# Extremal stability for configuration spaces

Ben Knudsen  Department of Mathematics, Northeastern University, USA
<EMAIL_ADDRESS>,  Jeremy Miller  Department of Mathematics, Purdue
University, USA<EMAIL_ADDRESS>and  Philip Tosteson  Department of
Mathematics, University of Chicago, Chicago, IL<EMAIL_ADDRESS>
###### Abstract.

We study stability patterns in the high dimensional rational homology of
unordered configuration spaces of manifolds. Our results follow from a general
approach to stability phenomena in the homology of Lie algebras, which may be
of independent interest.

Ben Knudsen was supported in part by NSF grant DMS-1906174

Jeremy Miller was supported in part by a Simons collaboration grant

Philip Tosteson was supported in part by NSF grant DMS-1903040.

##  1\. Introduction

The purpose of this paper is to investigate a stability phenomenon in the high
dimensional rational homology of the unordered configuration spaces of a
$d$-manifold $M$. Writing

$\mathrm{Conf}_{n}(M)=\\{(x_{1},\ldots,x_{n})\in M^{n}\,|\,x_{i}\neq
x_{j}\text{ for }i\neq j\\}$

for the ordered configuration space of $n$ points in $M$ and
$B_{n}(M)=\mathrm{Conf}_{n}(M)/S_{n}$ for the unordered configuration space,
classical results of McDuff, Segal, and Church show that the spaces $B_{n}(M)$
exhibit _rational homological stability_. We will restrict our attention to
the case of even $d$, since there is a simple closed form available for $d$
odd [BCT89].

######  Theorem 1.1 (Homological stability [McD75, Seg79, Chu12]).

Let $M$ be a manifold of finite type and even dimension $d\geq 2$. For $n$
sufficiently large with respect to $i$, the function $n\mapsto\dim
H_{i}(B_{n}(M);\mathbb{Q})$ is equal to a polynomial in $n$ of degree at most
$\dim H_{0}(M;\mathbb{Q})-1$.

Traditionally, Theorem 1.1 is stated in the case that $M$ is connected, where
it is the statement that the function $n\mapsto\dim
H_{i}(B_{n}(M);\mathbb{Q})$ is eventually constant; the general case follows
easily from the connected case and the Künneth theorem.

While homological stability is a stability pattern in low homological
dimension, extremal stability, our main theorem, is a pattern in low
homological _codimension_. Since $H_{i}(B_{n}(M))=0$ for
$i>\nu_{n}:=n(d-1)+1$, it is reasonable to think of $H_{\nu_{n}-i}(B_{n}(M))$
as the codimension $i$ homology of $B_{n}(M)$, at least generically.

######  Theorem 1.2 (Extremal stability).

Let $M$ be a manifold of finite type and even dimension $d\geq 2$. For $n$
sufficiently large with respect to $i$, the function $n\mapsto\dim
H_{\nu_{n}-i}(B_{n}(M);\mathbb{Q})$ is equal to a quasi-polynomial in $n$ of
degree at most $\dim H_{d-1}(M;\mathbb{Q}^{w})-1$ and period dividing $2$.

Here, we have written $\mathbb{Q}^{w}$ for the orientation sheaf of $M$.
Equivalently, Theorem 1.2 states that there are two polynomials $p_{{\rm
even}}^{i}(n)$ and $p_{\rm odd}^{i}(n)$, governing the codimension $i$
homology of $B_{n}(M)$ for even and odd $n$ respectively. See §2.3 for our
conventions on quasi-polynomials.

######  Example 1.3 ([DCK17]).

For $\Sigma$ a compact, orientable surface of genus $2$, we have

$\dim
H_{\nu_{n}}(B_{n}(\Sigma);\mathbb{Q})=\begin{cases}\frac{n^{3}+n^{2}+16}{16}\vspace{.1cm}&\quad
n\text{ even}\\\ \frac{n^{3}+n^{2}-9n-9}{16}&\quad n\text{ odd}.\end{cases}$

$n$$i$ZeroExtremal stabilityHomological stability$H_{i}(B_{n}(M),\mathbb{Q})$
Figure 1. Homological stability and extremal stability

Instances of extremal stability for configuration spaces were first noticed by
Maguire in the case $M=\mathbb{C}P^{3}$ [Mag]. The only other prior example of
extremal stability that we are aware of is the high dimensional cohomology
groups of congruence subgroups of $SL_{n}(\mathbb{Z})$ [MNP20].

If $H_{d-1}(M,\mathbb{Q}^{w})=0$, then Theorem 1.2 is just the statement that
the groups $H_{\nu_{n}-i}(B_{n}(M),\mathbb{Q})$ eventually vanish. In this
situation, $H_{\nu_{n}-i}(B_{n}(M),\mathbb{Q})$ should not be thought of as
the codimension $i$ homology of $B_{n}(M)$. In Theorem 4.9, we prove a more
refined stability result for manifolds with vanishing high degree homology
groups.

###  1.1. Stabilization maps

Extremal stability is induced by a family of maps of the form

$H_{i}(B_{n}(M);\mathbb{Q})\to H_{i+2d-2}(B_{n+2}(M);\mathbb{Q}).$

Heuristically, such a map is obtained as follows. Fixing a class $\alpha\in
H_{d-1}(M;\mathbb{Q})$, which we imagine as a $(d-1)$-parameter family of
configurations of a single point in $M$, we write $\alpha\otimes[v,v]\in
H_{2d-2}(B_{2}(M);\mathbb{Q})$ for the class obtained by replacing this single
point with a pair of orbiting points. Here, we think of $v$ as the class of a
single point in $\mathbb{R}^{d}$ and $[\cdot,\cdot]$ as the Browder bracket in
the homology of Euclidean configuration spaces.

We wish to define our stabilization map by superposition with the class
$\alpha\otimes[v,v]$; that is, we attempt to send the cycle depicted on the
left of Figure 2 to the cycle depicted on the right. Unfortunately, this cycle
does not lie in the configuration space, since it contains points of
intersection among distinct particles!

Figure 2. Extremal stabilization map

This issue can be resolved in the example depicted (where $i=1$) by performing
surgery in a Euclidean neighborhood of the intersection point, resulting in a
$(2d-1)$-chain in $B_{3}(M)$ whose boundary corresponds to the image of the
nested Browder bracket $[v,[v,v]]\in
H_{2d-2}(B_{3}(\mathbb{R}^{d});\mathbb{Q})$ under the coordinate embedding
$\mathbb{R}^{d}\subseteq M$. Since $[v,[v,v]]=0$ by the Jacobi relation, we
may choose a bounding chain, removing the point of intersection.

The key fact exploited in this construction is that $[v,v]$ lies in the center
of the Lie algebra formed by the homology of the configuration spaces of
$\mathbb{R}^{d}$. This algebraic observation points the way to the rigorous
definition of the desired extremal stabilization maps.

###  1.2. Transit algebras

We approach Theorem 1.2, a priori a topological statement, in an entirely
algebraic setting. As shown by the first author [Knu17], the total homology of
the configuration spaces of a manifold is the homology of a certain Lie
algebra—see Theorem 4.7 below—so stability phenomena in Lie algebra homology
give rise to stability phenomena in the homology of configuration spaces.

