X.

Example 6.4. A Zb) 'mudel' for H*(Q2Shf1; Zb)) .

Let C = T ( x ) with the standard shufee coproduct, where 1x1 = 2n for n 2 1. By

Proposition 6.2, QG = U(L7 a), where L = L(s-II), I = T+(x) . The differential a can

be given explicitlyi Denote by xk the generator s"(x@~); note that lxi[ = 2nk - 1. Then

Proposition 6 -2 further asserts the existence of an injective DGC quasi-isomorphism

(C, O) 7 C.(L, â), which dualizes to yield a surjective DCA quasi-isomorphism ?r :

C*(L,â) ? (C,O)~. But Cd = I'(y), Iyl = 2n7 so the model u : (AN7&) ? (I'(y),O)

of Proposition 4.8 lifts through ir (Lemma 4.3) to defùie a minimal model m : (AN, 6) 7

C ~ L , a). Recall that N = Zbl {vis w ~ } ~ ~ ~ luil = 2npi, and lwil = 2npi+l - 1. The acyclic closure

C = (AN @ SN, 6 ) has difïèrential G ( s w ~ ) = wj + p! cj svj+l+ $-'svj and G(svi) = vi. -

SO the fibre is given by SN), $1, where b(svi) = 0, b(sq-) = p! cj svj+i, with C, 1

(mod p) as in Proposition 4.6.

Applying [BI , we can compute the cohomoIogy BSS. Specifically, &(UL*) = H((UL8

FP)fl) = r(s N) ; that is,

where lsvil = 2npà - 1 and lswil = 2np'C' - 2, and &(swj) = -svj+l. So H ( U L 8 F,) n

UE, where E = Lab(q, y{), P ' ( X ~ + ~ ) = y+ It follows that & = Ea = - - - - - E, = A(svo),

andE2=E3=-- -= Ew = ~ L & ( x ~ ) .

Furthmore, since (r(y), O) is admissible and G*(L) 7 (r(y), O), Proposition 6.3

states that C*(L) is admissible. Therefore by Theorem E, H(L) E x K, where K is a

(2nO) + 1) - p - 3)-connected abelian ideal, and H(H(L) , ,ûl) Lab (za).

Remark 6.5. There is a chain algebra quasCisomorphism 0 : UL 3 C,(St2S2"+'; Zbl). I f

0 commutes with the natural coproducts up to DGA homotopy, then by [18], the above

work gives a new calculation of the algebra H*(f22S2n+1; F,) dong with its Bocksteins

(originally calculateci, 1 believe, by Browder in his Princeton thesis; see also [ I l ] ) .

Example 6.6. A Zbl rmodel' of 0S2"+'w) .

Let (C, a) = (A(w) @ T ( x ) , a(w) = prx) , where 1x1 = 2n, Iw 1 = 2n + 1, n, r 2 1.

Equip C with the natural coproduct, with w and x primitive. B y Proposition 6.2,

n(C, a) = U ( L , a), where L = L(s"I), I = A+(w) 8 T(x ) + A(w) O T+(x). Denote

by x k and wk the generators ss-'(1 @ xak) and Ë 1 ( w 8 xBk), respectively. Note that

Ixkl = 2nk - 1 and Iwkl = 2n(k + 1) . Then

As in Example 6.4, we have a sqjective DGA quasi-komorphism p : c*(L) % ((A, d),

where (A, d) = (C, a)$ = (A(u) @ II(u), d(v) = pru). Monifving the definition of (Ml 6)

(Section 4.2) by setting b(vo) = pr t , we get a mini'md model p : (AP, 6) ? (A(u) 0

ï ( v ) , d ) . Since p is surjective, p Iifts through p to define a minimal model m : (AP, 6) 7

C*(L, a). It foiiows fiom [18] that we may compute E(UL 8 Fp) and E((UL 8 F,)$) by

findïng the acyclic clmure for (hp, 6).

We form the acyclic closure as follows. Its unddying algebra is AP @ r(sP). To

compute the Merential on sP, we e s t compute the acyclic closure of (A, d) , then

use the model p. First, introduce the divided powers generator su of degree 2n. Set

SU) = U. Let U' = u -prsu. Then ( A 8 r (su) ,d) (AU @ r (su) ,d) 8 (r(vl),O).

