16.4805pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{I/I^{(2)}}$}}}}}}}{\hbox{\kern 44.09682pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 77.71313pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{T\widehat{\otimes}\overline{UL_{V_{I}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
37.10074pt\raise 5.0625pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.0625pt\hbox{$\scriptstyle{s\nu^{\vee}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 16.4805pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
40.4805pt\raise-39.13332pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
12.96333pt\raise-26.17778pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\widetilde{\gamma}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 6.61182pt\raise-5.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
67.51125pt\raise-25.28888pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.13889pt\hbox{$\scriptstyle{id\otimes 1}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 86.34944pt\raise-3.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces}}}}\ignorespaces,$

in which $\widetilde{\gamma}$ is the composite
$T\stackrel{{\scriptstyle\gamma}}{{\to}}I\to I/I^{(2)}$ and $s\nu^{\vee}$ is a
morphism of the $\overline{UL_{V_{I}}}$-modules defined in (ii). Thus by
hypothesis, $s\nu^{\vee}$ is an isomorphism. Therefore $\nu$ is an
isomorphism, and it follows (because $I$ is profree) that $\land
V_{I}\otimes\land^{\geq 1}Z_{I}\stackrel{{\scriptstyle\simeq}}{{\to}}S$. Now
it follows from the analogue of (117) in Proposition 22.2 that $\land
V_{I}\stackrel{{\scriptstyle\simeq}}{{\to}}A$.

But by definition, $\varphi:\land V\stackrel{{\scriptstyle\simeq}}{{\to}}A$ is
a minimal Sullivan model. Therefore we can choose $\varphi$ so that it factors
as

$\land V\stackrel{{\scriptstyle\cong}}{{\to}}\land V_{I}\to A.$

In particular, $\varphi$ extends to an isomorphism $\land V\otimes\land
Z_{I}\stackrel{{\scriptstyle\cong}}{{\to}}\land W$ and, by Theorem 7, this
implies that $(\gamma,\rho)$ is inert. $\square$

###  23 Properties of inert attaching maps

####  23.1 Pairs of dgl attaching maps

Suppose (§17.2) that $\gamma:(\overline{\mathbb{L}}_{T},0)\to(L,\partial_{L})$
is a dgl attaching map, and that

$T=T(1)\oplus T(2)$

decomposes $T$ as a direct sum of closed subspaces. This yields (§19) dgl
attaching maps

$\gamma(1):(\overline{\mathbb{L}}_{T(1)},0)\to(L,\partial_{L})\hskip
28.45274pt\mbox{and
}\gamma(2):(\overline{\mathbb{L}}_{T(2)},0)\to(L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{sT(1)},\partial_{L}+\partial_{\gamma(1)}).$

######  Proposition 23.1.

With the hypotheses and notation above,

> $\gamma$ is inert $\Longleftrightarrow$ $\gamma(1)$ and $\gamma(2)$ are both
> inert.

proof. Denote by

$j(1):L\to L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{sT(1)}\hskip
14.22636pt\mbox{ and
}j(2):L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{sT(1)}\to
L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{sT(1)}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{sT(2)}=L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{sT}$

the inclusions corresponding to $\gamma(1)$ and $\gamma(2)$. By definition,
$j:L\to L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{s(T)}$ is given by

$j=j(2)\circ j(1).$

In particular, if $\gamma(1)$ and $\gamma(2)$ are inert, then
$H(j)=H(j(2))\circ H(j(1))$ is surjective and by definition $\gamma$ inert.

In the reverse direction, suppose $\gamma$ is inert. Since $H(j)$ is
surjective, $H(j(2))$ is also surjective and therefore $\gamma(2)$ is inert.
To show that $\gamma(1)$ is inert we may assume (Theorem 6) that the
differential in $L$ is zero. In this case, since $\gamma$ is inert Theorem 6
asserts that ker$\,H(j)$ is freely generated by
$\gamma(\overline{\mathbb{L}}_{T})$. In other words, if $I$ is the closed
ideal generated by $\gamma(T)$ we have a commutative diagram

---
$\textstyle{\overline{\mathbb{L}}_{T}\,\widehat{\amalg}\,C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\gamma\,\widehat{\amalg}\,\lambda}$$\scriptstyle{0\,\widehat{\amalg}\,p\lambda}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{L/I.}$

Now write
$\overline{\mathbb{L}}_{T}\,\widehat{\amalg}\,C=\overline{\mathbb{L}}_{T(1)}\,\widehat{\amalg}\,(\overline{\mathbb{L}}_{T(2)}\,\widehat{\amalg}\,C)$.
Then this diagram shows that $I(1)$ is freely generated by $\gamma(T(1))$.
Therefore (Theorem 6) $\gamma(1)$ is inert. $\square$

####  23.2 Inertness of elements in a profree Lie algebra

######  Theorem 9.

Suppose $\overline{\mathbb{L}}_{T}$ is a profree Lie algebra in which
$T=T_{even}$. Then any non-zero morphism

$\gamma:\overline{\mathbb{L}}_{\mathbb{Q}x}\to\overline{\mathbb{L}}_{T}$

is an inert attaching map.

proof. We accomplish the proof in three steps.

_Step One. Reduction to the case $T=T_{0}$_.

Suppose $(\land W,d)$ is the quadratic model of a complete enriched Lie
algebra, $L$. While wedge degree provides a second gradation for the algebra
$\land W$, in general $(\land W,d)$ with this gradation is not a dga. However,
if $L=L_{even}$ then $W=W^{odd}$ and, with the wedge gradation $(\land W,d)$
is a quadratic Sullivan algebra.

For convenience, in this case we denote by $\widetilde{W}$ the graded vector
space defined by

  1. $\bullet$

$\widetilde{W}=W$ as a vector space, but

  2. $\bullet$

$\widetilde{W}=(\widetilde{W})^{1}$.

The gradation in the cdga $(\land\widetilde{W},d)$ is then the wedge degree.
Now denote the homotopy Lie algebra of $(\land\widetilde{W},d)$ by
$\widetilde{L}$. By construction, $W^{\vee}\subset\widetilde{W}^{\vee}$ and it
follows that

$sL=W^{\vee}\subset\widetilde{W}^{\vee}=s\widetilde{L}.$

This identifies $L$ as a closed sub Lie algebra of $\widetilde{L}$ in which
the inclusion ”forgets” the degrees in $L$.

######  Proposition 23.2.

With the hypotheses and notation above

  1. (i)

$L$ is profree if and only if $\widetilde{L}$ is profree.

  2. (ii)

If $\gamma:T\to L$ is a coherent linear map and
$\widetilde{\gamma}:T\to\widetilde{L}$ is the composition of $\gamma$ with the
inclusion $L\subset\widetilde{L}$ then

> $\gamma$ is inert $\Longleftrightarrow$ $\widetilde{\gamma}$ is inert.

proof. (i) According to Theorem 1, $L$ (respectively $\widetilde{L})$ is
profree if and only if
$W\cap\mbox{ker}\,d\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}H(\land^{\geq
1}W)$, respectively
$\widetilde{W}\cap\mbox{ker}\,d\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}H(\land^{\geq
1}\widetilde{W},d)$. But by definition the (non-degree preserving) isomorphism
$id:W\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\widetilde{W}$ extends
to a dga isomorphism $(\land
W,d)\stackrel{{\scriptstyle\cong}}{{\rightarrow}}(\land\widetilde{W},d)$. This
implies (i).

(ii) The Sullivan attaching maps corresponding to $\gamma$ and
$\widetilde{\gamma}$ are morphisms

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

which coincide if we forget degrees. Thus if either $\gamma$ or
$\widetilde{\gamma}$ is surjective, both are, and

$\mbox{ker}\,\rho\cap W=\\{w\in W\,|\,<w,sT>=0\\}\hskip 14.22636pt\mbox{ and
ker}\,\widetilde{\rho}\cap\widetilde{W}=\\{w\in\widetilde{W}\,|\,<w,sT>=0\\}.$

Now let $A=\mathbb{Q}\oplus\mbox{ker}\,\rho\cap W\oplus\land^{\geq 2}W$ and
let $\varphi:\land V\stackrel{{\scriptstyle\cong}}{{\to}}(A,d)$ b a minimal
Sullivan model. Forgetting degrees identifies $\land V$ as a minimal Sullivan
model of
$\widetilde{A}=\mathbb{Q}\oplus\mbox{ker}\,\widetilde{\rho}\cap\widetilde{W}\oplus\land^{\geq
2}\widetilde{W}$. Moreover, composing $\varphi$ with the inclusion $A\to\land
W$ yields

$\lambda:\land V\to\land W$

which is identified then with $\lambda:\land V\to\land\widetilde{W}$. In
particular,

$\lambda(V)\cap\land^{\geq 2}W=0\Longleftrightarrow\lambda(V)\cap\land^{\geq
2}\widetilde{W}=0,.$

This establishes (ii) and completes the proof of Step One.

$\square$

_Step Two. Reduction to the case $T=T_{0}$ and dim$\,T<\infty$._

By definition (§6), $T$ is equipped with a family of surjections
$\rho_{\alpha}:T\to T_{\alpha}$ onto finite dimensional vector spaces and

$\overline{\mathbb{L}}_{T}=\varprojlim_{\alpha}\overline{\mathbb{L}}_{T_{\alpha}}.$

Therefore, if $x\in\overline{\mathbb{L}}_{T}$ is non-zero then some
$x_{\alpha}=\overline{\mathbb{L}}_{\rho_{\alpha}}x$ is non-zero. On the other
hand, (§2), $T=\mbox{ker}\,\rho_{\alpha}\oplus Q$ with
$\rho_{\alpha}:Q\stackrel{{\scriptstyle\cong}}{{\to}}T_{\alpha}$. It is
immediate that
$\overline{\mathbb{L}}_{\mbox{ker}\,\rho_{\alpha}}\to\overline{\mathbb{L}}_{T}$
is inert and therefore by Proposition 23.1, $x$ is inert if and only if
$\rho_{\alpha}x$ is inert.

