RATIONALIZED EVALUATION SUBGROUPS OF A MAP II:
QUILLEN MODELS AND ADJOINT MAPS
GREGORY LUPTON AND SAMUEL BRUCE SMITH
Abstract.Let !: map(X, Y ; f) ! Y denote a general evaluation fibration.
Working in the setting of rational homotopy theory via differential grad*
*ed Lie
algebras, we identify the long exact sequence induced on rational homoto*
*py
groups by ! in terms of (generalized) derivation spaces and adjoint maps*
*. As
a consequence, we obtain a unified description of the rational homotopy *
*theory
of function spaces, at the level of rational homotopy groups, in terms o*
*f deriva-
tions of Quillen models and adjoints. In particular, as a natural extens*
*ion of
a result of Tanr'e, we identify the rationalization of the evaluation su*
*bgroups
of a map f :X ! Y in this setting. As applications, we consider a gener-
alization of a question of Gottlieb, within the context of rational homo*
*topy
theory. We also identify the rationalization of the G-sequence of f and *
*make
explicit computations of the homology of this sequence. In a separate re*
*sult of
independent interest, we give an explicit Quillen minimal model of a pro*
*duct
A x X, in the case in which A is a rational co-H-space.
1.Introduction
Let f :X ! Y be a based map of connected spaces. Let map (X, Y ) de-
note the space of unbased maps from X to Y and map (X, Y ; f) the path com-
ponent containing f. Evaluation at the basepoint of X determines a fibration
! :map(X, Y ; f) ! Y . The nth evaluation subgroup of f is then defined to be t*
*he
subgroup
Gn(Y, X; f) = Image{!# :ßn(map (X, Y ; f)) ! ßn(Y )}
of ßn(Y ). The nth Gottlieb group Gn(X) of a space X occurs as the special case
X = Y and f = 1. Because ! :map(X, X; 1) ! X can be identified with the
connecting map of the universal fibration for fibrations with fibre X, the Gott*
*lieb
group is an important universal object for fibrations with fibre X. The structu*
*re
of the Gottlieb group and its role in homotopy theory represents a broad area of
research having its beginnings, of course, with the papers of Gottlieb [Got65, *
*Got68,
Got69] (see [Opr95 ] for a recent survey and further references). A fundamental
obstacle to computing Gottlieb groups is their lack of functorality, in that ge*
*nerally
f# G*(X) 6 G*(Y ) for a map f :X ! Y . By widening our perspective so as to
include the evaluation subgroups of a map, we remedy this situation somewhat, in
that f :X ! Y always induces a map f# :Gn(X) ! Gn(Y, X; f).
____________
Date: January 15, 2004.
2000 Mathematics Subject Classification. 55P62, 55Q52.
Key words and phrases. Gottlieb group, rational homotopy theory, Quillen min*
*imal model,
adjoint map, coformal map, G-sequence.
1
2 GREGORY LUPTON AND SAMUEL BRUCE SMITH
In a previous paper [LS03], we have developed a basic framework within which
rational homotopy groups of function spaces and related topics_including ratio-
nalized evaluation subgroups_ may be studied. All results in that paper were
developed from the Sullivan minimal model point of view, that is, using DG alge-
bras. The current paper is intended both as a complement to and a continuation
of the earlier work. It is complementary, in that all results here, and the ba*
*sic
framework that we establish, are developed from the Quillen minimal model point
of view, that is, using DG Lie algebras. It continues the earlier work in that *
*we
present some different developments of the basic results. We refer to [LS03] fo*
*r a
general introduction to the themes of this paper, and for a survey of the exisi*
*ting
results in the area.
In [LS03, Th.2.1], we identified the rationalization of the induced homomorph*
*ism
!# :ßn(map (X, Y ; f)) ! ßn(Y ) in terms of the homology of derivation spaces of
the Sullivan models of the spaces X and Y . Our result extends, to a connected
component of a general function space, the isomorphism
ßn(map (X, X; 1)) Q ~=Hn(Der*(MX )),
which was first observed by Sullivan in [Sul78]. Here Der*(MX ) is the differen*
*tial
graded vector space of degree-lowering derivations of the Sullivan model of X.
Our results in [LS03] provide a general framework for studying certain long exa*
*ct
sequences of homotopy groups for function space components within the context of
Sullivan minimal models. In particular, they allow for an extension of the F'el*
*ix-
Halperin description of the rational Gottlieb groups (see [FH82 , Th.3]) to the*
* more
general setting of rational evaluation subgroups of a map.
The developments of this paper proceed in an analogous way to those of [LS03].
As a special case of Theorem 4.1, we obtain an isomorphism
ßn(map *(X, X; 1)) Q ~=Hn(Der*(LX )),
where (Der*(LX )) is the differential graded vector space of degree-raising der*
*iva-
tions of the Quillen model of X. More generally, the results of Section 4 lead *
*to
a complete picture, within the framework of Quillen models, of various long exa*
*ct
sequences of rational homotopy groups of function space components. In particul*
*ar,
in Theorem 5.2 we extend Tanr'e's description of the rationalized Gottlieb group
[Tan83, VII.4.(10)] to a description of the rationalized evaluation subgroup of*
* a
map.
We pursue two applications of these results here. We first consider a genera*
*l-
ization of a question of Gottlieb, in the context of rational homotopy theory. *
* It
is well-known that Gottlieb elements in G*(X) have vanishing Whitehead product
with all elements of ß*(X). Let [ , ]w denote the Whitehead bracket in ß*(X)
and let P*(X) denote the subgroup of ß*(X) consisting of homotopy elements with
vanishing Whitehead product with all elements of ß*(X)_the so-called Whitehead
center of ß*(X). Gottlieb's question asks about the difference between P*(X) and
its subgroup G*(X). More generally, given a map f :X ! Y set
Pn(Y, X; f) = {ff 2 ßn(Y ) | [ff, f# (fi)]w = 0 for allfi 2 ß*(X)}.
Then Gn(Y, X; f) is a subgroup of Pn(Y, X; f), and P*(X) occurs as the special *
*case
X = Y and f = 1. In Theorem 5.4, we identify the quotient P*(Y, X; f)=G*(Y, X; *
*f)
for general maps f :X ! Y between rational spaces via a particular commutative
diagram of adjoints and derivation spaces. On the other hand, in Theorem 5.9 we
QUILLEN MODELS AND ADJOINT MAPS 3
prove G*(Y, X; f) = P*(Y, X; f) when f :X ! Y is a coformal map of rational
spaces.
As a second application, we describe and study the rationalization of the so-
called G-sequence of a map as constructed by Lee and Woo [WL88b ]. This is a
chain complex of the form
f#
. ._.__//Gn(X)_____//Gn(Y, X; f)__//_Greln(Y, X;_f)//_Gn-1(X)____//. . .
which arises naturally in the context of function spaces and evaluation maps de-
scribed above. We identify the rationalization of the G-sequence in the framewo*
*rk
of derivation spaces and adjoint maps of Quillen models (Corollary 6.4). We giv*
*e,
in particular, a complete calculation of the homology of this sequence at a par*
*tic-
ular term, in the case of a single cell-attachment (Theorem 6.7). We also prove
the rationalized G-sequence is exact at the same particular term for coformal m*
*aps
(Theorem 6.8), as an extension of the equality above.
The paper is organized as follows. Section 2 contains purely algebraic defin*
*i-
tions and constructions used throughout the paper. Here, we describe the basic
framework of generalized derivation spaces and generalized adjoint maps, and the
exact sequences in homology which arise in this context. In Section 3 we obtain*
* a
first connection between generalized derivations and topology. In a self-contai*
*ned
argument, we describe an explicit Quillen minimal model for a product A x X in
terms of the Quillen models of the factors, in the case in which A is a rationa*
*l co-
H-space. The description depends on a particular class of generalized derivatio*
*ns.
While we believe the general result is of independent interest, we require it h*
*ere
mainly for the case in which A is a single sphere Sn and briefly for the case in
which A = Sn _ Sn. In Section 4 we connect the algebraic framework of Section 2
to the homotopy data of function spaces and evaluation maps. The main results
here are Theorem 4.1 and Theorem 4.4, in which we identify the rational homotopy
groups of function spaces, and more generally the long exact rational homotopy
sequence of the evaluation fibration ! :map(X, Y ; f) ! Y , in terms of homology
of derivation spaces of Quillen models. Sections 5 and 6 contain our examples a*
*nd
applications. In a technical appendix, we review some basic material concerning
DG Lie algebra homotopy theory and provide details of some results from this ar*
*ea
that we use for our proofs.
Throughout this paper, spaces X and Y will be based, simply connected CW
complexes of finite type. Given a map f :A ! B, either topological or algebraic,
f* denotes pre-composition by f and f* post-composition by f. We use "*" to
denote the constant map of spaces. We write ! generically for an evaluation map
! :map(X, Y ; f) ! Y , since it will be clear from the context which component *
*is
intended. We use H(f) to denote the map induced on homology by f, and, when
f is a map of spaces, f# for the map induced on homotopy groups. We write
f ~ g to denote that based maps f and g are homotopic via a based homotopy.
In some instances, we will consider homotopies that are either unbased_ such as
those in the unbased function space component map (X, Y ; f), or are relative to
some subspace. In such instances, we will specify the nature of the homotopy. We
will continue to introduce notation and set conventions throughout the paper, on
an as-needed basis and usually at the start of each section.
4 GREGORY LUPTON AND SAMUEL BRUCE SMITH
2.Lie Derivation Spaces
In this section, we consider a class of chain complexes over Q obtained from
generalized derivations of differential graded Lie algebras. Geometric content *
*will
be obtained when we apply these basic constructions to Quillen models in Sectio*
*n 4.
We first establish our notational conventions for this material.
By vector space, we mean a graded vector space of finite type over the ration*
*als.
Furthermore, all our vector spaces will be connected, that is, they will be pos*
*itively
graded. The degree of an element x 2 V is written |x|. The space of degree n
elements of V will be denoted Vn. A vector space generated by a single element
v will be written Qv. That is, if |v| = n, then (Qv)i is isomorphic to Q if i =*
* n
and is zero otherwise. The kth suspension of V , denoted by sk(V ), is the vect*
*or
space defined as (sk(V ))n = Vn-k. In particular, the desuspension of V denoted
s-1V is the vector space with (s-1V )n = Vn+1. A typical example of desuspension
that we consider is that of the reduced rational homology of a simply connected
space of finite type, denoted by s-1He*(X; Q). A vector space V equipped with
a differential, that is, a linear map d: V ! V of degree -1 that satisfies d2 =*
* 0,
will be called a DG (differential graded) vector space. Given a cycle x 2 Vn, a*
* DG
vector space, we write for the homology class in Hn(V, d) represented by x.
A Lie algebra L is a (graded) vector space L equipped with a bilinear, degree
zero map (the "bracket" multiplication) [ , ]: L x L ! L that satisfies
(i)Anti-symmetry: [ff, fi] = -(-1)|ff||fi|[fi, ff]
(ii)Jacobi identity: [ff, [fi, fl]] = [[ff, fi], fl] + (-1)|ff||fi|[fi, [ff,*
* fl]].
The motivating example here is ß*( X) Q, the rational homotopy of X with
the Samelson bracket [ , ].
A DG Lie algebra is a pair (L, d) where L is a Lie algebra and d is a vector *
*space
differential that satisfies the derivation law d([x, y]) = [d(x), y] + (-1)|x|[*
*x, d(y)].
A map of DG Lie algebras OE: (L, d) ! (L0, d0) is a homomorphism that respects
differentials, that is, satisfies OEd = d0OE. Given a map of DG Lie algebras, w*
*e may
pass to homology in the usual way. If the induced homomorphism H(OE) is an
isomorphism, then we say OE is a quasi-isomorphism. We will frequently use the
symbol "'" to denote the fact that a map is a quasi-isomorphism, especially in
diagrams. We write L(V ) for the free Lie algebra generated by the vector space
V . The coproduct (or "free product") of (DG) Lie algebras L and L0 is written
L t L0. We usually abuse notation somewhat and write L(V, W ) for the coproduct
L(V ) t L(W ) = L(V W ). Similarly, we will also write L(v) for L(Qv), L(v, w*
*) for
L(Qv, Qw) = L(Qv Qw) = L(Qv)tL(Qw), L(v, W ) for L(Qv, W ) = L(Qv W ) =
L(Qv)tL(W ), and so-forth. We commit another abuse of notation by saying that a
DG Lie algebra is free if the underlying Lie algebra is a free Lie algebra. To *
*reduce
parentheses, we usually write a free DG Lie algebra L(V ), d as L(V ; d). Fin*
*ally,
a DG Lie algebra (L, d) is minimal if L is free and the differential is decompo*
*sable,
that is, d(L) [L, L].
We assume the reader is familiar with the basic facts of rational homotopy th*
*eory
from the Quillen point of view, that is, using DG Lie algebra minimal models. G*
*ood
references for this material include [Tan83] and [FHT01 , Part IV]. Specificall*
*y,
we recall that each space X has a Quillen minimal model which is a minimal
DG Lie algebra (LX , dX ) whose isomorphism type is a complete invariant of the
rational homotopy type of X. As a Lie algebra, we have LX = L(s-1He*(X; Q)).