The homology of a Lie algebra $\mathfrak{g}$ supports two types of action
relevant to stability. First, given an Abelian quotient
$\mathfrak{g}\to\mathfrak{k}$, there results an action of
$\operatorname{Sym}(\mathfrak{k}^{\vee}[-1])$; in the example of configuration
spaces, this degree lowering action gives rise to the maps

$H_{*}(B_{n}(M),\mathbb{Q})\to H_{*}(B_{n-1}(M),\mathbb{Q})$

obtained from the inclusion ${\rm Conf}_{n}(M)\subseteq{\rm
Conf}_{n-1}(M)\times M$ by pairing with an element of $H^{*}(M)$ and applying
transfer. Second, given a central subalgebra
$\mathfrak{h}\subseteq\mathfrak{g}$, there results an action of
$\operatorname{Sym}(\mathfrak{h}[1])$; in the example of configuration spaces,
this degree raising action gives rise to the extremal stabilization maps
described heuristically in Section 1.1.

These two actions interact via Weyl relations, and we call the resulting
algebra a _transit algebra_. As shown in §4.2, transit algebras provide a
common source for classical homological stability and extremal stability.

###  1.3. Acknowledgments

We thank Megan Maguire for helpful conversations and for the initial
inspiration to investigate this question.

##  2\. Algebraic background

In this section, we detail our conventions on graded objects, algebras, and
modules.

###  2.1. Degrees and slopes

We work in the setting of bigraded vector spaces over $\mathbb{Q}$. Such a
vector space $V$ arrives equipped with a decomposition
$V=\bigoplus_{n,i\in\mathbb{Z}}V_{n,i}$. We write $\langle r\rangle$ and $[s]$
for the relevant shift operations, i.e.,

$(V\langle r\rangle[s])_{n,i}=V_{n-r,i-s}.$

The parameters $n$ and $i$ are called the weight and the (homological) degree,
respectively, and we write $w(v)=n$ and $d(v)=i$ for a bihomogeneous element
$v\in V$ of bidegree $(n,i)$.

Duals and tensor products of bigraded vector spaces are defined by the
stipulations

$(V^{\vee})_{n,i}=\mathrm{Hom}_{\mathbb{Q}}(V_{-n,-i},\mathbb{Q})\qquad\qquad(V\otimes
W)_{n,i}=\bigoplus_{a+b=n,c+d=i}V_{a,c}\otimes W_{b,d}.$

With this tensor product, the category of bigraded vector spaces is monoidal,
and we equip it with the symmetric monoidal structure whose symmetry
incorporates Koszul signs in the homological degree _but not in the weight._
Because of this symmetry, parity of degree will play an important role in what
follows, and we write

$V^{\epsilon}=\bigoplus_{n\in\mathbb{Z}}\,\bigoplus_{i\equiv\epsilon\,\mathrm{mod}\,2}V_{n,i}$

for $\epsilon\in\\{0,1\\}$, considered as a bigraded subspace of $V$.

We say that $V$ is of finite type if $V_{n,i}$ is finite dimensional for
$n,i\in\mathbb{Z}$. If $V$ is of finite type and vanishes in a cofinite set of
bidegrees, we say that $V$ is finite dimensional. We say that $V$ is bounded
below if $V_{n,i}=0$ whenever either $n$ or $i$ is a negative number of
sufficiently large absolute value. We say that $V$ is connected if
$V_{0,0}=0$.

We record the following simple observations regarding the interaction of these
finiteness properties with duals and tensor products.

######  Lemma 2.1.

Let $V$ and $W$ be bigraded vector spaces.

  1. (1)

If $V$ is of finite type or finite dimensional, then so is $V^{\vee}$.

  2. (2)

If $V$ and $W$ are finite dimensional, then so is $V\otimes W$.

  3. (3)

If $V$ and $W$ are bounded below, then so is $V\otimes W$.

  4. (4)

If $V$ and $W$ are bounded below of finite type, then so is $V\otimes W$.

We think of a bigraded vector space as a planar grid of vector spaces, with
the weight recorded on the horizontal axis and the degree on the vertical. Our
language reflects this idea; for example, we say that $V$ is first-quadrant if
$V_{n,i}=0$ for $n<0$ and $i<0$ (in which case $V^{\vee}$ is third-quadrant).
This picture also informs the following.

######  Definition 2.2.

Fix a bigraded vector space $V$.

  1. (1)

Let $0\neq v\in V$ be a bihomogeneous element of nonzero weight. The _slope_
of $v$ is the rational number $m(v)=d(v)/w(v)$.

  2. (2)

Suppose that $V$ is concentrated in nonzero weights. The _maximal slope_ of
$V$ is

$m_{\max}(V)=\max\\{C\in\mathbb{Q}\mid\exists v\in V:m(v)=C\\}$

(resp. minimal slope, $m_{\min}(V)$, $\min$).

Given $C\in\mathbb{Q}$, we write $V_{C}\subseteq V$ for the span of the
bihomogeneous elements of slope $C$, and similarly for $V_{<C}$ etc. If
$V=V_{C}$, then we say that $V$ is of slope $C$. Note that, if $V$ is of slope
$C$, then $V$ is concentrated in nonzero weight.

A related notion of slope will also be important in what follows.

######  Definition 2.3.

Fix a bigraded vector space $V$ and $C\in\mathbb{Q}$. A _ray of slope_ $C$ in
$V$ is a subspace of the form $\mathcal{R}=\bigoplus_{u\geq
0}V_{au,bu+i_{0}}$, where $C=b/a$ with $(a,b)=1$. The _graded dimension_ of
$\mathcal{R}$ is the function

$n\mapsto\begin{cases}\dim\mathcal{R}_{n,Cn+i_{0}}&\quad a\mid n\\\
0&\quad\text{otherwise}.\end{cases}$

At times, it will be convenient to work with a third grading, by polynomial
degree, which will always be non-negative. Bigraded vector spaces are regarded
as trigraded vector spaces concentrated in polynomial degree $1$. Duals and
tensor products of trigraded vector spaces are defined analogously, with
Koszul signs reflecting only the homological degree. Discussions of slope will
never involve the third grading.

###  2.2. Symmetric (co)algebras and their duals

Given a bigraded vector space $V$, we write
$\operatorname{Sym}^{k}(V)=(V^{\otimes k})_{S_{k}}$, where the symmetric group
acts via the symmetry of the symmetric monoidal structure introduced above.

######  Definition 2.4.

Let $V$ be a bigraded vector space. The _symmetric algebra_ on $V$ is the
trigraded vector space $\operatorname{Sym}(V)=\bigoplus_{k\geq
0}\operatorname{Sym}^{k}(V)$, equipped with the product given componentwise by
the dashed filler in the commuting diagram

$\textstyle{V^{\otimes k}\otimes
V^{\otimes\ell}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{V^{\otimes(k+\ell)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Sym}^{k}(V)\otimes\operatorname{Sym}^{\ell}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Sym}^{k+\ell}(V),}$

where the vertical arrows are the respective projections.