The left tensorand is acyclic, so it remains to compute the acyclic closure for (r(vf), O)

and then back-substitute. We essentially calculated thk acyclic closure in Example 6.4:

it i~ (r(vr, SV^, S W ~ ) , d) , where ~ ( s w ) = (d) and d(swj) = p! c,- sVj+i + (v') SV^,

with cj 1 (mod p). Thus the acyclic closure for A is (Au Q r(v) @ r(su, sui, swj), d )

k rk pi-& where d(su) = u, d(u) = pru, d(sui) = ~ ~ , - , ( - l ) p 7 (v)+(su), and d(-9uij) =

P! cj svj+l+~~o(-l)k~kl(p-l)!)7'pik(~)~k(Su)SZ1j Therefore, the qmi-i~~m~rphism

p "plies that a valid dinerential in the acyclic closure of (AP, 6) is given by d(st) =

Z, SU^) = ~ ~ - , ( - l ) k p r k @ c k k ( ) and d(swj) = p! cj svj+l + ~ f ! - ~ ( - i ) ~ ( p ' ~ / @ -

1) !) @ j , i , * rk (~~) suj where <niIr E A(vo . - - , vi) satisfies /L(@~,~) = . Taking the -

fibre Zbl ,)@ (AP @ I'(sP),d), we get ( I ' ( s ~ , s v ~ , s w ~ ) , d ) ~ where d(sz) = O, d(sui) =

-p 'p iy~ i (~~) ) md ~ ( s w ~ ) = p!svjtr - @"P'-/(P - I)!)sv# (SZ) - We now compute the cobomology BSS. Let Er = E,(UL~). Then El = H((UL 0

F)) = l?(sP) = I'(sz, sui, = ~ ( s z ) 8 (@C,j20A(~~i) @ I '(SW~)), where [SZ[ = 2n,

lsvi[ = 2npi - 1, and [scojl = 2 n P 1 - 2. hirthermore &(sw~) = -SVj+l- Therefore

E2 = A ( S V ~ ) ~ ~ ( S Z ) - Set 60 = sr, a0 = -suo. Then& = E3 = - - - = Er, and&(%) = bo-

It ~ ~ O W S from Proposition 6.1 that Ee = A(aj) 8 r(bj) , with = [aj- l~p'(bj- l)] ,

bj = [ ~ * ( b ~ - ~ ) ] ~ and &*-(aj) = bj- Since the aj and bj are stndly increasing in degree

with j , it follows that E, = 0 .

Dudy, set G = E ( U L ) . Then G = ULr, with Lr as ~OUOWS- L1 = Lab(w, xi, yj),

with P ' ( Z ~ + ~ ) = -y.. Thus L2 = Lab(w, xo)- Set eo = w, fo = 30. Then La = L3 =

- - = Lr, when p ( ~ ) = fo- Then L*' = L*(ej, f j ) , where ej = [<-II and fi = [tif] in H(ULr+j"), and p+j (e j ) = fi- Ag&, since the ej and fi are strictly increasing in

degree with j, E"" = 0.

Furthermore, C*(L,â) is admissible (since (C, B)t is), so by Theorem E, H(L)

L1 x K , and H(H(L) , ,ûl) H(L1, pl) = Lh(w, xO) -

Remark 6. 7. There is a DGA quasi-isomorphism 4 : U(L, 8) 7 C,(QS2"+'@'}; Zbl),

where S2*+'Cpc) is the homotopy fibre of the map pr : S2n+1 _+ S2"+l [IO]. Indeed,

mode1 the pr map (over Zb)) by (Aul, O ) + (Au, O ) , ul e pru, lui 1 = = 2n + 1. Form

the pushout diagram

where a(v) = ui and d(v) = pru. Then A = (Au 8 r(v), d) models the plocal cochains of

Sh+'(P.), in the sense that there is an algebra isomorphism E(A@F,) i B ( S 2 n C 1 ~ } ; Fp)

[20] and A gives the correct Bocksteins.

6.2 Counterexamples to rational behaviour

Example 6.8. This example concems the algebraic analogue of "looping" a fibration.