QUILLEN MODELS AND ADJOINT MAPS 5
The differential dX is determined by the topology of X in a more arcane way.
The Quillen model of X recovers the rational homotopy Lie algebra of X via a
Lie algebra isomorphism H*(LX , dX ) ~= ß*( X) Q. Furthermore, a map of
spaces f :X ! Y induces a map of Quillen models Lf: LX ! LY with H(Lf)
corresponding, via the above isomorphism, to the rationalization of the map ind*
*uced
on homotopy groups by f : X ! Y . We refer to this map of Quillen models as
the Quillen minimal model of the map f. Our basic reference for rational homoto*
*py
theory is [FHT01 ].
Let (L, d) be a DG Lie algebra. A derivation of degree n of L is a linear map
` :L ! L that raises degree by n and satisfies the rule
`([ff, fi]) = [`(ff), fi] + (-1)|`||ff|[ff, `(fi)].
When n = 1 we also require d O ` = -` O d. We write Dern(L) for the space
of derivations of degree n of L. The space Der*(L) has the structure of a DG
Lie algebra, with (commutator) bracket [`1, `2] = `1 O `2 - (-1)|`1||`2|`2 O `1*
*, and
differential defined as D(`) = [d, `]. The adjoint map ad: L ! Der*(L), given by
ad(x)(y) = [x, y], is now a map of DG Lie algebras.
At the expense of the Lie bracket, we can extend the notion of a derivation of
a DG Lie algebra to a derivation with respect to a map of Lie algebras, as foll*
*ows.
Given a map _ :(L, dL) ! (K, dK ) of DG Lie algebras, define a derivation of
degree n with respect to _, or simply a _-derivation of degree n, to be a linea*
*r map
` :L ! K that increases degree by n and satisfies
`([ff, fi]) = [`(ff), _(fi)] + (-1)n|ff|[_(ff), `(fi)]
for ff, fi 2 L. Let Dern(L, K; _) denote the space of all _-derivations of deg*
*ree
n from L to K. Next, define D : Dern(L, K; _) ! Dern-1(L, K; _) by D(`) =
dK O ` - (-1)|`|` O dL. The pair Der*(L, K; _), D is then a DG vector space. *
*The
adjoint map associated to _ is ad_:K ! Der*(L, K; _) where
ad_(ff)(fi) = [ff, _(fi)].
It is easy to check that ad_ is a map of DG vector spaces. We continue to write*
* ad
for the adjoint map associated to the identity 1: L ! L.
We now begin to consider homology of derivation spaces. At this point, we
introduce only what is needed for Section 4. Further notions concerning derivat*
*ion
spaces and their homology will be introduced as needed in subsequent sections.
To ease notation in what follows, we adopt the following conventions. If (V, d)*
* is
a DG vector space we suppress the differential when we write its nth homology
group, writing just Hn(V ). Furthermore, we indicate only the outermost degree.
For example, Hn(Der(L, K; _)) denotes the nth homology group of the DG vector
space (Der*(L, K; _), D).
Next, we construct a long exact homology sequence that we will show corre-
sponds to the long exact rational homotopy sequence of the evaluation fibration
(Theorem 4.4). Given a map OE: V ! W of DG vector spaces we will need the
relativization of OE. This is the DG vector space (Rel*(OE), ffi) given, in deg*
*ree n, by
Reln(OE) = Wn Vn-1
with differential of degree -1 defined as
ffi(w, v) = (OE(v) - dW (w), dV (v)).
6 GREGORY LUPTON AND SAMUEL BRUCE SMITH
The inclusion J : Wn ! Reln(OE) with J(wn) = (wn, 0), and the projection P :
Reln(OE) ! Vn-1 with P (wn, vn-1) = vn-1, give a short exact sequence of chain
complexes
0 ____//_W*_J__//Rel*(OE)P_//V*-1____//0.
This leads to a long exact sequence whose connecting homomorphism is H(OE).
Actually, J is really an ä nti-chain" map, in that ffiJ = -JdW . But J still in*
*duces
a homomorphism H(J) on homology, and gives rise to a long exact sequence in the
usual way. Thus we have a long exact homology sequence
H(P) H(OE) H(J)
. .!.Hn+1(Rel(OE)) ---! Hn(V ) ---! Hn(W ) ---! Hn(Rel(OE)) ! . ...
We refer to this sequence as the long exact homology sequence of OE.
Now apply this construction to the adjoint map ad_ :K ! Der*(L, K; _) from
above. We obtain a long exact homology sequence
H(J)
. ._.________//_Hn+1(Rel(ad_))BC_
_________H(P)_____________________
GFfflffl|
H(ad_) H(J)
Hn(K) ______//Hn Der(L, K; _)____//_Hn(Rel(ad_))BC_
_________H(P)_____________________
GFfflffl|
H(ad_)
Hn-1(K) ___________//_. . .
We call this sequence the long exact derivation homology sequence of _.
3.Quillen models for certain products
The Sullivan minimal model of a product of spaces is easily expressed as the
tensor product of the Sullivan minimal models of the spaces. For Quillen minimal
models, the situation is more complicated due to the fact that the direct sum of
minimal DG Lie algebras_which is the categorical product in this setting_is not
minimal. Tanr'e has described how to construct the Quillen minimal model of a
product of spaces A x X in terms of the Quillen minimal models of A and X (see
[Tan83, Prop.VII.1(2)]). For our purposes, we need explicit minimal models for *
*the
cases in which A = Sn and A = Sn _ Sn. In this section, we describe a Quillen
minimal model of A x X for the more general case in which A is any rational co-
H-space. Our description essentially makes Tanr'e's construction explicit for t*
*hese
cases, although our treatment here is self-contained.
Suppose X has Quillen minimal model L(W ; dX ), and that L(V ; d = 0) is the
Quillen minimal model of a simply connected, finite-type rational co-H-space A.
Suppose {vi}i2J is a (connected, finite type) basis for V , and |vi| = ni- 1 fo*
*r each
i 2 J. Topologically,Wthis corresponds to A being of the rational homotopy type*
* of
the wedge of spheres i2JSni. We construct a new minimal DG Lie algebra from
these data as follows: For each i 2 J, let Wi denote the ni-fold suspension of *
*W ,
that is, set Wi= sni(W ). Let ~: L(W ) ! L(W, V, i2JWi) denote the inclusion of
graded Lie algebras. For each i 2 J, let Si:W ! Widenote the ni-fold suspension
isomorphism and define a ~-derivation Si:L(W ) ! L(W, V, i2JWi) by extending
QUILLEN MODELS AND ADJOINT MAPS 7
Siusing the ~-derivation rule. Now define a differential @ on L(W, V, i2JWi) t*
*hat
extends the differentials on L(W ) and L(V ), by setting
@(Si(w)) = [vi, w] + (-1)nSi(dX (w))
for each generator w 2 W (and thus each generator of Wi). This definition is
equivalent to the identity ad(vi) = D(Si) in Derni-1(L(W ), L(W, V, i2JWi); ~).
We will show that L(W, V, i2JWi; @) is the Quillen minimal model of A x X.
First we check that the preceding formula defines a differential.
Lemma 3.1. The derivation @ of L(W, V, i2JWi) satisfies @ O @ = 0.
Proof.It is sufficient to check on generators. Further, since @ extends the dif*
*fer-
entials on L(W ) and L(V ), it is sufficient to check that (@)2(Si(w)) = 0 for *
*each
w 2 W and i 2 J. We compute as follows:
(@)2(Si(w))= @ ([vi, w] + (-1)niSi(dX (w)))
= (-1)ni-1[v, dX (w)] + (-1)ni@Si(dX (w))
= (-1)ni-1ad(vi)(dX (w)) + (-1)ni@Si(dX (w))
= SidX (dX (w)) = 0.
The penultimate step follows from the identity ad(vi) = D(Si) observed above,
which expands to give [vi, Ø] = @Si(Ø) - (-1)niSidX (Ø) for any Ø 2 L(W ), and
which here is applied to dX (w) 2 L(W ).
We will need the following technical point in our argument. The discussion he*
*re
is lifted from [FHT01 , Sec.22(f)]. Suppose that L(V ; d) is a free, but not ne*
*cessarily
minimal, connected DG Lie algebra. That is, suppose that the differential d may
have a non-trivial linear part. Since L(V ) is free, we can write d as the sum*
* of
two derivations d = d0 + d+ , where d0: V ! V is the linear part of the differe*
*ntial
d and d+ is the decomposable part of d that increases bracket length. Indeed, d0
itself is a differential_although d+ is generally not_and so we obtain a DG vec*
*tor
space (V, d0). Next, suppose OE: L(V ; d0) ! L(W ; d) is a morphism of free, bu*
*t not
necessarily minimal, connected DG Lie algebras. Again, because the Lie algebras
are free, we may write the linear map OE: V ! L(W ) as a sum OE = OE0 + OE+ , w*
*here
OE0: V ! W is the linear part of OE and OE+ is the decomposable part of OE that
increases bracket length. In this way, we obtain a morphism of DG vector spaces
OE0: (V, (d0)0) ! (W, d0),
which we refer to as the linearization of OE. The following result can be inter*
*preted
as a version of Whitehead's theorem in our context.
Lemma 3.2. Let OE: L(V ; d0) ! L(W ; d) be a morphism of connected free DG Lie
algebras. Let OE0: (V, (d0)0) ! (W, d0) be the linearization of OE. Then OE is *
*a quasi-
isomorphism of DG Lie algebras if, and only if, OE0 is a quasi-isomorphism of DG
vector spaces.
Proof.This is proved as [FHT01 , Prop.22.12].
We now come to the main point of the section.
Theorem 3.3. Let X be a simply connected space of finite type with Quillen mini-
mal model L(W ; dX ). Let A be a rational co-H-space of the rational homotopy t*
*ype
8 GREGORY LUPTON AND SAMUEL BRUCE SMITH
W
of the wedge of spheres i2JSni. Then L(W, V, i2JWi; @), as described above, *
*is
the Quillen minimal model of A x X.
Proof.Our starting point is the well-known fact that the direct sum of Quillen *
*min-
imal models gives a (non-minimal) DG Lie algebra model for the product [FHT01 ,
p.332,Ex.3]. In our case, this gives L(V ) L(W ; dX ) as a non-minimal model *
*for
A x X. We will show that the obvious projection
p: L(W, V, i2JWi; @) ! L(V ) L(W ; @)
is a quasi-isomorphism. Since the domain is a minimal DG Lie algebra, this is
sufficient to show that it is the Quillen minimal model of the product.
So consider the following commutative diagram of DG Lie algebra morphisms:
q
0________//K____i___//L(W, V, i2JWi;_@)__//L(V_)___//0
p0|| p|| 1||
|fflffli0 fflffl| q0 fflffl|
0_____//L(W ; dX_)__//_L(V ) L(W ;_dX_)_//L(V_)___//0
Here, q and q0 are the obvious (quotient) projections onto L(V ), and i and i0 *
*are
the inclusions of the kernels, so that the rows are short exact sequences of DG*
* Lie
algebras. We will argue that p0:K ! L(W ; dX ) is a quasi-isomorphism. First no*
*te
that, as a sub-DG Lie algebra of a connected, free DG Lie algebra, K is itself a
connected, free DG Lie algebra. Indeed, as a Lie algebra, we may write
K = L(W, iWi, [V, W ], i[V, Wi], [V, [V, W ]], i[V, [V, Wi]], . .;.@K )
or more succinctly K = L({adj(V )(W )}j 0, { iadj(V )(Wi)}j 0; @K ). In these
expressions, [V, W ] denotes the vector space spanned by brackets [v, w] with v*
* 2 V
and w 2 W , and so-forth, and ad0(V )(W ) denotes W , ad1(V )(W ) denotes [V, W*
* ],
ad2(V )(W ) denotes [V, [V, W ]], and so- forth. We now claim that (@K )0, the *
*linear
part of the differential in K, induces isomorphisms
(@K )0: i2Jadj(V )(Wi) ! adj+1(V )(W )
for each j 0. First recall that @K is simply the restriction of the differe*
*ntial
@ to the kernel of q, and that @(V ) = 0. Extending our notation a little fur-
ther, we can denote a typical spanning element of adj(V )(W ) as follows. Sup-
pose (vr1, vr2, . .,.vrj) 2 V jis a j-tuple. Then write ad(vr1, vr2, . .,.vrj)*
*(w) for
[vr1, [vr2, [. .,.[vrj-1, vrj]] . .].. Likewise for elements of adj(V )(Wi). No*
*w let w 2 W
be a typical element. From the definition of @ above, we have
@ ad(vr1, vr2, . .,.vrj)(Si(w))= ad(vr1, vr2, . .,.vrj)(@(Si(w)))
= ad(vr1, vr2, . .,.vrj, vi)(w)
ad(vr1, vr2, . .,.vrj)(SidX (w)).