This multiplication map furnishes the symmetric algebra with the structure of
a commutative algebra object in the category of trigraded vector spaces. It is
a universal commutative algebra in the sense of the commutative diagram

---
$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{A}}$$\textstyle{\operatorname{Sym}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exists!}$

of trigraded vector spaces. In particular, there is a canonical, natural map
of algebras

$\operatorname{Sym}(V\oplus
W)\to\operatorname{Sym}(V)\otimes\operatorname{Sym}(W)$

induced by the assignment $(v,w)\mapsto v\otimes 1+1\otimes w$, which is
easily seen to be an isomorphism. This map furnishes the symmetric algebra
functor with an oplax monoidal structure; in particular,
$\operatorname{Sym}(V)$ is canonically a bicommutative bialgebra with
comultiplication induced by the diagonal of $V$.

Given a bihomogeneous basis $\\{t_{i}\\}_{i\in I}$ for $V$, a bihomogeneous
basis for $\operatorname{Sym}^{k}(V)$ is provided by the set of equivalence
classes of degree $k$ monomials $t_{i_{1}}\cdots t_{i_{k}}$ under the
equivalence relation

$t_{i_{1}}\cdots t_{i_{j}}t_{i_{j+1}}\cdots
t_{i_{k}}\sim(-1)^{d(t_{i_{j}})d(t_{i_{j+1}})}t_{i_{1}}\cdots
t_{i_{j+1}}t_{i_{j}}\cdots t_{i_{k}}.$

In this way, we obtain a trihomogeneous basis for $\operatorname{Sym}(V)$,
which we refer to as the _monomial basis_. In this basis, multiplication is
given by concatenation of monomials and comultiplication by the usual shuffle
coproduct. It should be emphasized that a generator behaves as an exterior or
polynomial generator according to the parity of its homological degree alone.

In the presence of the above basis, we write $\partial_{t_{i}}\in V^{\vee}$
for the functional dual to $t_{i}$, i.e.,
$\partial_{t_{i}}(t_{j})=\delta_{ij}.$ Allowing these functionals to act
according to the usual rules of differential calculus, with appropriate Koszul
signs, defines a linear embedding

$\operatorname{Sym}(V^{\vee})\to\operatorname{Sym}(V)^{\vee}$

which is an isomorphism of trigraded vector spaces under appropriate
finiteness assumptions, e.g., if $V$ is bounded below of finite type.

In this situation, by adjunction, there is a one-to-one correspondence of sets
of maps

$\varphi:\operatorname{Sym}(V^{\vee})\otimes\mathcal{M}\to\mathcal{M}\qquad\iff\qquad\widetilde{\varphi}:\mathcal{M}\to\operatorname{Sym}(V)\otimes\mathcal{M}$

given explicitly by the formulas
$\widetilde{\varphi}(m)=\sum_{p}p\otimes\varphi(\partial_{p}\otimes m)$ and
$\varphi(\partial_{p}\otimes m)=\partial_{p}\cap\widetilde{\varphi}(m),$ where
$p$ ranges over the monomial basis. In this way, a
$\operatorname{Sym}(V^{\vee})$-module determines a
$\operatorname{Sym}(V)$-comodule, referred to as the _adjoint comodule_. If
$\mathcal{M}$ is of finite type (as a trigraded vector space), then this
comodule structure is in turn equivalent to the
$\operatorname{Sym}(V^{\vee})$-module structure on $\mathcal{M}^{\vee}$
determined by the map

$\operatorname{Sym}(V^{\vee})\otimes\mathcal{M}^{\vee}\cong(\operatorname{Sym}(V)\otimes\mathcal{M})^{\vee}\xrightarrow{\widetilde{\varphi}^{\vee}}\mathcal{M}^{\vee}.$

We refer to this module as the _dual adjoint module_.

###  2.3. Modules and growth

In what follows, we will be interested in the eventual growth rates of graded
dimensions of vector spaces.

######  Definition 2.5.

A _quasi-polynomial_ is an element of $\Pi[t]$, where $\Pi$ is the ring of
periodic functions from $\mathbb{Z}$ to $\mathbb{Q}$. The _period_ of a quasi-
polynomial is the least common multiple of the periods of its coefficients.

When working over a symmetric algebra with generators of fixed slope, rays of
the same slope exhibit predictable growth. As a matter of notation, we write

$\mathrm{lcm}(V)=\mathrm{lcm}\\{n\in\mathbb{Z}\mid\exists
i\in\mathbb{Z}:V_{n,i}\neq 0\\}.$

######  Lemma 2.6.

Let $V$ be a finite dimensional, first-quadrant bigraded vector space of slope
$C\in\mathbb{Q}$ and $\mathcal{M}$ a finitely generated
$\operatorname{Sym}(V)$-module. The graded dimension of any ray in
$\mathcal{M}$ of slope $C$ is eventually equal to a quasi-polynomial of degree
at most $\dim V^{0}-1$ and period dividing $\mathrm{lcm}(V^{0})$.

###### Proof.

Any ray in $\mathcal{M}$ is a $\operatorname{Sym}(V)$-submodule, since $V$ has
slope $C$, and $\operatorname{Sym}(V)$ is Noetherian, since $V$ is finite
dimensional, so we assume that $\mathcal{M}$ is itself a ray of slope $C$.
Applying the shift functor $[r]$ to this ray does not change its graded
dimension, so we may assume that $\mathcal{M}=\mathcal{M}_{C}$. Lastly, we may
assume that $V=V^{0}$; indeed,
$\operatorname{Sym}(V)\cong\operatorname{Sym}(V^{0})\otimes\operatorname{Sym}(V^{1})$,
and $\operatorname{Sym}(V^{1})$ is finite dimensional, since $V^{1}$ is so.
With these assumptions in place, the homological degree in
$\operatorname{Sym}(V)$ and in $\mathcal{M}$ is determined by the weight, so
the claim follows from the classical theory of the Hilbert function of a
graded module over a graded polynomial ring [AM16, Theorem 11.1]. ∎

We pair this result with a criterion for detecting finite generation over
symmetric algebras with generators of fixed slope.

######  Lemma 2.7.

Let $V$ be a finite dimensional bigraded vector space and $\mathcal{M}$ a
finitely generated $\operatorname{Sym}(V)$-module. If $m_{\max}(V^{0})\leq C$,
then every ray in $\mathcal{M}$ of slope $C$ is finitely generated over
$\operatorname{Sym}(V^{0}_{C})$.

###### Proof.

If $m_{\max}(V^{0})<C$, then $V_{C}^{0}=0$ by assumption, so
$\operatorname{Sym}(V_{C}^{0})=\mathbb{Q}$, the monoidal unit. A finitely
generated $\mathbb{Q}$-module is simply a finite dimensional bigraded vector
space, so the first claim implies the second.

