Now, for any minimal Sullivan algebra $\land V$ the identifications (14)
extend to natural bijections

$\displaystyle\pi_{n}\langle\land V\rangle=(V^{n})^{\vee}=s(L_{V})_{n-1}.$
(16)

This converts the Lie brackets in $L_{V}$ to the Whitehead products in
$\pi_{*}\langle\land V\rangle$:

######  Proposition 8.2.

Suppose $(\land V,d)$ is a minimal Sullivan algebra. Then

  1. (i)

For $n\geq 2$, (16) is a linear isomorphism. In particular,
$\pi_{n}\langle\land V\rangle$ is the rational vector space, $s(L_{V})_{n-1}$.

  2. (ii)

For $p,q\geq 2$, (16) identifies the Lie bracket
$(L_{V})_{p}\times(L_{V})_{q}\stackrel{{\scriptstyle[\,,\,]}}{{\longrightarrow}}(L_{V})_{p+q}$
with the Whitehead product $\pi_{p+1}\langle\land
V\rangle\times\,\pi_{q+1}\langle\land V\rangle\to\pi_{p+q+1}\langle\land
V\rangle$.

  3. (iii)

The composite

$\textstyle{G_{L}}$$\textstyle{(L_{V})_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\exp}$$\textstyle{\pi_{1}\langle\land
V\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$

is an isomorphism of groups, where $G_{L}$ is the fundamental group of $L_{V}$
defined in §1.2.

  4. (iv)

Finally (16) identifies the right adjoint representation of
$\pi_{1}\langle\land V\rangle$ in $\pi_{n}\langle\land V\rangle$ as

$\beta\bullet\exp\alpha=e^{-ad\,\alpha}(\beta),\hskip
14.22636pt\alpha\in(L_{V})_{0},\beta\in L_{V}.$

proof. Recall that $(\land V,d)=\varinjlim_{\alpha}(\land V_{\alpha},d)$ in
which the $V_{\alpha}$ are finite dimensional subspaces of $V$. It follows
that $\pi_{*}\langle\land V\rangle=\varprojlim_{\alpha}\langle\land
V_{\alpha}\rangle$. Now Proposition 8.2 follows respectively from [19, Theorem
1.4], [17, Proposition 13.16], [19, Theorem 2.4] and [19, Theorem 2.5].
$\square$

_To summarize, a Sullivan algebra $\land V$ computes $\pi_{*}\langle\land
V\rangle$ together with its Whitehead products, the group structure in
$\pi_{1}\langle\land V\rangle$, and the action of $\pi_{1}\langle\land
V\rangle$ on $\pi_{\geq 2}\langle\land V\rangle$._

Example. Denote by $\land V$ the minimal Sullivan model of $S^{1}\vee S^{3}$.
Then $V^{1}=\mathbb{Q}v$ while $V^{3}$ has a basis
$w_{0},w_{1},\dots,w_{n},\dots$ satisfying $dw_{0}=0$ and $dw_{n+1}=vw_{n}$.
Define $x\in(L_{V})_{0}$ by $<v,sx>=1$ and identifies $\pi_{3}\langle\land
V\rangle=(V^{3})^{\vee}$ as the space of infinite series

$g(t)=\sum_{n\geq 0}c_{n}t^{n}$

with $<w_{n},g(t)>=c_{n}$. Then the action of $\pi_{1}\langle\land V\rangle$
in $\pi_{3}\langle\land V\rangle$ is given by

$g(t)\bullet\exp x=e^{t}\cdot g(t).$

Finally, the connection between the properties of $H(\land V)$ and $L_{V}$ is
illustrated in

######  Proposition 8.3.

The following conditions on a minimal Sullivan algebra $(\land V,d)$ are
equivalent:

  1. (i)

$L_{V}=\varprojlim_{n\in\mathbb{N}}L(n)$ in which each $L(n)$ is a finite
dimensional nilpotent Lie algebra.

  2. (ii)

dim$\,V$ is at most countable.

  3. (iii)

dim$\,H(\land V,d)$ is at most countable.

proof. (i) $\Rightarrow$ (ii) $\Rightarrow$ (iii). Let $\land V_{n}$ be the
quadratic Sullivan model of $L(n)$. Then $(\land V,d_{1})=\varinjlim_{n}(\land
V_{n},d_{1})$ is at most countable. This proves (ii) and (ii) $\Rightarrow$
(iii) is obvious.

(iii) $\Rightarrow$ (i). Define subspaces $W_{n}\subset V$ by
$W_{0}=V\cap\mbox{ker}\,d$ and $W_{n+1}=V\cap d^{-1}(\land W_{n})$. It follows
from (iii) that each dim$\,W_{n}$ is at most countable. But because $(\land
V,d)$ is a Sullivan algebra it follows that $V=\cup_{n}W_{n}$. and so dim$\,V$
is at most countable. Thus because $(\land V,d)$ is a Sullivan algebra
$V=\cup_{\alpha}V_{\alpha}$ where dim$\,V_{\alpha}<\infty$ and $\land
V_{\alpha}$ is preserved by $d$. Since dim$\,V$ is at most countable this
implies that $V=\varinjlim_{n}V_{n}$ where $(\land V_{n},d)$ is a sub Sullivan
algebra and dim$\,V_{n}<\infty$; Thus $L_{V}=\varprojlim_{n}L(n)$, $L(n)$
denoting the homotopy Lie algebra of $(\land V_{n},d)$. $\square$

Corollary. If $(\land V,d)$ is the minimal Sullivan of a space $X$ and
satisfies the conditions of the Proposition then $H(X)$ is a graded vector
space of finite type.

####  8.1 Profree homotopy Lie algebras

Profree Lie algebras are characterized in Part II in terms of their quadratic
Sullivan models. This can, however, be extended to general minimal Sullivan
algebras (Proposition 8.4 below), and they arise, with their sub Lie algebras,
in a natural way as homotopy Lie algebras of wedges of spheres.

######  Proposition 8.4.

Denote by $L$ and by $(\land V,d)$ the homotopy Lie algebra and associated
quadratic Sullivan algebra of a minimal Sullivan algebra $(\land V,d)$. The
following conditions are then equivalent

  1. (i)

There is a quasi-isomorphism $(\land
V,d)\stackrel{{\scriptstyle\simeq}}{{\to}}\mathbb{Q}\oplus S$ with zero
differential in $S$ and $S\cdot S=0$.

  2. (ii)

The generating space $V$ can be chosen so that $d:V\to\land^{2}V$ and

$\mathbb{Q}\oplus(V\cap\mbox{ker}\,d)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}H(\land
V,d).$

  3. (iii)

$\mathbb{Q}\oplus
V\cap\mbox{ker}\,d_{1}\stackrel{{\scriptstyle\cong}}{{\to}}H(\land V,d_{1})$.

  4. (iv)

$L=\overline{\mathbb{L}}_{T}$ is profree, with $sT=H^{\geq 1}(\land
V)^{\vee}=S^{\vee}$ defining the enriched structure.

If they hold then $(\land V,d)\cong(\land V,d_{1})$ and, if $V$ is chosen to
satisfy (iii), then $V\cap\mbox{ker}\,d=V\cap\mbox{ker}\,d_{1}$.

proof. (i) $\Leftrightarrow$ (ii). First, let $S=S^{\geq 1}$ be any graded
vector space. Then $\mathbb{Q}\oplus S$, with zero differential and $S\cdot
S=0$ is a cdga, and its minimal model has the form

$(\land
V,d)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(\mathbb{Q}\oplus S,0).$

On the other hand, a successive adjoining of variables $w$ to $S$ constructs a
quadratic Sullivan algebra $(\land W,d_{1})$ such that $H^{[1]}(\land
W,d_{1})=S$ and $H^{[2]}(\land W)=0$. (Here as in Part II, $H^{[k]}(\land
W,d_{1})$ denotes the homology classes represented by cycles in $\land^{k}W$.)
Now Lemma 6.2 asserts that $H^{[k]}(\land W)=0$, $k\geq 2$. Thus division by
$\land^{\geq 2}W$ and by a direct summand of $S$ in $W$ defines a quasi-
isomorphism

$(\land W,d_{1})\stackrel{{\scriptstyle\simeq}}{{\to}}\mathbb{Q}\oplus S.$

Since minimal models are unique it follows that there is an isomorphism

$(\land W,d_{1})\cong(\land V,d).$

Now choose $V$ so this isomorphism takes
$W\stackrel{{\scriptstyle\cong}}{{\to}}V$.

Now, if (ii) holds, then with the given choice of $V$, division by
$\land^{\geq 2}V$ and by a direct summand of $V\cap\mbox{ker}\,d$ in $V$
defines a quasi-isomorphism

$(\land
V,d)\stackrel{{\scriptstyle\simeq}}{{\to}}\mathbb{Q}\oplus(V\cap\mbox{ker}\,d).$

This gives (i).

(i) $\Leftrightarrow$ (iii). If (i) holds, the observations above yield an
isomorphism $(\land W,d_{1})\cong(\land V,d)$. This induces an isomorphism
$(\land W,d_{1})\stackrel{{\scriptstyle\cong}}{{\to}}(\land V,d_{1})$, which
for appropriate choice of $V$ preserves wedge degree. It follows that with
this choice $(\land V,d)$ is quadratic and
$V\cap\mbox{ker}\,d=V\cap\mbox{ker}\,d_{1}$. This proves (iii), and the same
argument shows that (iii) $\Rightarrow$ (ii).

Finally, the assertion (iii) $\Leftrightarrow$ (iv) is Theorem 1. $\square$

Corollary 1. If $H^{\geq 1}(X)\neq 0$ then the homotopy Lie algebra $L_{X}$ is
profree if and only if $X$ is formal and $H^{\geq 1}(X)\cdot H^{\geq 1}(X)\neq
0$. (A space $X$ is _formal_ if its minimal Sullivan model is also a minimal
Sullivan model of $(H(X),0)$.)

Corollary 2. If cat$\,X=1$ then the homotopy Lie algebra, $L_{X}$ is profree.

proof. It follows from ([19, Formula (9.1) and Theorem 9.1]) that if
cat$\,X=1$ then its minimal Sullivan model satisfies condition (i) of the
Proposition. $\square$

Corollary 3. If a connected co-H-space, $X$, satisfies $H^{\geq 1}(X)\neq 0$,
then $L_{X}$ is profree.

proof. Let $U$ be a contractible open neighbourhood of a base point in $X$.
Then $X\vee U$ and $U\vee X$ form an open cover for $X\vee X$

Since X is a co-H-space, the diagonal map $\Delta:X\to X\times X$ factors up
to homotopy through $X\vee X$,

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\scriptstyle{f}$$\textstyle{X\vee
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\times
X.}$

The open sets $f^{-1}(X\vee U)$ and $f^{-1}(U\vee X)$ make then a covering of
$X$ by contractible open sets, and cat$\,X=1$. Now the result follows from
Corollary 2. $\square$

###  9 The Sullivan completion, $X_{\mathbb{Q}}$, of a space $X$

_Throughout §9 we consider a connected space, $X$ with a minimal Sullivan
model_

$\varphi:(\land
V,d)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}A_{PL}(X).$

Definition. The adjoint map

$|\varphi|:X\to X_{\mathbb{Q}}:=\langle\land V\rangle,$

is the _Sullivan completion_ of $X$.

######  Proposition 9.1.

  1. (i)

$\pi_{*}|\varphi|$ converts Whitehead products in $\pi_{*}(X)$ to
constructions in $L_{X}$ as set out in Proposition 8.2.

  2. (ii)

([19, Theorem 15.11]). If $X$ is a simply connected space and $H(X)$ is a
graded vector space of finite type then $\pi_{*}|\varphi|$ factors as

$\pi_{*}(X)\to\pi_{*}(X)\otimes\mathbb{Q}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\pi_{*}(X_{\mathbb{Q}}).$

The definition above depends on the choice for any space $X$ of a minimal
Sullivan model $\land V$. However, since two minimal Sullivan models $\land
V,\land V^{\prime}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}A_{PL}(X)$
are connected by a unique homotopy class of isomorphisms, it follows that for
any minimal model $\land
V^{\prime}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}A_{PL}(X)$, the
homeomorphisms $\langle\land
V^{\prime}\rangle\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}X_{\mathbb{Q}}$
are connected by unique homotopy classes of isomorphisms. This identifies the
vector spaces $V^{\prime}$, the groups $\pi_{*}\langle\land
V^{\prime}\rangle$, and the enriched Lie algebras $L_{V^{\prime}}$, and we
write

$V_{X}=V^{\prime},\pi_{*}(X_{\mathbb{Q}})=\pi_{*}\langle\land
V^{\prime}\rangle,\hskip 8.53581pt\mbox{and }L_{X}=L_{V^{\prime}}.$

Definition. _$L_{X}$ is the homotopy Lie algebra of $X$_.

In particular, (16) specializes to

$\displaystyle\pi_{*}(X_{\mathbb{Q}})=V_{X}^{\vee}=sL_{X}.$ (17)

Moreover, a continuous map $f:Y\to X$ has a unique homotopy class of Sullivan
representatives $\varphi:\land V_{Y}\leftarrow\land V_{X}$ between the
respective minimal Sullivan models. Now it follows from (16) that

$\displaystyle\pi_{*}(f_{\mathbb{Q}})=\pi_{*}\langle\varphi\rangle=sL_{f}.$
(18)

This makes $\pi_{*}(X_{\mathbb{Q}})$ effectively computable in many cases,
which in turn can provide information about $\pi_{*}(X)$. In particular,
Proposition 8.2 provides an explicit description of the Whitehead products in
$\pi_{*}(X_{\mathbb{Q}})$ and of the group structure of
$\pi_{1}(X_{\mathbb{Q}})$ in terms of the minimal model $(\land V,d)$ of $X$.

However $H(X_{\mathbb{Q}})$ remains mysterious, and it is unknown what
conditions are necessary and or sufficient for
$H^{n}(X_{\mathbb{Q}};\mathbb{Z})$, $n\geq 1$, to be a rational vector space.
We do have

######  Proposition 9.2.

Suppose $\varphi:(\land V,d)\stackrel{{\scriptstyle\simeq}}{{\to}}A_{PL}(X)$
is a minimal Sullivan model of a space $X$. Then $(\land V,d)$ is a retract of
the minimal Sullivan model of $X_{\mathbb{Q}}$, and so $H(X)$ is a retract of
$H(X_{\mathbb{Q}})$.

proof. Let $(\land
W,d)\stackrel{{\scriptstyle\simeq}}{{\to}}A_{PL}\langle\land V\rangle$ be the
minimal Sullivan model. Since (15) $A_{PL}|\varphi|\circ\iota_{\land
V}\sim\varphi$ the homotopy commutative diagram

$\textstyle{A_{PL}\langle\land
V\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A_{PL}|\varphi|}$$\textstyle{A_{PL}(X)}$$\textstyle{\land
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\land
V}}$$\textstyle{\land
W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\land
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\varphi}$

exhibits $(\land V,d)$ as a homotopy retract of $(\land W,d)$. $\square$

Corollary. Any continuous map $f:Y\to X_{\mathbb{Q}}$ extends to a map
$Y_{\mathbb{Q}}\to X_{\mathbb{Q}}.$

proof. Compose $f_{\mathbb{Q}}$ with the retraction
$(X_{\mathbb{Q}})_{\mathbb{Q}}\to X_{\mathbb{Q}}$. $\square$

Now observe that the retraction

$\xi:\land^{\geq 1}V\to V$

with kernel $\land^{\geq 2}V$ satisfies $\xi\circ d=0$ and so induces a map
$H(\xi):H^{\geq 1}(\land V)\to V$. On the other hand, denote by
$hur:\pi_{*}(X)\to H_{*}(X;\mathbb{Z})$ the Hurewicz homomorphism, and recall
that

$\pi_{*}(|\varphi|):\pi_{*}(X)\to\pi_{*}(X_{\mathbb{Q}})=V^{\vee}.$

######  Proposition 9.3.

([19, Proposition 1.19]) For $\alpha\in H^{\geq 1}(\land V)$ and
$\beta\in\pi_{*}(X)$,

$<H(\varphi)\alpha,hur\,\beta>=<H(\xi)(\alpha),\pi_{*}(|\varphi|)\beta>;$

Moreover, because of the naturality of the Hurewicz homomorphism, this follows
from a commutative diagram valid for any minimal Sullivan algebra, $\land V$.
Recall from §7 the natural morphism $\iota_{\land V}:\land V\to
A_{PL}\langle\land V\rangle$ adjoint to $id_{\langle\land V\rangle}$.
Proposition 8.2 specializes to (and follows from) the commutative diagram

(23)

in which the upper isomorphism is the isomorphism (14). Note that the
generating space $V$ can always be chosen so that $\xi:\mbox{ker}\,d\to
V\cap\mbox{ker}\,d$. In this case in the diagram above $H^{\geq 1}(\land
V)^{\vee}$ may be replaced by $(V\cap\mbox{ker}\,d)^{\vee}$.

####  9.1 The Sullivan completion of a free group, $G$

Suppose $b_{1},\dots,b_{s}$ freely generate a free group $G$. The classifying
space for $G$ is then

$X=S^{1}_{b_{1}}\vee\dots\vee S^{1}_{b_{s}},$

and $b_{i}$ is the homotopy class of the inclusion of $S^{1}_{b_{i}}$ in $X$.
In particular, $\pi_{*}(X)=\pi_{1}(X)=G$.

On the other hand, (Corollary 2 to Proposition 8.4), the minimal Sullivan
model of $X$ satisfies the conditions of Proposition 8.4. Therefore it is a
quadratic Sullivan algebra $(\land V,d_{1})$ and the inclusion

$j:V\cap\mbox{ker}\,d_{1}{\longrightarrow}\land^{\geq 1}V$

induces an isomorphism
$V\cap\mbox{ker}\,d_{1}\stackrel{{\scriptstyle\cong}}{{\to}}H^{1}(\land
V)=H^{1}(X)$. It follows that $V=V^{1}$.

Now recall that the quasi-isomorphism $\varphi:(\land
V,d_{1})\stackrel{{\scriptstyle\simeq}}{{\to}}A_{PL}(X)$ induces via
adjointness a continuous map

$|\varphi|:X\to\langle\land V\rangle=X_{\mathbb{Q}}.$

Definition. The group morphism
$\pi_{1}|\varphi|:G=\pi_{1}(X)\to\pi_{1}(X_{\mathbb{Q}})$ is the _Sullivan
completion_ of $G$.

We now recall three important constructions:

  1. $\bullet$

The _homotopy Lie algebra_ , $L_{V}$, of $(\land V,d_{1})$, a complete
enriched Lie algebra, defined (§8) via the quadratic differential, $d_{1}$. By
construction

$sL_{V}=V^{\vee}$

as graded vector spaces.