_Step Three. Proof when $T=T_{0}$ and dim$\,T<\infty$._

By definition, $\gamma$ is inert if and only if

$\overline{\mathbb{L}}_{T}\to
H(\overline{\mathbb{L}}_{T}\,\widehat{\amalg}\,{\mathbb{L}}_{\mathbb{Q}y})$

is surjective where the differential in
$\overline{\mathbb{L}}_{T}\,\widehat{\amalg}\,\mathbb{L}_{\mathbb{Q}y}$ is
defined by $\partial y=x$. Now fix a basis $x_{1},\dots,x_{n}$ of $T$ and
observe that
$\overline{\mathbb{L}}_{T}\,\widehat{\amalg}\,\mathbb{L}_{\mathbb{Q}y}$ is the
completion of the free graded Lie algebra,

$L=\mathbb{L}(x_{1},\dots,x_{n},y).$

Denote by $L(k)$ the subspace of $L$ of iterated Lie brackets of length $k$ in
the $x_{i}$ and $y$. Thus an element in
$(\overline{\mathbb{L}}_{T}\,\widehat{\amalg}\,\mathbb{L}_{\mathbb{Q}y})_{p}$
is an infinite sum of the form $z=\sum_{k\geq m}z_{k}$, with each $z_{k}\in
L(k)_{p}$. For convenience, given a fixed $m$, we denote the subspace of such
elements by $L(\geq m)_{p}$. In particular, for some $r\geq 1$,

$\partial y=\sum_{k\geq r}a_{k}$

with $a_{k}\in L(k)_{0}$ and $a_{r}\neq 0$.

Now consider the dgl $(L,\delta)$ with $\delta y=a_{r}$. For this dgl we can
modify the degrees without changing $L\to H(L,\delta)$, as follows: simply
assign deg $2$ to each $x_{i}$ and degree $2r+1$ to $y$. It follows from
([24], Theorem 3.12) that with this new gradation, $L\to H(L,\delta)$ is
surjective. This then remains true with the original gradation:

$L_{0}\to H(L,\delta)$

is also surjective and so $H_{\geq 1}(L,\delta)=0$.

Next, let $\omega$ be a cycle of degree $p\geq 1$ in
$\overline{\mathbb{L}}_{T}\,\widehat{\amalg}\,\mathbb{L}_{\mathbb{Q}y}$ and
write

$\omega=\sum_{k\geq m}\omega_{k},\hskip 11.38109pt\mbox{with
}\omega_{k}\in(L(k)_{p},$

and $\omega_{m}\neq 0$. Then $\delta\omega_{m}\in L(m+r-1)_{p-1}$ and

$\partial\omega-\delta\omega_{m}\in L(\geq m+r)_{p-1}.$

It follows that $\delta\omega_{m}=0$ and so $\omega_{m}$ is a
$\delta$-boundary:

$\omega_{m}=\delta\Phi_{m-r+1},\hskip 8.53581pt\mbox{some }\Phi_{m-r+1}\in
L(m-r+1)_{p+1}.$

But this in turn implies that

$\omega-\partial\Phi_{m-r+1}\in L(\geq m+1).$

Iterating this process yields a sequence $\Phi_{k-r+1}\in L(k-r+1)_{p+1}$ with

$\partial(\sum_{k}\Phi_{k-r+1})=\omega.$

$\square$

Corollary. Suppose $a,b\in\overline{\mathbb{L}}_{T}$ and $T=T^{even}$. If
$a\in\overline{\mathbb{L}}_{T}^{(2)}$, $b\in T$ and $a$ is not in the closed
ideal generated by $b$, then $b$ represents an inert element in
$\overline{\mathbb{L}}_{T}/I$, where $I$ is the closed ideal generated by $a$.

proof. It follows from the hypothesis that
$\overline{\mathbb{L}}_{T}=\mathbb{L}_{\mathbb{Q}b}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T^{\prime}}.$
Thus if $J$ is the closed ideal generated by $b$, then
$\overline{\mathbb{L}}_{T}/J\cong\overline{\mathbb{L}}_{T^{\prime}}$. Hence
(Theorem 10) $a$ represents an inert element in $\overline{\mathbb{L}}_{T}/J$.
Similarly, $b$ is inert in $\overline{\mathbb{L}}_{T}$. Thus (Proposition
22.1) $\mathbb{Q}a\oplus\mathbb{Q}b$ is an inert subspace of
$\overline{\mathbb{L}}_{T}$. Now (again by Proposition 22.1) $b$ represents an
inert element in $\overline{\mathbb{L}}_{T}/(a).$

$\square$

####  23.3 Free products

######  Proposition 23.3.

Suppose $Q\subset L$ and $Q^{\prime}\subset L^{\prime}$ are inert subspaces of
complete enriched Lie algebras. Then

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

is inert.
In particular, $Q\subset L\,\widehat{\amalg}\,L^{\prime}$ is inert.

proof. By hypothesis we have dgl’s
$(L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{N},\partial)$ and
$(H(L^{\prime}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{N^{\prime}},\partial^{\prime})$
with $\partial:N\stackrel{{\scriptstyle\cong}}{{\to}}Q$ and
$\partial^{\prime}:N^{\prime}\stackrel{{\scriptstyle\cong}}{{\to}}Q^{\prime}$
and

$H(L)\to H^{\prime}L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{N})\hskip
8.53581pt\mbox{ and }H(L^{\prime})\to
H(L^{\prime}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{N^{\prime}})$

both surjective.

Since $H(L\,\widehat{\amalg}\,L^{\prime})\cong
H(L)\,\widehat{\amalg}\,H(L^{\prime})$ this is a direct consequence of the
following commutative diagram in which $\psi$ is surjective (Proposition
10.1(iii)),

$\textstyle{H(L\,\widehat{\amalg}\,L^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{H(L\,\widehat{\amalg}\,L^{\prime}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{N}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{N^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{H(L)\,\widehat{\amalg}\,H(L^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{H(L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{N},\partial)\,\widehat{\amalg}\,H(L^{\prime}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{N^{\prime}},\partial^{\prime}).}$

$\square$

####  23.4 Semi-direct products and weighted Lie algebras

A closed ideal, $I$, in a complete enriched Lie algebra, $L$, determines (§3)
the weighted Lie algebra $g(I):=\oplus_{n\geq 1}I^{(n)}/I^{(n+1)}$. Then
$g(I)$ is an enriched Lie algebra with completion

$\overline{g(I)}=\prod_{n\geq 1}I^{(n)}/I^{(n+1)}.$

Moreover, the right adjoint representation of $L$ induces representations of
$L/I$ in each $I^{(n)}/I^{(n+1)}$. The resulting semi-direct product

$g(I,L):=g(I)\,\widehat{\times}\,L/I$

inherits a natural enriched structure with completion the semi-direct product
$\overline{g(I)}\,\widehat{\times}\,L/I$.

_Henceforth in this section we fix such a pair $I\subset L$ and assume further
that_

$I=\overline{\mathbb{L}}_{T}$

_is a profree Lie algebra._ Then Proposition 6.4 provides closed subspaces
$\overline{T(k)}\subset I^{(k)}$, $k\geq 1$, satisfying

$T=T(1),\hskip 8.53581ptI^{(k)}=\overline{T(k)}\oplus I^{(k+1)},\hskip
8.53581pt\mbox{and
}[\overline{T(k)},\overline{T(\ell)}]\subset\overline{T(k+\ell)}.$

This identifies $\overline{T(k)}=I^{(k)}/I^{(k+1)}$, and $g(I)=\oplus_{k\geq
1}\overline{T(k)}$. Moreover (Proposition 6.4(iii))

$\overline{g(I)}=\prod_{k\geq 1}\overline{T(k)}=\overline{\mathbb{L}}_{T}=I.$

In particular, the construction above defines the semi-direct product

$I\,\widehat{\times}\,L/I=\overline{g(I)}\,\widehat{\times}\,L/I.$

Finally suppose, in addition to the notation and hypotheses above, that $Q$ is
a closed subspace of $T$ and which generates the ideal
$I=\overline{\mathbb{L}}_{T}$. The inclusion extends to a morphism
$\psi:\overline{\mathbb{L}}_{Q}\rightarrow\overline{\mathbb{L}}_{T}$ and this
extends to the morphism

$\varphi:=\psi\,\widehat{\amalg}\,id_{L/T}:\overline{\mathbb{L}}_{Q}\,\widehat{\amalg}\,L/I\to\overline{\mathbb{L}}_{T}\,\widehat{\times}\,L/I.$

######  Proposition 23.4.

With the hypotheses and notation above the following conditions are equivalent
:

  1. (i)

$Q$ is inert in $L$

  2. (ii)

$Q$ is inert in $\overline{\mathbb{L}}_{T}\,\widehat{\times}\,L/I$.

  3. (iii)

$\varphi$ is an isomorphism.

proof. (i) $\Leftrightarrow$ (ii) By Theorem 8, $Q$ is inert in $L$ if and
only if the right adjoint representation of $L/I$ in $I/I^{(2)}$ induces an
isomorphism

$Q\,\widehat{\otimes}\,\overline{UL/I}\longrightarrow I/I^{(2)}.$

But by definition this is the right representation of $L/I$ in
$\overline{\mathbb{L}}_{T}/\overline{\mathbb{L}}_{T}^{(2)}$ resulting from the
semi-direct product, and so (i) $\Leftrightarrow$ (ii) follows from Theorem 8.