For the first claim, we may reduce to the case
$\mathcal{M}=\operatorname{Sym}(V)$ using shifts, sums, and quotients. As in
the proof of Lemma 2.6, we may further assume that $V=V^{0}$. Choosing a
bihomogeneous basis $\\{u_{1},\dots,u_{m}\\}$ for $V_{<C}$, we observe that
$\mathcal{M}$ is freely generated over $\operatorname{Sym}(V^{0}_{C})$ by the
set of monomials of the form $u_{1}^{j_{1}}\dots u_{m}^{j_{m}}$ with
$j_{1},\dots,j_{m}\in\mathbb{Z}_{\geq 0}$. Fixing a ray
$\mathcal{R}=\bigoplus_{t\geq 0}\mathcal{M}_{at,bt+i_{0}}$ with $b/a=C$ and
$(a,b)=1$, such a monomial lies in $\mathcal{R}$ if and only if

$\sum_{k=1}^{m}j_{k}(ad(u_{k})-bw(u_{k}))=ai_{0}.$

By assumption, we have $ad(u_{k})-bw(u_{k})<0$ for $1\leq k\leq m$, so only
finitely many monomials lie in $\mathcal{R}$. Since the product of a monomial
and an element of $\operatorname{Sym}(V_{C}^{0})$ lies in $\mathcal{R}$ if and
only if the monomial does, the claim follows. ∎

At times, an alternative to finite generation will be the relevant property.

######  Definition 2.8.

Let $V$ be a first-quadrant bigraded vector space and $\mathcal{M}$ a
$\operatorname{Sym}(V^{\vee})$-module. We say that $\mathcal{M}$ is _finitely
detected_ if there is a surjective linear map $q:\mathcal{M}\to N$ with finite
dimensional target such that, for every $m\in\mathcal{M}\setminus\\{0\\}$,
there is a monomial $p\in\operatorname{Sym}(V)$ with $q(\partial_{p}\cdot
m)\neq 0$.

Through duality, finite detection is closely related to other notions of
finiteness. For simplicity, we do not state the following results in the
greatest possible generality.

######  Lemma 2.9.

Let $V$ be a finite dimensional, first-quadrant bigraded vector space. The
following are equivalent.

  1. (1)

The $\operatorname{Sym}(V^{\vee})$-module $\mathcal{M}$ is finitely detected.

  2. (2)

The adjoint $\operatorname{Sym}(V)$-comodule structure on $\mathcal{M}$ is
finitely cogenerated.

  3. (3)

The dual adjoint $\operatorname{Sym}(V^{\vee})$-module structure on
$\mathcal{M}^{\vee}$ is finitely generated.

###### Proof.

Let $q:\mathcal{M}\to N$ be a surjection with finite dimensional target. The
composite

$\mathcal{M}\xrightarrow{\delta}\operatorname{Sym}(V)\otimes\mathcal{M}\xrightarrow{\mathrm{id}\otimes
q}\operatorname{Sym}(V)\otimes N$

is given by the formula $m\mapsto\sum_{p}p\otimes q(\partial_{p}\cdot m)$,
where $p$ ranges over the monomial basis for $\operatorname{Sym}(V)$. Since
$q$ cogenerates if and only if this composite is injective, the first claim
follows. Since the dual of an injection is surjective, the second claim
follows as well. ∎

It follows that finitely detected modules enjoy many of the properties of
finitely generated modules.

######  Lemma 2.10.

Let $V$ be a finite dimensional, first-quadrant bigraded vector space. Any
submodule or quotient of a finitely detected
$\operatorname{Sym}(V^{\vee})$-module is also finitely detected.

###### Proof.

The claim follows from Lemma 2.9 and Noetherianity of
$\operatorname{Sym}(V^{\vee})$. ∎

######  Lemma 2.11.

Let $V$ be a finite dimensional, first-quadrant bigraded vector space of slope
$C\in\mathbb{Q}$ and $\mathcal{M}$ a finitely detected
$\operatorname{Sym}(V^{\vee})$-module. The graded dimension of any ray in
$\mathcal{M}$ of slope $C$ is eventually equal to a quasi-polynomial of degree
at most $\dim V^{0}-1$ and period dividing $\mathrm{lcm}(V^{0})$.

###### Proof.

The claim follows form Lemma 2.6 and 2.9 after negating bidegrees. ∎

##  3\. Transits and transit algebras

In this section, we introduce a family of algebras acting on the homology of a
Lie algebra.

###  3.1. Lie algebras and transits

Throughout, the term “Lie algebra” refers to a Lie algebra in the category of
bigraded vector spaces detailed above. Explicitly, a Lie algebra is a bigraded
vector space $\mathfrak{g}$ equipped with a map of bigraded vector spaces
$[-,-]:\mathfrak{g}\otimes\mathfrak{g}\to\mathfrak{g}$, called the bracket of
$\mathfrak{g}$, satisfying the equations

  1. (1)

$[x,y]+(-1)^{d(x)d(y)}[y,x]=0$

  2. (2)

$(-1)^{d(x)d(z)}[[x,y],z]+(-1)^{d(y)d(x)}[[y,z],x]+(-1)^{d(z)d(y)}[[z,x],y]=0$

for bihomogeneous elements $x,y,z\in\mathfrak{g}$. A map of Lie algebras is a
map of bigraded vector spaces intertwining the respective brackets.

We emphasize that, while weight is additive under the bracket, all signs are
independent of weight. We write $\mathfrak{z}=\mathfrak{z}(\mathfrak{g})$ for
the center of the Lie algebra $\mathfrak{g}$ and
$\mathfrak{a}=\mathfrak{a}(\mathfrak{g})$ for its Abelianization.

######  Definition 3.1.

Let $\mathfrak{g}$ be a Lie algebra. A _transit_ of $\mathfrak{g}$ is a pair
of maps of Lie algebras

$\mathfrak{h}\xrightarrow{f}\mathfrak{g}\xrightarrow{g}\mathfrak{k}$

with $\mathfrak{h}$ and $\mathfrak{k}$ Abelian and $f$ central. We say the
transit is _null_ or _split_ if $gf$ is trivial or bijective, respectively,
and _exact_ if $f$ and $g$ form a short exact sequence. A _map of transits_
from $(f_{1},g_{1})$ to $(f_{2},g_{2})$ is a commutative diagram of Lie
algebras of the form

$\textstyle{\mathfrak{h}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\textstyle{\mathfrak{h}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}}$$\textstyle{\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\scriptstyle{g_{2}}$$\textstyle{\mathfrak{a}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{a}_{1}.}$

Maps of transits compose in the obvious way, forming a category. Note that a
Lie algebra admits an exact transit if and only if it is two-step nilpotent.

######  Example 3.2.

The pair $\mathfrak{z}\to\mathfrak{g}\to\mathfrak{a}$ given by inclusion and
projection is a transit, called the _universal transit_.

The term “universal” refers to the property of being a terminal object in the
category of transits.

######  Example 3.3.

Given transits $(f_{1},g_{1})$ and $(f_{2},g_{2})$ of $\mathfrak{g}$, the
_product transit_ is the pair

$\mathfrak{h}_{1}\times\mathfrak{h_{2}}\xrightarrow{f_{1}+f_{2}}\mathfrak{g}\xrightarrow{(g_{1},g_{2})}\mathfrak{k}_{1}\times\mathfrak{k}_{2}$

(we use that the $\mathfrak{h}_{i}$ are Abelian). We say the product is
_clean_ if $g_{2}f_{1}=g_{1}f_{2}=0$.

######  Lemma 3.4.

Every transit is isomorphic (non-canonically) to the clean product of a split
transit and a null transit.