  2. $\bullet$

The _group_ $G_{L_{V}}$ (§1.2) connected with $L_{V}$ by natural inverse
bijections

$\textstyle{L_{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exp}$$\textstyle{G_{L_{V}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\log}$

  3. $\bullet$

A _natural bijection of sets_ (12)

$\tau_{V}:\pi_{1}\langle\land
V\rangle\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}V^{\vee}.$

With $L_{V}$ and $V^{\vee}$ identified as above, ([19, Theorem 2.4]) provides
a commutative diagram,

(28)

in which $\sigma_{V}$ is a natural isomorphism of groups. We denote by
$\log_{G}$ the composite

$\log_{G}:G\longrightarrow
G_{L_{V}}\stackrel{{\scriptstyle\log}}{{\longrightarrow}}L_{V}.$

The constructions above combine to form the commutative diagram,

|
---|---
$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{V}\circ\pi_{1}|\varphi|}$$\textstyle{V^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\scriptstyle{j^{\vee}}$$\textstyle{sL_{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1)}$$\textstyle{(2)}$$\textstyle{G/G^{2}\otimes\mathbb{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{(V\cap\mbox{ker}\,d_{1})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{sL_{V}/L_{V}^{2},}$

which we now establish.

First, the isomorphism
$V\cap\mbox{ker}\,d_{1}\stackrel{{\scriptstyle\cong}}{{\to}}H^{1}(X)$ dualizes
to an isomorphism
$H_{1}(X;\mathbb{Q})\stackrel{{\scriptstyle\cong}}{{\to}}(V\cap\mbox{ker}\,d_{1})^{\vee}$.
The composite

$G\to V^{\vee}\stackrel{{\scriptstyle
j^{\vee}}}{{\longrightarrow}}(V\cap\mbox{ker}\,d_{1})^{\vee}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}H_{1}(X;\mathbb{Q})$

is a set map $G\to H_{1}(X;\mathbb{Q})$. It follows from Proposition 9.3 that
this is $hur\otimes\mathbb{Q}$, where $hur$ is the classical Hurewicz
homomorphism. Thus it induces an isomorphism
$G/G^{2}\otimes\mathbb{Q}\stackrel{{\scriptstyle\cong}}{{\to}}(V\cap\mbox{ker}\,d_{1})^{\vee}$.
This establishes $(1)$.

Second, the quasi-isomorphism
$\mathbb{Q}\oplus(V\cap\mbox{ker}\,d_{1})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(\land
V,d_{1})$ admits a cdga retraction $\rho:(\land
V,d_{1})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathbb{Q}\oplus(V\cap\mbox{ker}\,d_{1})$,
where $V\cap\mbox{ker}\,d_{1}$ is given the trivial product. Thus Proposition
8.4 identifies $L_{V}$ as a profree Lie algebra
$\overline{\mathbb{L}}_{T^{\prime}}$, in which
$\rho^{\vee}:(V\cap\mbox{ker}\,d_{1})^{\vee}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}T^{\prime}$.
Since $j^{\vee}\circ\rho^{\vee}=id_{(V\cap\mbox{ker}\,d_{1})^{\vee}}$ it
follows that ker$\,j^{\vee}$ is a direct summand of $T^{\prime}$. But for
$v\in V\cap\mbox{ker}\,d_{1}$, and for $x,y\in L_{V}$,

$<v,j^{\vee}s[x,y]>=<d_{1}j(v),s[x,y]>=0.$

Therefore $L_{V}^{(2)}\subset\mbox{ker}\,j^{\vee}$ and so
$L^{(2)}=\mbox{ker}\,j^{\vee}$. This establishes (2).

Next, let $\overline{J}$ denote the augmentation ideal of $\overline{UL_{V}}$
(§1.2) and recall that the right adjoint representation of $L_{V}$ extends to
a representation of $\overline{J}$.

######  Proposition 9.4.

(i) The elements $\log_{G}b_{i}$ are the basis of a subspace $T\subset L_{V}$
for which $L_{V}=\overline{\mathbb{L}}_{T}$.

(ii) If $c,d\in G$ then

$\log_{G}cd\in\log_{G}c+\log_{G}d+\frac{1}{2}[\log_{G}c,\log_{G}d]+[\log_{G}c,\log_{G}d]\cdot\overline{J},$

and

$\log_{G}(cdc^{-1}d^{-1})\in[\log_{G}c,\log_{G}d]+[\log_{G}c,\log_{G}d]\cdot\overline{J}.$

proof. (i) follows from (18), diagrams (1) and (2) above, and the commutative
diagram ([19, Theorem 2.2]),

$\textstyle{G_{L_{V}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\log}$$\textstyle{L_{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{L_{V}}/G_{L_{V}}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\log}$$\textstyle{L_{V}/L_{V}^{(2)},}$

and diagrams (1) and (2) above.

(ii) A naturality argument reduces this to the case $G$ is freely generated by
$c$ and $d$. In this case $T=\mathbb{Q}\log_{G}c\oplus\mathbb{Q}\log_{G}d$,
$[T,T=\mathbb{Q}[\log_{G}c,\log_{G}d]$ and
$L_{V}^{(3)}\subset[\log_{G}c,\log_{G}d]\cdot\overline{J}$. Thus (ii) follows
immediately from ([19, Proposition 2.4]). $\square$

####  9.2 Remarks

1\. Let $\varphi:\land V\to A_{PL}(X)$ and $\psi:\land W\to A_{PL}(Y)$ be
minimal Sullivan models. Then cf (13) we have bijections

$[Y,X_{\mathbb{Q}}]\cong[\land V,A_{PL}(Y)]\cong[\land V,\land W].$

2\. By [17, Proposition 17.13] Sullivan completions induce a morphism

$[\land V,\land W]\to[\langle\land W\rangle,\langle\land V\rangle].$

This morphism is not always injective. For instance, if $Y$ is an infinite
wedge of spheres $S^{3}$, and $X=S^{3}$, then $\land V=(\land u,0)$ with
deg$\,u=3$, and $\land W\simeq(H(Y),0)$. Therefore

$[\land V,\land W]\cong W^{3}\,\hskip 8.53581pt\mbox{and }[\langle\land
W\rangle,\langle\land V\rangle]\cong H^{3}\langle\land
W\rangle=(\pi_{3}\langle\land
W\rangle)^{\vee}=\left((W^{3})^{\vee}\right)^{\vee}.$

Suppose $\alpha\in H^{3}\langle\land W\rangle$ is in the kernel of
$H(|\psi|):H(\langle\land W\rangle)\to H(Y)$. This gives a map
$g:Y_{\mathbb{Q}}\to X_{\mathbb{Q}}$ with $g\circ|\psi|\sim*\sim*\circ|\psi|$.

3\. When $Y$ is a simply connected CW complex of finite type, then ([17,
Theorem 17.12]) $|\psi|:Y\to Y_{\mathbb{Q}}$ is an isomorphism in rational
cohomology and classical obstruction theory shows that in that case the
morphism $[\land V,\land W]\to[\langle\land W\rangle,\langle\land V\rangle]$
is a bijection.

4\. Let $X=\langle\land V\rangle$ be the spatial realization of the Sullivan
quadratic model of the profree Lie algebra $\overline{\mathbb{L}}_{T}$ where
$T=T_{1}$ is a countably infinite dimensional vector space. Then $H_{3}(X)\neq
0$ and $H_{4}(X)\neq 0$.

We only provide the proof for $H_{3}(X)$. The proof for $H_{4}(X)$ is similar.
Let $\omega\in\pi_{3}(X)$ be a homotopy class corresponding by suspension to
an element in $L_{2}$, but not in $[L_{1},L_{1}]$. Then $X$ contains a simply
connected finite CW complex $Y$ such that $\omega$ is in the image of
$\pi_{3}(Y)$. We denote this element by $\omega_{Y}\in\pi_{3}(Y)$. It follows
that $\omega_{Y}$ is not decomposable, and by Lemma 9.1 below, the image
$a_{Y}$ of $\omega_{Y}$ in $H_{3}(Y)$ is non zero.

For any finite sub CW complex $Z$ of $X$ containing $Y$, the element
$\omega_{Z}$ induces also a non-zero element $a_{Z}$ by the Hurewicz map, and
by naturalness of the Hurewicz map, the map $H_{3}(Y)\to H_{3}(Z)$ maps
$a_{Y}$ to $a_{Z}$. Now $X$ is the union of the finite sub CW complexes
$Z_{\alpha}$ containing $Y$. It follows that
$H_{3}(X)=\varinjlim_{\alpha}H_{3}(Z_{\alpha})$ and therefore the family
$a_{Z_{\alpha}}$ induces a nonzero element in $H_{3}(X)$. $\square$

######  Lemma 9.1.

Let $L$ be the homotopy Lie algebra of a finite type simply connected CW
complex, $Y$.

  1. (i)

If $L_{2}\neq[L_{1},L_{1}]$, then the Hurewicz map $\pi_{3}(Y)\to H_{3}(Y)$ is
non zero.

  2. (ii)

If $L_{3}\neq[L_{1},L_{2}]$, then the Hurewicz map $\pi_{4}(Y)\to H_{4}(Y)$ is
non zero.

Proof. Denote by $(\land V,d)$ the Sullivan minimal model of $Y$. Then $V$ is
a finite type vector space. Therefore the brackets $[\,,\,]:\land^{2}L_{1}\to
L_{2}$, and $L_{1}\otimes L_{2}\to L_{3}$ are dual to the differentials
$d:V^{3}\to\land^{2}V^{2}$, and $V^{4}\to V^{2}\otimes V^{3}$. Consider the
situation (i); the proof for (ii) is similar. Since $L_{2}\neq[L_{1},L_{1}]$,
$d$ is not injective. This implies that $H^{3}(\land V)\to H^{3}(\land
V/\land^{\geq 2}V)$ is non zero. But this last map is the dual of the Hurewicz
map. Thus if $\omega\in\pi_{3}(Y)\otimes\mathbb{Q}=(V^{3})^{\vee}$ with
$\omega(v)\neq 0$, then hur $(\omega)\neq 0$. $\square$

####  9.3 $\pi_{1}|\varphi|:\pi_{1}(X)\to\pi_{1}(X_{\mathbb{Q}})$

Recall that for any group, $G$, $G^{n}$ denotes the subgroup generated by
iterated commutators of length $n$, and that for any graded Lie algebra, $L$,
$L^{n}$ is the span of the iterated commutators of length $n$. In particular a
normal subgroup $Tor(G)$ is defined by

$Tor(G)=\\{a\in G\,|\,\mbox{for each $n$ there is a $k$ such that }a^{k}\in
G^{n}\\}.$

(Thus if $G$ is abelian, $Tor(G)$ is the subgroup of torsion elements.)

######  Proposition 9.5.

If $H_{1}(X;\mathbb{Z})$ is finitely generated then

$\mbox{ker}(\pi_{1}|\varphi|:\pi_{1}(X)\to\pi_{1}(X_{\mathbb{Q}}))=Tor(\pi_{1}(X)).$

proof. For simplicity we denote $\pi_{1}(X)$ by $G$. Then the classifying map
$X\to BG$ induces an isomorphism in $H_{1}(-;\mathbb{Z})$ and, by a theorem
([8, Theorem 5.2]) of H. Hopf, a surjection in $H_{2}(-;\mathbb{Q})$.
Therefore it induces an isomorphism in $H^{1}(-;\mathbb{Q})$ and an injection
in $H^{2}(-;\mathbb{Q})$. In particular the corresponding morphism of minimal
Sullivan models restricts to an isomorphism

$(\land V^{1},d)\stackrel{{\scriptstyle\cong}}{{\longleftarrow}}(\land
W^{1},d).$

This identifies $\pi_{1}(X_{\mathbb{Q}})$ with $\pi_{1}((BG)_{\mathbb{Q}})$
and reduces the Proposition to the case $X=BG$.

Denote by $L$ the component of degree $0$ in the homotopy Lie algebra of $BG$,
and by $G_{L}$ the group $\pi_{1}((BG)_{\mathbb{Q}})$. Since (by hypothesis)
$G/G^{2}$ is finitely generated it follows that each $G^{n}/G^{n+1}$ is
finitely generated. In particular,

$\mbox{ker}\,(G^{n-1}/G^{n}\to G^{n-1}/G^{n}\otimes_{\mathbb{Z}}\mathbb{Q})$

is a torsion group. On the other hand ([19, Theorem 7.5]),
$\log\circ\pi_{1}|\varphi|$ induces isomorphisms

$\psi_{n}:G^{n-1}/G^{n}\otimes_{\mathbb{Z}}\mathbb{Q}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}L^{n-1}/L^{n}.$

Now suppose $a\in\mbox{ker}\,\pi_{1}|\varphi|$ and that $a^{k}\in G^{n-1}$.
Let $b=$ image of $a^{k}$ in $G^{n-1}/G^{n}$. Then $\psi_{n}b$ is the image of
$\log\pi_{1}|\varphi|a^{k}=k\log\pi_{1}|\varphi|a=0$. Thus $b$ is in the
kernel of $G^{n-1}/G^{n}\to G^{n-1}/G^{n}\otimes_{\mathbb{Z}}\mathbb{Q}$, and
so

$a^{\ell k}=b^{\ell}\in G^{n}.$

It follows by induction that $a\in Tor(G)$.

In the reverse direction the same argument shows that if $a\in Tor(G)$, then
for each $n$ and some $k$,

$k\log\pi_{1}|\varphi|a=\log\pi_{1}|\varphi|a^{k}\in L^{n}.$

It follows that $\log\pi_{1}|\varphi|a\in\cap_{n}L^{n}=0$, and thus
$a\in\mbox{ker}\pi_{1}|\varphi|$. $\square$

####  9.4 The radical and center of $\pi_{*}(X_{\mathbb{Q}})$

As recalled in §6, the category, cat$\,(\land V)$, is the least $m$ (or
$\infty$) for which $\land V$ is a homotopy retract of $\land V/\land^{>m}V$.
Moreover ([19, Theorem 9.2]),

$\mbox{cat}\,(\land V)\leq\mbox{cat}\,X,$

where cat$\,X$ is the classical Lusternik-Schnirelmann category of $X$. On the
other hand, the _radical_ , rad$\,L_{V}$, of $L_{V}$ is the sum of the
solvable ideals of $L_{V}$. Thus combining ([12, Theorem 1]) with ([13,
Theorem C]) yields

######  Proposition 9.6.

Suppose cat$\,(X)<\infty$. Then

  1. (i)

The sum of the solvable ideals, rad$\,L_{V}$, is finite dimensional and

$\mbox{dim(rad}\,(L_{V})_{even})\leq\mbox{ cat}\,X.$

  2. (ii)

The union of the normal solvable subgroups of $\pi_{1}(X_{\mathbb{Q}})$ is a
normal solvable subgroup of solv length $\leq$ cat$\,(X)$.

proof. (i) is immediate from the remarks above. For the proof of (ii) note
that rad$(L_{V})_{0}$ is a closed ideal since it is finite dimensional. Thus
$\exp$(rad$(L_{V})_{0}$) is a normal subgroup of $\pi_{1}(X_{\mathbb{Q}})$
whose quotient is $\exp$($(L_{V})_{0}/\mbox{rad}(L_{V})_{0})$. Therefore,
since the center of $(L_{V})_{0}/\mbox{rad}(L_{V})_{0}$ is zero,
$\exp$($L_{V}/\mbox{rad}(L_{V})_{0})$ has no center and therefore no normal
solvable subgroup. $\square$

Now define $C_{*}(X_{\mathbb{Q}})\subset\pi_{*}(X_{\mathbb{Q}})$ to be the
subgroup of elements $\alpha$ satisfying

$\mbox{Wh}(\alpha,\beta)=0,\hskip 14.22636pt\beta\in\pi_{*}(X_{\mathbb{Q}}).$

It follows from Proposition 8.2 that $C_{*}(X_{\mathbb{Q}})$ corresponds under
$\exp$ to the centre of the homotopy Lie algebra $L_{V}$. Thus Proposition 9.6
has the

Corollary. If cat$\,X<\infty$ then the group $C_{*}(X_{\mathbb{Q}})$ is a
finite dimensional rational vector space.

####  9.5 The Gottlieb groups $G_{*}(X)$ and $G_{*}(X_{\mathbb{Q}})$

The _$n^{th}$ Gottlieb group_, $G_{n}(X)$, is the subgroup of $\pi_{n}(X)$
formed by the homotopy classes $[f]$ of maps $f:S^{n}\to X$ such that
$f\vee\mbox{id}:S^{n}\vee X\to X$ extends to the product $S^{n}\times X$. The
Whitehead products of $\alpha\in G_{*}(X)$ and $\beta\in\pi_{*}(X)$ are zero.
In particular, $G_{*}(X_{\mathbb{Q}})\subset C_{*}(X_{\mathbb{Q}})$.

More generally, if $g:Y\to X$ is a continuous map, then the $n^{th}$ Gottlieb
group of $g$, $G_{n}(g)$ is the subgroup of $\pi_{n}(X)$ formed by the
homotopy classes of maps $f:S^{n}\to X$ such that the map $f\vee g:S^{n}\vee
Y\to X$ extends to $S^{n}\times Y$.

Finally, recall from (13) that for any minimal Sullivan algebra $\land W$,

$\pi_{n}\langle\land W\rangle=[\land W,\mathbb{Q}\oplus\mathbb{Q}a_{n}]$

where $a_{n}$ is the fundamental cohomology class in $H^{n}(S^{n})$. We define
the _Gottlieb group_ ,

$G_{n}(\land W)\subset\pi_{n}\langle\land W\rangle$

to be the subgroup formed by those elements represented by morphisms
$\sigma:\land W\to\mathbb{Q}\oplus\mathbb{Q}a_{n}$ such that $\sigma\times
id:\land W\to(\mathbb{Q}\oplus\mathbb{Q}a_{n})\times_{\mathbb{Q}}\land W$
factors as

$\land W\to(\mathbb{Q}\oplus\mathbb{Q}a_{n})\otimes\land
W\to(\mathbb{Q}\oplus\mathbb{Q}a_{n})\times_{\mathbb{Q}}\land W.$

Note here that

$(\mathbb{Q}\oplus\mathbb{Q}a_{n})\times_{\mathbb{Q}}\land
W=\mathbb{Q}a_{n}\oplus\land W$

and that $\sigma$ is just the map

$W=\land^{\geq 1}W/\land^{\geq 2}W\to\mathbb{Q}a_{n}.$

######  Proposition 9.7.

  1. (i)

$\pi_{*}|\varphi|:G_{*}(X)\to G_{*}(X_{\mathbb{Q}})$.

  2. (ii)

$G_{*}(\land V)=G_{*}(|\varphi|)=G_{*}(X_{\mathbb{Q}})$.