Since L(W ; dX ) is the Quillen minimal model of X, dX (w) is decomposable in L*
*(W )
and thus SidX (w) is decomposable in L(W, Wi). From this, it follows that the l*
*ast
term displayed above, namely ad(vr1, vr2, . .,.vrj)(SidX (w)), is decomposable *
*in
K. We prove this assertion in Lemma 3.4 below. Assuming for the time being its
validity, it follows that the linear part of the differential in K induces isom*
*orphisms
(@K )0: adj(V )(Wi) ~=adj(V )ad(vi)(W ) for each i 2 J and each j 0, and hence
isomorphisms (@K )0: iadj(V )(Wi) ~=adj+1(V )(W ) for each j 0, as claimed.
Notice that as a consequence of this, we must have (@K )0 = 0 on each vector sp*
*ace
QUILLEN MODELS AND ADJOINT MAPS 9
of generators adj+1(V )(W ) in K, for j 0, since the linear part of a differe*
*ntial is
itself a differential. In any case, this latter fact also follows from Lemma 3.*
*4. Finally,
notice that (@K )0 = 0 on the vector space of generators W in K, since @ = dX
is decomposable on W . It now follows that the DG vector space (Q(K), (@K )0)
obtained by linearizing K may be written as a direct sum
i j
(Q(K), (@K )0) ~=(W, (@K )0 = 0) j 0 iadj(V )(Wi) adj+1(V )(W ) , (@K )0 ,
i j
in which each summand iadj(V )(Wi) ad j+1(V )(W ) , (@K )0 is an acyclic
DG vector space. It is now evident that H(Q(K), (@K )0) ~=W and that the lin-
earization of p0, that is, (p0)0: (Q(K), (@K )0) ! (W, @0 = 0), is a quasi-isom*
*orphism
of DG vector spaces. It follows from Lemma 3.2 that p0is a quasi-isomorphism of
DG Lie algebras.
Returning to the diagram of short exact sequences, we now have left and right
vertical arrows that are quasi-isomorphisms. Therefore, by passing to the induc*
*ed
diagram of long exact homology sequences and applying the five-lemma, we obtain
that p is a quasi-isomorphism. Hence L(W, V, i2JWi; @) is the Quillen minimal
model of A x X.
The proof of Theorem 3.3 will be completed when we establish the following
lemma:
Lemma 3.4. With notation as in the proof above, suppose Ø is a decomposable
element in K. Then [v, Ø] is also decomposable in K, for any v 2 V . In particu*
*lar,
if Ø is decomposable in L(W, Wi) for some i 2 J, and hence decomposable in K, t*
*hen
ad(vr1, vr2, . .,.vrj)(Ø) is decomposable in K for any j-tuple (vr1, vr2, . .,.*
*vrj) 2 V j.
Proof.Recall that K = L({adj(V )(W )}j 0, i{adj(V )(Wi)}j 0) is a sub-Lie alge-
bra of L(W, V, iWi), and observe that v 2 V is not a generator_is not even an
element_of K, so the statement is not entirely trivial. Without loss of general*
*ity,
we may assume that Ø is a monomial term. We argue by induction on the bracket
length in K of Ø. When Ø has length 2 in K, we have Ø = [Ø1, Ø2] with Ø1 and
Ø2 indecomposable monomials in K. From the Jacobi identity in L(W, V, iWi),
we may write [v, Ø] = [Ø1, [v, Ø2]] [Ø2, [v, Ø1]]. Now Ø1 is an element from*
* either
adj(V )(W ) or adj(V )(Wi) for some i 2 J and j 0, and likewise for Ø2. So [v*
*, Ø1]
is from adj+1(V )(W ) K or adj+1(V )(Wi) K, and likewise for [v, Ø2]. That *
*is,
if Ø1 and Ø2 are indecomposable in K, then [v, Ø] = [Ø1, [v, Ø2]] [Ø2, [v, Ø*
*1]] dis-
plays [v, Ø] as decomposable (and again of length 2) in K. Now assume inductive*
*ly
that the assertion is true for Ø of bracket length r in K. Let Ø be a monomial
of bracket length r + 1 in K. By judicious use of the Jacobi identity in K, we
may assume that Ø = [Ø1, Ø2] with Ø1 an indecomposable in K and Ø2 a decom-
posable monomial of bracket length r in K. Once again, the Jacobi identity in
L(W, V, iWi) yields [v, Ø] = [Ø1, [v, Ø2]] [Ø2, [v, Ø1]]. Our inductive hyp*
*othesis
implies that [v, Ø2] is (decomposable) in K, and the same observations as were *
*used
to start the induction show that [v, Ø1] is an (indecomposable) element in K. H*
*ence
[v, Ø] is decomposable in K, and the induction is complete. The result follows.
Since it is the main case we require here, we write out explicitly what this *
*gives
for the model of Sn x X, with a slight easing of notation.
10 GREGORY LUPTON AND SAMUEL BRUCE SMITH
Corollary 3.5. Suppose X has Quillen minimal model L(W ; dX ), let L(v) with
|v| = n - 1 and zero differential be the Quillen model of Sn, and set W 0= sn(W*
* ).
Let ~: L(W ) ! L(W, v, W 0) be the inclusion, and S :L(W ) ! L(W, v, W 0) be the
~-derivation that extends the linear map S(w) = w0. Define a differential @ on
L(W, v, W 0) by @(w) = dX (w), @(v) = 0, and
@(w0) = [v, w] + (-1)nS(dX (w)),
for each w 2 W . Then L(W, v, W 0; @) is the Quillen minimal model of Sn x X.
4. Lie Derivations and Homotopy Groups of Function Spaces
Say two maps of vector spaces f :U ! V and g :U0 ! V 0are equivalent if there
exist isomorphisms ff and fi which make the diagram
f
U _____//_V
ff~=|| fi~=||
|fflffl fflffl|
U0 __g__//V 0
commutative. We will extend this notion of equivalence in the obvious way to ex*
*act
sequences of vector spaces, and any other diagram of vector space maps. Given a*
*ny
map f :X ! Y , we have the homomorphism
j# 1: ßn map *(X, Y ; f) Q ! ßn map (X, Y ; f) Q
induced on rational homotopy groups by the fibre inclusion of the general evalu*
*ation
fibration map *(X, Y ; f)j__//map(X, Y ;_f)!_//Y. On the other hand, we have
the homomorphism
H(J): Hn Der(LX , LY ; Lf) ! Hn(Rel(adLf)
that forms part of the long exact homology sequence of the adjoint map adLf:LY !
Der(LX , LY ; Lf) of the Quillen minimal model Lf: LX ! LY of f. In Theorem 4.1,
which is the basic result of this section, we establish that these two homomorp*
*hisms
are equivalent. This result and one immediate consequence will occupy the remai*
*n-
der of this section.
The main step is to establish vector space isomorphisms
: ßn(map *(X, Y ; f)) Q ! Hn(Der(LX , LY ; Lf))
and
: ßn(map (X, Y ; f)) Q ! Hn(Rel(adLf)),
that give the equivalence. In the following, we assume a fixed choice of Quill*
*en
minimal model Lf: LX ! LY for f :X ! Y . Write LX = L(W ; dX ) and LY =
L(V ; dY ). To this end, we define group homomorphisms
0:ßn(map *(X, Y ; f)) ! Hn(Der(LX , LY ; Lf))
and
0:ßn(map (X, Y ; f)) ! Hn(Rel(adLf))
from the ordinary homotopy groups to the appropriate vector spaces, for n
2. Then the isomorphisms and are obtained as the rationalizations of these
homomorphisms.
QUILLEN MODELS AND ADJOINT MAPS 11
Define 0 as follows. Let ff 2 ßn(map *(X, Y ; f)) be represented by a map
a: Sn ! map*(X, Y ; f). Then the adjoint A: Sn xX ! Y of a has Quillen minimal
model LA :LSnxX ! LY . Recall from Corollary 3.5 that LSnxX = L(W, v, W 0; @).
Now, since a is a (based) map into the function space of based maps, we have
A O i1 = *: Sn ! Y and A O i2 = f :X ! Y . It follows that we may take the
Quillen minimal model of A to be a DG Lie algebra map
LA :L(W, v, W 0; @) ! LY
that satisfies LA (v) = 0 and LA (w) = Lf(w) for each w 2 W . See the appendix *
*for
justification of this last assertion. Now define a linear map `A :L(W ) ! LY th*
*at
increases degree by n as the composition
L(W )__S__//L(W, v, W_0)LA//_LY,
where S :L(W ) ! L(W, v, W 0) is the derivation from Corollary 3.5. A straightf*
*or-
ward check shows that `A is an Lf-derivation in Dern(LX , LY ; Lf). Furthermore,
we have dY `A = (-1)n`A dX and so `A is a cycle in the derivation space. Fi-
nally, we set 0(ff) = <`A > 2 Hn(Der(LX , LY ; Lf)). We will establish that 0*
*is a
well-defined homomorphism, and that its rationalization is an isomorphism, in
Theorem 4.1 below.
We define 0, and thus its rationalization , in a similar manner. Let ff 2
ßn(map (X, Y ; f)) be represented by a map a: Sn ! map (X, Y ; f). Then the
adjoint A: Sn x X ! Y of a still satisfies A O i2 = f :X ! Y , but the com-
position A O i1: Sn ! Y may give a non-trivial element of ßn(Y ). Correspond-
ingly, the Quillen minimal model of A satisfies LA (w) = Lf(w) for each w 2 W ,
but LA (v) 2 LY is now some non-trivial dY -cycle. As before, setting `A =
LA O S :L(W ) ! LY defines an Lf-derivation in Dern(LX , LY ; Lf). Recalling
the definition of Rel*(adLf) from Section 2, we obtain an element
(`A , LA (v)) 2 Reln(adLf).
Let ffi and D denote the differentials in Rel*(adLf) and Dern(LX , LY ; Lf) res*
*pec-
tively. Then we have ffi(`A , LA (v)) = adLf(LA (v)) - D(`A ), dY LA (v)) , a*
*nd
dY LA (v) = 0. We check that adLf(LA (v)) - D(`A ) = 0 2 Dern-1(LX , LY ; Lf).
For Ø 2 LX , we have
adLf(LA (v))(Ø) - D(`A )(Ø)
n
= [LA (v), Lf(Ø)] - dY LA O S(Ø) + (-1) LA O SdX (Ø)
n
= LA [v, Ø] + (-1) SdX (Ø) - dY LA S(Ø)
= LA @ S(Ø) - dY LA S(Ø)
= 0.
Thus (`A , LA (v)) is a cycle in the relative chain complex. Finally, we set 0*
*(ff) =
<`A , LA (v)> 2 Hn Rel(adLf) .
In the following result, we establish the basic properties of and .
Theorem 4.1. Suppose n 2. Then we have:
(A) 0and 0are well- defined homomorphisms;
(B) Their rationalizations and are isomorphisms;
12 GREGORY LUPTON AND SAMUEL BRUCE SMITH
(C) The following square is commutative:
ßn(map *(X, Y ; f)) __Q~=//_Hn(Der(LX , LY ; Lf))
j# Q|| H(J)||
|fflffl ~= fflffl|
ßn(map (X, Y ; f)) __Q_____//Hn(Rel(adLf))
Proof.Throughout the proof we will give full details for arguments concerning .
The arguments for are similiar, and we will simply indicate them without deta*
*ils.
We will need to use some facts about homotopy in the DG Lie algebra setting. The
most complete reference for this material is [Tan83, Ch.II.5]. The appendix to *
*this
paper contains a quick overview, and also provides careful justifications of so*
*me
technical details used in the following proof. We will make free use of the not*
*ation
concerning DG Lie algebra homotopy reviewed in the appendix.
(A) 0 is well-defined: Suppose that a ~ b: Sn ! map *(X, Y ; f) are two rep-
resentatives of the homotopy class ff. The adjoint of the (based) homotopy in
map*(X, Y ; f) from a to b gives a homotopy of their adjoints, A, B :Sn x X ! Y*
* ,
that is stationary on the subset Sn _ X Sn x X. Indeed, the homotopy of the
adjoints is stationary at the constant map on Sn and is stationary at f on X.
Consequently, the corresponding Quillen minimal models LA , LB :LSnxX ! LY
are homotopic via a DG Lie algebra homotopy
H: L(W, v, W 0)I ! LY
that satisfies H(L(v)I) = 0, H(L(sW, cW)) = 0, and H(w) = Lf(w) for each w 2 W
(see Lemma A.2 for details). Let oe :L(W, v, W 0)I ! L(W, v, W 0)I be the deriv*
*ation
of degree 1 defined in the appendix. Define a linear map : L(W ) ! LY of degree
n + 1 as the composition = H O oe O S. A straightforward check, using the fact
that H(sW, cW) = 0, shows that is an Lf-derivation. We will show that D =
`B -`A 2 Dern(LX , LY ; Lf). We have `B = HO~1OS and `A = HO~0OS = HOS.
Let J denote the ideal of L(W, v, W 0)I generated by sW cW. Since oe vanishes*
* on
the generators of J, and DI preserves the set of generators, J is stable under *
*the
composition oeDI. Furthermore, H is zero on J, since it is zero on the generato*
*rs.
We claim that (oeDI)r: W 0! L(W, v, W 0)I has image in J, for each r 2. First
observe that
oeDI(w0) = oe@(w0)= oe([v, w] + (-1)nSdX (w))
= [sv, w] + (-1)n[v, sw] + (-1)noeSdX (w).