(ii) $\Leftrightarrow$ (iii) First observe (Corollary 2 to Theorem 7) that $Q$
is inert in $\overline{\mathbb{L}}_{Q}$. Therefore (Proposition 23.3) $Q$ is
inert in $\overline{\mathbb{L}}_{Q}\,\widehat{\amalg}\,L/I$. The kernel of the
surjection

$\overline{\mathbb{L}}_{Q}\,\widehat{\amalg}\,L/I\to L/I$

is the ideal $J$ generated by $Q$. Since $Q$ is inert in the free product, and
since it generates the ideal $J$, it follows from Theorem 8 that the right
representation of $L/I$ in $J/J^{(2)}$ defines an isomorphism

$Q\,\widehat{\otimes}\,\overline{UL/I}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}J/J^{(2)}.$

On the other hand, since $\varphi$ is a morphism of enriched Lie algebras it
restricts to a morphism $J\to I$. This in turn induces a morphism

$\chi:J/J^{(2)}\to I/I^{(2)}$

of $L/I$-modules, in which $\chi_{|Q}=id_{Q}$. In view of the observations
above, $\chi$ may be identified with the morphism

$Q\,\widehat{\otimes}\,\overline{UL/I}\to I/I^{(2)}$

induced by the right representation of $L/I$ in $I/I^{(2)}$. Thus $Q$ is inert
in $L$ if and only if $\chi$ is an isomorphism.

Finally, since $J$ is the ideal generated by $Q$ it follows from Theorem 8
that $J$ is profree. Thus $\varphi:J\to I$ is a morphism of profree Lie
algebras, and is therefore an isomorphism if and only if it induces an
isomorphism
$J/J^{(2)}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}I/I^{(2)}$.
$\square$

##  Part VII Applications in topology

###  24 Characterization of rationally inert topological attaching maps

Recall (§19) that a topological attaching map

$g:\vee_{\alpha}S^{n_{\alpha}}\to X$

is _rationally inert_ if

$\pi_{*}(\iota_{X})_{\mathbb{Q}}:\pi_{*}(X_{\mathbb{Q}})\to\pi_{*}(X\cup_{g}\vee_{\alpha}D^{n_{\alpha}+1})_{\mathbb{Q}}$

is surjective. Moreover, if $\lambda:\land V\to\land W$ is a Sullivan
representative of $\iota_{X}:X\to X\cup_{g}D^{n_{\alpha}+1}$, then (18)

$\pi_{*}(\iota_{X})_{\mathbb{Q}}=\pi_{*}\langle\lambda\rangle=sL_{\lambda}.$

It follows that $g$ is rationally inert if and only if $L_{\lambda}$ is
surjective.

Next, let $L_{X}$ denote the homotopy Lie algebra of $X$, and recall that
associated with $g$ are two Lie algebra morphisms:

  1. $\bullet$

A dgl representative $\ell_{g}:(\overline{\mathbb{L}}_{T},0)\to(L,\partial)$
of $g$.

  2. $\bullet$

The desuspension $L_{g}:\overline{\mathbb{L}}_{T}\to L_{X}$ of
$\pi_{*}(g_{\mathbb{Q}})$.

It follows from (63) that $L_{g}=H(\ell_{g})$. Thus Proposition 19.1 and
Theorem 6 combine to yield the equivalence of the three assertions:

  1. $\bullet$

$g$ is rationally inert.

  2. $\bullet$

$\ell_{g}$ is inert.

  3. $\bullet$

$L_{g}$ is inert.

On the other hand recall from (97) the decomposition

$\land V\stackrel{{\scriptstyle\eta}}{{\longrightarrow}}\land V\otimes\land
Z\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\land W$

of the minimal Sullivan model, $\land W$, of $X$. This in turn ([17,
Proposition 17.9]) provides a fibration

(120)

By Proposition 17.6, if $\rho:\land W\to\mathbb{Q}\oplus S$ is surjective then
$\langle\land Z\rangle$ is homotopy equivalent to the homotopy fibre of
$(\iota_{X})_{\mathbb{Q}}$.

######  Theorem 10.

With the hypotheses and notation above, if $\rho$ is surjective then

$g$ is rationally inert $\Longleftrightarrow$ $\langle\land Z\rangle$ is
rationally wedge-like.

proof. If $g$ is rationally inert then (Theorem 7) $\land V\otimes\land
Z\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\land W$ and the homotopy
Lie algebra, $L_{Z}$, is profree. By definition (§10.4) $\langle\land
Z\rangle$ is rationally wedge-like.

In the reverse direction we need to show that if $g$ is not rationally inert
then $\langle\land Z\rangle$ is not rationally wedge-like. The proof is
essentially the same as the proof that (iii) $\Rightarrow$ (ii) in Theorem 7.
If $g$ is not rationally inert then $\pi_{*}(\iota_{X})_{\mathbb{Q}}$ is not
surjective. Thus in the long exact homotopy sequence for (120) the connecting
homomorphism

$\partial:\pi_{*}\langle\land V\rangle\to\pi_{*-1}\langle\land Z\rangle$

is not trivial. In particular for some $0\neq y\in\pi_{*-1}\langle\land
Z\rangle$, $y=\partial x$. A standard argument from classical homotopy theory
shows that Whitehead products

$Wh(\partial x,z),\hskip 8.53581ptz\in\pi_{*}\langle\land Z\rangle$

all vanish.

On the other hand, if $\langle\land Z\rangle$ is rationally wedge-like then we
would have $\langle\land Z\rangle\simeq\langle\land Q\rangle$ with the
homotopy Lie algebra $L_{Q}$ of $\land Q$ a profree Lie algebra. Thus $y$
would correspond to an element $y^{\prime}\in\pi_{*}\langle\land Q\rangle$ and
all the Whitehead products $y^{\prime}*z^{\prime}$,
$z^{\prime}\in\pi_{*}\langle\land Q\rangle$ would vanish. Now it would follow
from Proposition 8.2 that $y^{\prime}$ would correspond to a non-zero element
in the centre of $L_{Q}$. Since $L_{Q}$ is profree this would imply that
$L_{Q}$ is a free Lie algebra on a single generator.

Finally, since the isomorphism $\pi_{*}\langle\land
Q\rangle\cong\pi_{*}\langle\land Z\rangle$ preserves Whitehead products, it
would also follow that the homotopy Lie algebra $L_{Z}$ is a free Lie algebra
on a single generator, so that necessarily dim$\,H^{1}(\land Z)\leq 1$. In
this case the proof of (iii) $\Rightarrow$ (ii) in Theorem 7 shows that $g$ is
rationally inert. This establishes the Theorem. $\square$

###  25 Poincaré duality complexes

We say a CW complex $Y=X\cup_{f}D^{n+1}$ is a _rational Poincaré duality
complex_ if $H(Y)$ is a Poincaré duality algebra and the top class is in the
image of $H(Y,X)$. In this case it follows that $H^{\leq
n}(X)\stackrel{{\scriptstyle\cong}}{{\to}}H(X)$.

######  Theorem 11.

If $Y=X\cup_{f}D^{n+1}$ is a rational Poincaré duality complex and the algebra
$H(Y)$ requires at least two generators, then $f$ is rationally inert.

This Theorem was first established for simply connected spaces in [24, Theorem
5.1].

proof. Recall from §19 that $f$ is rationally inert if

$\pi_{*}(i_{\mathbb{Q}}):\pi_{*}(X_{\mathbb{Q}})\to\pi_{*}(X\cup_{f}D^{n+1})_{\mathbb{Q}}$

is surjective. Now let $\lambda:\land V\to\land W$ be a Sullivan
representative of $i:X\to X\cup_{f}D^{n+1}$ and then decompose $\land W$ as a
$\Lambda$-extension

$\textstyle{\land
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{\land
V\otimes\land
Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi}$$\textstyle{\land
Z}$

as described in (97). Then according to Theorem 7 $f$ is rationally inert if
and only if the homotopy Lie algebra, $L_{Z}$, is profree, i.e. (Theorem 1)

$\land Z\simeq(\mathbb{Q}\oplus S,0)$

with $S\cdot S=0$.

Now let $\land V\otimes\land U$ be the acyclic closure of $\land V$. Then we
have

$\land W\otimes\land U:=\land W\otimes_{\land V}(\land V\otimes\land
U)\simeq\land Z.$

Therefore we have only to prove that

$\land W\otimes\land U\simeq(\mathbb{Q}\oplus S,0)$

with $S\cdot S=0$.

For this, we first establish the following notation. Let $P$ be a direct
summand in $(\land V)^{n+1}$ of $(\land V)^{n+1}\cap\mbox{ker}\,d$. Then
division by $P$ and by $(\land V)^{>n+1}$ defines a surjective quasi-
isomorphism $\land V\stackrel{{\scriptstyle\simeq}}{{\to}}A$, and

$A^{n+1}=A^{n+1}\cap\mbox{Im}\,d\oplus\mathbb{Q}\omega,$

where $\omega$ is a cycle representing the top cohomology class of $Y$.

In the same way, let $P^{\prime}$ be a direct summand in $(\land W)^{n}$ of
$(\land W)^{n}\cap\mbox{ker}\,d$. Then division by $P^{\prime}$ and $(\land
W)^{>n}$ defines a surjective quasi-isomorphism $\land
W\stackrel{{\scriptstyle\simeq}}{{\to}}B$. Moreover, by construction $\varphi$
factorizes in a commutative diagram,

$\textstyle{\land
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\varphi}$$\textstyle{\land
W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\varphi}}$$\textstyle{B,}$

in which $\overline{\varphi}\omega=0$.

Now consider the commutative cdga diagram

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\varphi}}$$\scriptstyle{j}$$\textstyle{B}$$\textstyle{A\oplus\mathbb{Q}t\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{\simeq}$

in which $dt=\omega$, $t\cdot A^{\geq 1}=0$, $t^{2}=0$, and $\psi t=0$. Here
$\psi$ is a quasi-isomorphism and $j$ is a cdga representative of $\varphi$.
This provides a cdga quasi-isomorphism

$\land W\otimes\land U\simeq(A\oplus\mathbb{Q}t)\otimes_{A}(A\otimes_{\land
V}(\land V\otimes\land U))=(A\oplus\mathbb{Q}t)\otimes\land U.$

Thus from the short exact sequence of chain complexes

$0\to A\otimes\land U\to(A\oplus\mathbb{Q}t)\otimes\land
U\to\mathbb{Q}t\otimes\land U\to 0,$

we deduce a linear isomorphism

$H^{\geq 1}((A\oplus\mathbb{Q}t)\otimes\land
U)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathbb{Q}t\otimes\land U$

of graded vector spaces. It remains to show that we can lift a basis of
$\mathbb{Q}t\otimes\land U$ to cycles
$\Phi_{i}\in(A\oplus\mathbb{Q}t)\otimes\land U$ such that
$\Phi_{i}\cdot\Phi_{j}=0$.