  3. (iii)

$G_{*}(X_{\mathbb{Q}})\subset C_{*}(X_{\mathbb{Q}})$.

proof. (i) If $f:S^{n}\to X$ represents an element of $G_{*}(X)$ we may extend
$f\vee id_{X}$ to $g:S^{n}\times X\to X$. Now Proposition 8.3(iii) identifies
$(S^{n}\times X)_{\mathbb{Q}}=S^{n}_{\mathbb{Q}}\times X_{\mathbb{Q}}$. Thus
we obtain $g_{\mathbb{Q}}:S^{n}_{\mathbb{Q}}\times X_{\mathbb{Q}}\to
X_{\mathbb{Q}}$. It follows from (16) that $g_{\mathbb{Q}}$ restricts to
$|\varphi|\circ f:S^{n}\to X_{\mathbb{Q}}$. Thus $g_{\mathbb{Q}}:S^{n}\times
X_{\mathbb{Q}}\to X_{\mathbb{Q}}$ extends $|\varphi|\circ f\vee
id_{X_{\mathbb{Q}}}$.

(ii) We proceed in two steps.

Step One. $G_{*}(\land V)=G_{*}\langle\land V\rangle$.

Let $\land Z$ denote the minimal model of $S^{n}$; Then (13)

$\pi_{n}\langle\land V\rangle=\mbox{Hom}(V^{n},\mathbb{Q})=[\land V,\land
Z]=[\land V,A_{PL}(S^{n})].$

Thus an element in $G_{n}(\land V)$ is represented by a morphism $\sigma:\land
V\to\land Z$ for which $\sigma\times_{\mathbb{Q}}id:\land V\to\land
Z\times_{\mathbb{Q}}\land V$ factors up to homotopy as

$\textstyle{\land
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma\times_{\mathbb{Q}}id}$$\textstyle{\land
Z\otimes\land
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\land
Z\times_{\mathbb{Q}}\land V.}$

The identifications in Proposition 8.2 then convert this diagram into the
homotopy commutative diagram

$\textstyle{\langle\land V\rangle}$$\textstyle{\langle\land
Z\rangle\times\langle\land
V\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{n}\times\langle\land
V\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\langle\land
Z\rangle\vee\langle\land
V\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\langle\sigma\times_{\mathbb{Q}}id\rangle}$$\textstyle{S^{n}\vee\langle\land
V\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

in which the vertical arrows are the standard inclusions. In particular, the
map

$S^{n}\to\langle\land
Z\rangle\stackrel{{\scriptstyle\langle\sigma\rangle}}{{\longrightarrow}}\langle\land
V\rangle$

represents $[\sigma]\in\pi_{n}(\land V)$.

In the reverse direction, suppose $\tau\in G_{n}\langle\land V\rangle$. Then
we have the homotopy commutative diagram

$\textstyle{\langle\land V\rangle}$$\textstyle{S^{n}\times\langle\land
V\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{n}\vee\langle\land
V\rangle.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\tau,id)}$

Let $\land W$ be the minimal Sullivan model of $\langle\land V\rangle$. Then
the identifications of Proposition 8.2, together with the proof of Proposition
9.2, convert this to the homotopy commutative diagram

$\textstyle{\land
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{PL}(S^{n})\otimes
A_{PL}\langle\land
V\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\land
Z\otimes\land
W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\land
Z\otimes\land
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{PL}(S^{n})\times_{\mathbb{Q}}A_{PL}\langle\land
V\rangle}$$\textstyle{\land Z\times_{\mathbb{Q}}\land
W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\land
Z\times_{\mathbb{Q}}\land V.}$

Step Two. $G_{n}(|\varphi|)=G_{n}(X_{\mathbb{Q}}$).

The injection $G_{n}(X_{\mathbb{Q}})\subset G_{n}(\varphi_{X})$ is a direct
consequence of the following commutative diagram associated to an element
$[f]\in G_{n}(X_{\mathbb{Q}})$,

|
---|---
$\textstyle{S^{n}\vee
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id\vee|\varphi|}$$\textstyle{S^{n}\vee
X_{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\vee
id}$$\textstyle{X_{\mathbb{Q}}}$$\textstyle{S^{n}\times
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id\times|\varphi|}$$\textstyle{S^{n}\times
X_{\mathbb{Q}}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

For the reverse inclusion $G_{n}(|\varphi|)\subset G_{n}(X_{\mathbb{Q}})$
suppose $f:S^{n}\to X_{\mathbb{Q}}$ represents an element in
$G_{n}(|\varphi|)$. Then $|\varphi|\vee f$ factors as

$X\vee S^{n}\to X\times S^{n}\stackrel{{\scriptstyle
g}}{{\to}}X_{\mathbb{Q}}.$

Restricting $g_{\mathbb{Q}}$ to $X_{\mathbb{Q}}\times S^{n}$, and denoting by
$r:(X_{\mathbb{Q}})_{\mathbb{Q}}\to X_{\mathbb{Q}}$ the retraction
(Proposition 8.1(iii)), then gives the commutative diagram,

$\textstyle{X\vee
S^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\times
S^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{X_{\mathbb{Q}}}$$\textstyle{X_{\mathbb{Q}}\vee
S^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{\mathbb{Q}}\times
S^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{\mathbb{Q}}}$$\textstyle{(X_{\mathbb{Q}})_{\mathbb{Q}}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$

This gives $[f]\in G_{n}(X_{\mathbb{Q}})$.

(iii) This is immediate from the definitions and Proposition 9.6(ii).
$\square$

Recall next ([19, §3.3]) that a _minimal Sullivan extension_ is a morphism,

$\land W\to\land W\otimes\land V,\hskip 28.45274ptw\mapsto w\otimes 1,$

of Sullivan algebras in which $\land W$ and the quotient $(\land
V,d_{V})=\mathbb{Q}\otimes_{\land W}(\land W\otimes\land V)$ are minimal
Sullivan algebras. However the differential $d$ in $\land W\otimes\land V$ may
have a component $d_{0}$ of wedge degree $0$: $d_{0}:W\leftarrow V$. The dual
of $d_{0}$ is then a linear map

$\delta:W^{\vee}\to V^{\vee}$

of degree $-1$. Recall also that $V^{\vee}=\mbox{Hom}(V,\mathbb{Q})\supset
G_{*}(\land V)$.

######  Proposition 9.8.

With the notation and hypotheses above $\delta:W^{\vee}\to V^{\vee}$ is
injective, and

$\mbox{Im}\,\delta\subset G_{*}(\land V).$

In particular Im$\,\delta$ suspends to a subspace in the center of $L_{V}$.

proof. Fix $n$ and choose a cdga retraction

$\xi:(\land W,d)\to(\mathbb{Q}\oplus d_{0}V^{n},0)$

with $\land^{\geq 2}W\subset\mbox{ker}\,\xi$. Then extend $\mathbb{Q}\oplus
d_{0}V^{n}$ to an acyclic cdga

$\land U\otimes(\mathbb{Q}\oplus
d_{0}V^{n})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathbb{Q}$

with $U=U^{\geq n}$, and

$d:U^{n}\stackrel{{\scriptstyle\cong}}{{\to}}d_{0}V^{n}\hskip
14.22636pt\mbox{and }d:U^{>n}\to\land^{\geq 1}U\otimes d_{0}V^{n}.$

Now applying $-\otimes_{\land W}(\land W\oplus d_{0}V)$ defines a surjective
quasi-isomorphism

$\rho:\land U\otimes(\mathbb{Q}\oplus d_{0}V^{n})\otimes\land
V\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(\land V,d_{V}).$

Thus $\rho$ has a left inverse $\sigma:(\land V,d_{V})\to\land
U\otimes(\mathbb{Q}\oplus d_{0}V^{n})\otimes\land V$, and it is immediate from
the definitions that $\sigma$ may be chosen so that $\sigma v=1\otimes v$,
$v\in V^{<n}$ and

$\sigma v=1\otimes v-\alpha(d_{0}v),$

where $\alpha$ is inverse to the isomorphism
$U^{n}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}d_{0}V^{n}$. Division
by $d_{0}V^{n}$ and by $U^{>n}$ then yields the sequence of morphisms

$(\land V,d_{V})\to(\land U^{n},0)\otimes(\land V,d_{V})\to(\land
U^{n},0)\times_{\mathbb{Q}}(\land V,d_{V}).$

In particular, any morphism $\land U^{n}\to H(S^{n})$ induces a sequence

$(\land V,d_{V})\to H(S^{n})\otimes(\land V,d_{V})\to
H(S^{n})\times_{\mathbb{Q}}(\land V,d_{V}).$

In particular, the resulting morphism $(\land V,d_{V})\to H(S^{n})$ is an
element of $G_{n}(\land V)$.

Finally, this element of $G_{n}(\land V)$ is determined by the composite
$V^{n}\to U^{n}\to H^{n}(S^{n})$, and by construction the map $V^{n}\to U^{n}$
is given by $v\mapsto-\alpha(d_{0}v)$. This gives the commutative triangles

|
---|---
$\textstyle{U^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{d_{0}V^{n}}$$\textstyle{V^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}}$ and  |
---|---
$\textstyle{(U_{n})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(d_{0}V^{n})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{(V^{n})^{\vee},}$

This identifies the elements in Im$\,\delta$ as elements in $G_{n}(\land V)$
and, since $V^{n}\to d_{0}V^{n}$ is surjective, $\delta$ is injective.

$\square$

######  Proposition 9.9.

If $\land V$ is a minimal Sullivan algebra and cat$(\land V)<\infty$, then

  1. (i)

$G_{2n}(\land V)=0$, $n\geq 1$, and

  2. (ii)

dim $G_{*}(\land V)\leq$ cat$(\land V)$.

proof. (i) Let $B=\land a/(a^{2})$, with deg$\,a=2n$. We suppose that there
exists a morphism $\varphi:\land V\to B$ that is a representative of an
element of $G_{2n}(\land V)$. If $\varphi\neq 0$ there is an element $z_{0}\in
V^{2n}$ with $\varphi(z_{0})=a$.

Now let $(\land W,d):=\mathbb{Q}\otimes_{\land V^{<2n}}(\land V,d)$. By ([19,
Proposition 9.3]) cat$\,(\land W)\leq$ cat$(\land V)<\infty$. We denote by
$\overline{\varphi}:\land W\to B$ the map induced by $\varphi$. Since
$\overline{\varphi}$ is also in $G_{n}(\land W)$, we get a morphism

$\psi:\land W\to\land W\otimes B$

with $\psi(w)-w\in\land W\otimes B^{+}$ and $\psi(z_{0})=z_{0}+a.$ By
iteration this gives

$\psi(q):\land W\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}\land W\otimes
B\stackrel{{\scriptstyle\psi\otimes id}}{{\longrightarrow}}\land W\otimes
B\otimes B\to\dots\to\land W\otimes B^{\otimes^{q}}.$

In particular, $\psi(q)(z_{0}^{q})-a^{\otimes^{q}}\in\land^{+}W\otimes
B^{\otimes^{q}}$. By construction $d(z_{0})=0$. Since cat$\land W<\infty$,
there is $y\in\land W$ with $dy=z_{0}^{q}$. It follows that
$\psi(q)(z_{0}^{q})$ is a boundary which is impossible. This shows that
$\varphi=0$.

(ii). Decompose $\land V$ as a Sullivan extension $\land V=\land U^{\leq
n}\otimes\land Z^{\geq n}$ in which $G_{n}(\land V)\to G_{n}(\land Z)$ is
injective and $<v,x>=1$ for some $x\in G_{n}(\land Z)$ and some cycle $v\in
Z^{n}$. Since ([19, Proposition 9.3]) cat$(\land Z)\leq$ cat$(\land V)$, it is
sufficient to show that dim$\,G_{*}(\land Z)\leq$ cat$(\land Z)$. In other
words, without loss of generality we may suppose $V=V^{\geq n}$ and that $x\in
G_{n}(\land V)$ and $v$ is a cycle in $V^{n}$. Since (i) implies that $n$ is
odd we also have

$H(S^{n})=\land\widehat{v}\cong\land v.$

Moreover, since $x\in G_{n}(\land V)$, by definition we obtain the sequence of
morphisms

$\land
V\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}\land\widehat{v}\otimes\land
V\longrightarrow\land\widehat{v}\times_{\mathbb{Q}}\land V$

induced by $x$.

Now set $W=\\{w\in V\,|,<w,x>=0\\}$. Then $W=\cup_{k\geq 1}W(k)$ defined by
$W(1)=W\cap\mbox{ker}\,d$ and $W(k+1)=W\cap d^{-1}(\land v\otimes\land W(k))$.
Suppose next, by induction on $k$ that $\land W(k)$ is preserved by $d$ and
that

$\varphi w-w\in\widehat{v}\land v\otimes\land W(k),\hskip 28.45274ptw\in
W(k).$

iC1-. Finally, fix a basis element $z$ in a direct summand of $W(k)$ in
$W(k+1)$. Then by definition,

$dz=v\Psi_{1}+\Psi_{2}\hskip 14.22636pt\mbox{and }\varphi
z=\widehat{v}\land(v\Phi_{1}+\Phi_{2})+z,$

where $\Phi_{1},\Phi_{2},\Psi_{1}$ and $\Psi_{2}\in\land W(k)$. This yields

$d\varphi z=\widehat{v}\,v\,d\Phi_{1}-\widehat{v}\,d\Phi_{2}+dz$ $\varphi
dz=(\widehat{v}+v)\varphi\Psi_{1}+\varphi\Psi_{2}=v\,\Psi_{1}+\Psi_{2}+\widehat{v}\,\Psi_{1}+\widehat{v}\,v\,\Gamma,$

with $\Gamma\in\land W(k)$. Since $\varphi dz=d\varphi z$ and
$dz=v\Psi_{1}+\Psi_{2}$, it follows that

$\widehat{v}v\,d\Phi_{1}-\widehat{v}\,d\Phi_{2}=\widehat{v}\,\Psi_{1}+\widehat{v}v\Gamma.$

In other words, $\Psi_{1}=-d\Phi_{2}$.

Now extend the splitting to $W(k+1)$ by replacing $z$ by $z-v\Phi_{2}$. It is
straightforward to check that our conditions are satisfied. Thus by induction
we may write $\land V=\land v\otimes\land W$ with $\land W$ preserved by $d$.
It follows that $G_{*}(\land V)=\mathbb{Q}x\oplus G_{*}(\land W)$. Since by
([19, Corollary 9.3]), cat$(\land W)=$ cat$(\land V)-1$, (ii) follows by
induction on cat$(\land V)$. $\square$

###  10 Wedges of topological spaces

####  10.1 The minimal Sullivan model of $X\vee Y$

If $X$ and $Y$ are connected spaces then the surjections $H(X\vee Y)\to
H(X),H(Y)$ define an isomorphism $H(X\vee
Y)\stackrel{{\scriptstyle\cong}}{{\to}}H(X)\times_{\mathbb{Q}}H(Y)$. It
follows that Sullivan models $\land
W\stackrel{{\scriptstyle\simeq}}{{\to}}A_{PL}(X)$ and $\land
Q\stackrel{{\scriptstyle\simeq}}{{\to}}A_{PL}(Y)$ extend to a quasi-
isomorphism

$\land W\times_{\mathbb{Q}}\land
Q\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}A_{PL}(X\vee Y).$

This identifies a Sullivan model $\land
T\stackrel{{\scriptstyle\simeq}}{{\to}}\land W\times_{\mathbb{Q}}\land Q$ as a
Sullivan model for $X\vee Y$.

On the other hand, for any two Sullivan algebras $\land W$ and $\land Q$ the
surjection $\land W\otimes\land Q\to\land W\times_{\mathbb{Q}}\land Q$ extends
to a quasi-isomorphism

$\varphi:\land T:=\land W\otimes\land Q\otimes\land
R\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\land
W\times_{\mathbb{Q}}\land Q$

from a minimal Sullivan extension. Since $\land W\otimes\land Q\to\land
W\times_{\mathbb{Q}}\land Q$ is surjective we may choose $R\subset T$ so that
$\varphi(R)=0$. It follows that $\land T$ is a Sullivan algebra, minimal if
$\land W$ and $\land Z$ are minimal, and that in any case the quotient $(\land
R,\overline{d})$ is a minimal Sullivan algebra.

######  Proposition 10.1.

With the notation and hypotheses above,

  1. (i)

With appropriate choice of generating space $R\subset\land R$,

$R\cap\mbox{ker}\,\overline{d}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}H^{\geq
1}(\land R,\overline{d}).$

In particular (Proposition 8.4) the homotopy Lie algebra $L_{R}$ is profree.

  2. (ii)

If $\land W$ and $\land Q$ are quadratic Sullivan algebras then $\land T$ is
quadratic.

  3. (iii)

In general, the quadratic Sullivan algebra $(\land T,d_{1})$ for $\land T$ is
the Sullivan model of $(\land W,d_{1})\times_{\mathbb{Q}}(\land Q,d_{1})$.

Remark. This result is analogous to the fact that the usual fibre of the
injection $X\vee Y\to X\times Y$ is the join of $\Omega X$ and $\Omega Y$ and
thus a suspension. (But note that $(X\vee Y)_{\mathbb{Q}}$ may be different
from $X_{\mathbb{Q}}\vee Y_{\mathbb{Q}}$.)

proof. (i) Let $\land W\otimes\land U_{W}$ and $\land Q\otimes\land U_{Q}$
denote the respective acyclic closures. These, together with $\varphi:\land
T\stackrel{{\scriptstyle\simeq}}{{\land}}W\times_{\mathbb{Q}}\land Q$, yield
the quasi-isomorphisms

$\land R\stackrel{{\scriptstyle\simeq}}{{\longleftarrow}}\land T\otimes_{\land
W\otimes\land Q}(\land W\otimes\land U_{W}\otimes\land Q\otimes\land
U_{Q})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}A:=(\land
W\times_{\mathbb{Q}}\land Q)\otimes\land U_{W}\otimes\land U_{Q}.$

Now divide $A$ by the ideal generated by $W$ to obtain the short exact
sequence

$0\to\land^{\geq 1}W\otimes\land U_{W}\otimes\land U_{Q}\to A\to\land
Q\otimes\land U_{W}\otimes\land U_{Q}\to 0.$

Next, decompose the differential in $\land W\otimes\land U_{W}$ in the form
$d=d_{1}+d^{\prime}$ with $d_{1}(W)\subset\land^{2}W$, $d_{1}(U_{W})\subset
W\otimes\land U_{W}$, $d^{\prime}(W)\subset\land^{\geq 3}W$ and
$d^{\prime}(U_{W})\subset\land^{\geq 2}W\otimes\land U_{W}$. Then $d_{1}$ is a
differential and $(\land W\otimes\land U_{W},d_{1})$ is the acyclic closure of
$(\land W,d_{1})$. Choose a direct summand, $S$, of $d_{1}(\land^{\geq
1}U_{W})$ in $W\otimes\land U_{W}$. Then $I=(\land^{\geq 2}W\otimes\land
U_{W})\oplus S$ is acyclic for the differential $d_{1}$ and therefore also for
the differential $d$. Thus $J=I\otimes\land U_{Q}$ is an acyclic ideal in $A$
and $A\stackrel{{\scriptstyle\cong}}{{\to}}A/J$.