Furthermore, the only terms not in J that may appear in oeSdX (w) are terms in
the sub-Lie algebra L(W, sW 0) that have exactly one occurrence of an element f*
*rom
sW 0. On applying DI to such terms, the only terms still not in J that may appe*
*ar
in DIoeSdX (w) are terms in the sub-Lie algebra L(W, sW 0, cW)0that have exactly
one occurrence of an element either from sW 0, or from cW 0. Since oe is zero *
*on
sW 0 cW,0oeDIoeSdX (w) 2 J. Direct computation shows that both oeDI([sv, w])
and oeDI([v, sw]) are in J. That is, (oeDI)r(w0) 2 J, for each r 2. Consequen*
*tly,
QUILLEN MODELS AND ADJOINT MAPS 13
H vanishes on these terms. Finally, using this, we compute that
`B (w)= LB O S(w) = H O ~1(w0)
0 0 0 X 1 r 0
= H w + DIoe(w ) + oeDI(w ) + __(oeDI) (w )
r 2r!
= H(w0) + HDIoe(w0) + H [sv, w] + (-1)n[v, sw] + (-1)noeSdX (w)
= LA (w0) + dY Hoe(w0) + (-1)nHoeSdX (w)
= `A (w) + dY (w) - (-1)n+1 dX (w).
It follows that the difference of derivations `B - `A = D in Dern(LX , LY ; Lf*
*).
Hence 0is well- defined.
0 is a homomorphism: Let :Sn ! Sn _ Sn denote the usual pinching co-
multiplication. Given ff, fi 2 ßn(map *(X, Y ; f)), the sum ff + fi is the comp*
*osition
(ff | fi) O . Suppose ff, fi have adjoints A, B :Sn x X ! Y , respectively. *
*Let
i1, i2: Sn ! Sn _ Sn denote the inclusions, and let (A | B)f: (Sn _ Sn) x X ! Y
be the map defined by (A | B)fO(i1x1) = A and (A | B)fO(i2x1) = B. Then the
adjoint of ff+fi is C := (A | B)fO( x1): Sn xX ! Y . We focus on identifying t*
*he
Quillen minimal model of (A | B)f, and it will follow that 0fis a homomorphism.
The map (A | B)f: (Sn _ Sn) x X ! Y is determined, up to homotopy, as the
unique map F that makes the following diagram homotopy commutative:
Sn x XR
RRRR
i1x1|| RARRRRR
fflffl| F RRRR))R
(Sn _ Sn)OxOX__________//___________________Y55lll
llll
i2x1|| llllll
| lll B
Sn x X
Consequently, any DG Lie algebra map that makes the diagram
L(W, v, W 0;T@)
TTTT
j1|| TTLATTTTT
fflffl| TTTTT))T
L(W, v1, v2,OW1,OW2;_@)_______//____________________LY55jjj
jjjj
j2|| jjjjjjj
| jjjj LB
L(W, v, W 0; @)
commute up to DG homotopy is a Quillen model for (A | B)f. In this diagram,
L(W, v1, v2, W1, W2; @) is the Quillen model of (Sn _ Sn) x X as described in T*
*he-
orem 3.3, and j1, j2 the obvious inclusions that identify v with v1 and W 0with
W1, and v with v2 and W 0with W2 respectively. This characterization of (the
Quillen model of) (A | B)f is explained in detail in [LS03]. Now there is an
obvious choice for : L(W, v1, v2, W1, W2; @) ! LY , namely the map that makes
the diagram commute. Hence, this map is a Quillen model for (A | B)f. Fi-
nally, since :Sn ! Sn _ Sn has Quillen model L : L(v) ! L(v1, v2) given by
L (v) = v1+ v2, it follows that x 1: Sn x X ! (Sn _ Sn) x X has Quillen model
14 GREGORY LUPTON AND SAMUEL BRUCE SMITH
L x1: L(W, v, W 0; @) ! L(W, v1, v2, W1, W2; @) given by L x1(w0) = w1+w2. Thus
LC (w0) = O L x1(w0) = (w1 + w2) = `A (w) + `B (w).
Hence we have `C = LC O S = `A + `B and it follows that 0is a homomorphism.
0 is a well-defined homomorphism: This is established by making small ad-
justments to the preceding arguments for 0. In this case, the homotopy of the
adjoints A and B is stationary at f on X, but is not stationary on Sn. The corr*
*e-
sponding homotopy H of Quillen minimal models still satisfies H(L(sW, cW)) = 0,
and H(w) = Lf(w) for each w 2 W (see Lemma A.2 for details), but we must
allow for non-zero terms in H(L(v)I). Since v is a @-cycle, however, we have
~1(v) = v + bv2 L(W, v, W 0)I. Therefore, LB (v) = H O ~1(v) = H(v) + HDoe(v) =
LA (v)+dY (H(sv)). Since H still vanishes on sW cW, the composition = HOoeOS
still defines an Lf-derivation. Computing exactly as before, we find that `B (w*
*) =
`A (w) + D (w) + [H(sv), Lf(w)]. It follows that, in the relative complex, we h*
*ave
ffi(- , H(sv)) = (`B - `A , LB (v) - LA (v)). So 0is well-defined. To show tha*
*t 0
is a homomorphism, we can use precisely the same argument as for 0, and use the
additional identity
LC (v) = O L x1(v) = (v1 + v2) = LA (v) + LB (v)
at the final step.
(B) is a surjection: Suppose given ` 2 Dern(LX , LY ; Lf), a cycle derivati*
*on
of degree n. Define a Lie algebra map LA :L(W, v, W 0; @) ! LY by setting
LA (w) = Lf(w), LA (v) = 0 and LA (w0) = `(w)
for w 2 W . Just as in the definition of 0, LA O S is an Lf-derivation and by
construction we have LA O S = `. We check that LA commutes with differentials
as follows:
LA (@(w0))= LA ([v, w] + (-1)nS(dX (w))
= 0 + (-1)nLA (S(dX (w))
= (-1)n`(dX (w))
= dY (`(w))
= dY (LA (w0))
Let A: Sn x X ! YQ be the geometric realization of LA , from the correspondence
between (homotopy classes of) maps between rational spaces and DG Lie algebra
maps between Quillen models. Let i1: Sn ! Sn x X and i2: X ! Sn x X denote
the inclusions. Since LA O Li1= 0 and LA O Li2= Lf, we have A O i1 ~ * and
A O i2 ~ fQ. Altering the geometric realization A up to homotopy, we may assume
A O i1 = * and A O i2 = fQ. Thus, the adjoint a: Sn ! map *(X, YQ; fQ) of A
represents an element ff 2 ßn(map *(X, YQ; fQ)). Clearly, we have (ff) = <`>, *
*and
so is surjective.
is an injection: Since is a homomorphism, it is sufficient to check that
(ff) = 0 implies ff = 0 2 ßn(map *(X, YQ; fQ)). As before, write (ff) = <`A >
and suppose `A 2 Dern(LX , LY ; Lf) is a boundary so that `A = D( ) for 2
Dern+1(LX , LY ; Lf). Define a homotopy
G :L(W, v, W 0)I ! LY
by setting G = LA on L(W, v, W 0) (so G starts at LA ), G = 0 on L(sW, cW, sv, *
*bv),
and G(sw0) = (w) on generators w0 2 W 0. We then set G(cw0) = G(DIsw0) =
QUILLEN MODELS AND ADJOINT MAPS 15
dY G(sw0) = dY (w), for cw02 cW 0, and then extend G as a Lie algebra map, so
that G is a DG Lie algebra map. It is straightforward to check that G ends at
G O ~1(w) = Lf(w) and G O ~1(v) = 0 on L(W, v). Just as in the proof of (A) abo*
*ve,
we observe that because G is zero on L(sW, cW), the composition G O oe O S acts*
* as
a derivation in Dern+1(LX , LY ; Lf). Therefore, we have G O oe O S(dX w) = (d*
*X w)
for each w 2 W . Furthermore, again just as in part (A), we find that G is zero*
* on
terms (oeDI)r(w0), for each r 2. Therefore, for w02 W 0, this homotopy ends at
0 0 0 X 1 r 0
G O ~1(w0)= G w + DIoe(w ) + oeDI(w ) + __(oeDI) (w )
r 2r!
= G(w0) + GDIoe(w0) + Goe((-1)nSdX (w))
= G(w0) + dY Goe(w0) + (-1)nGoeSdX (w)
= `A (w) + dY (w) - (-1)n+1 dX (w),
which is zero. Hence G ends at the Quillen minimal model of the composition
f O p2: Sn x X ! Y . It follows that A ~ f O p2: Sn x X ! YQ. Taking adjoints,
we obtain that a ~ *: Sn ! map *(X, Y ; f). Actually, we only obtain this last
homotopy as a free homotopy by taking adjoints, since the homotopy between A
and f O p2 is based, but not necessarily relative to X. However, a based map fr*
*om
a sphere is based-homotopic to the constant map if it is freely homotopic to the
constant map [Spa89, p.27]. Thus is injective.
is an isomorphism: Once again, we need only make slight adjustments to the
preceding arguments for . Suppose given (`, y) a ffi-cycle in Reln(adLf). Th*
*en
dY (y) = 0 and D(`) = adLf(y), that is, we have
dY `(Ø) - (-1)n`dX (Ø) = [y, Lf(Ø)]
for Ø 2 LX . In this case, define LA :L(W, v, W 0; @) ! LY by LA (w) = Lf(w),
LA (v) = y, and LA (w0) = `(w). Now argue that is surjective following the sa*
*me
steps as were taken for .
To show injectivity of , suppose that (`A , LA (v)) 2 Reln(adLf) is a bounda*
*ry
in the relative complex, so that (`A , LA (v)) = ffi( , y). That is, `A = adLf(*
*y) -
dY +(-1)n+1 dX and dY (y) = LA (v). In this case, define the DG homotopy G by
setting G = LA on L(W, v, W 0), G = 0 on L(sW, cW), G(sv) = -y, G(bv) = -LA (v),
G(sw0) = (w), and G(cw0) = dY (w). With this definition, the same steps as we*
*re
used for may now be followed to show that is injective.
(C) The commutativity of the diagram follows directly from the definitions.
Notice that j :map*(X, Y ; f) ! map *(X, Y ; f) is the fibre inclusion of the e*
*val-
uation fibration. Therefore, if a is a representative of ff 2 ßn(map *(X, Y ; *
*f)),
we may also take a to be a representative of j# (ff) 2 ßn(map (X, Y ; f)). Con-
sequently, the Quillen minimal model of the adjoint of a representative of j# (*
*ff)
may be may be taken as LA with LA (v) = 0. Hence, before rationalizing, we have
0(j# (ff)) = <`A , 0> = = H(J)(<`A >) = H(J) O 0(ff).
Remark 4.2. The homomorphisms 0and 0, and hence their rationalizations, have
certain naturality properties. Post-composition by a map g :Y ! Z gives a map
g*: map*(X, Y ; f) ! map *(X, Z; g O f). On the other hand, the Quillen minimal
model of g induces a chain map (Lg)*: Der*(LX , LY ; Lf) ! Der*(LX , LZ; LgOf).
In this latter chain complex, we choose Lg O Lf as the Quillen minimal model for
16 GREGORY LUPTON AND SAMUEL BRUCE SMITH
g O f. If we denote the homomorphism of Theorem 4.1 by f*, so as to include
the original map in the notation, then we have the following commutative diagra*
*m:
f*
ßn map *(X, Y ; f) __Q____//Hn Der(LX , LY ; Lf)
(g*)# 1|| H((Lg)*)||
fflffl| gOf fflffl|
ßn map *(X, Z; g O f) _Q*_//_Hn Der(LX , LZ; LgOf)
Pre-composition by a map h: W ! X gives a similar naturality property of .
Furthermore, is natural in the same sense.
Schlessinger and Stasheff, and Tanr'e have constructed a (non-minimal) Quillen
model for the universal fibration
X ! B map*(X, X; 1) ! B map(X, X; 1)
in terms of Lie derivations and adjoint maps ([SS, Tan83]). Their result specia*
*l-
izes to identify the long exact sequence induced in rational homotopy groups by
the universal fibration, in the framework of Quillen models and derivations. The
following consequence of Theorem 4.1 extends this specialization of their resul*
*t, in
that it identifies the long exact rational homotopy sequence of a general evalu*
*ation
fibration.
We first make the following observation.
Lemma 4.3. Suppose given a diagram of vector spaces
jn+1 kn+1 in jn
Bn+1 _____//Cn+1____//An____//_Bn____//Cn
___
~=fin+1|| ~=fln+1|| _ffn______~=fin||~=fln||
fflffl| fflffl| fflffl____fflffl|fflffl|
Yn+1 _qn+1//_Zn+1rn+1//_Xnpn_//Yn_qn_//Zn
for each n 2 (that is, a äl dder with every third rung missing"). Suppose the
rows are exact, each fin and fln is an isomorphism, and fln Ojn = qn Ofin for e*
*ach n.
Then there exist isomorphisms ffn :An ! Xn, for n 2, which makes the entire
ladder commutative.
Proof.This is straightforward. See [LS03, Lem.3.1] for details.