###### Proof.

Set $\mathfrak{h}_{1}=\mathfrak{k}_{1}=\mathrm{im}(gf)$,
$\mathfrak{h}_{2}=\ker(gf)$, and $\mathfrak{k}_{2}=\mathrm{coker}(gf)$.
Choosing splittings, which are splittings of Lie algebras by Abelianness, the
product decomposition

$\mathfrak{h}_{1}\times\mathfrak{h}_{2}\cong\mathfrak{h}\xrightarrow{f}\mathfrak{g}\xrightarrow{g}\mathfrak{k}\cong\mathfrak{k}_{1}\times\mathfrak{k}_{2}$

has the desired properties by inspection. ∎

###  3.2. Transit algebras

In this section, we fix a Lie algebra $\mathfrak{g}$ and a transit
$\mathfrak{h}\xrightarrow{f}\mathfrak{g}\xrightarrow{g}\mathfrak{k}$. We write
$\langle-,-\rangle$ for the graded commutator of an algebra $\mathcal{A}$,
i.e., $\langle x,y\rangle=xy-(-1)^{d(x)d(y)}yx$ for bihomogeneous elements
$x,y\in\mathcal{A}$.

######  Definition 3.5.

The _transit algebra_ associated to the transit $(f,g)$ is the quotient

$W(f,g)=T(\mathfrak{h}[1]\oplus\mathfrak{k}[1]^{\vee})/I,$

where $T$ denotes the tensor algebra and $I$ the two-sided ideal generated by
the relations

$\displaystyle\langle x,y\rangle$ $\displaystyle=\langle\lambda,\mu\rangle=0$
$\displaystyle\langle\lambda,x\rangle$ $\displaystyle=\lambda(g(f(x)))$

for $x,y\in\mathfrak{h}[1]$ and $\lambda,\mu\in\mathfrak{k}[1]^{\vee}$.

The transit algebra extends in the obvious way to a functor from transits to
algebras. In particular, given an arbitrary transit $(f,g)$, there is a
canonical map of algebras

$W(f,g)\to W(\mathfrak{g}),$

where $W(\mathfrak{g})$ is the transit algebra of the universal transit.

######  Example 3.6.

If $(f,g)$ is null, then
$W(f,g)=\operatorname{Sym}(\mathfrak{h}[1]\oplus\mathfrak{k}[1]^{\vee})$.

######  Example 3.7.

If $(f,g)$ is split, then $W(f,g)$ is isomorphic to the Weyl algebra on the
vector space $\mathfrak{h}[1]$.

######  Lemma 3.8.

The transit algebra of a clean product is canonically isomorphic to the tensor
product of the transit algebras of the factors.

###### Proof.

Cleanness implies that $\langle\lambda,x\rangle=0$ for $x\in\mathfrak{h}_{i}$
and $\lambda\in\mathfrak{k}_{j}$ when $i\neq j\in\\{1,2\\}$, and the claim
follows. ∎

We anticipate that the following simple result will be fundamental to future
applications, although we make no use of it in what follows.

######  Proposition 3.9.

Transit algebras are Noetherian.

###### Proof.

By Lemmas 3.4 and 3.8 and Examples 3.6 and 3.7, it suffices to show that the
tensor product of a finitely generated symmetric algebra and a finitely
generated Weyl algebra is Noetherian, which follows by repeated application of
[MR01, Thm. 2.9]. ∎

###  3.3. Action on homology

The role of the transit algebras introduced above is as a source of extra
structure on the homology of Lie algebras.

######  Definition 3.10.

The _Chevalley–Eilenberg complex_ of the Lie algebra $\mathfrak{g}$ is the
bigraded vector space
$\mathrm{CE}(\mathfrak{g})=\operatorname{Sym}(\mathfrak{g}[1])$ equipped with
the differential $\partial$ determined as a coderivation by the formula

$\partial(xy)=(-1)^{d(x)}[x,y].$

The _Lie algebra homology_ of $\mathfrak{g}$ is the bigraded vector space
$H_{*}^{\mathrm{Lie}}(\mathfrak{g})=H_{*}(\mathrm{CE}(\mathfrak{g}),\partial)$.

######  Remark 3.11.

Our notation is abusive in that the symbols $x$, $y$, and $[x,y]$ refer to the
corresponding elements of $\mathfrak{g}[1]$, while $d(x)$ is the degree of $x$
as an element of $\mathfrak{g}$.

The implicit claim that $\partial^{2}=0$ is equivalent to the Jacobi identity
(2) above. Note that $\partial\equiv 0$ if $\mathfrak{g}$ is Abelian.

######  Theorem 3.12.

For any transit $(f,g)$ of $\mathfrak{g}$, there is a canonical, functorial
action of $W(f,g)$ on $H_{*}^{\mathrm{Lie}}(\mathfrak{g})$.

###### Proof.

It suffices to consider the universal case. By Abelianness, addition equips
$\mathfrak{z}$ with the structure of a commutative monoid with respect to the
Cartesian monoidal structure on Lie algebras. For the same reason, the
composite map

$\mathfrak{z}\times\mathfrak{g}\subseteq\mathfrak{g}\times\mathfrak{g}\xrightarrow{+}\mathfrak{g}$

is a map of Lie algebras, equipping $\mathfrak{g}$ with the structure of a
$\mathfrak{z}$-module. The Chevalley–Eilenberg complex is a symmetric monoidal
functor, so we obtain an action of
$\mathrm{CE}(\mathfrak{z})=\operatorname{Sym}(\mathfrak{z}[1])$ on
$\mathrm{CE}(\mathfrak{g})$, hence on its homology (we again use that
$\mathfrak{z}$ is Abelian).

On the other hand, the diagonal equips any Lie algebra with the structure of
cocommutative comonoid with respect to the Cartesian monoidal structure, and
the composite map

$\mathfrak{g}\xrightarrow{\Delta}\mathfrak{g}\times\mathfrak{g}\xrightarrow{\pi\times\mathrm{id}}\mathfrak{a}\times\mathfrak{g}$

equips $\mathfrak{g}$ with the structure of an $\mathfrak{a}$-comodule, where
$\pi$ is the projection to the quotient. Applying the Chevalley–Eilenberg
complex and dualizing, we obtain an action of
$\operatorname{Sym}(\mathfrak{a}[1]^{\vee})\subseteq\mathrm{CE}(\mathfrak{a})^{\vee}$
on $\mathrm{CE}(\mathfrak{g})$, hence on its homology (we use that
$\mathfrak{a}$ is Abelian).

It remains to check that these two actions descend to an action of
$W(\mathfrak{g})$, for which it suffices to verify the last relation. To
begin, we note that, at the level of underlying graded objects, the first
action is given by the composite

$\operatorname{Sym}(\mathfrak{z}[1])\otimes\operatorname{Sym}(\mathfrak{g}[1])\subseteq\operatorname{Sym}(\mathfrak{g}[1])\otimes\operatorname{Sym}(\mathfrak{g}[1])\xrightarrow{m}\operatorname{Sym}(\mathfrak{g}[1]),$

where $m$ is the usual multiplication (note that $m$ is not itself a chain
map), while the second action is given by the composite (cap product)

$\operatorname{Sym}(\mathfrak{a}[1]^{\vee})\otimes\operatorname{Sym}(\mathfrak{g}[1])\xrightarrow{\operatorname{Sym}(\pi^{\vee})\otimes\delta}\operatorname{Sym}(\mathfrak{g}[1]^{\vee})\otimes\operatorname{Sym}(\mathfrak{g}[1])\otimes\operatorname{Sym}(\mathfrak{g}[1])\xrightarrow{\mathrm{ev}\cdot\mathrm{id}}\operatorname{Sym}(\mathfrak{g}[1]),$

where $\delta$ is the usual comultiplication.