On the other hand, consider the short exact sequence

$0\to d(\land U_{W})\otimes\land U_{Q}\to A/J\to\land Q\otimes\land
U_{W}\otimes\land U_{Q}\to 0$

in which $d(\land U_{W})\otimes\land U_{Q}$ is an ideal with trivial
multiplication and trivial differential. It follows from the long homology
sequence that

$\mathbb{Q}\oplus d(\land U_{W})\otimes\land^{\geq
1}U_{Q}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}A/J.$

Since $\land R\simeq A\simeq A/J$, this completes the proof of (i).

(ii) Assign $\land W$ and $\land Q$ wedge degree as a second degree and assign
$U_{W}$ and $U_{Q}$ second degree $0$. Then $(\land W\otimes\land
U_{W},d_{1})$ and $(\land Q\otimes\land U_{Q},d_{1})$ are the respective
acyclic closures of $(\land W,d_{1})$ and $(\land Q,d_{1})$, and $d_{1}$
increases the second degree by 1. Now ${\varphi}$ and ${T}$ may be constructed
so that ${R}$ is equipped with a second gradation for which $d$ increases the
second degree by one and ${\varphi}$ is bihomogeneous of degree zero.

The argument in the proof of (i) now yields a sequence of bihomogeneous quasi-
isomorphisms connecting

$\mathbb{Q}\oplus\left(d_{1}(\land^{+}U_{W})\otimes\land^{+}U_{Q}\right)\simeq\land{R}.$

Thus $H^{\geq 1}(\land{R})$ is concentrated in second degree 1. Denote by
$S\subset R$ the subspace of elements of second degree 1. Then $(\land S,d)$
is a quadratic Sullivan algebra. Moreover by (i), any cycle
$\Phi\in\land^{2}S$ is a boundary, necessarily of an element $v\in S$ of wedge
degree 1. Thus $H(\land S)$ has no homology of wedge degree 2. Now Lemma 6.2
implies that $S\cap\mbox{ker}\,d=H^{\geq 1}(\land S)$. But if $S\neq R$ then
for some $z\in R$, $dz\in\land S$. This implies that for some $\Psi\in\land
S$, $z-d\Psi$ is a cycle and therefore is in $S$. This gives $S=R$. Thus
$\land T$ is quadratic and $\varphi$ preserves wedge degrees.

(iii) Denote by $d=\sum_{i\geq 1}d_{i}$ the differentials in $\land W$ and
$\land Q$, where $d_{i}$ increases the wedge degree by $i$. Then (ii)
constructs a quasi-isomorphism

$\varphi:(\land T,d_{1})=(\land W\otimes\land Q\otimes\land
R,d_{1})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(\land
W,d_{1})\times_{\mathbb{Q}}(\land Q,d_{1})$

in which $\varphi$ is surjective and preserves wedge degree. In particular,

$H(\mbox{Ker}\,\varphi,d_{1})=0.$

Now we deform $d_{1}$ to a differential $d=\sum_{i\geq 1}d_{i}$ in $\land T$
with the following properties

  1. $\bullet$

$d$ restricts to the original differentials in $\land W$ and $\land Q$.

  2. $\bullet$

$d_{i}:R\to\mbox{Ker}\,\varphi$, $i\geq 2$.

Write $R=\cup_{\neq\geq 0}R_{n}$, with $R_{0}=R\cap d_{1}^{-1}(\land
W\otimes\land Q)$ and for $n>0$,

$R_{n}=R\cap d_{1}^{-1}(\land W\otimes\land Q\otimes\land R_{n-1}).$

We define the derivations $d_{2},\dots,d_{q},...$ on $R$ by induction on $n$
with the property that

$d_{i}(R)\subset(\land^{+}R\otimes\land(W\otimes\land
Q))\,\cap\,\land^{i+1}(R\oplus W\oplus Q)$

and $(d_{1}+\sum_{j\geq 2}d_{j})^{2}=0$ on $R_{n}$.

So suppose this is done on $R_{n-1}$ and $z\in R_{n}$. We construct the
$d_{p}$ by induction on $p$. So suppose $d_{2},\dots,d_{p-1}$ constructed such
that for $2\leq q\leq p-1$, we have

$(d_{1}d_{q}+d_{2}d_{q-1}+\dots+d_{q}d_{1})z=0.$

Then let $R(z)=(d_{p}d_{1}+d_{p-1}d_{2}+\dots+d_{2}d_{p-1})(z)$. Since
$d_{1}R(z)=0$ and $R(z)\in\mbox{Ker}\,\varphi$, there is an element
$u\in\mbox{Ker}\,\varphi$ with $R(z)=-d_{1}(u)$. We define $d_{p}(x)=u$. It
follows by induction on $p$ that $d_{p}$ increases wedge degree by $p$. Thus
$d_{p}z=0$ if $p>n$. We set $d=\sum_{i\geq 1}d_{i}$.

It is immediate from the construction that $d^{2}=0$ and that the two required
properties are satisfied. In particular $(\land T,d)$ is a Sullivan algebra
and since $\varphi$ is a quasi-isomorphism with respect to $d_{1}$,

$\varphi:(\land T,d)\to(\land W,d)\times_{\mathbb{Q}}(\land Q,d)$

is a Sullivan model. Finally, the associated quadratic Sullivan algebra is by
(ii) the Sullivan model of $(\land W,d_{1})\times_{\mathbb{Q}}(\land
Q,d_{1})$.

$\square$

Example 1. If $\land W$ and $\land Q$ are minimal Sullivan algebras then
(Proposition 10.1) the surjection $\land W\otimes\land Q\to\land
W\times_{\mathbb{Q}}\land Q$ extends to a quasi-isomorphism

$\land T=\land W\otimes\land Q\otimes\land
R\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\land
W\times_{\mathbb{Q}}\land Q$

in which $\land T$ is a minimal Sullivan algebra and the homotopy Lie algebra,
$L_{R}$, is profree. Thus $\langle\land R\rangle$ is rationally wedge-like.

2. If one of the spaces $X$ and $Y$ has rational homology of finite type then the homotopy fibre of $(X\vee Y)_{\mathbb{Q}}\to(X\times Y)_{\mathbb{Q}}$ is rationally wedge-like.

In fact, in this case if $\land W$ and $\land Q$ are respectively the minimal
Sullivan models of $X$ and $Y$ then the inclusion $\land W\otimes\land
Q\to\land T$ is a Sullivan representative for the inclusion $i:X\vee Y\to
X\times Y$. This identifies $i_{\mathbb{Q}}$ as the map $\langle\land
W\otimes\land Q\rangle\leftarrow\langle\land T\rangle$ and it follows from
([17, Proposition 17.9]) that its homotopy fibre is $\langle\land R\rangle$.

####  10.2 Free products and the homotopy Lie algebra of $X\vee Y$

The category of enriched Lie algebras has free products. In fact the classical
construction of the free product, $L\amalg L^{\prime}$, of two graded Lie
algebras extends naturally to enriched Lie algebras, $(L,\\{I_{\alpha}\\})$
and $(L^{\prime},\\{I^{\prime}_{\beta}\\})$, with the enriched structure given
by the surjections

$\xi_{\alpha,\beta,n}:L\,\amalg\,L^{\prime}\to
L_{\alpha}\,\amalg\,L^{\prime}_{\beta}\,/\,(L_{\alpha}\,\amalg\,L^{\prime}_{\beta})^{n}.$

Its completion will be denoted by $L\,\widehat{\amalg}\,L^{\prime}$, and
almost by definition,

$L\,\widehat{\amalg}\,L^{\prime}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\overline{L}\,\widehat{\amalg}\,\overline{L^{\prime}}.$

Definition. $L\,\widehat{\amalg}\,L^{\prime}$ is the _free product_ of $L$ and
$L^{\prime}$.

Remarks. 1. As in the classical case $L\,\widehat{\amalg}\,L^{\prime}$ is
equipped with inclusions
$\textstyle{L,L^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\,\widehat{\amalg}\,L^{\prime}}$,
and here if $L$ and $L^{\prime}$ are complete the images are closed sub Lie
algebras. Moreover these inclusions extend to a _linear_ inclusion

$\textstyle{L\oplus
L^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\,\widehat{\amalg}\,L^{\prime}}$

into a closed subspace of the free product.

2. If $C\subset L$ and $C^{\prime}\subset L^{\prime}$ are closed subspaces, we denote by

$C\oplus C^{\prime}\subset C\,\widehat{\amalg}\,C^{\prime}\subset
L\,\widehat{\amalg}\,L^{\prime}$

the closure of the subspace of iterated Lie brackets of sequences of elements
alternating between $C$ and $C^{\prime}$.

######  Lemma 10.1.

(i). With the notation above, any two morphisms $f:(L,\\{I_{\alpha}\\})\to E$
and $g:(L^{\prime},\\{I_{\beta}^{\prime}\\})\to E$ into a complete enriched
Lie algebra extend uniquely to a morphism

$L\,\widehat{\amalg}\,L^{\prime}\to E.$

This identifies $L\,\widehat{\amalg}\,L^{\prime}$ as a coproduct in the
category of complete Lie algebras, and characterizes
$L\,\widehat{\amalg}\,L^{\prime}$ up to natural isomorphism.

(ii). Suppose a profree Lie algebra $L=\overline{\mathbb{L}}_{T}$, and
$T=T(1)\oplus T(2)$ with $T(1)$ and $T(2)$ closed subspaces. Then
$\overline{\mathbb{L}}_{T(1)}$ and $\overline{\mathbb{L}}_{(T(2)}$ are the
closures of the Lie algebras generated by $T(1)$ and $T(2)$, and

$\overline{\mathbb{L}}_{T(1)}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T(2)}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\overline{\mathbb{L}}_{T}.$

proof. (i) Let $\\{J_{\gamma}\\}$ denote the enriched structure for $E$. Then
for each $\gamma$ there are indices $\alpha(\gamma),\beta(\gamma)$ such that
$f$ and $g$ factor to yield morphisms

$f_{\gamma}:L_{\alpha(\gamma)}\to E_{\gamma}\hskip 14.22636pt\mbox{and
}g_{\gamma}:L^{\prime}_{\beta(\gamma)}\to E_{\gamma}.$

Moreover, since $E_{\gamma}^{n(\gamma)}=0$, some $n(\gamma)$,
$f_{\gamma}\amalg g_{\gamma}$ factors to yield a morphism

$h_{\gamma}:(L_{\alpha(\gamma)}\amalg
L^{\prime}_{\beta(\gamma)})/(L_{\alpha(\gamma)}\amalg
L^{\prime}_{\beta(\gamma)})^{n}\longrightarrow E_{\gamma}.$

Since $E$ is complete these extend to

$h:=\varprojlim_{\gamma}h_{\gamma}:L\,\widehat{\amalg}\,L^{\prime}\to E.$

The uniqueness is immediate.

(ii) The inclusion $T(1)$, $T(2)\to T$ extend to a morphism
$\overline{\mathbb{L}}_{T(1)}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T(2)}\to\overline{\mathbb{L}}_{T}$,
with inverse given by the extension to $\overline{\mathbb{L}}_{T}$ of
$T=T(1)\oplus
T(2)\to\overline{\mathbb{L}}_{T(1)}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T(2)}$.
$\square$

######  Proposition 10.2.

Denote by $L_{X}$ and $L_{Y}$ the homotopy Lie algebras of connected spaces
$X$ and $Y$. Then the inclusions $X,Y\to X\vee Y$ induce an isomorphism

$L_{X\vee
Y}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}L_{X}\,\widehat{\amalg}\,L_{Y}.$

In particular, the correspondence $X\mapsto L_{X}$ preserves coproducts.

Proposition 10.2 is an immediate consequence of the explicit description in
Proposition 10.3 below of the homotopy Lie algebra of a fibre product $\land
W\times_{\mathbb{Q}}\land Q$ of any two minimal Sullivan algebras. Recall from
Proposition 10.1 that the natural surjection $\land W\otimes\land Q\to\land
W\times_{\mathbb{Q}}\land Q$ extends to a minimal Sullivan model

$\varphi:\land T:=\land W\otimes\land Q\otimes\land
R\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\land
W\times_{\mathbb{Q}}\land Q.$

Denote the corresponding homotopy Lie algebras (§8) by $L_{W},L_{Q},L_{T}$ and
$L_{R}$. In particular, this identifies $L_{T}$ as the homotopy Lie algebra of
a minimal Sullivan model of $\land W\times_{\mathbb{Q}}\land Q$. The
inclusions $\land W,\land Q\to\land W\times_{\mathbb{Q}}\land Q$ then lift to
unique based homotopy classes of morphisms $\land W,\land Q\to\land T$, and
thus induce morphisms $L_{T}\to L_{W},L_{Q}$. These provide a surjection
$L_{T}\to L_{W}\times L_{Q}$.

######  Proposition 10.3.

With the hypotheses and notation above,

  1. (i)

$\varphi$ induces a quasi-isomorphism $(\land T,d_{1})\to(\land
W,d_{1})\times_{\mathbb{Q}}(\land Q,d_{1})$ of associated quadratic Sullivan
algebras.

  2. (ii)

The surjection $L_{T}\to L_{W}\times L_{Q}$ factors as

$\textstyle{L_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\sigma}$$\textstyle{L_{W}\,\widehat{\amalg}\,L_{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{W}\times
L_{Q},}$

and $\sigma$ is natural with respect to morphisms $\land W\to\land W^{\prime}$
and $\land Q\to\land Q^{\prime}$.

  3. (iii)

If $f:L\to L^{\prime}$ and $g:E\to E^{\prime}$ are morphisms of complete
enriched Lie algebras, then

$f\,\widehat{\amalg}\,g:L\,\widehat{\amalg}\,E\to
L^{\prime}\,\widehat{\amalg}\,E^{\prime}$

is surjective if and only if $f$ and $g$ are surjective.

Corollary. If $L$ and $L^{\prime}$ are complete enriched Lie algebras then the
kernel of

$L\,\widehat{\amalg}\,L^{\prime}\to L\times L^{\prime}$

is profree. Moreover, if $L$ and $L^{\prime}$ are both profree then
$L\,\widehat{\amalg}\,L^{\prime}$ is also profree.

proof of the Corollary. Apply Proposition 10.1 to identify the kernel as a
profree Lie algebra $L_{R}$. If now $L$ and $L^{\prime}$ are profree then by
Proposition 8.4 there are quasi-isomorphisms

$\land W\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathbb{Q}\oplus
S\hskip 14.22636pt\mbox{and }\land
W^{\prime}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathbb{Q}\oplus
S^{\prime}$

with $S\cdot S=0=S^{\prime}\cdot S^{\prime}$, and vanishing differentials in
$S$ and $S^{\prime}$. Thus $\land W\times_{\mathbb{Q}}\land
W^{\prime}\simeq\mathbb{Q}\oplus S\oplus S^{\prime}$, and its homotopy Lie
algebra, $L\,\widehat{\amalg}\,L^{\prime}$, is profree by Proposition 8.4.

$\square$

proof of Proposition 10.3. (i) As recalled at the start of §8, $L_{T},L_{W}$
and $L_{Q}$ depend only on the quadratic Sullivan algebras $(\land T,d_{1})$,
$(\land W,d_{1})$ and $(\land Q,d_{1})$. Moreover, by Proposition 10.1,

$(\land T,d_{1})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(\land
W,d_{1})\times_{\mathbb{Q}}(\land Q,d_{1}).$

Therefore for any quadratic Sullivan algebra $(\land V,d_{1})$, the based
homotopy classes of morphisms satisfy

$\begin{array}[]{ll}[(\land V,d_{1}),(\land T,d_{1})]_{*}&=[(\land
V,d_{1}),(\land W,d_{1})\times_{\mathbb{Q}}(\land Q,d_{1})]*\\\ &=[(\land
V,d_{1}),(\land W,d_{1})]_{*}\times[(\land V,d_{1}),(\land
Q,d_{1})]*.\end{array}$

(ii) Proposition 4.1 translates this equality to give

${\mathcal{C}}(L_{T},L_{V})={\mathcal{C}}(L_{W},L_{V})\times{\mathcal{C}}(L_{Q},L_{V}),$

where ${\mathcal{C}}(-,-)$ denotes the set of morphisms in the category of
enriched Lie algebras. This identifies $L_{T}=L_{W}\,\widehat{\amalg}\,L_{Q}$.
The naturality is a standard diagram chasing.

(iii) To show that if $f$ and $g$ are surjective then $f\,\widehat{\amalg}\,g$
is surjective, it is sufficient to consider the case
$g=id_{E}:E\stackrel{{\scriptstyle=}}{{\longrightarrow}}E$. In this case we
have the row-exact commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{L\,\widehat{\amalg}\,E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\,\widehat{\amalg}\,id_{E}}$$\textstyle{L\times
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\times
id_{E}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{\prime}\,\widehat{\amalg}\,E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{\prime}\times
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

and so it is also sufficient to show that $h$ is surjective.

Now let $\land W$, $\land W^{\prime}$ and $\land Q$ be, respectively the
quadratic models of $L$, $L^{\prime}$ and $E$, and let $\varphi:\land
W^{\prime}\to\land W$ be the morphism corresponding to $f$. Then the
restriction $\varphi:W^{\prime}\to W$ dualizes to $f$, and so $\varphi$ is
injective. In particular, $\varphi$ extends to an inclusion

$\varphi:\land W^{\prime}\otimes\land U_{W^{\prime}}\to\land W\otimes\land
U_{W}$

of the acyclic closures, restricting to an inclusion $U_{W^{\prime}}\to
U_{W}$.

On the other hand, the diagram above is the diagram of homotopy Lie algebras
associated with a commutative diagram,

$\textstyle{\land W\otimes\land
Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\land
W\otimes\land Q\otimes\land
R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\land
R}$$\textstyle{\land W^{\prime}\otimes\land
Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi\otimes
id}$$\textstyle{\land W^{\prime}\otimes\land Q\otimes\land
R^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\land
R^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{\sigma}$

which identifies $L_{\sigma}$ with $h$.

Moreover, the computations in the proof of Proposition 10.1 identify

$H(\sigma)=\varphi:\mathbb{Q}\oplus d(\land U_{W^{\prime}})\to\mathbb{Q}\oplus
d(\land U_{W}).$

It follows that $H(\sigma)$ is injective. But since $L_{R}$ and
$L_{R^{\prime}}$ are profree, Lemma 4.1 identifies

$L_{R}/L_{R}^{(2)}=H^{\geq 1}(\land R)^{\vee}\hskip 14.22636pt\mbox{and
}L_{R^{\prime}}/L_{R^{\prime}}^{(2)}=H^{\geq 1}(\land R^{\prime})^{\vee}.$

It follows that $L_{\sigma}:L_{R}\to L_{R^{\prime}}$ induces a surjection
$L_{R}/L_{R}^{(2)}\to L_{R^{\prime}}/L_{R^{\prime}}^{(2)}$. Now Lemma 6.1 (i)
implies that $L_{\sigma}$ is surjective.