Now consider the evaluation fibration map *(X, Y ; f)j_//map(X, Y ;_f)!//_Y
for a map f :X ! Y . By going one step back in the Barratt-Puppe sequence, we
obtain a fibration
j
Y __@__//map*(X, Y ;_f)__//map(X, Y ; f)
and a long exact sequence in homotopy
@# j# !# @#
(1) . ._._//_ßn (map*(X, Y ;_f))//_ßn map (X, Y_;_f)//_ßn-1(_Y_)//. . .
Here, we are identifying ßn-1( Y ) with ßn(Y ) in the usual way. Notice that wh*
*en
X = Y and f = 1 the long exact sequence (1) is equivalent to that of the univer*
*sal
fibration [DZ79 ].
QUILLEN MODELS AND ADJOINT MAPS 17
Theorem 4.4. Let f :X ! Y be a map between simply connected CW complexes
of finite type with X finite. Then the rationalization of the long exact homot*
*opy
sequence (1), as far as the term ß1 Y Q, is equivalent to the long exact de*
*riva-
tion homology sequence of the Quillen minimal model Lf: LX ! LY of f, that
is,
H(adLf) H(J) H(P) H(adLf)
. ._.__//Hn Der(LX , LY ; Lf)_//_Hn(Rel(adLf)___//Hn-1(LY )___//. . .
as far as the term H1(LY ).
Proof.Replace the diagrams
jn
Bn _____//Cn
~=|fin| ~=fln||
fflffl| fflffl|
Yn _qn__//Zn
of Lemma 4.3 with the diagrams
j# Q
ßn(map *(X, Y ; f)) __Q_//_ßn(map (X, Y ; f)) Q
~=|| ~=||
fflffl| H(J) fflffl|
Hn Der(LX , LY ; Lf)_______//_Hn(Rel(adLf)
for n 2. Then with the top row replaced by the rationalization of the long ex*
*act
homotopy sequence (1), and the bottom row by the long exact derivation homology
sequence of Lf: LX ! LY , Lemma 4.3 obtains isomorphisms
ffn :ßn( Y ) Q ! Hn(LY )
with appropriate commutativity properties for n 2. At the bottom end, we have
a diagram
j# Q !# Q
ß2(map *(X, Y ; f)) __Q_//_ß2(map (X, Y ; f))___Q//_ß1( Y ) Q
___
~=|| ~=|| _ff1______
fflffl| H(J) fflffl| fflffl____
H2 Der(LX , LY ; Lf)_______//_H2(Rel(adLf)H(P)____//_H1(LY )
Here, we may use the standard identification of ß1( Y ) Q with H1(LY ) for our
last isomorphism ff1. Then we need only check that the final square commutes. F*
*or
this, we regard the standard identification ff1: ß1( Y ) Q ! H1(LY ) as follo*
*ws.
Suppose a: S2 ! Y is a representative of j 2 ß1( Y ) Q = ß2(Y ) Q. Then we
have La: LS2 ! LY and we set ff1(j) = La(u), where LS2 = L(u) with |u| = 1.
Now suppose given i 2 ß2(map (X, Y ; f)) Q with adjoint G: S2 x X ! YQ. Then
!# (i) = [G O i1] 2 ß2(Y ) Q. Hence, we have H(P ) O (i) = H(P )<`G , LG (v)*
*> =
= = ff O w# (i). Thus we have the equivalence asserted.
Remark 4.5. The last part of the proof above extends to give ffnO(!# Q) = H(P *
*)O
for all n 2, if ffn is taken to be the standard identification ffn :ßn( Y )*
* Q !
Hn(LY ). However, we have chosen to argue as above to avoid the details involved
in showing the remaining squares of the diagram commute with the standard choice
18 GREGORY LUPTON AND SAMUEL BRUCE SMITH
of ffn. Unfortunately, this means that we have not established a natural equiva*
*lence
of sequences in Theorem 4.4.
5.Evaluation Subgroups and Gottlieb's Question
As an immediate consequence of Theorem 4.4, we identify the rationalized eval-
uation subgroup of a map f :X ! Y in terms of Quillen models. Let _ :K ! L
be a DG Lie algebra map.
Definition 5.1. The nth evaluation subgroup of the DG Lie algebra map _ :L ! K
is the subgroup
Gn(K, L; _) = ker{H(ad_): Hn-1(K) ! Hn-1(Der(L, K; _))}
of Hn-1(K). Notice the shift in degrees. We specialize this to define the Gottl*
*ieb
group of a DG Lie algebra following Tanr'e. The nth Gottlieb group of the DG Lie
algebra (L, d) is the subgroup
Gn(L) = ker{H(ad): Hn-1(L) ! Hn-1(Der(L))}
of Hn-1(L).
Theorem 5.2. Let f :X ! Y be a map between simply connected CW complexes
of finite type with X finite. Then the rationalization of the evaluation subgro*
*up of
f is isomorphic to the evaluation subgroup of the Quillen model Lf: LX ! LY of
f as in Definition 5.1. That is, for n 2 we have
Gn(Y, X; f) Q ~=ker{H(adLf): Hn-1(LY ) ! Hn-1(Der(LX , LY ; Lf))}.
Proof.From the long exact homotopy sequence of the evaluation fibration, the
rationalized evaluation subgroup Gn(Y, X; f) Q corresponds to the kernel of
@ :ßn-1( Y ) Q ! ßn-1(map *(X, Y ; f)) Q. The result follows from Theo-
rem 4.4.
Specializing to the identity map we recover
Corollary 5.3 ([Tan83, Cor.VII.4(10)]). Let X be a simply connected, finite com-
plex. Then the rationalization of the Gottlieb group of X is the Gottlieb group*
* of
the Quillen model (LX , dX ) of X as in Definition 5.1. That is, for n 2 we h*
*ave
Gn(X) Q ~=ker{H(adLX ): Hn-1(LX ) ! Hn-1(Der(LX ))}.
We apply these identifications to address Gottlieb's question on the differen*
*ce
between the Whitehead centralizer P*(X) and the Gottlieb group G*(X). In or-
dinary homotopy theory, constructing spaces with G*(X) 6= P*(X) represents a
challenging problem. Ganea gave the first example of inequality in [Gan68 ]. See
[Opr95 ] for a recent reference and some interesting examples of G1(X) 6= P1(X)
with X a finite complex.
Recall the definition of the generalized Whitehead center of a map Pn(Y, X; f)
ßn(Y ) from the introduction. From the identification of the Samelson product in
ß*( Y ) Q with the product in H(LY ), we have
Pn(YQ, XQ; fQ) ~=ker{adH(Lf):Hn-1(LY ) ! Dern-1(H(LX ), H(LY ); H(Lf))}.
Notice that although the relative evaluation subgroup behaves well with respect*
* to
rationalization, in the sense that Gn(Y, X; f) Q = G*(YQ, XQ; fQ) (at least f*
*or X
finite), the inclusion P*(X) Q P*(XQ) is usually strict.
QUILLEN MODELS AND ADJOINT MAPS 19
Rationally, the difference between the nth evaluation subgroup of a map and t*
*he
generalized Whitehead center can be described precisely. The difference is gove*
*rned
by the "induced derivation" map
I :Hn(Der(LX , LY ; Lf)) ! Dern(H(LX ), H(LY ); H(Lf))
which we now introduce.
A D-cycle ` 2 Dern(L, K; _) commutes (in the graded sense) with the differen-
tials of L and K, and so induces a map H(`) 2 Dern(H(L), H(K); H(_)) defined
by H(`)(<,>) = <`(,)> for , a cycle of L. If ` is a D-boundary then it carries *
*cycles
of L to boundaries of K. Thus we obtain a linear map
I :Hn(Der(L, K; _)) ! Dern(H(L), H(K); H(_))
given by I(<`>) = H(`) for ` a cycle of Dern(L, K; _).
Now consider the commutative diagram
H(adLf)
Hn(LY )______________//_VHn(Der(LX , LY ; Lf))
VVVV
VVVVVV I|
adH(Lf)VVVVVV++VVV fflffl||
Dern(H(LX ), H(LY ); H(Lf)).
Theorem 5.4. Let f :X ! Y be a map between simply connected CW complexes
of finite type with X a finite complex. For n 1, we have
_Pn+1(YQ,_XQ;_fQ)_~ker(I) \ im(H(ad )).
Gn+1(YQ, XQ; fQ)= Lf
Proof.The map H(adLf) induces a map
________ ker(adH(Lf))
H(adLf): ____________ker(H(ad! ker(I) \ im(H(adLf))
Lf))
that is easily checked to be an isomorphism. The result follows from Theorem 5.2
and the above discussion, in which we identify ker(H(adLf)) with Gn+1(YQ, XQ; f*
*Q),
and ker(adH(Lf)) with Pn+1(YQ, XQ; fQ).
Using these notions, it is straightforward to give examples of maps f :X ! Y
which give an inequality G*(YQ, XQ ; fQ) 6= P*(YQ, XQ; fQ).
Example 5.5. Suppose that f :X ! Y is a rationally trivial map. Then Lf =
0: LX ! LY . It follows that adLf = 0: LY ! Der*(LX , LY ; Lf). In this case,
we have G*(YQ, XQ ; fQ) = P*(YQ, XQ; fQ) = ß*(Y ) Q. However, suppose that
f# Q = 0: ß*(X) Q ! ß*(Y ) Q, or even just that im(f# Q) P*(YQ). Then
adH(Lf)= 0: H*(LY ) ! Der*(H(LX ), H(LY ); H(Lf)) and so P*(YQ, XQ; fQ) =
ß*(Y ) Q. However, the equality G*(YQ, XQ ; fQ) = ß*(Y ) Q would only hold *
*if
H(adLf) = 0, and there is no particular reason why this should be so. To illust*
*rate
this last case, consider f :CP 2! S4 obtained by pinching out the 2-cell of CP *
*2.
This map has Quillen minimal model
Lf: L(x1, x3; dX ) ! L(u3; dY = 0)
with Lf(x1) = 0 and Lf(x3) = u3. Here, the subscript of a generator denotes its
degree and dX (x1) = 0, dX (x3) = [x1, x1]. Now adLf(u3) 2 Der3(LX , LY ; Lf) is
defined by adLf(u3)(x1) = 0 and adLf(u3)(x3) = [u3, u3]. On the other hand, D =
0 in Der3(LX , LY ; Lf). Therefore, H(adLf)( 6= 0 and G4(YQ, XQ ; fQ) = 0.
20 GREGORY LUPTON AND SAMUEL BRUCE SMITH
We next give a class of examples for which the equality G*(YQ, XQ; fQ) =
P*(YQ, XQ; fQ) holds. For this, we review the notion of coformality and some
terminology associated with this concept. Suppose that a minimal DG Lie algebra
L(V ; d) has a second (or ü pper") grading on the generating subspace V = i 0V*
* i.
This extends to a second grading of L(V ) in the obvious way, and we write L(V *
*)i
for the sub-vector space of L(V ) consisting of all elements of L(V ) of second*
* grad-
ing equal to i. We also write V (i)for the sub-vector space of V consisting of *
*all
elements of V of second grading less than or equal to i. Then we say that L(V ;*
* d)
is a bigraded minimal DG Lie algebra if the differential decreases second degree
homogeneously by one, that is, if d(V 0) = 0 and d(V i) L(V )i-1 for i 1. *
*If
L(V ; d) is a bigraded minimal DG Lie algebra, then the second grading passes to
homology, making H(L(V ; d)) a bigraded Lie algebra. We write Hi(L(V ; d)) for
the sub-vector space of H(L(V ; d)) consisting of homology classes represented *
*by
cycles of upper degree equal to i, and we have H(L(V ; d)) = i 0Hi(L(V ; d)).
Definition 5.6. Let L(V ; d) be a bigraded minimal DG Lie algebra in the above
sense. We say L(V ; d) is coformal if Hi(L(V ; d)) = 0 for i > 0, so that H(L(V*
* ; d)) =
H0(L(V ; d)). We say that a space X is a coformal space if its Quillen minimal *
*model
is coformal.
Equivalently, we may define L(V ; d) to be coformal if there exists a quasi-
isomorphism æ: L(V ; d) ! (H(L), d = 0). The equivalence of these definitions
is established by the notion of a bigraded model in the DG Lie algebra setting_*
*see
[HS79 , Sec.3] for the Sullivan model setting. There are many interesting examp*
*les of
coformal spaces: Moore spaces and more generally rational co-H-spaces, including
suspensions; some homogeneous spaces; products and wedges of coformal spaces.
This notion of coformality extends to a map. Suppose that OE: L(V ; d) !
L(W ; d0) is a map of bigraded minimal DG Lie algebras as defined above. If
OE(V i) L(W )i for each i 0, then we say that OE is a bigraded map.
Definition 5.7. A map OE: L(V ; d) ! L(W ; d0) of bigraded minimal DG Lie al-
gebras is a coformal map if both L(V ; d) and L(W ; d0) are coformal, and OE is*
* a
bigraded map (with respect to the second gradings that display the coformality *
*of
L(V ) and L(W )). A map of coformal spaces f :X ! Y is a coformal map if its
Quillen minimal model Lf: LX ! LY is a coformal map of bigraded minimal DG
Lie algebras.