Let R be a principal ideal domain containing 1/2- A minimal relative Sullivan alge-

bm is a CDGA of the form (B 8 AW, d ) , where (B, d) is a connected sub-CDGA, and

the quotient R 8 ( ~ , 4 (B 8 AW, d ) = (AW, 2) is an R-minimal Sullivan algebra We

think of (B, d ) -t (B 8 hW, d) as modelling a fibration, and c d (B, d) the buse of the

minimal relative S u l l i . algebra Suppose (B, d ) = (IUT, d ) is an R-minimal Sullivan

algebra. Then (AX 63 A w d ) is itself an R-minimal Sullivan algebra. We may take a

constnictible acyclic closure of the morphism (M, d) -+ ( A X 8 AW, d ) which then in-

duces (I'(sX), D) + (I'(sX) @ r ( s W ) , D) with fibre (QsW), b) . If R is a field, then the

differentials above Mnish ([l?, Theorem 5-21 in characteristic zero, [12, Proposition 1-91

in positive characteristic). Therefore ( ~ ( s X ) 8 I'(sW), O ) "- ( ~ ( s X ) , O ) 8 ( ~ ( s W ) , O), so

the 'looped' fibration splits.

Now let R = Zb), p odd, be our ground ring. We will give a minimal Sullivan

algebra over Zk) which does not exhibit the above behaviour. Let (B @ AW, d ) =

(A(v, ~ 1 . . . 8 A(+), d ) , where dm = pu, duj = uj-iv for j = 2 . . .p, and lul[ = 2n.

The acyclic closure of (B 63 AW, d) is the DGA C = (A@, ul . . . +) 8 l'(su, sui. . . sy), D) k with dinerential D(sv) = v and D(suk) = ~ j = o ( - ~ ) i ~ k - j ~ ' ( ~ ~ ) , where uo = p- Note

that C restricts to an acyclic dosure for ((B, d), so i : (B, d) + (B @ AW, d ) extends to

an inclusion of acyciic closures.

Applying Z@) @i -, we get

with fibre (r(s%), O).

For j = 2.. . p - 1, replace suj by suj + ( - l ) ~ j - l s u l ~ ~ - l ( s v ) . Then the difkrential

in (6.1) is dehed simply by b(sui ) = pv and b ( s y ) = -pyP(sv), with s u ~ , . - . , S(LP-1

cycles- Since d ( s y ) is not a boundary in I'(su, sul.. . s ~ + ~ ) , (r (sv, S U I , . . . , sy), 8) SU, sul - . s+-=), D) B (r(s+), 0)-

Example 6.9. Let k be a field. Consider the following diagram of CDGA'S over k,

where rn is a quasi-isomorphism and (AX, d) is a Su l l i van algebra. If Q c k, then the

Lifting Lemma [13, Chapter 121 implies that there exïsts a morphism $ : (M, d) + (B, d )

such that m$ ci p. This example demonstrates that the lifting lemma fa&, in general,

in non-zero CharactenStic-

Let k = Fp. Let (A, d) = (A@, v ) , du = u), with lu[ = 2n. Let (B , d ) = (A(ul, q), O ) ,

with lull = 2np, lul 1 = 2np + 1, and define a quasi-isomorphism m : (B, d) 3 (A, d )

by m(ul) = uP, m ( q ) = uP%. Let (AX, d) be the minimal Sullivan algebra defineci

by X = F,{X, y, z), 1x1 = 2n@ + l), [y[ = 2np, [zl = 2n + 1; & = yz. Define a DGA

6.3 Behaviour of the Bockstein spectral sequence

Example 6.1 0. This example shows that {Er(L)} may coiIapse while { E ( U L ) ) does not.

It is essentiaily Example 6.6 with the algebra laid bare.

Let L = Lab(e, f) over Zbl on generators e and f of degrees 2n-1 and 2n, respectively.

Set 0 f = pe. Applying Proposition 6.1 recursively, we have ET(UL) = I'(szr, sy,) and

Er(uL) = U&(G, fr), with 1 ~ 1 = Iszrl = 2np' - 1, Ifrl = lsyvl = 2npr7 P r ( s ~ r ) = syr,

and PT ( fr ) = +, while the sequence G ( L ) collapses after the ikst term.

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