In the reverse direction note that the maps $id_{L}$ and $E\to 0$ define a
retraction $L\,\widehat{\amalg}\,E\to L$. The corresponding retraction
$L^{\prime}\,\widehat{\amalg}\,E^{\prime}\to L^{\prime}$ fits in the
commutative diagram,

$\textstyle{L\,\widehat{\amalg}\,E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\,\widehat{\amalg}\,g}$$\textstyle{L^{\prime}\,\widehat{\amalg}\,E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{L^{\prime}.}$

Therefore if $f\,\widehat{\amalg}\,g$ is surjective so is $f$ and, similarly,
so is $g$.

$\square$

Example. Free products of completions of weighted Lie algebras

Let $E$ and $E^{\prime}$ be weighted enriched Lie algebras with enriched
structures given by morphisms $E\to E_{\alpha}$ and $E^{\prime}\to
E_{\beta}^{\prime}$. Denote by $L$ and $L^{\prime}$ the completions of $E$ and
$E^{\prime}$. The weight decompositions in $E$ and $E^{\prime}$ then extend in
the standard way to a weight decomposition in $E\,\amalg\,E^{\prime}$. It then
follows from §2.1 that $L\,\widehat{\amalg}\,L^{\prime}$ is the completion of
$E\,\amalg\,E^{\prime}$.

####  10.3 Limits and colimits of complete enriched Lie algebras

######  Proposition 10.4.

The category $\overline{\mathcal{C}}$ of complete enriched Lie algebras is
complete and cocomplete : i.e., each diagram of complete enriched Lie algebras
has a direct and an inverse limit.

proof. The construction of the coproduct of two complete enriched Lie algebras
can be generalized to any family $\\{L(i)\\}_{i\in I}$ of complete enriched
Lie algebras. In fact, denote by $A_{i}$ the family of ideals defining the
enriched structure in $L(i)$. For each finite subset $J\subset I$, we denote
by $p_{J}:\amalg_{i\in I}L(i)\to\amalg_{i\in J}L(i)$ the projection defined by
$p_{J}(L(i))=0$ for $i\not\in J$. Then for $J\subset I$, finite, for
$\alpha_{i}\in A_{i}$ for $i\in J$, and for an integer $n$, we denote by
$K_{J,(\alpha_{i}),n}$ the kernel of the projection

$\textstyle{\amalg_{i\in
I}L(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{J}}$$\textstyle{\amalg_{i\in
J}L(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\amalg_{i\in
J}L(i)_{\alpha_{i}}/(\amalg_{i\in J}L(i)_{\alpha_{i}})^{n}.}$

These ideals define an enriched structure on $\amalg_{i\in I}L(i)$ and its
completion $\overline{\amalg_{i\in I}L(i)}$ is the coproduct of the the $L(i)$
is the category of complete enriched Lie algebras.

Now let $(L(i),f_{j})$ be a diagram in $\overline{\mathcal{C}}$. Its colimit
in the category of Lie algebras is the quotient of $\amalg_{i\in I}L(i)$ by an
ideal $J$ associated to the morphisms $f_{j}$. Denote by $\iota:\amalg_{i\in
I}L(i)\to\overline{\amalg_{i\in I}L(i)}$ the natural injection and by
$\overline{J}$ the closure of the ideal generated by $\iota(J)$ in
$\overline{\amalg L(i)}$. Then by the Corollary to Lemma 2.2, the quotient
$\overline{\amalg L(i)}/\overline{J}$ inherits a complete enriched structure.
This is clearly the colimit of the diagram $(L(i),f_{j})$ in
$\overline{\mathcal{C}}$.

On the other hand the usual product of the complete enriched Lie algebras
$L(i)$ is complete for the family of the ideals which are the kernels of the
surjections

$\textstyle{\prod_{i\in
I}L(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{J}}$$\textstyle{\prod_{i\in
J}L(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\prod\rho(i)_{\alpha_{i}}}$$\textstyle{L(i)_{\alpha_{i}}}$

associated to the finite subsets $J\subset I$ and the surjections
$\rho(i)_{\alpha_{i}}:L(i)\to L(i)_{\alpha_{i}}.$

Now consider a diagram $(L(i),f_{j})$ in $\overline{\mathcal{C}}$. Its limit
in the category of Lie algebras is the sub Lie algebra $S\subset\prod L(i)$
consisting of families $a_{i}\in L(i)$ such that for any morphism
$f_{j}:L(i_{0})\to L(i_{1})$ we have $f_{j}(a_{i_{0}})=a_{i_{1}}$. Since for
any morphism $f:L(1)\to L(2)$ of complete enriched Lie algebras, $f$ is a
limit of morphisms $f_{\alpha}:L(1)_{\beta(\alpha)}\to L(2)_{\alpha}$, the sub
Lie algebra $S$ is closed and is the limit of the diagram in
$\overline{\mathcal{C}}$.

$\square$

####  10.4 Wedges of spheres

Suppose $\\{S^{n_{\alpha}}\\}_{\alpha\in\mathcal{S}}$ is a collection of
spheres in which $\mathcal{S}$ is a linearly ordered set and each
$n_{\alpha}\geq 1$. For each finite subset $\sigma\subset{\mathcal{S}}$ we
write $\sigma=\\{\sigma_{1},\dots,\sigma_{r}\\}$ with
$\sigma_{1}<\dots<\sigma_{r}$, and set $|\sigma|=r$. If $\sigma\subset\tau$,
so that $|\sigma|\leq|\tau|=q$, the inclusion defines an inclusion

$j_{\sigma,\tau}:S^{n_{\sigma_{1}}}\vee\dots\vee
S^{n_{\sigma_{r}}}\longrightarrow S^{n_{\tau_{1}}}\vee\dots\vee
S^{n_{\tau_{q}}}.$

Definition. A _wedge of spheres_ is a direct limit of the form

$X=\vee_{\alpha}S^{n_{\alpha}}:=\varinjlim_{\sigma_{1}<\dots<\sigma_{r}}\,\,S^{n_{\sigma_{1}}}\vee\dots\vee
S^{n_{\sigma_{r}}}.$

Remark. It is immediate from the definition that a wedge of spheres is a co-H-
space. Thus (Corollary 3 to Proposition 8.4) its minimal Sullivan model,
$\land V$, is quadratic and its homotopy Lie algebra is profree. Thus we may
assume $d:V\to\land^{2}V$ and write $V=S\oplus T$ with

$S=V\cap\mbox{ker}\,d\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}H^{\geq
1}(\land V).$

Now let ${\mathcal{S}}=\\{\alpha\\}$ index a linearly ordered basis
$\\{v_{\alpha}\\}$ of $S$. This then defines a wedge

$X=\vee_{\alpha}S^{n_{\alpha}}$

of spheres.

######  Proposition 10.5.

A minimal Sullivan algebra, $\land V$, is the Sullivan model of a wedge of
spheres, $X$, if and only if

  1. (i)

The homotopy Lie algebra, $L_{V}$, is profree, and

  2. (ii)

$H(\land V)$ is the dual of a graded vector space.

Moreover, if $H(\land V)$ is a graded vector space of finite type then $\land
V$ determines $X$ up to homeomorphism.

proof. As observed above a wedge of spheres, $X$, has a profree homotopy Lie
algebra. Moreover $H(\land V)=H(X)=H_{*}(X;\mathbb{Q})^{\vee}$.

In the reverse direction suppose $\land V$ satisfies (i) and (ii), so that
$L_{V}$ is profree and $H^{\geq 1}(\land V)=T^{\vee}$. Then by Proposition
8.4,

$\land V\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(\mathbb{Q}\oplus
T^{\vee},0).$

Choose a linearly ordered basis $\\{x_{i}\\}$ of $T$ and set
$n_{i}=\mbox{deg}\,x_{i}$ and ${\mathcal{S}}_{X}=\\{i\\}$. Then set

$X=\vee_{i\in{\mathcal{S}}_{X}}S^{n_{i}}_{i}.$

By construction, $H_{\geq 1}(X;\mathbb{Q})=T.$

Now let $\land W$ be the minimal Sullivan model of $X$. Since $X$ is a wedge
of spheres,

$\land W\simeq(\mathbb{Q}\oplus H^{\geq 1}(X),0)=(\mathbb{Q}\oplus
T^{\vee},0)=(\mathbb{Q}\oplus H^{\geq 1}(\land V),0)\simeq\land V.$

Since $\land W$ and $\land V$ are minimal Sullivan algebras, it follows that
$\land V\cong\land W$. Therefore $\land V$ is the minimal Sullivan model of
$X$.

Finally, suppose $H(\land V)$ is a graded vector space of finite type, and
that $\land V$ is the minimal Sullivan model of a second wedge of spheres $Y$.
Then $H^{\geq 1}(Y)$ and $H^{\geq 1}(X)$ are isomorphic vetor spaces and so
$H_{\geq 1}(Y;\mathbb{Q})\cong H_{\geq 1}(X;\mathbb{Q})$. Since $Y$ is a wedge
of spheres

$Y=\vee_{j\in{\mathcal{S}}_{Y}}S^{n_{j}}$

and the number of indices $j\in{\mathcal{S}}_{Y}$ with $n_{j}=n$ is
dim$\,H_{n}(Y;\mathbb{Q})=$ dim$\,H_{n}(X;\mathbb{Q})$. Therefore
${\mathcal{S}}_{X}$ and ${\mathcal{S}}_{Y}$ are isomorphic sets and $X\cong
Y$. $\square$

These constructions provide an explicit description of the morphism (§9),

$\pi_{*}|\varphi|:\pi_{*}(Y)\to\pi_{*}(Y_{\mathbb{Q}})$

when $Y=\vee_{\sigma}S^{n_{\sigma}}$ is a wedge of spheres of finite type.
When $Y$ is a wedge of circles, this extends diagrams (1) and (2) in §9.1.

######  Proposition 10.6.

With the hypotheses and notation above, denote by
$\sigma\in\pi_{n_{\sigma}}(S^{n_{\sigma}})$ the homotopy class represented by
id${}_{S^{n_{\sigma}}}$. Then the homotopy Lie algebra $L_{Y}$ of $Y$ can be
written $\overline{\mathbb{L}}_{T}$ in which $\pi_{*}|\varphi|$ maps the set
$\\{\sigma\\}$ bijectively to a basis of $T$.

proof. Here (Proposition 8.4) $L_{Y}=\overline{\mathbb{L}}_{T}$ with
$sT=(V\cap\mbox{ker}\,d_{1})^{\vee}$. Because $Y$ is a wedge of finite type,

$(V\cap\mbox{ker}\,d_{1})^{\vee}=H^{\geq 1}(Y)^{\vee}=H_{\geq
1}(Y;\mathbb{Q}).$

However, a basis of $H_{\geq 1}(Y;\mathbb{Q})$ is provided by the images of
the homotopy classes $\sigma$ under the Hurewicz homomorphism. Since the
Hurewicz homomorphism is natural, the Proposition follows from Proposition
9.3. $\square$

Definition. A space $X$ is _rationally wedge-like_ if it is homotopy
equivalent to a spatial realization $\langle\land V\rangle$ with the homotopy
Lie algebra of $\land V$ a profree Lie algebra.

Remark. It follows from Proposition 10.5 that if $\vee S^{n}_{\alpha}=X$ is a
wedge of spheres and $\land V$ is its minimal Sullivan model then

$(\vee S^{n}_{\alpha})_{\mathbb{Q}}=X_{\mathbb{Q}}=\langle\land V\rangle$

is rationally wedge-like.

###  11 Sullivan rational spaces

Let $\varphi_{X}:\land V\stackrel{{\scriptstyle\simeq}}{{\to}}A_{PL}(X)$ be a
minimal Sullivan model for a connected space, $X$, and let $L_{X}$ denote the
homotopy Lie algebra of $\land V$. Recall (§9) that the adjoint map

$|\varphi_{X}|:X\to\langle\land V\rangle:=X_{\mathbb{Q}}$

is called the _Sullivan completion_ of $X$.

Definition. A path connected space, $X$, is a _Sullivan rational space_ if the
Sullivan completion $|\varphi_{X}|:X\to X_{\mathbb{Q}}$ is a homotopy
equivalence.

Remarks. 1. The condition that $|\varphi_{X}|$ is a homotopy equivalence
implies that the minimal Sullivan model of $X$ directly computes both
$H(X_{\mathbb{Q}})$ and $\pi_{*}(X_{\mathbb{Q}})$.

2. If $X$ is Sullivan rational then $X_{\mathbb{Q}}\simeq(X_{\mathbb{Q}})_{\mathbb{Q}}$ and $X_{\mathbb{Q}}$ is Sullivan rational.

Recall that ([40]) Quillen’s rationalization for simply connected spaces
assigns to each such space a map $Y\to\mathbb{Q}(Y)$ which induces
isomorphisms

$H_{\geq 1}(Y)\otimes\mathbb{Q}\stackrel{{\scriptstyle\cong}}{{\to}}H_{\geq
1}(\mathbb{Q}(Y))\hskip 14.22636pt\mbox{and
}\pi_{*}(Y)\otimes\mathbb{Q}\stackrel{{\scriptstyle\cong}}{{\to}}\pi_{*}(\mathbb{Q}(Y)).$

In particular, $\mathbb{Q}(\mathbb{Q}(Y))=\mathbb{Q}(Y)$.

Sullivan’s completion, $X_{\mathbb{Q}}$, is analogous to Quillen’s
rationalization, and also to the Bousfield-Kan completion
$\mathbb{Q}_{\infty}(X)$ \- cf [7]. Moreover, if $H(X)$ has finite type then
$\mathbb{Q}_{\infty}(X)$ and $X_{\mathbb{Q}}$ are homotopy equivalent ([6]).
But it is not always true that every $X_{\mathbb{Q}}$ is Sullivan rational.
For instance ([28]), $H^{2}((S^{1}\vee S^{1})_{\mathbb{Q}})$ has uncountable
dimension, where as if $X_{\mathbb{Q}}$ is Sullivan rational by the Corollary
to Theorem 2 below, $H(X_{\mathbb{Q}})$ has finite type.

Remarks 1. If $\varphi_{X}:\land V\to A_{PL}(X)$ is a minimal Sullivan model
then $X$ is Sullivan rational if and only if each

$\pi_{k}|\varphi_{X}|:(V^{k})^{\vee}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\pi_{k}(X).$

2. Let $Y$ be a Sullivan rational space and let $X$ be a connected space. Since $f_{\mathbb{Q}}\circ\varphi_{X}=\varphi_{Y}\circ f$, and $\varphi_{Y}$ is a homotopy equivalence, it follows that $f\mapsto f_{\mathbb{Q}}$ yields an injection $[X,Y]\to[X_{\mathbb{Q}},Y_{\mathbb{Q}}].$

Now recall from §7 that if $\land V$ is a Sullivan algebra then adjoint to
$id_{\langle\land V\rangle}$ is a morphism

$\iota_{\land V}:\land V\to A_{PL}\langle\land V\rangle.$

######  Lemma 11.1.

Suppose for some minimal Sullivan algebra, $\land V$, that $\iota_{\land
V}:\land V\to A_{PL}\langle\land V\rangle$ is a quasi-isomorphism. Then

  1. (i)

$\langle\land V\rangle$ is Sullivan rational.

  2. (ii)

A basis of $V$ is at most countable.

  3. (iii)

$H(\land V)$ is a graded vector space of finite type.

proof. (i) By hypothesis $\iota_{\land V}$ is a Sullivan model for
$\langle\land V\rangle$. Since its adjoint is the homotopy equivalence
$id_{\langle\land V\rangle}$, $\langle\land V\rangle$ is Sullivan rational.

(ii). It follows from (17) that the surjection
$V^{\vee}\to(V\cap\mbox{ker}\,d)^{\vee}$ decomposes as the composite

$\textstyle{V^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$hur$\textstyle{H_{*}\langle\land
V\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(V\cap\mbox{ker}\,d)^{\vee}.}$

Thus $H_{*}\langle\land V\rangle\to(V\cap\mbox{ker}\,d)^{\vee}$ is surjective.
Dualizing gives injections $(V^{n}\cap\mbox{ker}\,d)^{\vee\vee}\to H^{n}(\land
V).$

Now assume by induction that $V^{<n}$ has an at most countable basis and let
$Z$ be the space of cycles in $(\land V)^{n}$. Then $Z\subset\land
V^{<n}\oplus V^{n}$, and division by $V^{n}\cap\mbox{ker}\,d$ gives an exact
sequence

$0\to V^{n}\cap\mbox{ker}\,d\to Z\to\land V^{<n}.$

Since $(V^{n}\cap\mbox{ker}\,d)^{\vee\vee}$ embeds in $H^{n}(\land V)$, it
follows that dim$\,V^{n}\cap\mbox{ker}\,d<\infty$.

On the other hand, $V^{n}=\cup_{p\geq 0}V^{n}_{p}$ with $V^{n}_{0}=V^{n}\cap
d^{-1}(\land V^{<n})$ and $V_{p+1}^{n}=V^{n}\cap d^{-1}(\land
V^{<n}\otimes\land V^{n}_{p})$. Since dim$\,V^{n}\cap\mbox{ker}\,d<\infty$, it
follows by induction on $p$ that each $V^{n}_{p}$, and hence $V^{n}$ itself,
has an at most countable basis.

(iii) It follows from (i) that each $H^{n}(\land V)$ has an at most countable
basis. But since $H^{n}(\land V)=H^{n}(\langle\land
V\rangle)=H_{n}\langle\land V\rangle^{\vee}$, this implies that each
dim$\,H^{n}(\land V)<\infty$. $\square$

######  Lemma 11.2.

Suppose $\land V$ is a minimal Sullivan algebra. If $V^{1}=0$ then the
following conditions are equivalent:

  1. (i)

$\iota_{\land V}:\land V\to A_{PL}\langle\land V\rangle$ is a quasi-
isomorphism.

  2. (ii)

Each dim$\,V^{k}<\infty$.

  3. (iii)

Each dim$\,H^{k}(\land V)<\infty.$

  4. (iv)

Each dim$\,\pi_{k}<\land V>$ is finite.

proof. (i) $\Leftrightarrow$ (ii). This is [19, Theorem 1.5]

(ii) $\Leftrightarrow$ (iii). This follows because $d:V^{k}\to\land V^{<k}$.

(ii) $\Leftrightarrow$ (iv) This is because $\pi_{k}=(V^{k})^{\vee}$.
$\square$

######  Lemma 11.3.

Suppose $G$ is a discrete group and its classifying space, $BG$, is Sullivan
rational. Then

  1. (i)

Each $G^{n}/G^{n+1}$ is a finite dimensional rational vector space.