Before going further we first eliminate two special cases. First if $V^{1}=0$
the argument of ([24, §5]) shows that $(A\oplus\mathbb{Q}t)\otimes\land U$ is
rationally wedge-like, and so $f$ is rationally inert. (Note that in [24] it
is assumed that $X$ is simply connected; however the proof of this assertion
relies only on the fact that $V^{1}=0$.) Secondly, if $n=1$ then
$X\simeq_{\mathbb{Q}}S^{1}_{1}\vee\dots\vee S^{1}_{2q}$ and so $Y$ is
rationally equivalent to an oriented Riemann surface. In this case Theorem 12
is established in [15].

Thus we may now assume that $n\geq 2$ and that $A^{1}$ contains a non-zero
cycle $v$. Since $H(A)$ is a Poincaré duality algebra there is a cycle $w\in
A^{n}$ such that $wv=\omega$. The first step for the proof is then

######  Lemma 25.1.

With the hypotheses and notation above, $A^{n+1}\otimes\land U\subset
d(A^{n}\otimes\land U)$.

proof: Choose $\overline{v}\in U^{0}$ so that $d\overline{v}=v$. Since $\land
V$ is a minimal Sullivan algebra, $V$ is the union of an increasing sequence
of subspaces $V(0)\subset\dots\subset V(q)\subset\dots$ in which
$V(0)=\mathbb{Q}v$ and $d:V(q+1)\to\land V(q)$. It follows that $U$ is the
union of an increasing sequence of subspaces $U(0)\subset\dots\subset
U(q)\subset\dots$ in which $U(0)=\mathbb{Q}\overline{v}$ and

$d:U(q+1)\to A^{\geq 1}\otimes\land U(q).$

We show by induction on $q$ that

$A^{n+1}\otimes\land U(q)\subset d(A^{n}\otimes\land U(q))$

First note that any $z\in A^{n+1}$ has the form $z=dy+\lambda wv$, some
$\lambda\in\mathbb{Q}$. Thus

$z\otimes 1=d(y\otimes 1)+(-1)^{n}d(\lambda w\otimes\overline{v})\in
d(A^{n}\otimes\land U(0)).$

Then for $r\geq 1$,

$z\otimes\overline{v}^{r}=d(y\otimes\overline{v}^{r}+\frac{(-1)^{n}\lambda}{r+1}w\otimes\overline{v}^{r+1})-(-1)^{n}ryv\otimes\overline{v}^{r-1}.$

It follows by induction on $r$ that $A^{n+1}\otimes\land U(0)\subset
d(A^{n}\otimes\land U(0)).$

Now fix a direct summand, $T$, of $U(q)$ in $U(q+1)$, and assume by induction
that for some $s$,

$A^{n+1}\otimes\land U(q)\otimes\land^{\leq s}T\subset d(A^{n}\otimes\land
U(q)\otimes\land^{\leq s}T).$

Then write $\Phi\in A^{n+1}\otimes\land U(q)\otimes\land^{\leq s+1}T$ as
$\Phi=\sum\Phi_{i}\otimes\Psi_{i}$ with $\Phi_{i}\in A^{n+1}\otimes\land U(q)$
and $\Psi_{i}\in\land^{\leq s+1}T$. By the hypothesis $\Phi_{i}=d\Omega_{i}$
with $\Omega_{i}\in A^{n}\otimes\land U(q)$. Therefore

$\sum\Phi_{i}\otimes\Psi_{i}=d(\sum\Omega_{i}\otimes\Psi_{i})-(-1)^{n}\sum\Omega_{i}\land
d\Psi_{i}.$

The first term is in $d(A^{n}\otimes\land U(q)\otimes\land^{\leq s+1}T)$. On
the other hand, $d\Psi_{i}\in A^{\geq 1}\otimes\land U(q)\land^{\leq s}T$ and
so the second term is in $A^{n+1}\otimes\land U(q)\otimes\land^{\leq s}T$. By
hypothesis, the second term is contained in $d(A^{n}\otimes\land
U(q)\otimes\land^{\leq s}T)$. This closes the induction. $\square$

Now, let $\Phi\in\land U$. Then

$t-(-1)^{n}w\overline{v}\in(A\oplus\mathbb{Q}t)\otimes\land U$

is a cycle, and

$d((t-(-1)^{n}w\overline{v})\Phi)=-w\overline{v}\,d\Phi\in A^{n+1}\otimes\land
U.$

By Lemma LABEL:l24.1, $w\overline{v}\,d\Phi=d\Psi$ for some $\Psi\in
A^{n}\otimes\land U$. Thus $(t-(-1)^{n})w\overline{v})\Phi+\Psi$ is a cycle
projecting to $t\otimes\Phi$ in $\mathbb{Q}t\otimes\land U$. Thus such cycles
map to a basis of $\mathbb{Q}t\otimes\land U$. But because $n\geq 2$, $2n>n+1$
and so the product of any two of those cycles is zero. This shows that $f$ is
rationally inert. $\square$

###  26 Whitehead’s problem

A famous unsolved problem of JHC Whitehead [46] asks: is a subcomplex of an
aspherical two-dimensional CW complex aspherical ? As observed by Anick [2] it
is sufficient to consider the case that both subcomplexes share the same
$1$-skeleton and base point. The problem then reduces to the question: If $X$
is a finite 2-dimensional connected CW complex and
$X\cup\left(\vee_{k=1}^{p}D_{k}^{2}\right)$ is aspherical, is $X$ aspherical ?

In [2] Anick provides a positive answer to an analogous question for simply
connected rational spaces. Here we have a positive answer for Sullivan spatial
completions of connected spaces.

######  Theorem 12.

If $X$ is a connected CW complex and $(X\cup\vee_{k=1}^{p}D^{2})_{\mathbb{Q}}$
is aspherical then $X_{\mathbb{Q}}$ is aspherical.

proof: The obvious induction reduces the statement to the case $p=1$. Then,
since $\pi_{*}((X\cup_{f}D^{2})_{\mathbb{Q}})\cong V^{\vee}$ as sets where
$\land V$ is the minimal Sullivan model of $X\cup_{f}D^{2}$, our hypothesis
simply implies that $V=V^{1}$. Let $\varphi:(\land V,d)\to(\land W,d)$ be a
Sullivan representative for the inclusion $i:X\to X\cup_{f}D^{2}$. Since
$H^{1}(X\cup_{f}D^{2})\to H^{1}(X)$ is injective, it follows that $\varphi$ is
injective and so $\land W$ decomposes as $\land V\otimes\land Z$, with
$Z=Z^{\geq 1}$. In particular $f$ is rationally inert. Moreover, it follows
from Theorem 7 together with (36) that

$H^{\geq 1}(\land Z,\overline{d})\cong\mathbb{Q}b\otimes\land U,$

where deg$\,b=1$ and $\land V\otimes\land U$ is the acyclic closure of $\land
V$. Since $V=V^{1}$, $U=U^{0}$ and $H^{\geq 1}(\land
Z,\overline{d})=H^{1}(\land Z,\overline{d})$. This in turn implies $Z=Z^{1}$
and $X_{\mathbb{Q}}$ is aspherical.

$\square$

###  27 Homotopy Lie algebra of a 2-cone

A $2$-cone is a space $Y=\vee_{j}S^{n_{j}}\cup_{g}\vee_{i}D^{k_{i}+1}$
constructed from an attaching map $g:\vee_{i}S^{k_{i}}\to
X=\vee_{j}S^{n_{j}}$. We denote by

$\iota_{X}:X=\vee_{j}S^{n_{j}}\to X\cup_{g}\vee_{i}D^{k_{i}+1}=Y$

the inclusion.

######  Proposition 27.1.

Suppose $\iota_{X}:X\to Y$ is a two cone, and denote by $E$ the image of
$L_{X}$ in $L_{Y}$. Assume that $H^{\geq 1}(g)=0$ and that
$\pi_{*}(g_{\mathbb{Q}})$ is injective. Then there is a retraction $r:L_{Y}\to
E$ whose kernel is a profree Lie algebra $K$, i.e. there is a short exact
sequence

$0\to K\to L_{Y}\to E\to 0.$

In particular, the attaching map $g$ is rationally inert if and only if $K=0$,
and otherwise $L_{Y}$ contains a profree Lie algebra.

proof. Let $(\land W,d)$ be a quadratic model of $X$ and let $\rho:W\to S$ the
map associated to $g$. Then (Proposition 17.5) a cdga model of $Y$ is given by
the cdga

$(\land W\oplus s^{-1}S,D).$

We equip $\land W\oplus s^{-1}S$ with a new gradation $\land W\oplus
s^{-1}S=\oplus_{k\geq 0}(\land W\oplus s^{-1}S)^{(k)}$ with $W=W^{(1)}$ and
$s^{-1}S=(s^{-1}S)^{(2)}$. Then $D$ increases this gradation by $1$. Therefore
the cohomology of $\land W\oplus s^{-1}S$ inherits a new gradation

$H^{*}(\land W\oplus s^{-1}S,D)=\oplus_{p\geq 0}H^{(p)}.$

The short exact sequence

$0\to(s^{-1}S,0)\to(\land W\oplus s^{-1}S,D)\to(\land W,d)\to 0$

shows that $H^{(p)}=0$ for $p>2$. On the other hand, clearly,
$H^{(0)}=\mathbb{Q}$ and, since $X$ is a wedge of spheres, $H^{(1)}=H^{\geq
1}(X)$.