Equivalently, we may define OE: L(V ; d) ! L(W ; d0) to be coformal if there *
*exist
quasi-isomorphisms æ: L(V ; d) ! H(L(V ; d)) and æ0:L(W ; d0) ! H(L(W ; d0))
such that the diagram
L(V ; d)___OE__//L(W ; d0)
æ|'| ' |æ0|
fflffl| fflffl|
H(L(V ; d))H(OE)//_H(L(W ; d0))
is DG homotopy commutative.
Remark 5.8. Suppose given a map of DG Lie algebras : L ! L0. By constructing
DG Lie algebra minimal models of L and L0, and then using the standard lifting
QUILLEN MODELS AND ADJOINT MAPS 21
lemma, we obtain a DG homotopy commutative diagram
L(V ; d)_OE//_L(W ;.d0)
æ '|| ' |æ0|
fflffl| fflffl|
L ___=H(OE)_//L0
In this way, it is always possible to realize a Lie algebra map as the homomorp*
*hism
induced on homology by a coformal map. Notice, however, that there may be many
other DG Lie algebra maps L(V ; d) ! L(W ; d0) that induce the same homomor-
phism on homology as H(OE). Coformality, therefore, distinguishes a unique DG
homotopy class of maps from amongst the various realizations. From this point of
view, a coformal map is the simplest realization of its induced homomorphism on
homology.
Theorem 5.9. Let f :X ! Y be a coformal map between CW complexes of finite
type with X finite. Then P*(YQ, XQ; fQ) = G*(YQ, XQ; fQ).
Proof.Suppose LX = L(W ; dX ) and LY = L(V ; dY ) are coformal, and that Lf
is bigraded. Take ff 2 Hn(LY ). With reference to Theorem 5.4, we show that if
I O H(adLf)(ff) = 0, then H(adLf)(ff) = 0. Since Y is coformal, we may assume
ff = <,> for a dY -cycle , 2 L(V )0. Observe that adLf(,) 2 Dern(LX , LY ; Lf)*
* is
then a D-cycle that preserves upper degree. If I O H(adLf)(ff) = 0, then for ea*
*ch
dX -cycle Ø 2 L(W ), we have adLf(,)(Ø) = dY (j) for some j 2 L(V ). We now use
this to construct ` 2 Dern+1(LX , LY ; Lf) such that D(`) = adLf(,).
Since X is coformal, we have W = i 0W i, and each w 2 W 0is a dX -cycle.
Therefore, we have adLf(,)(w) = dY (j) for some j 2 L(V ). Furthermore, since
LY is bigraded, we may choose j 2 L(V )1. Use this to define a linear map
`0: W 0! L(V )1 and extend to an Lf-derivation `0 2 Dern+1(L(W 0), LY ; Lf).
By construction, we have D(`0)(Ø) = dY (`0(Ø)) = adLf(,)(Ø) for Ø 2 L(W 0).
Assume inductively that `m 2 Dern+1(L(W (m)), LY ; Lf) is defined, increasing
upper degree homogeneously by 1, and satisfying D(`m ) = adLf(,) on L(W (m)).
For w 2 W m+1, consider the element adLf(,)(w)+(-1)n+1`m (dX w). Since adLf(,)
is a D-cycle, and dX w 2 L(W )m L(W (m), we compute that
dY (adLf(,)(w) + (-1)n+1`m (dX w)) = dY adLf(,)(w) + (-1)n+1dY `m (dX w)
= (-1)nadLf(,)(dX w) + (-1)n+1adLf(,)(dX w)
+ adLf(,)dX (dX w)
= 0.
Since adLf(,)(w) + (-1)n+1`m (dX w) is a dY -cycle in L(V )m+1 , we again use t*
*he
coformality of LY _specifically, that H+ (LY ) = 0_to conclude that there exists
some i 2 LY with dY (i) = adLf(,)(w) + (-1)n+1`m (dX w). Furthermore, we may
choose i 2 L(V )m+2 . Clearly, i may be chosen so as to depend linearly on w. S*
*o use
this to define a linear map `m+1 :W m+1! L(V )m+2 with `m+1 (w) = i, and extend
`m to an Lf- derivation `m+1 2 Dern+1(L(W (m+1)), LY ; Lf). By construction, we
have
D`m+1 (w)= dY `m+1 (w) - (-1)n+1`m+1 (dX w)
= dY i - (-1)n+1`m (dX w) = adLf(,)(w)
22 GREGORY LUPTON AND SAMUEL BRUCE SMITH
for w 2 W m+1 and since D(`m+1 ) is an Lf-derivation, this gives D(`m+1 )(Ø) =
adLf(,)(Ø) for Ø 2 L(W (m+1)). This completes the induction and gives an Lf-
derivation ` 2 Dern+1(L(W ), LY ; Lf) that satisfies D(`) = adLf(,). The result
follows.
The following special case is well-known.
Corollary 5.10. Let X be a simply connected, finite CW complex. If X is coforma*
*l,
then P*(XQ) = G*(XQ).
Proof.Restrict Theorem 5.9 to the case f = 1: X ! X.
6.The Rationalized G-Sequence
In this section, we identify the rationalization of the G-sequence mentioned *
*in
the introduction. Suppose given a map f :X ! Y . Then we have the commutative
square
(2) X _______f______// Y
@|| |@|
fflffl| f* fflffl|
map *(X, X; 1)____//map*(X, Y ; f),
in which the vertical maps are the connecting maps arising from the evaluation
fibrations ! :map(X, X; 1) ! X and ! : map (X, Y ; f) ! Y as in Section 4. The
maps f and f* lead to long exact homotopy sequences and the vertical maps give
homomorphisms of corresponding terms, yielding a homotopy ladder
(3)
( f)# j p
. ._.___p___//_ßn( X)____________//_ßn( Y_)_______//_ßn( f)___//_. . .
@# || @#|| @#||
bp fflffl| (f*)# fflffl| bj fflffl|bp
. ._.__//_ßn(map *(X, X;_1))_//ßn(map *(X, Y ;_f))_//ßn(f*)___//. . .
in the usual way. Whenever we have such a ladder, with exact rows, there is an
associated "kernel sequence," that is, a sequence obtained by restricting the m*
*aps
in the top row to the kernels of the vertical rungs. The G-sequence of the map f
may be defined, with a shift in degree, as the kernel sequence of the above hom*
*otopy
ladder. A portion of this construction is shown here:
( f)# j p
. ._.p___//_Gn+1(X)"_`____//_Gn+1(Y,"X;`f)_//_Greln+1(Y,"X;`f)//_. . .
| | |
| | |
p fflffl| ( f)# fflffl| j fflffl|p
. ._.____//_ßn( X)___________//ßn( Y_)________//ßn( f)_____//. . .
|@#| |@#| ||
bp fflffl| (f*)# fflffl| bj fflffl|bp
. ._.//_ßn(map *(X, X;_1))//_ßn(map *(X, Y_;_f))//_ßn(f*)__//. . .
Note that the maps in the G-sequence are just the restrictions of the maps in t*
*he
long exact homotopy sequence of the map f : X ! Y . Thus compositions of
consecutive maps in the G-sequence are trivial. However, the sequence of kernel*
*s of
a commutative ladder of exact sequences need not be exact, and so the G-sequence
QUILLEN MODELS AND ADJOINT MAPS 23
is a chain complex (of Z-modules). The original description given in [WL88b , L*
*W93 ]
(see also [LS03, Sec.1]) represents the G-sequence as an image sequence, in a w*
*ay
obviously equivalent to the above.
Below, we construct a chain complex associated to any DG Lie algebra map that
we will show corresponds to the rationalized G-sequence. Observe that the relat*
*ive
chain complex is a functorial construction. That is, given a commutative square
OE
V _____//_W
|ff| |fi|
fflffl|Offlffl|E0
V 0____//_W 0
of DG vector space maps we obtain a DG vector space map (fi, ff): Rel(OE) ! Rel*
*(OE0)
given by (fi, ff)(w, v) = (fi(w), ff(v)). This leads to a commutative diagram o*
*f short
exact sequences
0_____//W*__J__//Rel*(OE)P_//V*-1____//0
|fi| (fi,ff)|| ff||
fflffl|J fflffl|P fflffl|
0_____//W*0____//Rel*(OE0)_//_V*0-1__//0,
and hence to a commutative ladder of long exact homology sequences.
Now suppose given a map _ :L ! K of DG Lie algebras. Apply the above to
the commutative square
(4) L _____________//_K
|ad| |ad_|
fflffl|_* fflffl|
Der(L, L; 1)___//_Der(L, K; _)
of DG vector spaces. Note that the relative term Rel*(_*) is given by
Reln(_*) = Dern(L, K; _) Dern-1(L, L; 1)
with differential ffi(`1, `2) = (_ O `2 - D(`1), D(`2)). We obtain the followin*
*g com-
mutative ladder of long exact homology sequences:
(5)
H(P) H(_) H(J) H(P)
. ._._______//Hn(L)_____________//Hn(K)_________//_Hn(Rel(_))___//_. . .
|H(ad)| |H(ad_)| H(ad_,ad)||
H(Pb) fflffl| H(_*) fflffl| H(Jb) fflffl|H(Pb)
. ._.__//_Hn(Der(L, L; 1))_//_Hn(Der(L, K; _))__//Hn(Rel(_*))___//_. . .
To obtain unambiguous notation, we have written bJ:Dern(L, K; Lf) ! Reln(_*)
and bP: Reln(_*) ! Dern-1(L) for the usual inclusion and projection maps in the
lower sequence. We refer to this ladder as the (horizontal) homology ladder ari*
*sing
from the diagram (4).
Theorem 6.1. Let f :X ! Y be a map between simply connected CW complexes
of finite type, with X finite. The rationalization of the homotopy ladder (3), *
*down
24 GREGORY LUPTON AND SAMUEL BRUCE SMITH
to the rung @# 1: ß2( Y ) Q ! ß2(map *(X, Y ; f)) Q, is equivalent to the
homology ladder arising from
Lf
(6) LX ________________//LY
|ad| adLf||
fflffl| (Lf)* fflffl|
Der(LX , LX ;_1)__//Der(LX , LY ; Lf),
down to the rung H(adLf): H2(LY ) ! H2(Der(LX , LY ; Lf)).
Proof.Let Y :ßn(map *(X, Y ; f)) Q ! Hn(Der(LX , LY ; Lf)) be the isomor-
phism defined in the proof of Theorem 4.1. Let X be the isomorphism obtained
in the same way, by specializing to the case in which Y = X and f = 1.
Consider the (vertical) homotopy ladder arising from (2), with exact columns *
*the
long exact sequences of the evaluation fibration sequences X ! map*(X, X; 1) !
map(X, X; 1) and Y ! map *(X, Y ; f) ! map (X, Y ; f). From Theorem 4.4 ap-
plied to each column, we obtain an equivalence between this ladder and the vert*
*ical
homology ladder arising from (6). It follows that, with the above notation, we *
*have
commutative cubes
H(Lf)
Hn(LX8)8____________________//Hn(LY8)8
ffXrrrr | ffYqqqqq |
rrr~=rr H(ad)|| qqq~=q ||
ß ( X) rQ ________|______//_ß Y q Q |H(adLf)|
n |( f)# 1 n |
| | | |
| | |@# |
|| fflffl| || |fflffl
@#| Hn Der(LX , LX ; 1)___|____//Hn Der(LX , LY ; Lf)
| rr99 H((L|f)*) q88q
| Xrrr | Yqqq
| rr~=r | qqq~=
fflffl|rr fflffl|qq
ßn map *(X, X; 1) Q(f*)#/1/_ßn map *(X, Y ; f) Q
for n 2. Here, ffX and ffY denote the isomorphisms that are obtained from
Lemma 4.3. Strictly speaking, we cannot conclude an equivalence of ladders from
Theorem 4.4, due to the non-natural choices made in Lemma 4.3. However, an easy
extension of Lemma 4.3 to the setting of an equivalence of ladders overcomes th*
*is
difficulty_see [LS03, Lem.3.9] for details.
Switching now to horizontal ladders, we may use top and bottom faces of the
displayed cubes, together with the extension of Lemma 4.3 referred to already, *
*to
obtain the desired equivalence of ladders.
Remark 6.2. Notice that Theorem 6.1 contains a description of the long exact
rational homotopy sequences of a general map f, and of the induced map f*. See
[LS03, Th.3.3, Th.3.5] for the corresponding descriptions of these sequences in*
* terms
of Sullivan minimal models.
In Definition 5.1, we have defined evaluation subgroups of a map of DG Lie
algebras, and of a DG Lie algebra. Our cast of characters that appear in the
G-sequence is completed by the following:
QUILLEN MODELS AND ADJOINT MAPS 25
Definition 6.3. Let _ :L ! K be a DG Lie algebra map. The nth relative
evaluation subgroup Greln(K, L; _) of _ is the subgroup
Greln(K, L; _) = ker{H(ad_, ad): Hn-1(Rel(_)) ! Hn-1(Rel(_*))}
of Hn(Rel(_)). The G-sequence of _ is the sequence of kernels from the commuta-
tive ladder (5). That is, the sequence
H(P) H(_) H(J) H(P)
. ._.__//_Gn(L)___//_Gn(K, L; _)__//_Greln(K, L;__)//_... .