Fix nonzero bihomogeneous elements $x\in\mathfrak{z}[1]$ and
$\lambda\in\mathfrak{a}[1]^{\vee}$ and a monomial
$p\in\operatorname{Sym}(\mathfrak{g}[1])$. Writing
$\delta(p)=\sum_{i}p_{i}\otimes p_{i}^{\prime}$, and abusively identifying
$\lambda$ and $\operatorname{Sym}(\pi^{\vee})(\lambda)$, we calculate that

$\displaystyle\langle\lambda,x\rangle\cdot p$
$\displaystyle=\sum_{i}\left[\lambda(xp_{i})p_{i}^{\prime}+(-1)^{|x||p_{i}|}\lambda(p_{i})xp_{i}^{\prime}-(-1)^{|\lambda||x|}\lambda(p_{i})xp_{i}^{\prime}\right]$
$\displaystyle=\lambda(x)p+\sum_{i}\left[(-1)^{|x||p_{i}|}\lambda(p_{i})xp_{i}^{\prime}-(-1)^{|\lambda||x|}\lambda(p_{i})xp_{i}^{\prime}\right]$
$\displaystyle=\lambda(x)p+\sum_{i:|\lambda|+|p_{i}|=0}\left[(-1)^{|x||p_{i}|}\lambda(p_{i})xp_{i}^{\prime}-(-1)^{|\lambda||x|}\lambda(p_{i})xp_{i}^{\prime}\right]$
$\displaystyle=\lambda(x)p,$

where we use that $\lambda$ vanishes on monomials of polynomial degree
different from $1$ and on those of bidegree different from
$(-w(\lambda),-d(\lambda))$. Since $p$ was arbitrary, this calculation
establishes the desired relation. ∎

##  4\. Stability phenomena

In this section, we give examples of stability phenomena in Lie algebra
homology arising from the transit algebra action introduced above. We then
deduce our results about the homology of configuration spaces.

###  4.1. Transit algebras and stability

The purpose of this section is to demonstrate that the transit algebra actions
described in Theorem 3.12 give rise to stability phenomena in the homology of
Lie algebras. For simplicity, we work under the following standing
assumptions.

######  Assumption 4.1.

The Lie algebra $\mathfrak{g}$ is finite dimensional with $\mathfrak{g}[1]$
first-quadrant and connected.

There are two versions of the following result. We state the version for the
maximal slope; the version for the minimal slope is obtained by reversing
inequalities and interchanging $\min$ and $\max$.

######  Proposition 4.2.

Let $g:\mathfrak{g}\to\mathfrak{k}$ be an Abelian quotient, and suppose that
we have the inequalities

$\displaystyle m_{\max}(\mathfrak{k}[1]^{0})$ $\displaystyle\leq C$
$\displaystyle m_{\max}(\ker g[1]^{0})$ $\displaystyle<C.$

Each ray of slope $C$ in $H_{*}^{\mathrm{Lie}}(\mathfrak{g})$ is finitely
detected over $\operatorname{Sym}(\mathfrak{k}^{\vee}[-1]^{0}_{C})\subseteq
W(0,g).$ In particular, the graded dimension of such a ray is eventually equal
to a quasi-polynomial of degree at most $\dim(\mathfrak{k}[1]^{0}_{C})-1$ and
period dividing $\mathrm{lcm}(\mathfrak{k}[1]^{0}_{C})$.

###### Proof.

Any ray of slope $C$ in $H_{*}^{\mathrm{Lie}}(\mathfrak{g})$ is a
$\operatorname{Sym}(\mathfrak{k}^{\vee}[-1]_{C}^{0})$-submodule and a
subquotient of the corresponding ray in $\mathrm{CE}(\mathfrak{g})$;
therefore, by Lemmas 2.9, 2.10, and 2.11, it suffices to show that every ray
of slope $C$ in the dual adjoint module $\mathrm{CE}(\mathfrak{g})^{\vee}$ is
finitely generated.

The action of $\operatorname{Sym}(\mathfrak{k}^{\vee}[-1]^{0}_{C})$ on
$\mathrm{C}\mathrm{E}(\mathfrak{g})^{\vee}$, which respects the differential,
extends along the monomorphism $g^{\vee}$ to an action of
$\operatorname{Sym}(\mathfrak{g}^{\vee}[-1])$, which does not respect the
differential. Under this action, $\mathrm{CE}(\mathfrak{g})^{\vee}$ is free of
rank $1$, hence finitely generated. Since
$m_{\max}(\mathfrak{g}^{\vee}[-1]^{0})\leq C$, it follows from Lemma 2.7 that
each ray of slope $C$ is finitely generated over
$\operatorname{Sym}(\mathfrak{g}^{\vee}[-1]_{C}^{0})$. Our assumption implies
that $\mathfrak{g}[1]^{0}_{C}\cap\ker g[1]^{0}=0$, so
$\mathfrak{g}^{\vee}[-1]^{0}_{C}=\mathfrak{k}^{\vee}[-1]^{0}_{C}$. ∎

In a sense, the next result is dual to Proposition 4.2. Again, there are two
versions, only one of which we state.

######  Proposition 4.3.

Let $f:\mathfrak{h}\to\mathfrak{g}$ be a central subalgebra, and suppose that
we have the inequalities

$\displaystyle m_{\max}(\mathfrak{h}[1]^{0})$ $\displaystyle\leq C$
$\displaystyle m_{\max}(\operatorname{coker}f[1]^{0})$ $\displaystyle<C.$

Each ray of slope $C$ in $H_{*}^{\mathrm{Lie}}(\mathfrak{g})$ is finitely
generated over $\operatorname{Sym}(\mathfrak{h}[1]^{0}_{C})\subseteq W(f,0).$
In particular, the graded dimension of such a ray is eventually equal to a
quasi-polynomial of degree at most $\dim(\mathfrak{h}[1]^{0}_{C})-1$ and
period dividing $\mathrm{lcm}(\mathfrak{h}[1]^{0}_{C})$.

###### Proof.

Since each ray of slope $C$ in $\mathrm{CE}(\mathfrak{g})$ is a submodule with
subquotient the corresponding ray of slope $C$ in
$H_{*}^{\mathrm{Lie}}(\mathfrak{g})$, it suffices by Noetherianity and Lemma
2.6 to show that each ray of slope $C$ in $\mathrm{CE}(\mathfrak{g})$ is
finitely generated.

The action of $\operatorname{Sym}(\mathfrak{h}[1]_{C}^{0})$, which respects
the differential, extends along the inclusion of $\mathfrak{h}$ to an action
of $\operatorname{Sym}(\mathfrak{g}[1])$, which does not respect the
differential. Under this action, $\mathrm{CE}(\mathfrak{g})$ is free of rank
$1$, hence finitely generated. Since $m_{\max}(\mathfrak{g}[1]^{0})\leq C$, it
follows from Lemma 2.7 that each ray of slope $C$ is finitely generated over
$\operatorname{Sym}(\mathfrak{g}[1]_{C}^{0})$, but
$\mathfrak{g}[1]^{0}_{C}=\mathfrak{h}[1]^{0}_{C}$ by our assumption. ∎

The same method yields the following mild extension.