  2. (ii)

$G=\varprojlim_{n}G/G^{n}$.

proof. Let $\land V$ be the minimal Sullivan model of $BG$. Then

$(V^{n})^{\vee}=\pi_{n}\langle\land
V\rangle=\pi_{n}(BG)=\left\\{\begin{array}[]{ll}0&n\geq 2\\\
G&n=1\end{array},\right.$

Thus $V=V^{1}$, and (cf Proposition 8.2)

$G=\pi_{1}\langle\land V^{1}\rangle=G_{L}$

where $L=L_{0}$ is the homotopy Lie algebra of $\land V$. Now the Lemma
follows from Proposition 8.2. $\square$

######  Proposition 11.1.

Suppose $\varphi:\land V\to A_{PL}(X)$ is a Sullivan model of a connected
space, $X$. Then

  1. (i)

$H(|\varphi|):H(X)\stackrel{{\scriptstyle\cong}}{{\longleftarrow}}H(X_{\mathbb{Q}})\,\Longleftrightarrow\,H(\iota_{\land
V}):H(\land V)\stackrel{{\scriptstyle\cong}}{{\rightarrow}}H(\langle\land
V\rangle)$.

  2. (ii)

When the conditions of (i) hold then $\langle\land V\rangle$ is Sullivan
rational, $X_{\mathbb{Q}}\to(X_{\mathbb{Q}})_{\mathbb{Q}}$ is an isomorphism
of simplicial sets, and $X_{\mathbb{Q}}$ is Sullivan rational.

proof. (i) According to (15),

$\varphi=A_{PL}(|\varphi|)\circ\iota_{\land V}.$

Since $H(\varphi)$ is an isomorphism this establishes (i).

(ii) By definition, $X_{\mathbb{Q}}=\langle\land V\rangle$. Therefore (i)
$\Rightarrow$ (ii) follows from Lemma 11.1. $\square$

Now suppose that

(33)

be a commutative cdga diagram, in which

  1. $\bullet$

The upper row is a $\Lambda$-extension decomposition of a minimal Sullivan
algebra $\land V=\land W\otimes\land Z$.

  2. $\bullet$

The lower row is an application of $A_{PL}(-)$ to a fibration
$\textstyle{B}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$
of path connected spaces.

######  Theorem 2.

Suppose in (33) that $F$ is Sullivan rational and that $H(\varphi_{F})$ is an
isomorphism. Then

  1. (i)

The holonomy representation of $\pi_{1}(B)$ in each $H^{k}(F)$ is nilpotent.

  2. (ii)

$H(\varphi_{B})$ is an isomorphism $\Leftrightarrow$ $H(\varphi_{X})$ is an
isomorphism.

  3. (iii)

If $H(\varphi_{B})$ and $H(\varphi_{X})$ are isomorphisms then $B$ and $X$ are
Sullivan rational.

proof. (i) The adjoint map $\langle\varphi_{B}\rangle:B\to\langle\land
W\rangle$ induces a morphism of fundamental groups, and ([19, Theorem 4.2])
the isomorphism $H(\varphi_{F}):H^{k}(\land
Z)\stackrel{{\scriptstyle\cong}}{{\to}}H^{k}(F)$ is equivariant with respect
to the two corresponding holonomy representations. The first is locally
nilpotent ([19, Proposition 4.4]) and the second is the dual of a
representation of $\pi_{1}(F)$ in $H_{*}(F)$. It follows that both
representations are nilpotent.

(ii) Since $\land Z$ is the minimal Sullivan model of a Sullivan rational
space it follows from Lemma 11.1 that $H(\land Z)$ is a graded vector space of
finite type. Thus, if $H(\varphi_{B})$ is an isomorphism, ([19, Theorem 5.1])
provides a Sullivan algebra $(\land V^{\prime},d^{\prime})=(\land
W\otimes\land Z,d^{\prime})$ and a commutative cdga diagram

$\textstyle{\land
W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{B}}$$\scriptstyle{\simeq}$$\textstyle{(\land
W\otimes\land
Z,d^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{\simeq}$$\textstyle{\land
Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{F}}$$\scriptstyle{\simeq}$$\textstyle{A_{PL}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A_{PL}(p)}$$\textstyle{A_{PL}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{PL}(F).}$

It is straightforward to extend the inclusion of $\land W$ to a lift of
$\varphi_{X}$ through $\varphi^{\prime}$ and to conclude that $H(\varphi_{X})$
is an isomorphism.

On the other hand, suppose $H(\varphi_{X})$ is an isomorphism. Then extend
$\varphi_{B}$ to a quasi-isomorphism, $\varphi_{B}^{\prime}:\land
W\otimes\land
W^{\prime}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}A_{PL}(B)$, from a
minimal $\Lambda$-extension. By [19, Theorem 5.1] this extends to a
commutative cdga diagram

$\textstyle{\land W\otimes\land
W^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\land
W\otimes\land W^{\prime}\otimes\land
Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\land
Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{F}}$$\textstyle{A_{PL}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{PL}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{PL}(F).}$

Since $\varphi_{X}:\land W\otimes\land Z\to A_{PL}(X)$ is also a quasi-
isomorphism it follows that $W^{\prime}=0$ and $H(\varphi_{B})$ is an
isomorphism.

(iii) In (33) we are now assuming that each of $H(\varphi_{B})$,
$H(\varphi_{X})$ and $H(\varphi_{F})$ are isomorphisms. On the other hand
taking the adjoints of $\varphi_{B}$, $\varphi_{X}$ and $\varphi_{F}$ produces
the commutative diagram

$\textstyle{\langle\land W\rangle}$$\textstyle{\langle\land W\otimes\land
Z\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\langle\land
Z\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\langle\varphi_{B}\rangle}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\langle\varphi_{X}\rangle}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\langle\varphi_{F}\rangle}$

According to ([17, Proposition 17.9]) the upper row is also a fibration. Thus
it follows from the 5-lemma that if any two of
$\pi_{*}\langle\varphi_{B}\rangle$, $\pi_{*}\langle\varphi_{X}\rangle$ and
$\pi_{*}\langle\varphi_{F}\rangle$ are isomorphisms so is the third. $\square$

######  Proposition 11.2.

If $\land V$ is a Sullivan algebra and each dim$\,V^{l}<\infty$ then
$\iota_{\land V}$ is a quasi-isomorphism.

proof. First observe that if deg$\,v=1$ then

$[S^{k},\langle\land v\rangle]=[\land
v,A_{PL}(S^{k})]=\left\\{\begin{array}[]{ll}\mathbb{Q}&\mbox{if }k=1\\\
0&\mbox{if }k>1.\end{array}\right.$

Thus $\pi_{k}\langle\land v\rangle=\mathbb{Q}$ if $k=1$ and $0$ if $k>1$. It
follows that

$H^{k}(\langle\land v\rangle)=\left\\{\begin{array}[]{ll}\mathbb{Q}&\mbox{if
}k=1\\\ 0&\mbox{if }k>1\end{array}\right.$

Moreover the identity of $\langle\land v\rangle$ is adjoint to $\iota_{\land
v}$ and so $\iota_{\land v}$ is not null homotopic; Therefore

$H(\iota_{\land v}):\land
v\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}A_{PL}\langle\land
v\rangle.$

Now we proceed by induction on dim $V^{1}$. If $V^{1}=0$ the Proposition is
proved in ([19, Theorem 1.5]). Otherwise we have a $\Lambda$-extension
decomposition of the form $\land V=\land v\otimes\land W$. In this case the
Proposition follows from Theorem 2. $\square$

##  Part IV Enriched dgl’s and their homotopy theory

###  12 Enriched dgl’s and semi-quadratic Sullivan algebras

Definitions.

  1. (1)

A _differential graded Lie algebra_ (dgl), $(L,\partial)$ is a graded Lie
algebra $L=\\{L_{k}\\}_{k\geq 0}$ together with a differential
$\partial:L_{k}\to L_{k-1}$ which is also a right derivation:

$\displaystyle\partial^{2}=0\hskip 14.22636pt\mbox{and
}\partial[x,y]=(-1)^{deg\,y}[\partial x,y]+[x,\partial y].$ (34)

  2. (2)

An _enriched dgl_ (edgl) is a dgl, $(L,\partial)$, together with an inverse
system of morphisms
$\rho_{\alpha}:(L,\partial)\to(L_{\alpha},\partial_{\alpha})$ such that

    1. $\bullet$

each $(L_{\alpha},\partial_{\alpha})$ is a finite dimensional nilpotent dgl,
and

    2. $\bullet$

$L\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\varprojlim_{\alpha}L_{\alpha}$.

In particular, _$L$ is a complete enriched Lie algebra_ and $\partial$ is a
coherent linear map.

  3. (3)

An enriched dgl, $(L,\partial)$ is _minimal_ if $\partial:L\to L^{(2)}$.

  4. (4)

A _morphism of enriched dgl’s_ is a morphism of enriched Lie algebras which
commutes with the differentials. In particular, morphisms are coherent. We
denote the set of morphisms $(L,\partial)\to(L^{\prime},\partial^{\prime})$ by
$\mbox{dgl}((L,\partial),(L^{\prime},\partial^{\prime}))$.

  5. (5)

A _dgl ideal_ in an enriched dgl, $L$, is a closed differential ideal; in
which case (Lemma 1.1) $L/I$ is an enriched dgl.

Remarks.

  1. 1)

For simplicity we will frequently denote an enriched dgl simply by
$(L,\partial)$.

  2. 2)

The convention that $\partial$ be a right derivation arises from the fact that
in applications $L$ frequently acts from the right.

The 1-1 correspondence in §4 between complete enriched Lie algebras and
quadratic Sullivan algebras extends to enriched dgl’s, $(L,\partial)$ as
follows. First, let $(\land V,d_{1})$ be the quadratic Sullivan model of $L$.
Recall from §4 that the inverse limit $sL=\varprojlim_{\alpha}sL_{\alpha}$ is
the dual of a unique direct limit $V=\varinjlim_{\alpha}V_{\alpha}$, where
$(\land V_{\alpha},d_{\alpha})=C^{*}(L_{\alpha})$. Then

$(\land V,d_{1})=\varinjlim_{\alpha}\,(\land V_{\alpha},d_{1,\alpha})$

is by definition the quadratic model of $L$.

On the other hand, each $\partial_{\alpha}$ is the dual (up to suspension) of
a unique differential $d_{0,\alpha}$ in $V_{\alpha}$ and, by Proposition 2.1,
$\partial$ is the dual of the differential
$d_{0}=\varinjlim_{\alpha}d_{0,\alpha}$ in $V$:

$<d_{0}v,sx>=<v,s\partial x>,\hskip 28.45274ptv\in V,x\in L.$

Finally, we extend $d_{0}$ to a derivation in $\land V_{0}$. Now a minimal
Sullivan model $(\land W,d)\stackrel{{\scriptstyle\simeq}}{{\to}}(\land
V,d_{0})$ induces an isomorphism
$W\stackrel{{\scriptstyle\cong}}{{\to}}H(V,d_{0})$. Let $L_{W}$ denote the
homotopy Lie algebra of $\land W$. Then there are natural isomorphisms

$\displaystyle\left.\begin{array}[]{l}s(L,\partial)=\varprojlim_{\alpha}s(L_{\alpha},\partial_{\alpha})=\left[\varinjlim_{\alpha}(V_{\alpha},d_{0,\alpha}\right]^{\vee}=(V,d_{0})^{\vee}\\\
H(L,\partial)=s^{-1}H(V,d_{0})^{\vee}=s^{-1}W^{\vee}=L_{W},\end{array}\right\\}$
(37)

which in particular provide an isomorphism $H(L,\partial)\cong L_{W}$ of
complete enriched Lie algebras. It also follows from (34) that
$(d_{0}+d_{1})^{2}=0$. Thus Lemma 12.1(i) immediately below shows that $(\land
V,d_{0}+d_{1})$ is a Sullivan algebra.

Definition. (i) A _semi-quadratic Sullivan algebra_ is a Sullivan algebra of
the form $(\land V,d=d_{0}+d_{1})$ in which $d_{0}:V\to V$ and
$d_{1}:V\to\land^{2}V$.

(ii) The semi-quadratic Sullivan algebra corresponding to an enriched dgl $L$
is its _semi-quadratic Sullivan model_ , and is denoted $\land V_{L}$.

(iii) A cdga morphism $\varphi:(\land V,d_{0}+d_{1})\to(\land W,d_{0}+d_{1})$
is _bihomogeneous_ if $\varphi:V\to W$.

######  Lemma 12.1.

Let $(\land W,d_{0}+d)$ be a cdga in which $d_{0}:W\to W$, $d:W\to\land^{\geq
2}W$ and $d^{2}=0$. Then $(\land W,d_{0}+d)$ is a Sullivan algebra if and only
if $(\land W,d)$ is a Sullivan algebra.

proof. Suppose first that $(\land W,d_{0}+d)$ is a Sullivan algebra. Then
$\land W=\varinjlim_{n}\land W_{n}$ in which $d_{0}+d:W_{n+1}\to\land W_{n}$.
But this implies that $d:W_{n+1}\to\land W_{n}$ and so $(\land W,d)$ is a
Sullivan algebra.

In the reverse direction, suppose $(\land W,d)$ is a Sullivan algebra. If
$P\subset W$ is finite dimensional then $P\subset$ some $W^{\prime}$ with
dim$\,W^{\prime}<\infty$ and $\land W^{\prime}$ preserved by $d$. It follows
that $\land(W^{\prime}+d_{0}W^{\prime})$ is preserved by both $d$ and $d_{0}$.
Therefore it is sufficient to prove that $(\land W,d_{0}+d)$ is a Sullivan
algebra when dim$\,W<\infty$.

In this case we set $W_{0}=W\cap\mbox{ker}\,d$. Then $\land W$ decomposes as
the $\Lambda$-extension

$(\land W_{0},d_{0})\to(\land W_{0}\otimes\land Z,d_{0}+d)\to(\land
Z,\overline{d_{0}}+\overline{d})$

of cdga’s and the result follows by induction on dim$\,W$. $\square$

######  Proposition 12.1.

  1. (i)

The correspondence $(L,\partial)\leadsto(\land V,d_{0}+d_{1})$ is a bijection
between enriched dgl’s and semi-quadratic Sullivan algebras.

  2. (ii)

This correspondence identifies the morphisms
$f:(L,\partial)\to(L^{\prime},\partial^{\prime})$ with the bihomogeneous
morphisms $\varphi:(\land V,d_{0}+d_{1})\leftarrow(\land
V^{\prime},d_{0}^{\prime}+d_{1}^{\prime})$ via

$\mbox{suspension of }f=(\varphi|_{V^{\prime}})^{\vee}.$

In particular $\varphi:(\land V,d_{1})\leftarrow(\land
V^{\prime},d_{1}^{\prime})$ is the morphism of quadratic Sullivan algebras
corresponding to $f:L\to L^{\prime}$.

  3. (iii)

$(L,\partial)$ is minimal if and only if $d_{0}(V\cap\mbox{ker}\,d_{1})=0$.

  4. (iv)

If $I\subset L$ is a dgl ideal in an enriched dgl, $L$, then the morphisms
$\land V_{L/I}\to\land V_{L}\to\land V_{I}$ decompose $\land V_{L}$ as a
Sullivan extension

$\land V_{L}=\land V_{L/I}\otimes\land V_{I}.$

Conversely, if an inclusion $\land W\to\land V_{L}$ of semi-quadratic Sullivan
algebras decomposes $\land V_{L}$ as a Sullivan extension $\land W\otimes\land
Z$ then the enriched dgl corresponding to $\land Z$ is a dgl ideal $I\subset
L$ and $\land W=\land V_{L/I}$.

proof.

(i) This is immediate.

(ii) The correspondence associates to a morphism $f$ the map
$\varphi:V\leftarrow V^{\prime}$, via the equation

$<\varphi v^{\prime},sx>=<v^{\prime},sf(x)>.$

In this case $\varphi\circ d_{0}=d_{0}^{\prime}\circ\varphi$ and $\varphi\circ
d_{1}=d_{1}^{\prime}\circ\varphi$. The bijection is a straightforward
consequence of the definitions and Propositions 2.1 and 4.1.

(iii) This follows because (Lemma 4.1) the isomorphism $sL\cong V^{\vee}$
induces an isomorphism $s(L/L^{(2)})\cong(V\cap\mbox{ker}\,d_{1})^{\vee}$.

(iv) This is immediate from the definitions and §4.1. $\square$

Definition. If $(\land V,d_{0}+d_{1})$ is the semi-quadratic Sullivan algebra
corresponding to an enriched dgl $(L,\partial)$ then it is the _semi-quadratic
Sullivan model_ of $(L,\partial)$ and $(L,\partial)$ is the _enriched dgl
model_ of $(\land V,d_{0}+d_{1})$.

Remark. In working with enriched dgl’s and their semi-quadratic models it is
important to note that

  1. 1)

Not every morphism is bihomogeneous, and

  2. 2)

The semi-quadratic Sullivan model of a minimal enriched dgl is a Sullivan
algebra, but is not minimal unless $d_{0}=0$.

Finally, if $(\land V,d_{0}+d_{1})$ is a semi-quadratic Sullivan algebra then
it is immediate that $\land V(n):=\land V^{\leq n}\otimes\land d_{0}V^{n}$ is
preserved by both $d_{0}$ and $d_{1}$. Thus

$(\land V,d_{0}+d_{1})=\varinjlim_{n}\,(\land V(n),d_{0}+d_{1}).$

######  Proposition 12.2.

  1. (i)

A bihomogeneous morphism $\varphi:(\land V,d_{0}+d_{1})\to(\land
W,d_{0}+d_{1})$ of semi-quadratic Sullivan algebras restricts to morphisms

$\varphi(n):(\land V(n),d_{0}+d_{1})\to(\land W(n),d_{0}+d_{1}),\hskip
14.22636ptn\geq 0.$

  2. (ii)

If $\psi\sim\varphi:(\land V,d_{0}+d_{1})\to(\land W,d_{0}+d_{1})$ then

$\varphi(n)\sim\psi(n),\hskip 14.22636ptn\geq 0.$

  3. (iii)

A bihomogeneous morphism $\varphi:(\land V,d_{0}+d_{1})\to(\land
W,d_{0}+d_{1})$ is a quasi-isomorphism if and only if each $\varphi(n)$ is a
quasi-isomorphism.

proof. (i) Since $\varphi$ preserves degrees it maps $\land V^{\leq n}$ to
$\land W^{\leq n}$. But if $v\in V^{n}$ then

$\varphi(d_{0}v)=\varphi(dv-d_{1}v)=d\varphi v-\varphi d_{1}v=d_{0}\varphi
v+d_{1}\varphi v-\varphi d_{1}v\in W(n),$

and so $\varphi:V(n)\to W(n)$.