Now, let $\varphi:(\land
V,d)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(\land W\oplus
s^{-1}S,D)$ be its Sullivan minimal model. We can suppose $(\land V,d)$
equipped with a new gradation preserved by $\varphi$. In particular, $(\land
V^{(1)},d)$ is a sub cdga and its homotopy Lie algebra is isomorphic to $E$.
The injection $(\land V^{(1)},d)\to(\land V,d)$ induces the surjection
$L_{X}\to E$.

Denote by $(\land V^{(1)}\otimes\land U,d)$ be the acyclic closure of $(\land
V^{(1)},d)$ equipped with the new gradation given by $U=U^{(0)}$. Then let
$\land Z$ be the Sullivan minimal model of $(\land V\otimes\land U,d)$. Then
$Z=\oplus_{p\geq 2}V^{(p)}$, and since $H(\land V\otimes\land
U,d)=\mathbb{Q}\oplus H^{(2)}(\land V\otimes\land U,d)$, the cdga $(\land
Z,d)$ is quasi-isomorphic to $(\mathbb{Q}\oplus(Z\cap\mbox{ker}\,d),0)$. Thus
$L_{Z}$ is profree and we get the required extension

$0\to L_{Z}\to L_{Y}\to E\to 0.$

$\square$

###  28 Wedges of circles

####  28.1 Inertness

For simplicity we say that $[f]\in\pi_{1}(X)$ is rationally inert if
$f:S^{1}\to X$ is rationally inert.

######  Theorem 13.

If $X$ is a wedge of at least two circles then any non-zero $[f]\in\pi_{1}(X)$
is rationally inert or, equivalently, $(X\cup_{f}D^{2})_{\mathbb{Q}}$ is
aspherical.

proof: It follows from Proposition 10.6 and its proof that the homotopy Lie
algebra, $L_{X}$ is profree and concentrated in degree zero. Thus by Theorem
10, every non-zero element in $L_{X}$ is inert. It is therefore sufficient to
show that $\pi_{1}(X)\to\pi_{1}(X_{\mathbb{Q}})$ is injective. Moreover, any
element of $\pi_{1}(X)$ is in the image of $\pi_{1}(Y)$ for some finite sub
wedge of circles $Y\subset X$. Then $Y$ (and hence $Y_{\mathbb{Q}}$) are
retracts of $X$ and $X_{\mathbb{Q}}$ and so it is sufficient to prove that
each element $[g]\in\pi_{1}(Y)$ is rationally inert.

For this, denote $G=\pi_{1}(Y)$, so that
$G_{\mathbb{Q}}=\pi_{1}(Y_{\mathbb{Q}})$. According to [19, Theorem 7.5],
$G^{n}/G^{n+1}\otimes\mathbb{Q}\stackrel{{\scriptstyle\cong}}{{\to}}G^{n}_{\mathbb{Q}}/G^{n+1}_{\mathbb{Q}}$.
But by [22], $G^{n}/G^{n+1}$ is a free abelian group, and hence
$G^{n}/G^{n+1}\to G^{n}_{\mathbb{Q}}/G^{n+1}_{\mathbb{Q}}$ is injective. Since
$G$ is a free group, $G\to\varprojlim_{n}(G/G^{n})_{\mathbb{Q}}$ is injective
and the image of $[g]$ in $G_{\mathbb{Q}}$ is non-zero.

Finally since any non-trivial $[f]\in\pi_{*}(X)$ is rationally inert it
follows by definition that

$\pi_{1}(X_{\mathbb{Q}})\to\pi_{*}((X\cup_{f}D^{2})_{\mathbb{Q}})$

is surjective. Since $\pi_{*}(X_{\mathbb{Q}})$ is concentrated in degree 1,
$(X\cup_{f}D^{2})_{\mathbb{Q}}$ is aspherical.

$\square$

####  28.2 A connected finite CW complex, $Y$, whose homotopy Lie algebra,
$L_{Y}$ is not nilpotent and satisfies dim$\,(L_{Y}/L_{Y}^{2})_{1}=\infty$.

Step One. Construction of $Y$ and a first dgl model

In [32] Lemaire constructs a simply connected finite CW complex, $X$, whose
homotopy Lie algebra, $L_{X}$, is not finitely generated. Since $L_{X}$ is a
graded vector space of finite type, the same is true for $L_{X}/L_{X}^{2}$,
and so the generators occur in infinitely many degrees.

Lemaire’s example is constructed by attaching seven cells to a wedge of five
spheres. Here we use a similar process to construct a CW complex, $Y$ by
attaching seven $2$-cells $D_{a_{i}}^{2}$ to a wedge of five circles
$S^{1}_{b_{j}}$ via a map

$g:\vee_{i=1}^{7}S^{1}_{a_{i}}\longrightarrow\vee_{j=1}^{5}S^{1}_{b_{j}}.$

The respective fundamental groups are free groups $G_{a}$ and $G_{b}$ freely
generated by generators $a_{1},\dots,a_{7}$ and $b_{1},\dots,b_{5}$.

We denote the commutators in $G_{b}$ by

$(c,d)=cdc^{-1}d^{-1}.$

Then $g$ is defined by the homomorphism $\pi_{1}(g)$ given explicitly by

$a_{1}\mapsto(b_{1},b_{3}),a_{2}\mapsto(b_{1},b_{4}),a_{3}\mapsto(b_{2},b_{3}),a_{4}\mapsto(b_{2},b_{4}),\hskip
8.53581pt\mbox{and }$
$a_{5}\mapsto(b_{5},b_{1}b_{3}^{-1}),a_{6}\mapsto(b_{5},b_{1}b_{4}^{-1}),a_{7}\mapsto(b_{5},b_{2}b_{3}^{-1}).$

Now set $X=\vee_{j=1}^{5}S^{1}_{b_{j}}$ and define

$Y=X\cup_{g}\vee_{i=1}^{7}D^{2}_{a_{i}}.$

As usual the attachment is denoted by $\iota_{X}:X\to
X\cup_{g}\vee_{i=1}^{7}D^{2}_{a_{i}}=Y.$

We now combine Proposition 9.4 and the Example in §17.1 to construct a first
dgl model of $Y$. As in §9.1, denote by $\log_{G_{b}}$ the map

$\log_{G_{b}}:G_{b}\longrightarrow
G_{L_{X}}\stackrel{{\scriptstyle\log}}{{\longrightarrow}}L_{X},$

recalling that $\log$ is a bijection of sets. Then (Proposition 9.4(i))
$L_{X}$ is a profree Lie algebra $\overline{\mathbb{L}}_{T(b)}$ in which the
elements $\log_{G_{b}}(b_{j})$ are a basis of $T(b)$. Then (§17.1) a dgl
representative of $\iota_{X}$ is given by

$(L_{X},0)\to(L_{X}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{sT(a)},\partial),$

in which the elements $\log_{G_{a}}(a_{i})$ are a basis of $T(a)$ and

$\partial\,s\log_{G_{a}}(a_{i})=\log_{G_{b}}(\pi_{1}(g)(a_{i})).$

_In particular,
$(L_{X}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{sT(a)},\partial)$ is a dgl
model for $Y$. _

To simplify notation we set

$\log_{G_{b}}b_{j}=x_{j}\hskip 14.22636pt\mbox{ and
}\log_{G_{a}}(a_{i})=y_{i}.$

Then the dgl morphism above can be written as

$(L_{X},0)\to
L_{X}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{sT(a)}=\overline{\mathbb{L}}(x_{1},\dots
x_{5})\,\widehat{\amalg}\,\overline{\mathbb{L}}(sy_{1},\dots,sy_{7})$

in which $\partial(sy_{i})=\log_{G_{b}}(\pi_{1}(g)(a_{i}))$. Specifically,

$\partial sy_{1}=\log_{G_{b}}(b_{1},b_{3}),\,\partial
sy_{2}=\log_{G_{b}}(b_{1},b_{4}),\,\partial
sy_{3}=\log_{G_{b}}(b_{2},b_{3}),\,\partial sy_{4}=\log_{G_{b}}(b_{2},b_{4})$

and

$\partial sy_{5}=\log_{G_{b}}(b_{5},b_{1}b_{3}^{-1}),\,\partial
sy_{6}=\log_{G_{b}}(b_{5},b_{1}b_{4}^{-1}),\hskip 8.53581pt\mbox{and }\partial
sy_{7}=\log_{G_{b}}(b_{5},b_{2}b_{3}^{-1}).$

Step Two. A second dgl model of $Y$.

We define a second dgl model,
$(L_{X}\,\widehat{\amalg}\,\overline{\mathbb{L}}(sy_{1},\dots,sy_{7}),\partial^{\prime})$
by setting

$\partial^{\prime}sy_{1}=[x_{1},x_{3}],\partial^{\prime}sy_{2}=[x_{1},x_{4}],\partial^{\prime}sy_{3}=[x_{2},x_{3}],\partial^{\prime}sy_{4}=[x_{2},x_{4}],\hskip
8.53581pt\mbox{and}$
$\partial^{\prime}sy_{5}=[x_{5},x_{1}-x_{3}],\partial^{\prime}sy_{6}=[x_{5},x_{1}-x_{4}]\hskip
8.53581pt\mbox{and }\partial^{\prime}sy_{7}=[x_{5},x_{2}-x_{3}].$

We then identify this as a dgl model for $Y$ by constructing a commutative
diagram of dgl morphisms

|
---|---
$\textstyle{(L_{X}\,\widehat{\amalg}\,\overline{\mathbb{L}}(sy_{1},\dots,sy_{7}),\partial)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\psi}$$\textstyle{(L_{X},0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(L_{X}\,\widehat{\amalg}\,\overline{\mathbb{L}}(sy_{1},\dots,sy_{7}),\partial^{\prime}).}$

For this, as in §9.1, denote by $\overline{J}$ the augmentation ideal of
$\overline{UL_{X}}$. Then the left and right adjoint representations of
$L_{X}$ extend to representations of $\overline{UL_{X}}$. In particular, if
$u\in L_{X}$ and $\alpha\in\overline{J}$ then

$u\cdot\alpha=\alpha^{t}\cdot u,$

where if $\alpha=\alpha_{1}\dots\alpha_{k}$ with $\alpha_{i}\in L_{X}$ then

$\alpha^{t}=(-1)^{k}\alpha_{k}\dots\alpha_{1}.$

More generally, if $\alpha=\sum\alpha_{k}$, with
$\alpha_{k}\in\overline{J}^{k}$ then $\alpha^{t}=\sum\alpha_{k}^{t}$.