######  Corollary 4.4.

Let $\mathfrak{g}$ be as in Proposition 4.3 and consider a semidirect product
$\widetilde{\mathfrak{g}}=\mathfrak{g}\rtimes\mathfrak{l}$ with $\mathfrak{l}$
free and finitely generated. If $\mathfrak{l}$ centralizes every odd degree
element of $\mathfrak{h}$ having slope $C$ in $\mathfrak{h}[1]$, then the
conclusion of Proposition 4.3 holds for $\widetilde{\mathfrak{g}}$.

###### Proof.

The action of $\operatorname{Sym}(\mathfrak{h}[1]^{0}_{C})$ on
$\mathrm{CE}(\mathfrak{g})$ described in the proof of Proposition 4.3 extends
to an action on the underlying bigraded vector space of
$\mathrm{CE}(\widetilde{\mathfrak{g}})$, the latter being a sum of bigraded
shifts of the former, and our assumption on the relationship between
$\mathfrak{l}$ and $\mathfrak{h}$ guarantees that this action is compatible
with the differential. This action is compatible with the filtration by
polynomial degree in elements of $\mathfrak{l}$, so the
Lyndon–Hochschild–Serre spectral sequence

$H_{*}^{\mathrm{Lie}}(\mathfrak{l};H_{*}^{\mathrm{Lie}}(\mathfrak{g}))\implies
H_{*}^{\mathrm{Lie}}(\widetilde{\mathfrak{g}})$

is a spectral sequence of
$\operatorname{Sym}(\mathfrak{h}[1]^{0}_{C})$-modules. By Noetherianity and
Lemma 2.6, it suffices to show that each ray of slope $C$ in the initial page
of this spectral sequence is finitely generated.

Writing $V$ for the finite dimensional bigraded vector space generating
$\mathfrak{l}$ as a free Lie algebra, the page in question is the homology of
the complex

$H_{*}^{\mathrm{Lie}}(\mathfrak{g})\otimes V\to
H_{*}^{\mathrm{Lie}}(\mathfrak{g})$

of $\operatorname{Sym}(\mathfrak{h}[1]^{0}_{C})$-modules. Each ray of slope
$C$ in this complex is a finite sum of bigraded shifts of rays of slope $C$ in
$H_{*}^{\mathrm{Lie}}(\mathfrak{g})$, each of which has already been shown to
be finitely generated. The claim follows by Noetherianity. ∎

######  Remark 4.5.

The same method of proof establishes the analogous conclusion for iterated
semidirect products with free Lie algebras. Details are left to the reader.

Our final corallary uses the action of both halves of the transit algebra to
establish that rays are free modules in certain cases.

######  Corollary 4.6.

Let $\mathfrak{h}\xrightarrow{f}\mathfrak{g}\xrightarrow{g}\mathfrak{k}$ be a
split transit, and suppose that we have the inequalities

$\displaystyle m_{\max}(\mathfrak{h}[1]^{0})$ $\displaystyle\leq C$
$\displaystyle m_{\max}(\operatorname{coker}f[1]^{0})$ $\displaystyle<C.$

Each ray of slope $C$ of $H^{\mathrm{Lie}}_{*}(\mathfrak{g})$ is a finitely
generated free module over
$\operatorname{Sym}(\mathfrak{h}[1]^{0}_{C})\subseteq W(f,g)$.

###### Proof.

Finite generation follows from 4.3. By Example 3.7 and Theorem 3.12, the
action of $\operatorname{Sym}(\mathfrak{h}[1]^{0}_{C})$ extends to an action
of the Weyl algebra on $\mathfrak{h}[1]^{0}_{C}$. It is well known that a
finitely generated module over a polynomial ring whose action extends to the
Weyl algebra is free—briefly, choosing a minimal set of generators and a
relation $R$ of minimal degree, we note that, unless $R$ is constant
(contradicting the first requirement of minimality), the derivative of $R$ is
a relation of strictly smaller degree. ∎

###  4.2. Application to configuration spaces

Fix a $d$-manifold $M$ of finite type and write

$\mathfrak{g}_{M}=H_{c}^{*}(M;\mathbb{Q}^{w})\otimes v\oplus
H_{c}^{*}(M;\mathbb{Q})\otimes[v,v],$

where $v$ and $[v,v]$ are formal parameters of bidegree $(1,d-1)$ and
$(2,2d-2)$, respectively, and cohomology is regarded as concentrated in
negative degrees and weight $0$ (the reader is reminded that $\mathbb{Q}^{w}$
denotes the orientation sheaf). This bigraded vector space becomes a Lie
algebra with bracket defined to be

$[\alpha\otimes v,\beta\otimes v]=(-1)^{d(\beta)(d-1)}\alpha\beta\otimes[v,v]$

and zero otherwise, where $\alpha$ and $\beta$ are multiplied via the twisted
cup product.

######  Theorem 4.7 ([Knu17]).

Let $M$ be a manifold of even dimension $d$. There is an isomorphism of
bigraded vector spaces

$\bigoplus_{n\geq 0}H_{*}(B_{n}(M);\mathbb{Q})\cong
H_{*}^{\mathrm{Lie}}(\mathfrak{g}_{M}).$

Through Theorem 4.7, stability phenomena in Lie algebra homology become
stability phenomena in the homology of configuration spaces. As a matter of
notation, we write $\mathfrak{h}_{M}=H_{c}^{*}(M;\mathbb{Q})\otimes[v,v]$ and
$\mathfrak{k}_{M}=H_{c}^{*}(M;\mathbb{Q}^{w})\otimes v$, considered as Abelian
Lie algebras. Thus, we have the exact transit
$\mathfrak{h}_{M}\to\mathfrak{g}_{M}\to\mathfrak{k}_{M}$. We observe that, for
a degree $i$ cohomology class $\alpha$, as elements of $\mathfrak{g}_{M}[1]$,
we have

$\displaystyle d(\alpha\otimes v)=m(\alpha\otimes v)$ $\displaystyle=d-i$
$\displaystyle d(\alpha\otimes[v,v])$ $\displaystyle=2d-i-1$ $\displaystyle
m(\alpha\otimes[v,v])$ $\displaystyle=d-\frac{i+1}{2}.$

Note that $\mathfrak{g}_{M}$ satisfies Assumption 4.1.

As a first example, we give a proof of classical homological stability.

###### Proof of Theorem 1.1.

Since $M$ is a manifold, we have $0\leq i\leq d$ in the equations above, so we
have the inequalities

$\displaystyle m_{\min}(\mathfrak{h}_{M}[1])$
$\displaystyle\geq\frac{d-1}{2}>0$ $\displaystyle
m_{\min}(\mathfrak{k}_{M}[1])$ $\displaystyle\geq 0$

(we use that $d>1$). The claim follows after invoking the minimal slope
version of Proposition 4.2 with $C=0$ and noting that
$\mathfrak{k}_{M}[1]_{0}^{0}\cong H_{c}^{d}(M;\mathbb{Q}^{w})\cong
H_{0}(M;\mathbb{Q})\neq 0$ by Poincaré duality. ∎

We turn now to extremal stability, our main result.