(ii) This follows in the same way as (i).

(iii) Since $\varphi$ is a quasi-isomorphism of Sullivan algebras it has a
homotopy inverse $\widehat{\varphi}:\land V\leftarrow\land W$. It follows from
(i) and (ii) that $\widehat{\varphi}$ and $\varphi$ restrict to homotopy
inverses between each $V(n)$ and $W(n)$.

$\square$

####  12.1 Homology of a semi-quadratic Sullivan algebra

######  Lemma 12.2.

If $(\land W,d_{0}+d_{1})$ is a semi-quadratic Sullivan algebra then

$(\land W,d_{0}+d_{1})=\varinjlim_{\alpha}(\land W_{\alpha},d_{0}+d_{1})$

is the limit of sub semi-quadratic Sullivan algebras $(W_{\alpha}\subset W)$
for which each $H(W_{\alpha},d_{0})\to H(W,d_{0})$ is injective.

proof. A sub cdga $(\land W_{\beta},d_{0}+d)$ of $\land W$ will be called
_regular_ if dim$\,W_{\beta}<\infty$ and if

$H(W_{\beta},d_{0})\to H(W,d_{0})$

is injective. We show by induction on $n$ that any finite dimensional subspace
$P\subset W^{\leq n}$ is contained in a regular cdga of the form $(\land
V^{\leq n}\oplus d_{0}V^{n}).$

Assuming this for some $n$, let $P\subset W^{\leq n+1}$ be finite dimensional.
It follows from (i) that $P\subset Z=Z^{\leq n+1}+d_{0}(Z^{\leq n+1})$, where
dim$\,Z<\infty$ and $\land Z$ is preserved by $d_{0}$ and $d_{1}$. Now choose
a subspace $S\subset W^{n}$ so that $S\cap Z^{n}=0$ and

$0\to S\stackrel{{\scriptstyle d_{0}}}{{\to}}H^{n+1}(Z,d_{0})\to
H^{n+1}(W,d_{0})$

is exact. By our induction hypothesis there is a regular cdga of the form
$\land(V^{\leq n}\oplus d_{0}V^{n})$ and for which

$S\oplus Z^{\leq n}\subset V^{\leq n}.$

Now, set $V^{n+1}=Z^{n+1}+d_{0}V^{n}$. We show that $\land(V^{\leq n+1}\oplus
d_{0}V^{n+1})$ is regular. Since $P\subset W^{\leq n+1}$ this will complete
the proof.

But since $\land Z$ and $\land(V^{\leq n}\oplus d_{0}V^{n})$ are both
preserved by $d_{0}$ and by $d_{1}$ it follows that so is $\land(V^{\leq
n+1}\oplus d_{0}V^{n+1})$. Moreover, for $k\leq n$ the map $H^{k}(V^{\leq
n+1}\oplus d_{0}V^{n+1})\to H^{k}(W)$ coincides with the injection
$H^{k}(V^{\leq n}\oplus d_{0}V^{n})\to H^{k}(W)$. Also, a cycle in $V^{n+1}$
is homologous to a cycle in $Z^{n+1}$. Thus if it is a boundary in $W^{n+1}$,
it is in $d_{0}(S)$ and hence a boundary in $W^{n+1}$.

$\square$

Corollary. If $(L,\partial)$ is complete and dim$\,H(L)/[H(L),H(L)]<\infty$
then $H(L)$ is pronilpotent.

proof. This follows directly from Proposition 3.1. $\square$

######  Proposition 12.3.

The enriched structure in an enriched dgl can be represented by surjections
$\rho_{\beta}:(L,\partial)\to(L_{\beta},\partial_{\beta})$ for which
$(H(L),\\{H(\rho_{\beta})\\})$ is a complete enriched Lie algebra.

proof. An enriched structure in $(L,\partial)$ is represented by surjections
$\rho_{\beta}:L\to L_{\beta}$ which are dual to inclusions
$j_{\beta}:(V_{\beta},d_{0}+d_{1})\to(V,d_{0}+d_{1})$. By Lemma 12.2 we may
assume that each $H(j_{\beta}):H(V_{\beta},d_{0})\to H(V,d_{0})$ is injective.
Then $H(\rho_{\beta})=s^{-1}H(j_{\beta})^{\vee}$ is surjective and
$(H(L),\\{H(\rho_{\beta})\\})$ is an enriched Lie algebra. Since $L$ is
complete, $(L,\partial)=\varprojlim_{\beta}(L_{\beta},\partial_{\beta})$. Thus
it follows from Proposition 1.1 that $H(L)=\varprojlim\,H(L_{\beta})$ is
complete. $\square$

######  Proposition 12.4.

Suppose $\varphi:(\land V,d_{0}+d_{1})\leftarrow(\land
V^{\prime},d^{\prime}_{0}+d^{\prime}_{1})$ is the bihomogeneous morphism
corresponding to a morphism $f:(L,\partial)\to(L^{\prime},\partial^{\prime})$
of enriched dgl’s. Then

  1. (i)

$H(\varphi,d_{0}):(\land H(V,d_{0}),H(d_{1}))\leftarrow(\land
H(V^{\prime},d_{0}^{\prime}),H(d_{1}))$ is the bihomogeneous morphism
corresponding to $H(f):(H(L),0)\to(H(L^{\prime}),0)$.

  2. (ii)

$H(f)$ is an isomorphism $\Leftrightarrow$ $H(\varphi,d_{0})$ is an
isomorphism.

  3. (iii)

If $H(\varphi,d_{0})$ is an isomorphism then $H(\varphi,d_{0}+d_{1})$ is an
isomorphism.

proof. (i) It follows from (37) that $H(\varphi,d_{0})^{\vee}$ is the
suspension of $H(f):L_{V}\to L_{V^{\prime}}$, where $L_{V}$ and
$L_{V^{\prime}}$ are the homotopy Lie algebras of $\land H(V,d_{0})$ and
$\land H(V^{\prime},d^{\prime}_{0})$.

(ii) This is immediate from (37).

(iii) Filtering by wedge degree produces a spectral sequence converging from
$H(\land V,d_{0})$ to $H(\land V,d_{0}+d_{1})$, and this establishes (ii).
$\square$

######  Proposition 12.5.

Suppose $\varphi:(\land V,d_{0}+d_{1})\to(\land W,D)$ is a quasi-isomorphism
from a semi-quadratic Sullivan algebra to a minimal Sullivan algebra.

  1. (i)

Then $\varphi$ is surjective and restricts to surjective dga morphisms

$(\land^{\geq
k}V,d_{0}+d_{1})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(\land^{\geq
k}W,D).$

  2. (ii)

The morphism of wedge-filtration spectral sequences induced by $\varphi$
begins with the isomorphism of $E_{1}$-terms

$(\land
H(V,d_{0}),H(d_{1}))\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}(\land
W,D_{1}),$

where $D_{1}$ is the quadratic component of $D$.

  3. (iii)

The isomorphism, (ii), of quadratic Sullivan algebras identifies the
isomorphism $H(L,\partial)\cong L_{W}$ of (37) as an isomorphism of enriched
Lie algebras.

proof. (i) Since $\varphi$ is a quasi-isomorphism of Sullivan algebras it has
a homotopy inverse, $\psi:(\land
W,D)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(\land V,d_{0}+d_{1})$.
Thus $\varphi\circ\psi$ is a quasi-isomorphism of minimal Sullivan algebras,
and is therefore an isomorphism. In particular it restricts to dga
automorphisms of each $\land^{\geq k}W$. It follows that $\varphi$ restricts
to surjective dga morphisms $\land^{\geq k}V\to\land^{\geq k}W$, $k\geq 0$.

(ii) Decompose $V$ as

$V=Z\oplus S\oplus d_{0}S,$

in which ker$\,d_{0}=Z\oplus d_{0}S$ and
$d_{0}:S\stackrel{{\scriptstyle\cong}}{{\to}}d_{0}S$. Filtration by wedge
degree then shows that the inclusions $Z,S$ and $d_{0}S$ in $\land V$ extend
to an isomorphism

$\chi:\land Z\otimes\land S\otimes\land
d_{0}S\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\land V$

of graded algebras. It follows that a differential, $d_{Z}$, in $\land Z$ is
determined by the requirement that $\chi$ be a dga isomorphism. Since
$\varphi$ is a quasi-isomorphism its restriction, $\chi_{Z}$, to $(\land
Z,d_{Z})$ is a quasi-isomorphism of minimal Sullivan algebras, and hence an
isomorphism.

On the other hand $\chi$ also induces a morphism of the spectral sequences
induced by filtration by wedge degrees, and the $E_{0}$-term is an isomorphism

$(\land Z,0)\otimes(\land(S\oplus
d_{0}S),d_{0})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}(\land
V,d_{0}).$

Thus the $E_{1}$\- morphism is an isomorphism

$(\land Z,(d_{Z})_{1})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}(\land
H_{0}(V),H(d_{1})).$

Finally, filtering $\land Z$ and $\land W$ by wedge degrees converts
$\chi_{Z}$ into a morphism of spectral sequences for which the $E_{1}$-term is
an isomorphism $(\land
Z,(d_{Z})_{1})\stackrel{{\scriptstyle\cong}}{{\to}}(\land W,D_{1})$. This,
combined with the isomorphism above, establishes (ii).

(iii) is immediate from the definitions.

$\square$

####  12.2 Free products

The differentials in enriched dgl’s $(L^{\prime},\partial^{\prime})$ and
$L^{\prime\prime},\partial^{\prime\prime})$ extend uniquely to a differential,
$\partial,$ in the free product
$L^{\prime}\,\widehat{\amalg}\,L^{\prime\prime}$ (recalled in §10.2). Then
$(L=L^{\prime}\,\widehat{\amalg}\,L^{\prime\prime},\partial)$ is an enriched
dgl, the _free product_ of $(L^{\prime},\partial^{\prime})$ and
$(L^{\prime\prime},\partial^{\prime\prime})$.

######  Proposition 12.6.

  1. (i)

Any two morphisms,
$\iota^{\prime},\iota^{\prime\prime}:(L^{\prime},\partial^{\prime}),(L^{\prime\prime},\partial^{\prime\prime})\to(E,\partial_{E})$
extend uniquely to a morphism

$(L^{\prime}\,\widehat{\amalg}\,L^{\prime\prime},\partial)\to(E,\partial_{E}).$

  2. (ii)

If $(\land W,d_{0}+d_{1})$ and $(\land Q,d_{0}+d_{1})$ are the semi-quadratic
models of $(L^{\prime},\partial^{\prime})$ and
$(L^{\prime\prime},\partial^{\prime\prime})$, then there is a bihomogeneous
quasi-isomorphism

$\textstyle{(\land
V,d_{0}+d_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{(\land
W,d_{0}+d_{1})\times_{\mathbb{Q}}(\land Q,d_{0}+d_{1}).}$

  3. (iii)

Denote by $(L,\partial)$ the edgl corresponding to $(\land V,d_{0}+d_{1})$.
The edgl morphisms $L^{\prime},L^{\prime\prime}\to L$ corresponding to the
surjections $\land V\to\land W,\land Q$ then extend to an isomorphism

$(L^{\prime},\partial^{\prime})\,\widehat{\amalg}\,(L^{\prime\prime},\partial^{\prime\prime})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}(L,\partial).$

proof. (i) This follows from Lemma 10.1 since by construction the morphism
$L^{\prime}\,\widehat{\amalg}\,L^{\prime\prime}\to E$ is compatible with the
differentials.

(ii) The proof of Proposition 10.1 provides a quasi-isomorphism

$\varphi:(\land V,d_{1})\to(\land W,d_{1})\times_{\mathbb{Q}}(\land Q,d_{1})$

in which $(\land V,d_{1})$ is a quadratic Sullivan algebra and $\varphi:V\to
W\oplus Q$. Let $L$ be the homotopy Lie algebra of $(\land V,d_{1})$. Then
(Proposition 10.3) the surjections $\land V\to\land W,\land Q$ induce an
isomorphism,

$L^{\prime}\,\widehat{\amalg}\,L^{\prime\prime}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}L,$

of enriched Lie algebras.

In particular, $\partial^{\prime}$ and $\partial^{\prime\prime}$ extend to a
differential, $\partial$, in $L$. This in turn defines a differential $d_{0}$
in $\land V$ for which $(\land V,d_{0}+d_{1})$ is a semi-quadratic Sullivan
algebra and $\varphi\circ d_{0}=d_{0}\circ\varphi$. Filtering by degree $-$
wedge produces a morphism of spectral sequences converging from
$H(\varphi,d_{1})$ to $H(\varphi,d_{0}+d_{1})$. Thus

$\varphi:(\land V,d_{0}+d_{1})\to(\land
W,d_{0}+d_{1})\times_{\mathbb{Q}}(\land Q,d_{0}+d_{1})$

is a quasi-isomorphism.

(iii) This is immediate from the construction in (ii). $\square$

###  13 Profree dgl’s

As in Part II, $\mathbb{L}_{T}$ denotes the free graded Lie algebra freely
generated by a subspace $T=T_{\geq 0}$. At the start of Part II a complete
enriched structure in $T$ is extended to a natural enriched Lie algebra
structure in $\mathbb{L}_{T}$, and for which $T$ is a closed subspace. Then,
by definition, a _profree Lie algebra_ is a complete enriched Lie algebra of
the form $L=\overline{\mathbb{L}}_{T}$. In this case $L$ is the direct sum,

$L=T\oplus L^{(2)},$

in which $T$ is closed in $L$. Finally Theorem 1, together with Lemma 6.2,
establishes the equivalence of the following three conditions on a complete
enriched Lie algebra, $L$, and its quadratic model $(\land V,d_{1})$:

$\displaystyle\left.\begin{array}[]{ll}(i)&L=\overline{\mathbb{L}}_{T}\mbox{
is profree}.\\\ (ii)&H(\land
V,d_{1})=\mathbb{Q}\oplus(V\,\cap\mbox{ker}\,d_{1}).\\\ (iii)&H^{[2]}(\land
V,d_{1})=0.\end{array}\right\\}$ (41)

(Here $H(\land V,d_{1})=\oplus_{k}H^{[k]}(\land V,d_{1})$ is the decomposition
induced by wedge degree.)

Definition. A _profree dgl_ is a dgl $(L,\partial)$ in which
$L=\overline{\mathbb{L}}_{T}$ is profree.

######  Proposition 13.1.

  1. (i)

An enriched dgl $(L,\partial)$ admits a surjective quasi-isomorphism from a
profree dgl $(\overline{\mathbb{L}}_{T},\partial)$.

  2. (ii)

If $\rho:(E,\partial)\to(F,\partial)$ is a surjective quasi-isomorphism of
enriched dgl’s, then any morphism
$g:(\overline{\mathbb{L}}_{T},\partial)\to(F,\partial)$ from a profree dgl
lifts to provide a commutative diagram

$\textstyle{(E,\partial)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{(\overline{\mathbb{L}}_{T},\partial)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{h}$$\textstyle{(F,\partial).}$

  3. (iii)

Any profree dgl $(L,\partial)$ is the free product
$(L,\partial)=(L^{\prime},\partial^{\prime})\,\widehat{\amalg}\,(L^{\prime\prime},\partial^{\prime\prime})$
of a profree dgl satisfying $H(L^{\prime})=0$ and a minimal profree dgl
$(L^{\prime\prime},\partial^{\prime\prime})$.

proof. (i). Let $(\land V,d_{0}+d_{1})$ be the semi-quadratic model of
$(L,\partial)$. We construct by induction on $n$ semi-quadratic Sullivan
models of the form $(\land V\otimes\land W(n),d_{0}+d_{1})$ with $W(0)\subset
W(1)\subset\dot{W}(n)\subset W(n+1)\dots$, and such that

  1. (1)

The induced map $H^{[2]}(\land V\otimes\land W(n),d_{1})\to H^{[2]}(\land
V\otimes\land W(n+1),d_{1})$ is zero.

  2. (2)

The induced map $H(V\oplus W(n),d_{0})\to H(V\oplus W(n+1),d_{0})$ is an
isomorphism.

Here $H^{[2]}$ denotes as before the component of the cohomology of wedge
degree 2. Suppose $\land V\otimes\land W(n)$ has been defined. Then let
$Z=s^{-1}H^{[2]}(\land V\otimes\land W(n),d_{1})$, $W(n+1)=W(n)\oplus Z\oplus
d_{0}(Z)$, and for $x=s^{-1}\Omega\in Z$, we choose a $d_{1}$-cycle $\omega$
with $[\omega]=\Omega$, and we define $d_{1}(x)=\omega$ and
$d_{1}d_{0}x=-d_{0}\omega$.

Denote $W=\cup_{n}W(n)$. Since $H^{[2]}=0$, by Lemma 6.2, $(\land
V\otimes\land W,d_{0}+d_{1})$ is the semi-quadratic model of a profree dgl
$(\overline{\mathbb{L}}_{T},\partial)$, and the injection $\land V\to\land
V\otimes\land W$ corresponds to a surjective quasi-isomorphism
$(\overline{\mathbb{L}}_{T},\partial)\to(L,\partial)$.

(ii) Since $T$ is closed and $\rho$ is a surjective coherent map there is a
coherent linear map $\alpha:T\to E$ such that $\rho\circ\alpha=id$. Moreover,
since $\rho$ is a coherent surjective quasi-isomorphism,
$H(\mbox{ker}\,\rho)=0$. Thus there is a coherent linear map
$\tau:\mbox{ker}\,\rho\cap\mbox{ker}\,\partial\to\mbox{ker}\,\rho$ such that
$\partial\circ\tau=id$.

Now assume $h$ has been defined in $T_{<k}$, and therefore also in
$\overline{\mathbb{L}}_{T_{\leq k}}$. Then
$\sigma\circ\partial_{T}-\partial\circ\alpha:T_{k}\to\mbox{ker}\,\rho\cap\mbox{ker}\,\partial$.
Extend $h$ to $T_{k}$ by setting

$hx=\alpha x+\tau(h\partial_{T}-\partial\alpha)(x).$

It is immediate that the extension to $\overline{\mathbb{L}}_{T_{\leq n}}$ is
a morphism and that $\rho\circ h=g$.

(iii). Denote by $\overline{\partial}$ the quotient differential in
$L/L^{(2)}$. Then (cf. Proposition 2.1)

$L/L^{(2)}=S\oplus\overline{\partial}S\oplus H,$

where $S,\overline{\partial}S$ and $H$ are closed subspaces,
$\overline{\partial}:S\stackrel{{\scriptstyle\cong}}{{\to}}\overline{\partial}S$,
and $\overline{\partial}(H)=0$. Now the isomorphism $T\cong L/L^{(2)}$ from a
closed direct summand of $L^{(2)}$ yields a decomposition

$T=P\oplus\partial P\oplus Q,$

in which $\partial:P\stackrel{{\scriptstyle\cong}}{{\to}}\partial P$ and
$\partial:Q\to L^{(2)}$. It follows that
$L=\overline{\mathbb{L}}_{(T\oplus\partial P\oplus Q)}$.