Similarly, if $\alpha=\alpha_{1}\dots\alpha_{k}\in\overline{J}$, with
$\alpha_{i}\in L_{X}$, we define $\alpha\cdot sy_{j}\in
L_{X}\,\widehat{\amalg}\,\overline{\mathbb{L}}(sy_{j})$ by setting

$\alpha\cdot sy_{j}=[\alpha_{1},[\alpha_{2},\dots[\alpha_{k},sy_{j}]\dots]].$

It is then immediate that for any $u\in L_{X}$, $\alpha\in\overline{J}$ and
$1\leq i\leq 7,$

$[u\cdot\alpha,sy_{i}]=[u,\alpha\cdot sy_{i}].$

We now define $\psi$. It is immediate from Proposition 9.4(ii) that for some
$\alpha_{1},\dots,\alpha_{4}\in\overline{J}$,

$\partial sy_{1}=[x_{1},x_{3}]+[x_{1},x_{3}]\cdot\alpha_{1},\hskip
14.22636pt\partial sy_{2}=[x_{1},x_{4}]+[x_{1},x_{4}]\cdot\alpha_{2},$

and

$\partial sy_{3}=[x_{2},x_{3}]+[x_{2},x_{3}]\cdot\alpha_{3},\hskip
14.22636pt\partial sy_{4}=[x_{2},x_{4}]+[x_{2},x_{4}]\cdot\alpha_{4}.$

We define

$\psi(sy_{i})=sy_{i}+\alpha_{i}\cdot sy_{i},\hskip 14.22636pt1\leq i\leq 4.$

Next, observe that for some $\beta$ and $\alpha_{5}\in\overline{J}$,

$\log_{G_{b}}(b_{1}b_{3}^{-1})=x_{1}-x_{3}-\frac{1}{2}[x_{1},x_{3}]+[x_{1},x_{3}]\cdot\beta,$

and

$(x_{5},\log_{G_{b}}(b_{1}b_{3}^{-1}))=[x_{5},\log_{G_{b}}(b_{1}b_{3}^{-1})]+[x_{5},\log_{G_{b}}(b_{1},b_{3}^{-1})]\cdot\alpha_{5}.$

It follows that

$[x_{5},\log_{G_{b}}(b_{1}b_{3}^{-1})]=[x_{5},x_{1}-x_{3}]+\frac{1}{2}[x_{1},x_{3}],x_{5}]-[x_{1},x_{3}]\cdot\beta
x_{5}.$

This gives

$\partial
sy_{5}=[x_{5},x_{1}-x_{3}]+[x_{1},x_{3}]\cdot\beta_{5}^{\prime}=[x_{5},x_{1}-x_{3}]+\beta_{5}\cdot[x_{1},x_{3}],$

for some $\beta^{\prime}_{5}$ and $\beta_{5}$ in $\overline{J}$. Similarly,

$\partial sy_{6}=[x_{5},x_{1}-x_{4}]+\beta_{6}\cdot[x_{1},x_{4}]\hskip
8.53581pt\mbox{and }\partial
sy_{7}=[x_{5},x_{2}-x_{3}]+\beta_{7}\cdot[x_{2},x_{3}].$

Now we define

$\psi(sy_{5})=sy_{5}+\beta_{5}\cdot
sy_{1},\psi(sy_{6})=sy_{6}+\beta_{6}sy_{2},\psi(sy_{7})=sy_{7}+\beta_{7}\cdot
sy_{3}.$

Then $\partial^{\prime}\psi=\psi\partial$, and it is immediate that $\psi$ is
an isomorphism.

Step Three. dim$\,(L_{Y}/L_{Y}^{2})_{1}=\infty$.

We first consider the uncompleted dgl
$(L,d^{\prime})=(\mathbb{L}(x_{i},sy_{j}),d^{\prime})$. Following the
computations of [32, page 118, 119], the dimension of
$(H(L,d^{\prime})/H(L,d^{\prime})^{2})_{1}$ is infinite.

Denote by $L_{s}^{r}$ the vector space generated by the Lie brackets
containing exactly $r$ elements $sy_{j}$ and $s$ elements $x_{i}$. Since
$d(sy_{j})\subset\mathbb{L}^{2}(x_{i})$, the dgl $L$ is the sum of the finite
dimensional complexes $C_{n}=\oplus_{2r+s=n}L_{s}^{r}$. Moreover $\sum_{r\geq
r_{0}}C_{r}$ is an ideal and we denote by $L(r_{0})$ the quotient dgl
$L/(\sum_{r\geq r_{0}}C_{r})$. Thus for any integer $k$ there is a $N$ such
that dim$\,(L(N)/L(N)^{2})_{1}>k$.

It follows that its completion $\overline{\mathbb{L}}(x_{i},sy_{j})$
decomposes as the product $\prod_{n}C_{n}$, so that its homology is
$\prod_{n}H_{*}(C_{n})$ is the completion of $H(L)$.

Therefore

$L_{Y}=H(\overline{\mathbb{L}}(x_{i},sy_{j}),d)\cong\prod_{n}H(C_{n}).$

Remark now that $\prod_{r\geq r_{0}}H(C_{r})$ is an ideal in $L_{Y}$ whose
quotient is $L(r_{0})$. Therefore if dim $(L_{Y}/L_{Y}^{2})_{1}\leq k$ for
some integer $k$, then for any $r_{0}$, dim $(L(r_{0})/L(r_{0})^{2})\leq k$,
which is not true. It follows that dim$\,(L_{Y}/L_{Y}^{2})_{1}=\infty$.

Step Four. $L_{Y}$ is not a pronilpotent Lie algebra.

This follows from the next Proposition.

Let $(L,\partial)$ be an enriched dgl satisfying dim$\,L/L^{2}<\infty$. Denote
by $\widehat{H(L)}$ the pronilpotent completion of $H(L)$,

$\widehat{H(L)}=\varprojlim_{n}H(L)/H(L)^{n}.$

######  Proposition 28.1.

There is a natural injection $j:\widehat{H(L)}\to H(L)$ that is an isomorphism
if and only if dim$\,H(L)/H(L)^{2}<\infty$.

proof. Recall that the complete enriched structure of $H(L)$ is given by the
ideals $I^{n}=\mbox{ker}\,H(L)\to H(L/L^{n})$. The pronilpotent completion
$\widehat{H(L)}$ can be viewed as the set of the series $\sum\alpha_{i}$ with
$\alpha_{i}\in H(L)^{i}$. Since $\alpha_{i}=[a_{i}]$ with $a_{i}\in L^{i}$,
$\alpha_{i}\in I^{i}$ and the series $\sum\alpha_{i}$ is a well defined
element in $H(L)$. The induced morphism $j:\widehat{H(L)}\to H(L)$ is
injective because it is an injection when followed by the projections $H(L)\to
H(L)/I^{n}$.

When dim$\,H(L)/H(L)^{2}<\infty$, then for each integer $n$ there is $p(n)$
such that $I^{p(n)}\subset H(L)^{n}$. It follows that each series
$\sum\alpha_{i}$, with $\alpha_{i}\in I^{i}$ belongs to $\widehat{H(L)}$,
which proves the surjectivity in that case.

Now suppose dim$\,H(L)/H(L)^{2}=\infty$. There exists then a sequence of
indecomposable elements $\alpha_{n}\in H(L)$ with $\alpha_{n}\in I^{n}$. It
follows that $\sum\alpha_{n}\in H(L)$, but not in $\widehat{H(L)}$. $\square$

Example. Recall that the fundamental group of $S^{1}_{a}\vee S^{1}_{b}$ is a
free group on two generators, $a,b$. Denote by $X$ the space obtained by
adding 2 cells of dimension 2 to $S^{1}_{a}\vee S^{1}_{b}$ to oblige $a$ and
$b$ to commute with the commutator $aba^{-1}b^{-1}$. Then $\pi_{1}(X)$ is a
metabelian group and the Lie algebra associated to its Malcev completion is
the quotient of the completion of the free Lie algebra on 2 elements $a,b$ by
the relations $[a,[a,b]],[b,[a,b]]$.

Denote by $(\land W,d)$ the minimal model of $S^{1}_{a}\vee S^{1}_{b}$ and let
$(\land V,d)\stackrel{{\scriptstyle\simeq}}{{\to}}(\land W\oplus s^{-1}P,D)$
be the minimal model of $X$ as in Proposition 17.5(ii). It follows that
$V^{1}$ has a basis with 3 elements $x,y,t$ with $dx=dy=0$, $dt=xy$, and
$(\land V,d)$ is quasi-isomorphic to $\land(x,y,t)/xyt$. Decompose as above
$\land V=\land V^{1}\otimes\land Z$, and denote by $\land V^{1}\otimes\land U$
the acyclic closure of $\land V^{1}$. The short exact sequence

$0\to\mathbb{Q}xyt\to\land(x,y,t)\to\land(x,y,t)/(xyt)\to 0$

shows that

$(\land
Z,\overline{d})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(\mathbb{Q}\oplus
s^{-1}(xyt\otimes\land U),0).$

It follows that $L_{Z}$ is a profree Lie algebra.