###### Proof of Theorem 1.2.

Assume first that $M$ has no compact component. Then $0<i\leq d$ in the
equations above, so

$\displaystyle m_{\max}(\mathfrak{h}_{M}[1])$ $\displaystyle\leq d-1$
$\displaystyle m_{\max}(\mathfrak{k}_{M}[1])$ $\displaystyle\leq d-1$

We observe that $d(\alpha\otimes v)$ is even if and only if $i$ is even, in
which case the second inequality is strict; thus, we may apply Proposition 4.3
with $C=d-1$. The claim in this case follows upon noting that
$\mathfrak{h}[1]_{d-1}^{0}\cong H_{c}^{1}(M;\mathbb{Q})\cong
H_{d-1}(M;\mathbb{Q}^{w})$ by Poincaré duality.

In general, we proceed by induction on the number of compact components of
$M$. Writing $\dot{M}$ for the complement of a point in $M$ lying in a compact
component, we have the exact sequence of Lie algebras

$\mathfrak{g}_{\dot{M}}\to\mathfrak{g}_{M}\to\mathfrak{l},$

where $\mathfrak{l}$ is free on one generator of bidegree $(1,d-1)$ or
$(2,2d-2)$ according to whether the component in question is orientable. In
either case, this sequence splits by freeness, expressing $\mathfrak{g}_{M}$
as a semidirect product. Since $\mathfrak{l}$ centralizes every element of
$\mathfrak{h}_{M}=H_{c}^{*}(M;\mathbb{Q})\otimes[v,v]$, the claim follows from
Corollary 4.4 (see Remark 4.5). ∎

We note the following structural statement, which we have established in the
proof of Thorem 1.2.

######  Theorem 4.8.

Let $M$ be as in Theorem 1.2. Then every ray of slope $d-1$ in
$\bigoplus_{n\geq 0}H_{*}(B_{n}(M))$ is a finitely generated
$\operatorname{Sym}(H_{d-1}(M;\mathbb{Q}^{w})\langle 2\rangle[2d-2])$ module.
If $H_{d}(M;\mathbb{Q})=0$, then these modules are free.

###### Proof.

The proof of Theorem 1.2 uses the central subalgebra
$H^{1}_{c}(M)\otimes[v,v]\subseteq\mathfrak{h}_{M}\subseteq\mathfrak{h}_{M}$.
By Poincaré duality $H^{1}_{c}(M)\cong H_{d-1}(M;\mathbb{Q}^{w})$. It is shown
that every slope $d-1$ ray is a finitely generated
$\operatorname{Sym}(H^{1}_{c}(M)\langle 2\rangle[1])$ module.

When $H^{0}_{c}(M;\mathbb{Q}^{w})\cong H_{d}(M;\mathbb{Q})=0,$ we have that

$\mathfrak{j}_{M}:=\bigoplus_{i>0}H^{i}_{c}(M;\mathbb{Q}^{w})\otimes
v\oplus\bigoplus_{i\neq 1}H^{i}_{c}(M;\mathbb{Q})\otimes[v,v]$

is an ideal— in fact
$[\mathfrak{g}_{M},\mathfrak{g}_{M}]\subseteq\mathfrak{j}_{M}$ since any
element of $\mathfrak{g}_{M}$ of the form $\alpha\otimes[v,v]$ is central, and
any pair of elements $\beta_{1}\otimes v,\beta_{2}\otimes
v\in\mathfrak{g}_{M}$ with $d(\beta_{i})>0$ satisfies $[\beta_{1}\otimes
v,\beta_{2}\otimes v]\in\mathfrak{j}_{M}$.

Further,
$H^{1}_{c}(M)\otimes[v,v]\to\mathfrak{g}_{M}\to\mathfrak{g}_{M}/\mathfrak{j}_{M}$
is a split transit. Hence as in the proof of Corollary 4.6, the action of
$\operatorname{Sym}(H_{d-1}(M;\mathbb{Q}^{w})[1]\langle 2\rangle)$ extends to
an action of the Weyl algebra. Therefore every ray of slope $d-1$ is a
finitely generated free module. ∎

###  4.3. Vanishing and a loose end

If $H_{d-1}(M;\mathbb{Q})=0$, then the conclusion of Theorem 1.2 is that
$H_{\nu_{n}-i}(B_{n}(M);\mathbb{Q})$ is eventually zero, i.e., the homology of
the configuration spaces of $M$ has maximal radial slope strictly less than
$d-1$. In such a situation, the question of extremal stability should concern
rays of smaller slope. For simplicity, we assume that $M$ is orientable.

######  Theorem 4.9.

Let $M$ be an orientable manifold of finite type and even dimension $d\geq 2$.
Fix $r\geq 3$, and suppose that $H_{d-s}(M;\mathbb{Q})=0$ for $0<s<r$. For $n$
sufficiently large with respect to $i$, the function $n\mapsto\dim
H_{i+n(d-1-\lfloor\frac{r}{2}\rfloor)}(B_{n}(M);\mathbb{Q})$ is equal to a
quasi-polynomial in $n$ of period dividing two, and degree at most ${\dim
H_{d-r}(M;\mathbb{Q}^{w})-1}$ if $r$ is odd and ${\dim
H_{d-r-1}(M;\mathbb{Q}^{w})-1}$ if $r$ is even.

###### Proof.

The proof mirrors that of Theorem 1.2, with the reduction to the non-compact
case proceeding unchanged. We aim to apply Proposition 4.3 with
$C=d-1-\lfloor\frac{r}{2}\rfloor$. Since $M$ is orientable, twisted and
untwisted cohomology coincide, and $H_{c}^{s}(M;\mathbb{Q})\cong
H_{d-s}(M;\mathbb{Q})=0$ for $0\leq s<r$. Thus, we have

$\displaystyle m_{\max}(\mathfrak{h}_{M}[1])$ $\displaystyle\leq
d-\frac{r+1}{2}$ $\displaystyle m_{\max}(\mathfrak{k}_{M}[1])$
$\displaystyle\leq d-r<C,$

where we use that $r\geq 3$. We observe that $d(\alpha\otimes[v,v])$ is even
if and only if $i$ is odd, so

$m_{\max}(\mathfrak{h}_{M}[1]^{0})\leq\begin{cases}d-\frac{r+1}{2}&\quad
r\text{ odd}\\\ d-\frac{r+2}{2}&\quad r\text{ even},\end{cases}$

as desired. As before, the claim follows from Poincaré duality. ∎

In the case $r=2$, the situation is unclear, and we ask the following.

######  Question 4.10.

Suppose that $H_{d-1}(M;\mathbb{Q})=0$. For $i\in\mathbb{N}$, is the Hilbert
function

$n\mapsto\dim H_{{n(d-2)+i}}(B_{n}(M);\mathbb{Q})$

eventually a quasi-polynomial?

This question has an affirmative answer under the further assumption that
$H_{d-2}(M;\mathbb{Q})=0$, by Theorem 4.9, and it is not hard to show that the
same is true when $H_{d-3}(M;\mathbb{Q})=0$ using Proposition 4.2.

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