Corollary 6.4. Let f :X ! Y be a map between simply connected CW complexes
of finite type, with X finite. Then the rationalization of the G-sequence of f
f# Q j
. ._.__//Gn(X)_____//Gn(Y, X; f) _Q__//_Greln(Y, X; f)___Q//_. . .
down to the term G3(Y, X; f) Q is equivalent to the G-sequence of the Quillen
model Lf: LX ! LY of f,
H(Lf) H(J)
. ._.___//Gn(LX )___//_Gn(LY , LX ;_Lf)_//Greln(LY , LX_;_Lf)//_. . .
down to the term G3(LY , LX ; Lf).
Proof.This follows immediately from Theorem 6.1, since equivalent ladders have
equivalent kernel (and image) sequences.
In particular, we can add the following to Theorem 5.2
Corollary 6.5. Let f :X ! Y be a map between simply connected CW com-
plexes of finite type with X finite. Then the rationalized relative evaluation*
* sub-
group Greln(Y, X; f) is isomorphic to the relative Gottlieb group of the Quille*
*n model
Lf: LX ! LY of f as in Definition 6.3. That is, for n 4
Greln(Y, X; f) Q ~=ker{H(adLf, ad): Hn-1(Rel(Lf)) ! Hn-1(Rel(L*f))}.
Remark 6.6. With a little more work, the equivalence of Corollary 6.4, and hence
Corollary 6.5, may be extended to the Grel3(Y, X; f) term. Under our hypotheses,
G2(X) Q = 0, and there seems little to be gained by trying to extend Corollary *
*6.4
beyond this point (cf. the remarks on the low-end terms in [LS03, Rem.3.11]).
As an application of the above, we consider the question of exactness of the
G-sequence for cellular extensions. The first example of inexactness for a cell*
*ular
inclusion of finite complexes was given in [WL88a ]. Observe that rational exam*
*ples
of non-exactness delocalize, since exactness in the ordinary setting implies ex*
*actness
after rationalization.
We focus on the rationalized G-sequence at the Gn(X) term. Following Lee and
Woo [LW93 ], define the !-homology of a map f :X ! Y at this term by setting
Ha!n(Y, X; f) = ker{f#_:Gn(X)_!_Gn(Y,_X;_f)}_.
im{p: Greln+1(Y, X; f) ! Gn(X)}
For a single cell-attachment, the following is a complete result for the lowest*
* degree
in which the (rationalized) G-sequence can be non-exact at the Gn(X)-term.
Theorem 6.7. Let X be a finite CW complex of dimension n, and Y = X [ffem+1
for some ff 2 ßm (X). Suppose the following three conditions hold:
(1) ffQ 6= 0;
26 GREGORY LUPTON AND SAMUEL BRUCE SMITH
(2) ffQ 2 Gm (XQ);
(3) Y is not rationally equivalent to a point.
Then the G-sequence of the inclusion i: X ! Y is non-exact at the Gm (X) term,
and Ha!m(Y, X; i) Q = Q. Conversely, if any of (1)-(3) do not hold, then
Ha!m(Y, X; i) Q = 0, that is,
p Q i# Q
Grelm+1(Y, X; i)___Q_//Gm (X) Q____//Gm (Y, X; i) Q
is exact.
Proof.Clearly, the kernel of i# Q: ßm (XQ) ! ßm (YQ) is the subspace
of ßm (XQ). If ffQ = 0, then i# Q : ßm (XQ) ! ßm (YQ) is injective, and if
ffQ 62 Gm (XQ), then the restriction of i# Q to Gn(XQ) of i# is injective. In *
*either
case, Ha!m(Y, X; i) Q = 0 by definition.
So suppose that (1) and (2) hold. We show that Grelm+1(Y, X; i) Q = 0 if (3) *
*also
holds. For this, we use Corollary 6.5 and show that H(adLi, ad): Hm (Rel(Li)) !
Hm (Rel((Li)*)) is injective.
We may write the Quillen model of i: X ! Y as an inclusion Li:LX ! LY ,
with LX = L(W ), LY = L(W ) t L(y), and differentials dY |L(W ) = dX and
dY (y) = Ø 2 L(W ) a cycle of degree (m-1) in LX whose homology class represents
ffQ via the isomorphism Hm-1 (LX ) ~=ßm (X) Q (cf. [FHT01 , Sec.24(d)]).
Any cycle i 2 Relm (Li) = (LY )m (LX )m-1 may be written i = (~y +
,, dY (~y + ,)) for ~ 2 Q and , 2 (LX )m . Suppose that H(adLi, ad)(*) = 0.
Then (adLi, ad)(i) = ffi(', `) for some (', `) 2 Relm+1((Li)*). In particular,*
* we
have adLi(~y + ,) = (Li)*(`) - D' 2 Derm (LX , LY ; Li). We now use (3), if
necessary, to choose an indecomposable w 2 W such that Ø and w are linearly in-
dependent in LX . (A choice is only necessary when dY (y) is indecomposable. If*
* Ø
is decomposable, then any indecomposable of W will do.) On this indecomposable,
we evaluate as follows:
(7) adLi(~y + ,)(w) = ~[y, w] + [,, w]
and
(8) ((Li)*(`) - D')(w) = LiO `(w) - dY '(w) + (-1)m+1 'dX (w)
We claim that all terms of (8) are independent of [y, w]. First, LiO `(w) 2 L(W*
* ).
Next, '(w) 2 (L(W ) t L(y))m+1+|w| may contain terms in L(W ), quadratic terms
[y, w0] for w02 W or [y, y] (if m is odd), or terms involving y of bracket-leng*
*th at
least three. Now dY ([y, w0]) = [Ø, w0] [y, dY (w0)], and dY ([y, y]) = 2[Ø, *
*y] (if m is
odd). By choice, Ø and w are linearly independent, and also dX is decomposable.
Hence, all terms of dY '(w) are linearly independent of [y, w]. Finally, dX (w*
*) is
decomposable and in L(W )|w|-1. Hence, 'dX (w) is in the ideal of L(W ) t L(y)
generated by elements of W of degree |w| - 2. This proves the claim, and shows
that if (7) and (8) agree, which must be the case if H(adLi, ad)(**) = 0, then
we have ~ = 0. We have shown that the kernel of H(adLi, ad): Hm (Rel(Li)) !
Hm (Rel((Li)*)) consists of classes represented by cycles of the form i = (,, d*
*Y (,))
for , 2 (LX )m . Since dY (,) = dX (,), we may write ffi(0, ,) = (,, dX (,) = *
*i.
That is, H(adLi, ad): Hm (Rel(Li)) ! Hm (Rel((Li)*)) is injective and hence by
Corollary 6.5, Grelm+1(Y, X; i) Q = 0. This establishes the first assertion *
*of the
theorem.
QUILLEN MODELS AND ADJOINT MAPS 27
The second assertion will be established when we handle the case in which (1)
and (2) hold, but (3) does not, that is, in which Y 'Q *. In this case, we must*
* have
X 'Q Sn, m = n, and ff 2 ßn(Sn) non-trivial. The Quillen model of Y is then
L(w, y) with |w| = n - 1, |y| = n, and dY (y) = w. (Notice that this is precise*
*ly
the case in which we cannot choose a second indecomposable of W independent
of Ø.) In addition, since we are assuming that (2) holds, we must have m odd,
and hence |w| is even and L(w) is an abelian Lie algebra. It is easy to see th*
*at
(y, w) 2 Relm(Li) is a cycle that cannot be a ffi-boundary. Define an Li-deriva*
*tion
' 2 Derm+1 (LX , LY ; Li) by setting '(w) = -1_2[y, y]. A straightforward check
shows that (adLi, ad)(y, w) = (adLi(y), 0) = ffi(OE, 0) in Relm((Li)*). That is*
*,
represents an element in Grelm(LY , LX ; Li). Furthermore, H(P )() = ,*
* and
so H(P ): Grelm(LY , LX ; Li) ! Gm (LX ) is onto. The result follows.
We conclude with an example of vanishing rational !-homology. In the following
result, we use the ideas discussed before Theorem 5.9, concerning the notion of*
* a
coformal map.
Theorem 6.8. Let f :X ! Y be a coformal map between CW complexes of finite
type, with X finite. Then Ha!n(Y, X; f) Q = 0, that is,
p Q f# Q
Greln+1(Y, X; f)___Q_//Gn(X) Q_____//Gn(Y, X; f) Q
is exact, for each n 3.
Proof.We will use Corollary 6.4 and show that ker{H(Lf)} im{H(P )}. From
Definition 5.7, we assume that both LX and LY admit upper (second) gradings with
the properties described in Definition 5.6, and that Lf preserves upper degrees*
*. Let
ff = <,> 2 Gn(LX ) satisfy H(Lf)(ff) = 0. We assume , is of upper degree zero in
the bigraded model for LX . Furthermore, Lf(,) = dY (y) for some y 2 (LY )n+1 t*
*hat
we may assume is of upper degree 1. Since ff is Gottlieb, ad(,) = D(_) for some
derivation _ 2 Dern+1(LX , LX ; 1) and using the coformality of LX again, we may
assume _ increases upper degree homogeneously by 1. The pair (y, ,) 2 Reln+1(Lf)
is a ffi-cycle that satisfies P (y, ,) = ,. We now show that (y, ,) represents*
* an
element in Greln+1(LY , LX ; Lf), that is, we show the pair (adLf(y), ad(,)) bo*
*unds in
Reln+1((Lf)*). Set = (Lf)*(_) - adLf(y), a derivation in Dern+1(LY , LX ; Lf).
It is direct to check that D( ) = dY O - (-1)n+1 O dX = 0. Moreover,
increases upper degree homogeneously by 1. Now adapt the proof of Theorem 5.9
to the current situation, by replacing the derivation adLf(,) in that proof by *
*_. The
inductive argument used there now results in a derivation ` 2 Dern+2(LX , LY ; *
*Lf),
constructed in the same way only increasing upper degree by 2, that satisfies *
* =
D(`). Then the pair (`, _) 2 Reln+2((Lf)*) satisfies ffi(`, _) = (adLf(y), ad(,*
*)).
Appendix A. Some DG Lie Algebra Homotopy Theory
In this appendix, we present some DG Lie algebra homotopy theory. Our main
references for this material are [Tan83, Ch.II.5] and part IV of [FHT01 ]. For *
*com-
pleteness and convenience, we recall the basic notions here. The main point of *
*the
appendix is to provide details for some facts used in a crucial way to establis*
*h our
main results. Since this is a technical appendix, we assume a greater degree of
familiarity with techniques from rational homotopy theory than in the main body
of the paper.
28 GREGORY LUPTON AND SAMUEL BRUCE SMITH
The algebraic notion of homotopy that we use here is "left homotopyö f DG
Lie algebra maps, defined in terms of a suitable cylinder object for a free DG *
*Lie
algebra L(V ; d). This is another free DG Lie algebra denoted L(V )I, together *
*with
inclusions ~0, ~1: L(V ) ! L(V )I and a projection p: L(V )I ! L(V ) that toget*
*her
satisfy pO~i= 1 for i = 0, 1. As a DG Lie algebra, we have L(V )I = L(V, sV, bV*
*; D),
with sV the suspension of V , and bVan isomorphic copy of V . The differential D
in L(V )I extends d, so that L(V ) is a sub-DG Lie algebra, and is defined on t*
*he
other generators as D(sv) = bvand D(bv) = 0, for v 2 V . The inclusion ~0: L(V *
*) !
L(V )I is the obvious inclusion of the sub-DG Lie algebra L(V ). The inclusion
~1: L(V ) ! L(V )I, on the other hand, is more involved. Define a derivation
oe 2 Der1(L(V )I, L(V )I; 1) on generators by oe(v) = sv, oe(sv) = 0, and oe(bv*
*) = 0,
then define ` 2 Der0(L(V )I, L(V )I; 1) as ` = [D, oe] = D O oe + oe O D. Obse*
*rve
that ` is a cycle, and is locally nilpotent. Therefore, exponentiating ` gives *
*a (DG)
automorphism exp(`): L(V )I ! L(V )I. Finally, define ~1 = exp(`) O ~0. The
projection p is defined in the obvious way as p(v) = v, p(sv) = 0, and p(bv) = *
*0.
Given a pair of DG Lie algebra maps OE, _ :L(V ) ! L, we say OE is homotopic to*
* _
if there exists a map H: L(V )I ! L such that H O ~0 = OE and H O ~1 = _. In th*
*is
case, we say H is a (DG) homotopy from OE to _.
In addition to the notation established above, which we take as fixed for this
appendix, we will use the following conventions for DG Lie algebra maps: We will
generally suppress differentials from our notation. Recall the model for Sn x X
from Section 3. We will use J to denote either inclusion L(W ) ! L(W, v, W 0) or
L(W, v) ! L(W, v, W 0), and JI to denote the corresponding inclusions of cylind*
*er
objects. Namely, in the first case, JI(w) = w, JI(sw) = sw, and JI(wb) = bw, for
w 2 W . We also fix some notation for maps of spaces that we use in this append*
*ix.
Let i1: Sn ! Sn x X and i2: X ! Sn x X denote the inclusions, and ß1 the
projection onto the first factor of a product of spaces.