###  29 Sequential cell attachments

A sequence $g_{k}:S^{n_{k}}\to X$ defines a map $g:\vee S^{n_{k}}\to X$. This
in turn yields a morphism

$(L,\partial)\longrightarrow(L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{Q},\partial+\partial_{\gamma})$

of enriched dgl’s, the dgl model for $X\cup_{f}\vee_{k}D^{n_{k}+1}$. Here
$sQ=H^{\geq 1}(\vee_{k}S^{n_{k}})^{\vee}.$ Thus if each $S^{n_{k}}$ is a
circle the $sQ=H_{1}(\vee_{k}S^{n_{k}};\mathbb{Q})^{\vee\vee}$. If there are
infinitely many circles then $Q$ is the dual of a vector space of uncountable
dimension, and this may not be a useful model for the construction of
computable topological spaces. Therefore we introduce the

Definition. _The sequential cell attachment_ associated with the sequence
$(g_{k})$ is the sequence of inclusions

$\varphi_{k}:X\to X_{k}:=X\cup_{g(k)}\,\vee_{\ell\leq k}D^{n_{\ell}+1},$

in which $g(k)=\\{g_{\ell}\,|\,\ell\leq k\\}$.

Here an enriched dgl representative for $\varphi_{k}$ is provided by the
morphism

$(L,\partial)\to(L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T_{k}},\partial+\partial_{\gamma}),$

where $sT_{k}=H^{\geq 1}(\vee_{\ell=1}^{k}\,S^{n_{k}})^{\vee}=H_{\geq
1}(\vee_{\ell=1}^{k}S^{n_{\ell}};\mathbb{Q})$ is finite dimensional. The
spatial realization of this morphism is then (Proposition 16.3) the map

$X_{\mathbb{Q}}\to(X_{k})_{\mathbb{Q}}.$

In particular corresponding to the sequence $\to X_{k}\to X_{k+1}\to$ we have
the sequence

$\to L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T_{k}}\to
L\,\widehat{\amalg}\,\overline{\mathbb{L}}_{T_{k+1}}\to.$

This in turn yields the sequence

$X_{\mathbb{Q}}\to\dots\to(X_{k})_{\mathbb{Q}}\to(X_{k+1})_{\mathbb{Q}}\to\dots\to(\cup_{k}X_{k})_{\mathbb{Q}},$

and we set

${Y}=\varinjlim(X_{k})_{\mathbb{Q}}.$

Note that ${Y}$ may not be the Sullivan completion of a topological space !
However we do have

######  Proposition 29.1.

With the hypotheses and notation above

  1. (i)

$\pi_{*}({Y})=\varinjlim\pi_{*}(X_{k})_{\mathbb{Q}}$ and

  2. (ii)

$H_{*}({Y};\mathbb{Q})\cong
T=\varinjlim_{k}T_{k}=H_{*}(\vee_{k}S^{n_{k}};\mathbb{Q})$.

proof. (i) follows because ${Y}$ is the limit of an increasing sequence of CW
complexes while (ii) follows from Proposition 13.2 because
$\partial_{\gamma}:T_{k}\to L.$ $\square$

####  29.1 A connected CW complex, $Y$, for which
$\pi_{*}(Y_{\mathbb{Q}})=\pi_{1}(Y_{\mathbb{Q}})$,
dim$\,H_{1}(Y;\mathbb{Q})=3$, dim $H_{2}(Y;\mathbb{Q})=\infty$ and
$H_{p}(Y;\mathbb{Q})=0$ for $p>2$.

Denote by $X=S^{1}_{a}\vee S^{1}_{b}\vee S^{1}_{c}$ a wedge of three circles.
Then $\pi_{1}(X)$ is the free group, $G$, on generators $a,b,c$ defined by the
inclusions of the circles. Thus (Proposition 9.4(i)) the profree dgl
$(L_{X},0)=(\overline{\mathbb{L}}(x,y,z),0)$ is a dgl model for $X$ where
$x,y,z=\log_{G}a,\log_{G}b,\log_{G}c$.

Now use $\overline{ad}(-)$ to denote commutation with elements in $G$,

$\overline{ad}(u)(v)=(u,v)=uvu^{-1}v^{-1}.$

Then a sequential cell attachment

$X\longrightarrow Y=\varinjlim_{r}(X\cup_{g}\vee_{n=1}^{r}D_{n}^{2})$

is defined by
$\pi_{1}(g)[S_{n}^{1}]=\overline{ad}^{n}(a)\overline{ad}^{n}(b)(c)$. A dgl
representative for this attachment is then provided (Proposition 9.4(i) and
§17.1) by the sequence

$(L_{X},0)\to\varinjlim_{r}(L_{X}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{sT(r)},\partial)$

where $T(r)=\oplus_{n=1}^{r}\mathbb{Q}z_{n}$ and $\partial
sz_{n}=\log_{G}(\overline{ad}^{n}(a)\overline{ad}^{n}(b)(c))$. Since
$\partial:sT(r)\to L_{X}^{2}$ it follows from Proposition 13.2 that

$\mbox{dim}\,H_{p}(X\cup_{g}\vee_{n=1}^{r}D_{n}^{2};\mathbb{Q})=\left\\{\begin{array}[]{ll}3&\mbox{if
}p=1\\\ r&\mbox{if }p=2\\\ 0&\mbox{if }p>2\end{array}\right.$

It follows from the obvious direct limit argument that

$\mbox{dim}\,H_{p}(Y;\mathbb{Q})=\left\\{\begin{array}[]{ll}3&\mbox{if }p=1\\\
r&\mbox{if }p=2\\\ 0&\mbox{if }p>2\end{array}\right.$

Denote by $ad(-)$ the conjugation in $L_{X}:ad_{x}(y)=[x,y]$. Then for some
element $\gamma_{n}\in L_{X}^{>2n+1}$ we have

$\log_{G}(\overline{ad}^{n}(a)\overline{ad}^{n}(b)(c))={ad}^{n}(x){ad}^{n}(y)(z)+\gamma_{n}.$

Now consider the dgl morphism

$(L_{X},0)\to(L_{X}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{sT(r)},\partial^{\prime}),$

where

$\partial^{\prime}sz_{n}=ad^{n}(x)ad^{n}(y)(z).$

Write $L=L_{X}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{sT(r)}$ and suppose
we have proved that $H_{q}(L,\partial^{\prime})=0$ for $q>1$. We introduce an
extra gradation in $L$ by putting $x,y$ and $z$ in degree $1$ and $z_{n}$ in
degree $2n+1$. We denote $L_{(n)}$ the component of $L$ in the new degree $n$.
Then $\partial^{\prime}:L_{(n)}\to L_{(n)}$ and
$(\partial-\partial^{\prime})(L_{(n)})\subset\oplus_{q>n}L_{(q)}$.

Suppose $\omega$ is a $\partial$-cycle in $L^{\geq 2}$, and write
$\omega=\sum_{q\geq q_{0}}\omega_{q}$ with $\omega_{q}\in L_{(q)}$. Then
$\omega_{q_{0}}$ is a $\partial^{\prime}$-cycle and so a
$\partial^{\prime}$-boundary. We write
$\omega_{q_{0}}=\partial^{\prime}(\beta_{0})$ with $\beta_{0}\in L_{(q_{0})}$.
Then denote $\omega^{\prime}=\omega-\partial(\beta_{0})$. We have
$\omega^{\prime}=\sum_{q>q_{0}}\omega^{\prime}_{q}$, and
$\omega^{\prime}_{q_{0}+1}$ is a $\partial^{\prime}$-cycle, so a boundary, and
there is $\beta_{1}\in L_{(q_{0}+1)}$ with
$\partial^{\prime}\beta_{1}=\omega^{\prime}_{q_{0}+1}$. We continue and
construct in this way a sequence of elements $\beta_{q}\in L_{(q_{0}+q)}$. The
series $\sum\beta_{q}$ is a well defined element in $L$ and

$\partial(\sum\beta_{q})=\omega.$

This shows that $H_{\geq 2}(L,\partial)=0$.

Now we prove that $H_{\geq 2}(L,\partial^{\prime})=0$. Let $(\land W,d)$ be
the quadratic Sullivan model of $L_{X}=\overline{\mathbb{L}}(x,y,z)$.
Proposition 6.4 equips $(\land W,d)$ with a second gradation $W=\oplus_{n\geq
0}W(n)$ satisfying

$W(0)=W\cap\mbox{ker}\,d\hskip 14.22636pt\mbox{and
}d:W(n)\to(\land^{2}W)(n-1).$

Then denote by $w_{x},w_{y}$ and $w_{z}$ the basis of $W(0)$ which is the dual
basis for $sx,sy$ and $sz$.

Next set $\alpha_{n}=ad^{n}(x)ad^{n}(y)(z)$. Then $\alpha_{n}$ determines a
cdga morphism $\rho_{n}:\land W\to\mathbb{Q}\oplus\mathbb{Q}a_{n}$ and
satisfying

$\rho_{n}(W)\neq 0\hskip 14.22636pt\mbox{and }\rho_{n}(W(k))=0\hskip
5.69054pt\mbox{ if }k\neq 2n.$

Set $S(r)=\oplus_{n=1}^{r}\mathbb{Q}a_{n}$. Then

$\rho(r)=\oplus_{n=1}^{r}\rho_{n}:\land W\to\mathbb{Q}\oplus S(r)$

is well defined and surjective.

Now we introduce a third gradation in $\land W$ by assigning $W(k)$ the new
degree $2k+3$. Then $d$ increases this new degree by $1$, so that $\land W$ is
identified with a quadratic Sullivan algebra $\land Z$ in which
$W(k)=Z^{2k+3}$ and $Z=Z^{\geq 3}$. In particular, $a_{n}$ has new degree
$2(2n)+3$. Thus $\rho(r)$ with this new grading becomes a surjective morphism

$\rho^{\prime}(r):\land Z\to\mathbb{Q}\oplus S^{\prime}(r).$

Now $\land
Z\stackrel{{\scriptstyle\simeq}}{{\to}}\mathbb{Q}\oplus(Z\cap\mbox{ker}\,d)$,
and $Z\cap\mbox{ker}\,d=W(0)$ but with new degree 3. It follows that the
homotopy Lie algebra of $\land Z$ is the free graded Lie algebra on the
elements $x,y,z$, but regarded as having degree 3. We now show that the
sequence $\alpha_{n}$, regarded as elements in $L_{\land Z}$, is inert.