Moreover, filtering the Lie bracket length provides a spectral sequence
converging from $\overline{\mathbb{L}}_{Q}$ to $H(L)$. This implies that
division by $T$ and $\partial T$ yields a quasi-isomorphism

$\rho:(L,\partial)\stackrel{{\scriptstyle\simeq}}{{\to}}(\overline{\mathbb{L}}_{Q},\partial_{Q})$

where $\partial_{Q}$ is the quotient differential.

Finally, by (ii) $\rho$ admits a splitting
$\sigma:(\overline{\mathbb{L}}_{Q},\partial_{Q})\to(L,\partial)$ and it is
immediate from the construction that

$id\,\widehat{\amalg}\,\sigma:\overline{\mathbb{L}}_{P\oplus\partial
P)}\,\widehat{\amalg}\,(\overline{\mathbb{L}}_{Q},\partial_{Q})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}(L,\partial).$

Since $\partial:Q\to\mathbb{L}_{Q}^{(2)}$, this completes the proof.

$\square$

Given a profree dgl, $(\overline{\mathbb{L}}_{T},\partial)$, division by
$\overline{\mathbb{L}}_{T}^{(2)}$ is a coherent surjection

$\rho:(\overline{\mathbb{L}}_{T},\partial)\to(T,\overline{\partial}).$

######  Proposition 13.2.

Suppose $(\overline{\mathbb{L}}_{T},\partial)$ is a profree dgl with semi-
quadratic Sullivan model $(\land V,d_{0}+d_{1})$, and let $\xi:\land^{\geq
1}V\to V$ denote the surjection (§9).

  1. (i)

Any coherent chain map $(T,\overline{\partial})\to(L,\partial)$ into an
enriched complete dgl extends uniquely to a morphism
$(\overline{\mathbb{L}}_{T},\partial)\to(L,\partial)$.

  2. (ii)

$H(V\cap\mbox{ker}\,d_{1},d_{0})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}H^{\geq
1}(\land V,d_{0}+d_{1})$.

  3. (iii)

The isomorphism (37),
$s(\overline{\mathbb{L}}_{T},\partial)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}(V,d_{0})^{\vee}$,
extends to the commutative diagram

$\textstyle{s(\overline{\mathbb{L}}_{T},\partial)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{s\rho}$$\textstyle{(V,d_{0})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi^{\vee}}$$\textstyle{s(T,\overline{\partial})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{(V\cap\mbox{ker}\,d_{1},d_{0})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{(\land^{\geq
1}V,d_{0}+d_{1})^{\vee}.}$

  4. (iv)

$(\overline{\mathbb{L}}_{T},\partial)$ is minimal if and only if
$\overline{\partial}=0$, or equivalently, if
$d_{0}|_{V\cap\mbox{ker}\,d_{1}}=0$.

  5. (v)

$H(\overline{\mathbb{L}}_{T},\partial)=0$ if and only if
$\overline{\mathbb{L}}_{T}^{(2)}$ has a closed direct summand of the form
$T=T(1)\oplus\partial(T(1))$ where
$\partial:T(1)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\partial T(1)$.

Corollary. If $(\overline{\mathbb{L}}_{T},\partial)$ is minimal then $sT\cong
H^{\geq 1}(\land V,d_{0}+d_{1})^{\vee}$.

proof of Proposition 13.2. (i) This is an immediate consequence of the
definition of a profree dgl.

(ii) Filtration of $\land V$ by degree $-$ wedge degree results in a spectral
sequence converging to $H(\land V,d_{0}+d_{1})$. Its $E_{1}$-term and
$E_{2}$-term are respectively $H(\land V,d_{1})$ and
$(\mathbb{Q}\oplus(V\cap\mbox{ker}\,d_{1}),d_{0})$. This proves (ii).

(iii) By Lemma 4.1, the inclusion $V\cap\mbox{ker}\,d_{1}\to V$ is dual to
$\overline{\mathbb{L}}_{T}\to\overline{\mathbb{L}}_{T}/\overline{\mathbb{L}}_{T}^{(2)}$,
since $(\land V,d_{1})$ is the quadratic model of $\overline{\mathbb{L}}_{T}$.
This gives (iii).

(iv) By definition, $(\overline{\mathbb{L}}_{T},\partial)$ is minimal if and
only if
$\partial:\overline{\mathbb{L}}_{T}\to\overline{\mathbb{L}}_{T}^{(2)}$. This
gives (iv).

(v) If $H(\overline{\mathbb{L}}_{T},\partial)=0$ it follows from the
isomorphism $s(\overline{\mathbb{L}}_{T},\partial)\cong(V,d_{0})^{\vee}$ that
$H(V,d_{0})=0$. This then implies that $(\land V,d_{0}+d_{1})$ is a
contractible Sullivan algebra, and therefore has zero homology in positive
degrees. Now (ii) and (iii) combine to give $H(T,\overline{\partial})=0$.

Write $T=T(1)\oplus\overline{\partial}T(1)$ and observe that it follows that

$\overline{\mathbb{L}}_{T}=\overline{\mathbb{L}}_{T(1)}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{\overline{\partial}T(1)}.$

Now it follows from the Corollary to Lemma 3.1 that

$\overline{\mathbb{L}}_{T(1)}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{\partial
T(1)}\to\overline{\mathbb{L}}_{T}$

is an isomorphism, and this is clearly an isomorphism of dgl’s.

$\square$

######  Proposition 13.3.

Let $f:(L,\partial)\to(L^{\prime},\partial^{\prime})$ be a morphism of profree
dgl’s with corresponding bihomogeneous morphism

$\varphi:(\land V,d_{0}+d_{1})\longleftarrow(\land
V^{\prime},d^{\prime}_{0}+d^{\prime}_{1})$

of their semi-quadratic Sullivan algebras. Then the following three conditions
are equivalent

  1. (i)

$H(\overline{f})$ is an isomorphism

  2. (ii)

$H(f)$ is an isomorphism

  3. (iii)

$H(\varphi)$ is an isomorphism.

proof. It follows from the diagram in Proposition 13.2(iii) that
$H(\overline{f})$ is an isomorphism if and only if $H(\varphi)$ is an
isomorphism. Moreover, by Proposition 13.2(iii), $H(f)$ is an isomorphism if
and only if $H(\varphi,d_{0})$ is an isomorphism. Finally, $\varphi:(\land
V^{\prime},d_{0}+d_{1})\to(\land V,d_{0}+d_{1})$ induces a morphism of minimal
Sullivan algebras of the form

$\psi:(\land H(V^{\prime},d_{0}),\widehat{d})\longrightarrow(\land
H(V,d_{0}),\widehat{d}).$

Here the linear map induced from $\psi$ by division by $\land^{\geq 2}$ is
$H(\varphi,d_{0})$. Therefore,

$\begin{array}[]{ll}H(\varphi,d_{0})\mbox{ is an isomorphism
}&\Longleftrightarrow\psi\mbox{ is an isomorphism}\\\ &\Longleftrightarrow
H(\psi)\mbox{ is an isomorphism }\\\ &\Longleftrightarrow H(\varphi)\mbox{ is
an isomorphism.}\end{array}$

$\square$

Corollary. If $L$ and $L^{\prime}$ are minimal then the conditions of
Proposition 13.3 are equivalent to each of the conditions

$(i)f\mbox{ is an isomorphism}\hskip 14.22636pt(ii)\varphi\mbox{ is an
isomorphism.}$

proof.  If $L$ and $L^{\prime}$ are minimal then
$\overline{f}=H(\overline{f})$ is an isomorphism. Thus $f$ maps a closed
direct summand, $T$, of $L^{(2)}$ isomorphically onto a direct summand,
$T^{\prime}$, of $(L^{\prime})^{(2)}$. Since $f$ is coherent, $T^{\prime}$ is
closed and $L^{\prime}$ is the closure of $\mathbb{L}_{T^{\prime}}$. It
follows that $f$ is an isomorphism. Since, in particular, $\varphi:(\land
V^{\prime},d_{1})\to(\land V,d_{1})$ is the morphism of quadratic Sullivan
algebras corresponding to $f$ it follows that $\varphi$ is also an
isomorphism. $\square$

###  14 Profree extensions and homotopy

####  14.1 Profree extensions

Definition.

  1. 1.

A _profree extension_ is an inclusion of the form
$i:(L,\partial)\to(L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S},\partial)$
of enriched dgl’s.

  2. 2.

The _quotient_ of a profree extension is the profree dgl
$(\overline{\mathbb{L}}_{S},\overline{\partial})$ obtained by dividing by the
ideal generated by $L$. The profree extension is _minimal_ if the quotient is
a minimal profree dgl.

  3. 3.

The _secondary gradation_ in $L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}$
is that uniquely determined by assigning to an iterated bracket of Lie
elements $x_{i}\in L$ and $y_{i}\in S$ the secondary degree,
$\sum\mbox{deg}\,y_{i}$.

  4. 4.

The _spectral sequence_ of the profree extension is the spectral sequence
obtained by the decreasing filtration by secondary degree.

  5. 5.

Two profree extensions $L\to L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}$,
$L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T}$ define a single profree
extension $L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S\oplus T}$ and

$L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S\oplus
T}=L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T}=L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}.$

Profree extensions have the five important properties listed in Proposition
14.1 below. The first three follow immediately from the definitions. The
fourth follows from the same argument as the proof of Proposition 13.1(ii).
The fifth follows from the definition (§10.2) of free products.

######  Proposition 14.1.

Let $(L,\partial)\to(L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S},\partial)$
be a profree extension.

  1. (i)

The spectral sequence of the extension converges from
$H(L)\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}$ to
$H(L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S})$.

  2. (ii)

A morphism $f:(L,\partial)\to(L^{\prime},\partial^{\prime})$ of enriched dgl’s
determines a unique differential in
$L^{\prime}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}$ for which
$f\,\widehat{\amalg}\,id_{\overline{\mathbb{L}}_{S}}$ is a morphism of profree
extensions.

  3. (iii)

If $f$ is a quasi-isomorphism so is
$f\,\widehat{\amalg}\,id_{\overline{\mathbb{L}}_{S}}$.

  4. (iv)

In a commutative diagram of enriched dgl morphisms,

$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\iota}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{F,}$

suppose that $\iota$ is the inclusion of a profree extension and that $\rho$
is a surjective quasi-isomorphism. Then there is a morphism

$h:L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}\longrightarrow E$

such that $h\circ\iota=f$ and $g\circ h=g$.

  5. (v)

The natural map
$L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}\to\varprojlim_{n}L/L^{(n)}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}$
is an isomorphism.

######  Proposition 14.2.

Suppose $\varphi:(E,\partial_{E})\to(F,\partial_{F})$ is a morphism of
enriched dgl’s. Then

  1. (i)

$\varphi$ factors as

$\textstyle{\varphi:(E,\partial_{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{(E\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S},\partial)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{(F,\partial_{F})}$

in which $\iota$ is the inclusion of a profree extension and $\rho$ is a
surjective quasi-isomorphism.

  2. (ii)

$\varphi$ also factors as

$\textstyle{\varphi:(E,\partial_{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\textstyle{(E\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T},\partial^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{(F,\partial_{F})}$

in which $\lambda$ is the inclusion of a minimal profree extension and
$\gamma$ is a quasi-isomorphism.

proof. (i) Proposition 13.1 gives a surjective quasi-isomorphism
$\sigma:(\overline{\mathbb{L}}_{S(0)},\partial)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(F,\partial)$.
By Proposition 12.6 (i) this extends to the morphism

$\textstyle{\rho(0)=\varphi\,\widehat{\amalg}\,\sigma:(E,\partial_{E})\,\widehat{\amalg}\,(\overline{\mathbb{L}}_{S(0)},\partial)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(F,\partial_{F}).}$

By construction, both $\varphi\,\widehat{\amalg}\,\sigma$ and
$H(\varphi\,\widehat{\amalg}\,\sigma)$ are surjective.

Next we construct $S(1)=\oplus_{n\geq 1}S(1)_{n}$ and a sequence of profree
extensions

$E\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S(0)}\,\widehat{\amalg}\,L(n)=\left[E\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S(0)}\,\widehat{\amalg}\,L(n-1)\right]\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S(1)_{n}}.$

These will be equipped with morphisms

$\rho(n):E\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S(0)}\,\widehat{\amalg}\,L(n)\longrightarrow
F$

with $\rho(n)$ extending $\rho(n-1)$ and $H_{<n}(\rho(n))$ an isomorphism.
Then $\overline{\mathbb{L}}_{S(1)}=\varinjlim_{n}L(n)$ and the $\rho(n)$
define a quasi-isomorphism

$\textstyle{\rho:E\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{F}$

with $S=S(0)\oplus S(1)$.

The construction is by induction on $n$. If $\rho(n-1)$ is constructed we may,
since $\rho(n-1)$ is coherent, write

$\mbox{ker}\,\rho(n-1)\cap\mbox{ker}\,\partial=C\oplus\partial(\mbox{ker}\,\rho(n-1))$

in which $C=C_{\geq n-1}$ is a closed subspace. Set $S(1)_{n}=sC_{n-1}$ and
define $\partial|_{S(1)_{n}}$ to be the isomorphism
$S(1)_{n}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}C_{n-1}$. Then
extend $\rho(n-1)$ to $\rho(n)$ by setting $\rho(n)(S(1)_{n})=0$.

(ii) Denote by
$\rho:(E\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S},\partial)\to F$ the
construction of (i), and denote by $\partial_{S}$ the quotient differential in
$\overline{\mathbb{L}}_{S}$. By Proposition 13.1(iii) we may write

$(\overline{\mathbb{L}}_{S},\partial_{S})=(\overline{\mathbb{L}}_{T},\partial_{T})\,\widehat{\amalg}\,\overline{\mathbb{L}}_{(U\oplus\partial_{S}U)}$

with $\partial_{S}:U\stackrel{{\scriptstyle\cong}}{{\to}}\partial_{S}U$ and
$(\overline{\mathbb{L}}_{T},\partial_{T})$ a minimal profree dgl. Filtering by
the closed ideals $E^{(n)}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}$ shows
(Proposition 14.1 (v)) that

$E\,\widehat{\amalg}\,(\overline{\mathbb{L}}_{T}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{(U\oplus\partial
U)})\longrightarrow E\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}$

is an isomorphism of dgl’s.

This provides the commutative diagram of dgl morphisms

|  |  |
---|---|---|---
$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(E\,\widehat{\amalg}\,\overline{\mathbb{L}}_{U\oplus\partial
U})\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\psi}$$\scriptstyle{\cong}$$\textstyle{E\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{F}$$\textstyle{E\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\scriptstyle{h}$$\textstyle{E\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T}}$

in which the spectral sequence shows that $\psi$ is a quasi-isomorphism. Thus
$h$ exists by Proposition 14.1(iv). In this case $h$ must also be a quasi-
isomorphism, and the composite

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{F}$

is the factorization of $\varphi$.

$\square$

Next suppose that

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{E}}$$\textstyle{E\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{E}}$$\scriptstyle{\simeq}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{F}}$$\textstyle{E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{\prime}\widehat{\amalg}\,\overline{\mathbb{L}}_{S^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{E^{\prime}}}$$\scriptstyle{\simeq}$$\textstyle{F^{\prime}}$

is a commutative diagram of morphisms of enriched dgl’s. Then extend the lower
row to

$\textstyle{E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{\prime}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{P\oplus\partial
P}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\alpha}$$\textstyle{F^{\prime}}$

by setting $P=F^{\prime}_{\geq 1}$,
$\partial:P\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\partial P$, and
requiring $\alpha|_{P}$ to be the inclusion.

######  Proposition 14.3.

With the notation and hypotheses above there is a morphism
$\gamma:E\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}\to
E^{\prime}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{P\oplus\partial
P}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S^{\prime}}$ for which the
diagram,

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{E}}$$\textstyle{E\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\alpha_{E}}$$\scriptstyle{\simeq}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{F}}$$\textstyle{E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{\prime}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{P\oplus\partial
P}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\simeq}$$\textstyle{F^{\prime},}$

commutes.

proof. Assume by induction that $\gamma$ is defined in $S_{<n}$, and let $x$
be a basis element in $S_{n}$. Then $\partial x\in
E\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S_{<n}}$ and

$\alpha\gamma\,\partial x=\partial\gamma_{F}\alpha_{E}x.$

Since $\gamma\partial x$ is a cycle and $\alpha$ is a quasi-isomorphism it
follows that

$\gamma\partial x=\partial y,\hskip 14.22636pt\mbox{some }y\in
E^{\prime}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{P\oplus\partial
P}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S^{\prime}}.$

In particular, $\partial\alpha y=\gamma_{F}\alpha_{E}\partial
x=\partial\gamma_{F}\alpha_{E}x$. Since $\alpha$ is a quasi-isomorphism

$\alpha y-\gamma_{F}\alpha_{E}x=\alpha z+\partial w$

for some cycle $z$ and some element $w\in F^{\prime}$. But necessarily
deg$\,w\geq 1$, so that $w=\alpha(u)$ for some $u\in P$. Thus

$\alpha(y-z-\partial u)=\gamma_{F}\alpha_{E}x,$

and we may extend $\gamma$ by setting $\gamma x=y-z-\partial u$. $\square$

######  Proposition 14.4.

Suppose

$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{\overline{\mathbb{L}}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi}$$\textstyle{L^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{\prime}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathbb{L}}_{S^{\prime}}}$

is a commutative diagram connecting two profree extensions. If any two of
$\varphi,\psi$ and $\chi$ are quasi-isomorphisms, then so is the third.

proof. We proceed in a number of Steps.

_Step One. If $\overline{\mathbb{L}}_{S}=\overline{\mathbb{L}}_{S^{\prime}}$
and $\psi=\varphi\,\widehat{\amalg}\,id_{\overline{\mathbb{L}}_{S}}$, then
$\psi$ is a quasi-isomorphism if and only if $\varphi$ is a quasi-
isomorphism._

If $\varphi$ is a quasi-isomorphism it follows from the associated spectral
sequence that $\varphi\,\widehat{\amalg}\,id_{\overline{\mathbb{L}}_{S}}$ is a
quasi-isomorphism.

In the reverse direction, suppose
$\varphi\,\widehat{\amalg}\,id_{\overline{\mathbb{L}}_{S}}$ is a quasi-
isomorphism. Factor $\varphi$ as (Proposition 14.2)

$L\to
L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}L^{\prime}$

in which $L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T}$ is a minimal profree
extension. This then extends to

$L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}\to
L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}\to
L^{\prime}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S},$

in which the second morphism is a quasi-isomorphism by the argument above.
Since the hypothesis $L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}\to
L^{\prime}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}$ is also a quasi-
isomorphism, it follows that

$L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}\to
L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T}=L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{S}$