The basic correspondence between the notions of homotopy in the topological
and algebraic settings is as follows: Maps f, g :X ! Y of rational spaces are
homotopic if and only if their Quillen models Lf, Lg: LX ! LY are homotopic in
the sense just defined. For the proof of Theorem 4.1, however, we need finer de*
*tail
than this basic correspondence. Consider the argument to show 0is well-defined,
for example. Suppose a, b: Sn ! map *(X, Y ; f) are homotopic representatives of
the same homotopy element, and that their adjoints are A, B :Sn x X ! Y . Then
A and B are homotopic relative to Sn _X. Indeed, if H denotes the homotopy from
A to B that is adjoint to the homotopy from a to b, then the following diagram
commutes:
(9) (Sn _ X) x I_ß1__//Sn _ X
(i1|i2)x1|| |(*|f)|
fflffl| fflffl|
Sn x X x I___H____//Y
For our argument that 0 is well- defined, it is crucial that we may assume the
DG Lie algebra homotopy that corresponds to H is restricted in a certain way.
Specifically, if H denotes the DG Lie algebra homotopy that corresponds to H,
QUILLEN MODELS AND ADJOINT MAPS 29
then we require that the following diagram commute:
(10) L(W, v)I___p__//_L(W, v)
JI || |(Lf|0)|
fflffl| fflffl|
L(W, v, W 0)I_H___//_LY
We also need a similar fact for homotopy elements of the unbased mapping space
to establish that 0is well-defined. Whilst this translation is intuitively pla*
*usible,
there are some technical details to be checked.
Our starting point for this is the corresponding result in the Sullivan model
setting, which we have proved in our earlier paper. We assume familiarity with
the usual notation in that setting, that is, rational homotopy theory from the *
*DG
algebra point of view (see [FHT01 ]). The algebraic notion of homotopy that we
used in [LS03] is "right homotopy" defined in terms of a suitable path object f*
*or a
free DG algebra. For (V ), this consists of maps
(11) (V )__j___// (V ) (t,_dt)p0,p1//_ (V )
that satisfy piO j = 1 for i = 0, 1. Here, j is the inclusion j(Ø) = Ø 1. T*
*he
projections are defined by pi(t) = i, pi(dt) = 0, and pi(Ø) = Ø for Ø 2 (V ). *
*Given
a pair of DG algebra maps OE, _ : (W ) ! (V ), we say OE is homotopic to _ if *
*there
exists a map G : (W ) ! (V ) (t, dt) such that p0 O G = OE and p1 O G = _. *
*In
this case, we say G is a (DG) homotopy from OE to _.
We fix more notation: Let MX denote the Sullivan minimal model of a space
X. Recall that the minimal model of a product of spaces is the tensor product of
their minimal models, so that MSn MX is a Sullivan model for Sn x X. Let
Mf: MY ! MX denote a Sullivan minimal model of a map f :X ! Y . Let
q1: MSn MX ! MSn and q2: MSn MX ! MX denote the projections.
Lemma A.1. Let f :X ! Y be a map with a fixed choice of Sullivan model
Mf: MY ! MX . Let A, B :Sn x X ! Y be maps that restrict to A O i2 =
B O i2 = f :X ! Y .
(i)Suppose A ~ B via a homotopy H relative to X, that is, suppose the
diagram
(12) X x I __ß1___//X
i2x1|| |f|
fflffl| fflffl|
Sn x X x I__H__//Y
commutes. Then there exists a homotopy G :MY ! MSn MX (t, dt)
from MA to MB that is "relative to MX ," in the sense that the diagram
(13) MY __G__//MSn MX (t, dt)
Mf || |q2|1
fflffl| fflffl|
MX ____j___//MX (t, dt)
(strictly) commutes.
30 GREGORY LUPTON AND SAMUEL BRUCE SMITH
(ii)Suppose further that A and B restrict to A O i1 = B O i1 = *: Sn ! Y , a*
*nd
the homotopy H is also relative to Sn, so that (9) commutes. Then there
exists a homotopy G :MY ! MSn MX (t, dt) from MA to MB that
is "relative to MSn_X ," in the sense that the diagram
MY _____G______//_MSn MX (t, dt)
(",Mf)|| (q1|1,q2|1)
fflffl| fflffl|
MSn MX _j_j_//MSn (t, dt) MX (t, dt)
(strictly) commutes (": MY ! MSn denotes the map that is zero in posi-
tive degrees).
Proof.The first point is proved in [LS03, Lem.A.2]. The argument given there is
readily adapted to prove the second point.
We will carefully translate this result into the Quillen model setting via the
so-called Quillen functor L . This is a functor from the category of different*
*ial
graded algebras to the category of differential graded Lie algebras (see [FHT01*
* ,
Sec.22(e)] or [Tan83, I.1.(7)] for details). Since we only use general properti*
*es of
this functor, we do not recall its definition here. We do recall that it prese*
*rves
quasi-isomorphisms and, as a consequence of its definition, takes an injective *
*DG
algebra map to a surjective DG Lie algebra map.
Sullivan and Quillen models of a map are related via the Quillen functor. Sup-
pose f :X ! Y has Sullivan minimal model Mf: MY ! MX . Applying L gives
L(Mf): L(MX ) ! L (MY ). Now suppose æX :LX ! L (MX ) and æY :LY !
L(MY ) are given minimal models. Then any map Lf: LX ! LY such that æY OLf
and L(Mf) O æX are homotopic is a Quillen minimal model for f. Recall that we
obtain such maps as follows: There is a standard way of converting the quasi-
isomorphism æY into a surjective quasi-isomorphism. Namely, let E(L (MY )) de-
note the acyclic, free DG Lie algebra L(V, s-1V ), with V a vector space isomor*
*phic
to L (MY ) and d(v) = s-1v for v 2 V . Let cæY:LY t E(L (MY )) ! L(MY ) be
the map that extends æY on LY , and maps cæY(v) = v, cæY(s-1v) = dv for v 2 V .
Then we may lift L(Mf) O æX through the surjective quasi-isomorphism cæY,
OEf ß
L(W )______//__________LY t_E(L/(MY/))_LY
ppp
æX|'| ' cæY||ppæpppp
fflffl| fflfflfflffl|Yxxppp
L(MX ) _________//L(MY )
L(Mf)
to obtain a map OEf that satisfies cæYO OEf = L(Mf) O æX (see [Tan83, II.5.(13)*
*] or
[FHT01 , Prop.22.11] for the standard lifting lemma). A Quillen minimal model f*
*or
f is then obtained by composing with the projection ß to give Lf = ß O OEf.
Homotopies are also related via the Quillen functor. However, since it is co*
*n-
travariant, the Quillen functor translates a right homotopy of DG algebra maps
into a left homotopy of DG Lie algebra maps, in the following way. By applying L
to (11), we obtain maps
L(p0),L (p1) L (j)
L (MX )________//LMX (t, dt) ________//L(MX )
QUILLEN MODELS AND ADJOINT MAPS 31
that satisfy L (j) O L(pi) = L (1) = 1 for i = 0, 1. Since j is an injective q*
*uasi-
isomorphism, L (j) is a surjective quasi-isomorphism. Since L (j) O L(p0) O æX =
æX = L(j) O L(p1) O æX :LX ! L(MX ), the linear difference L(p1) O æX - L(p0) O
æX :LX ! L (MX (t, dt)) has image contained in the DG ideal ker(L (j)) of
L(MX (t, dt)). Now suppose LX = L(W ). Since L (j) is a surjective quasi-
isomorphism, ker(L (j)) is an acyclic DG ideal. Hence, by a standard argument
([Tan83, Prop.II.5(4)]), we may construct a left-homotopy G: L(W )I ! L(MX
(t, dt)) from L(p0) O æX to L(p1) O æX that in addition satisfies G L(sW, cW)
ker(L (j)) (this last point is key). That is, we have the following commutative
diagram,
L(W )TTT
| TTTLT(p0)OæXTTT
~0| TTT
fflffl|G TTTT))
L(W )I __________//_L(MX55 (t, dt))
OO jjjjj
~1| jjjjj
|| jjjjjjLj(p1)OæX
L(W )
for which G L(sW, cW) ker(L (j)). Finally, suppose that G :MY ! MX
(t, dt) is a right homotopy of DG algebra maps from OE to _. Then L (G) O
G: L(W )I ! L(MY ) is a left homotopy of DG Lie algebra maps from L(OE) O æX
to L(_) O æX .
We now come to the main point of the appendix.
Lemma A.2. Let f :X ! Y be a map with a fixed choice of Quillen model
Lf: LX ! LY . Let A, B :Sn x X ! Y be maps that restrict to A O i2 = B O i2 =
f :X ! Y . Suppose L(W ) is the Quillen minimal model of X, and L(W, v, W 0) is
the Quillen model of Sn x X (see Corollary 3.5).
(i)Suppose A ~ B via a homotopy H relative to X, that is, suppose the dia-
gram (12) commutes. Then there exists a DG homotopy H: L(W, v, W 0)I !
LY from LA to LB that is "relative to LX ," in the sense that the diagram
L(W )I ___p___//_L(W )
JI|| Lf||
fflffl| fflffl|
L(W, v, W 0)IH___//LY
(strictly) commutes.
(ii)Suppose further that A and B restrict to A O i1 = B O i1 = *: Sn ! Y ,
and the homotopy H is also relative to Sn, so that (9) commutes. Then
there exists a DG homotopy H: L(W, v, W 0)I ! LY from LA to LB that is
"relative to LSn_X ," in the sense that the diagram (10) (strictly) comm*
*utes.
32 GREGORY LUPTON AND SAMUEL BRUCE SMITH
Proof.(i) In the Sullivan model setting, path objects for MX and MSn MX are
related as in the commutative diagrams
j pi
MSn MX ____//_MSn MX (t, dt)___//_MSn MX
q2|| q2|1| q2||
fflffl| fflffl| fflffl|
MX ______j_____//MX (t, dt)___pi_____//MX
for i = 0, 1. Applying L to this diagram, and following the construction of the*
* map
G as described above, we obtain homotopies GX :L(W )I ! L(MX (t, dt)) and
GSnxX :L(W, v, W 0)I ! L(MSn MX (t, dt)). By constructing GSnxX so as
to extend GX on L(W )I, we may assume these maps are compatible, in the sense
that L(q2 1)OGX = GSnxX OJI: L(W )I ! L(MSn MX (t, dt)). Combining
this with the diagram obtained from applying L to (13), we obtain a commutative
diagram
L(j)
L(W )I_____GX____//L(MX (t, dt))______//L(MX )
JI|| |L(q2|1) |L|(Mf)
fflffl| fflffl| fflffl|
L(W, v, W 0)IGSnxX//_L(MSn MX (t, dt))//_L(MY )
L (G)
Furthermore, from the construction of GX , we have L(j) O GX (sW, cW) = 0 so th*
*at
L(j) O GX = æX O p. It remains to check that we can preserve the properties of *
*this
diagram in lifting the homotopies to LY .
So suppose that Lf is a given Quillen model that arises in the way described
above from OEf: L(W ) ! LY t E(L (MY )). With the above ingredients, we obtain
a diagram
OEfOp
L(W )I___________//LY t E(L (MY ))
0 _____55__________
JI|| __G__________'|cæY|__
fflffl|________ fflfflfflffl|
L(W, v, W 0)I__________//_L(MY ),
L(G)OGSnxX
which commutes since cæYOOEfOp = L(Mf)OæX Op = L(Mf)OL (j)OGX . Therefore,
we may lift as indicated through the surjective quasi-isomorphism cæYto obtain *
*the
homotopy G0 that satisfies cæYO G0 = L (G) O GSnxX . Finally, define H = ß O G0.
This is a homotopy that starts at a Quillen model for A, and ends at one for B,
since we have
æY O G O ~i= æY O ß O G0O ~i
~ cæYO G0O ~i= L(G) O GSnxX O ~i
= L(G) O L(pi) O æSnxX = L(piO G) O æSnxX
which equals L(A) O æSnxX for i = 0 and L(B) O æSnxX for i = 1. That is, G O *
*~0
and G O ~1 may be taken as Quillen models for A and B, respectively.
(ii) The argument for (i) is easily adapted to establish (ii). In this case, *
*begin
by constructing a homotopy
GSn_X :L(W, v)I ! L(MX (t, dt)) t L(MSn (t, dt))
QUILLEN MODELS AND ADJOINT MAPS 33
so that (j t j) O GSn_X = (æX t æSn) O p: L(W, v)I ! L(MX ) t L(MSn). Then
extend L (q2 1) | L (q1 1) O GSn_X to GSnxX :L(W, v, W 0)I ! L (MSn
MX (t, dt)). The argument now proceeds as before.
Notice that in either case, Lemma A.2 shows that we may choose a Quillen model
LA of A that restricts to Lf on L(W ) (the restriction equals Lf, and is not ju*
*st
homotopic to it). This justifies a fact that we relied upon for the definition *
*of the
maps and in Theorem 4.1.
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Department of Mathematics, Cleveland State University, Cleveland OH 44115
E-mail address: G.Lupton@csuohio.edu
Department of Mathematics, Saint Joseph's University, Philadelphia, PA 19131
E-mail address: smith@sju.edu
*