Recall from [3, Theorem 1.4] that the sequence $\alpha_{n}$ is inert in
$L_{\land T}$ if and only if it is inert in its universal enveloping algebra.
Moreover following [3, Theorem 2.1] if for some ordering of $a,b,c$ the family
$w_{i}$ of the terms of maximum height of the $\alpha_{i}$ satisfies the two
following conditions, then the family $\alpha_{i}$ is inert. The conditions
are

(1) No $w_{i}$ is a submonomial of any other $w_{j}$.

(2) No $w_{i}$ ”overlaps with” any other $w_{j}$, i.e. is $w_{i}=uv$ and
$w_{j}=bw$ then $v=1$.

In our situation let $x>y>z$. Then the maximum height in $\alpha_{n}$ is
$x^{n}y^{n}z$ and this family satisfies the two conditions above.

It follows that the sequence $\alpha_{n}$ is inert in $\land Z$. This implies
that we can suppose the induced morphism $\lambda^{\prime}:\land
V^{\prime}\to\land Z$ satisfies $\lambda^{\prime}(V^{\prime})\subset Z$.

Now return to the original gradings. Since $\land V\cong\land Z$ and
$\lambda=\lambda^{\prime}$, it follows that $\lambda:V\to W$ is injective, and
by Proposition 19.1 this shows that the sequence $\alpha_{n}$ is inert in
$L_{X}$. In particular, for each $r$,

$L_{X}\to
H(L_{X}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{sT(r)},\partial^{\prime})\hskip
8.53581pt\mbox{and }L_{X}\to
H(L_{X}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{sT(r)},\partial)$

are surjective.

Since dim$\,T(r)<\infty$ it follows from Proposition 16.3 that

$X_{\mathbb{Q}}\to(X\cup_{g}\vee_{n=1}^{r}D^{2}_{n})_{\mathbb{Q}}$

is the spatial realization of $L_{X}\to
H(L_{X}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{sT(r)})$. This identifies
the surjection $L_{X}\to
H(L_{X}\,\widehat{\amalg}\,\overline{\mathbb{L}}_{sT(r)})$ with

$\pi_{*}(X_{\mathbb{Q}})\to\pi_{*}((X\cup_{g}\vee_{n=1}^{r}D_{n}^{2})_{\mathbb{Q}}).$

Since $L_{X}=(L_{X})_{0}$, $\pi_{*}(X_{\mathbb{Q}})=\pi_{1}(X_{\mathbb{Q}})$
and so $\pi_{\geq 2}(X\cup_{g}\vee_{n=1}^{r}D_{n}^{2})_{\mathbb{Q}}=0$.
Passage to direct limits shows that
$\pi_{*}(Y_{\mathbb{Q}})=\pi_{1}(Y_{\mathbb{Q}})$. $\square$

### References

  * [1] D. Anick, _Non-commutative algebras and their Hilbert series_ , J. Algebra 78 (1982), 120-140.
  * [2] D. Anick, _A rational homotopy analogue of Whitehead’s problem_ , in:Algebra, Algebraic Topology and its Applications, Lecture Notes in Math. 1183, Springer (1986), 28-31.
  * [3] D. Anick, _Inert sets and the Lie algebra associated to a group_ , J. Algebra 111 (1987), 154-165.
  * [4] M.G. Barratt and J. Milnor, _An example of anomalous singular homology_ , Proceedings AMS 13 (1962), 293-297.
  * [5] N. Bourbaki, _Eléments de Mathématique, Théorie des Ensembles, tome 3_ , 1970, Hermann.
  * [6] A.K. Bousfield and V.K. Gugenheim, _On PL de Rham theory and rational homotopy type_ , Mem. Amer. Math. Soc. 179 (1976).
  * [7] A.K. Bousfield and D.M. Kan, _Homotopy Limits, completions and localizations_ , Lecture Notes in Math. 304, Springer, 1972.
  * [8] K. Brown, _Cohomology of groups_ , Graduate Texts in Math. 87, Springer Verlag
  * [9] P. Bubenik, _Free and Semi-Inert Cell Attachments_ , Trans. Amer. Math. Soc. 357 (2005), 4533-4553.
  * [10] U.Buijs, Y. Félix, A. Murillo and D. Tanré, _Lie models in Topology_ , Birkhauser 2020.
  * [11] E. Dyer and A. Vasquez, _Some small aspherical spaces_ , J. Austr. Math. Soc. 16 (1973), 332-352.
  * [12] Y. Félix and S. Halperin, _The Sdepth of a homotopy Lie algebra_ , Geometry, Topology and Mathematical Physics Journal, 1 (2018), 4-26.
  * [13] Y. Félix and S. Halperin, _Malcev completions, LS category and depth_ , Bol. Soc. Math. Mex. 23 (2017), 267-288.
  * [14] Y. Félix and S. Halperin, _The depth and LS category of a topological space_ , Math. Scand., 123 (2018), 220-238.
  * [15] Y. Félix and S. Halperin, _The depth of a Riemann surface and of a right-angled Artin group_ , Journal of Homotopy and Related Structures 15 (2020, 223-248.
  * [16] Y. Félix and S. Halperin, _Rational homotopy theory via Sullivan spaces: a survey_ , ICCM Notices of the International Congress of Chinese Mathematicians, 5 (2017), 14-36.
  * [17] Y. Félix, S. Halperin and J.-C. Thomas, _Rational Homotopy Theory_ , Graduate Texts in Math. 201, Springer verlag, 2001.
  * [18] Y. Félix, S. Halperin and J.C. Thomas, _The ranks of the homotopy groups of a space of finite LS category_ , Expos. Math. 25 (2007), 67-76.
  * [19] Y. Félix, S. Halperin and J.-C. Thomas, _Rational Homotopy Theory II_ , World Scientific, 2015.
  * [20] Y. Félix and J.-M. Lemaire, _On the Pontryagin algebra of the loops on a space with a cell attached_ , Internat. J. Math. 2 (1991), 429-438.
  * [21] Y. Félix and J.-C. Thomas, _Module d’holonomie d’une fibration_ , Bull. Math. Soc. France 113 (1985), 255-258.
  * [22] M. Hall, _A basis for free Lie rings and higher commutators in free groups_ , Proc. Amer. Math. Soc. 1(1950) 575-581.
  * [23] S. Halperin, A. Gomez-Tato and D. Tanré, _Rational homotopy theory for non-simply connected spaces_ , Trans. Amer. Math. Soc. 352 (2000), 1493-1525.
  * [24] S. Halperin and J.M. Lemaire, _Suites inertes dans les algèbres de Lie graduées_ , Math. Scand 61 (1987), 39-67.
  * [25] S. Halperin and J.-M. Lemaire, _The fibre of a cell attachment_ , Proc. Roy. Soc. Edinburgh, 38 (1995), 295-311.
  * [26] K. Hess and J.-M. Lemaire, _Nice and lazy cell attachments_ , J. of Pure and Applied Algebra 112 (1996), 29-39.
  * [27] S. Halperin and J.D. Stasheff, _Obstructions to homotopy equivalences_ , adv. in Math. 32 (1979), 233-279.
  * [28] S. Ivanov and R. Mikhailov, _A finite $\mathbb{Q}$-bad space_, Geometry and Topology 23 (2019), 1237-1249.
  * [29] Thomas Jech, _Set Theory, the third millennium edition, revised and expanded_ , 2003, Springer Monographs in Mathematics.
  * [30] M. Lazard, _Sur les groupes nilpotents et les anneaux de Lie_ , Ann. Sc. Ec. Norm. Sup. 71 (1959), 101-190.
  * [31] A. Lazarev and M. Markl, _Disconnected rational homotopy theory_ , Adv. Math. 283 (2015), 303-361.
  * [32] J.-M. Lemaire, _A finite complex whose rational homology is not finitely generated_ , in Lecture Notes in Mathematics 196 (1971), 114-120.
  * [33] J.-M. Lemaire, _”Autopsie d’un meurtre” dans l’homologie d’une algèbre de chaînes_ , Ann. Scient. Ec. Norm. Sup. 11 (1978), 93-100.
  * [34] R.C. Lyndon, _Cohomology theory of groups with a single defining relation_ , Ann. of Math. 52 (1950), 650-656.
  * [35] J.P. May, _Simplicial objects in algebraic topology_ , Chicago Univ. Press (1967).
  * [36] J. Milnor and J.C. Moore, _On the structure of Hopf algebras_ , Annals of Math. 81 (1965), 211-264.
  * [37] S. Papadima and A. Suciu, _Algebraic invariants for right-angled Artin groups_ , Math. Annalen 334 (2006), 533-555.
  * [38] A. Putman, _One-relator groups_ , preprint 2018.
  * [39] J.P. Pridham, _Pro-algebraic homotopy types_ , Arxiv 2008
  * [40] D. Quillen, _Rational homotopy theory_ , Ann. of Math. 90 (1969), 205-295.
  * [41] J.P. Serre, _Lie algebras and Lie groups_ , Benjamin Inc. (1965).
  * [42] A.I. Shirshov, _Subalgebras of free Lie algebras_ , Mat. Sbornik 33 (1975), 441-452.
  * [43] R. Solovay, _$2^{\aleph_{0}}$ can be anything it ought to be_, Symposium on the Theory of Models, North-Holland Publish Co (1965), pg 435.
  * [44] D. Sullivan, _Infinitesimal computations in topology_ , Publ. IHES 47 (1977), 269-331.
  * [45] C. Watkiss, unpublished pager (1976).
  * [46] J.H.C. Whitehead, _On adding relations to homotopy groups_ , Ann. Math. 42 (1941), 409-428.

Institut de Mathématique et de Physique, Université Catholique de Louvain, 2,
Chemin du cyclotron, 1348 Louvain-La-Neuve, Belgium<EMAIL_ADDRESS>
Department of Mathematics, Mathematics Building, University of Maryland,
College Park, MD 20742, United States<EMAIL_ADDRESS>