August 28, 2024

# The Differential Topology of the
Thurston Spine of Teichmüller Space

Ingrid Irmer  SUSTech International Center for Mathematics
Southern University of Science and Technology
Shenzhen, China  Department of Mathematics
Southern University of Science and Technology
Shenzhen, China<EMAIL_ADDRESS>
###### Abstract.

This paper shows that there is a mapping class group-equivariant deformation
retraction of the Teichmüller space of a closed, orientable surface onto a
cell complex of dimension equal to the virtual cohomological dimension of the
mapping class group. The image of the deformation retraction is a subcomplex
of the CW complex first described by Thurston – the Thurston spine. The
Thurston spine is the set of points in Teichmüller space corresponding to
hyperbolic surfaces for which the set of shortest geodesics (the systoles)
cuts the surface into polygons. As in the case for punctured surfaces, there
is a mapping class group-equivariant cell decomposition of a bordification of
Teichmüller space implicit in the argument.

###### Contents

  1. 1 Introduction
  2. 2 Definitions and conventions
  3. 3 Thurston’s deformation retraction
  4. 4 Strata with filling systoles
  5. 5 Cells and Morse theory
    1. 5.1 Cell decompositions parametrised by length functions
  6. 6 The deformation retraction of the Thurston spine

##  1\. Introduction

In this paper it will be shown that the Teichmüller space of a closed,
orientable surface of genus $g\geq 2$ has a mapping class group-equivariant
deformation retraction onto a CW complex of dimension equal to the virtual
cohomological dimension of the mapping class group, namely $4g-5$, see [10].

In genus 2, the existence of a mapping class group-equivariant deformation
retraction onto a CW complex of dimension 3 follows from a computation by
Schmutz, see Theorem 44 of [24]. This has also been independently verified by
a SageMath calculation of the Author, using Rivin’s angle coordinates on a
sphere with six cone points. When the genus is greater than or equal to 2 and
the surface has at least one puncture, a mapping class group-equivariant
deformation retraction to a CW complex of dimension equal to the virtual
cohomological dimension of the mapping class group was given in [10]. This
result has had a number of applications; for example, it was used in [18] to
prove a conjecture of Witten’s about intersection theory on the moduli space
of Riemann surfaces with punctures. It is well known that attempts at
generalising results to surfaces without boundary and/or punctures often run
into serious technical difficulties. In this case, determining whether a
deformation retraction onto a CW complex of dimension equal to the virtual
cohomological dimension actually exists is listed as the first open question
in [6]. The interested reader is referred to [6] and Chapter 3.3 of [11] for a
survey of the background and applications of this question.

Recall that the Teichmüller space $\mathcal{T}_{g}$ of a closed surface of
genus $g$ is the space of hyperbolic structures on a fixed topological surface
of genus $g$. A more detailed definition will be given in Subsection 2. The
Thurston spine $\mathcal{P}_{g}$ is the subspace of $\mathcal{T}_{g}$ where
the surface is cut into polygons by the set of shortest geodesics (the
systoles). Thurston showed that $\mathcal{P}_{g}$ is a CW complex; more
details can be found in [11]. The dimension of the Thurston spine for
arbitrary genus is not known. In [4], by generalising an example due to
Schmutz, it was shown that the dimension can increase with genus faster than
$4g-5$.

######  Theorem 1.

There is a mapping class group-equivariant deformation retraction of the
Thurston spine of a closed orientable surface of genus $g$ onto a subcomplex
of dimension equal to $4g-5$.

In his preprint [29] from 1985, Thurston described the construction of a
deformation retraction of $\mathcal{T}_{g}$ onto $\mathcal{P}_{g}$. This
construction will be discussed in detail in Section 3. In [15] a list of
questions was made that would need to be answered for Thurston’s construction
to be considered a complete proof. These questions are resolved in Section 3.
In an unpublished preprint [5], false accusations were made, claiming that
[29] contains serious gaps, and insinuating that this preprint had gaps as a
result. It was pointed out to the author of [5] that the accusation taken from
[15] was dealt with in what has now been renamed Lemma 8, and the second
accusation arises from a misunderstanding of Morse Theory on the part of the
author of [5]. A deformation retraction analogous to Thurston’s flow, obtained
by taking the flow of a smooth vector field with isolated zeros, does not in
general give rise to a deformation retraction onto the set of points at which
the vector field vanishes. The reason for this is that there can be flowlines
of the vector field from one critical point to another. Although these errors
were pointed out to the author of [5], he has refused to retract his
accusations.

The Teichmüller space $\mathcal{T}_{g}$ is contractible, and by Fricke’s
theorem, the mapping class group acts properly discontinuously on it. A
specific construction and a discussion of “nice” spaces on which the mapping
class group acts, are given in [16]. One characterisation of moduli space is
as the quotient of $\mathcal{T}_{g}$ by the action of the mapping class group.
Studying mapping class group-equivariant deformation retractions of
Teichmüller space is therefore intimately connected with questions about the
virtual cohomological dimension of the mapping class group and about the
problem of finding a space of the lowest possible dimension on which the
mapping class group acts properly discontinuously. The virtual cohomological
dimension gives a lower bound on this dimension. As the Thurston spine is the
image of a mapping class group-equivariant deformation retraction of
Teichmüller space, Theorem 1 together with the construction in [29] shows that
this lower bound is achieved.

Systoles and topological Morse theory. A systole on a hyperbolic surface is a
curve whose geodesic representative has length less than or equal to that of
any other geodesic on the surface. Every hyperbolic surface has finitely many
systoles. The systole function $f_{sys}:\mathcal{T}_{g}\rightarrow\mathbb{R}$
is a piecewise smooth function whose value at any point is equal to the length
of the systoles. It was shown in [1] that the systole function is a
topological Morse function. Topological Morse functions were first defined in
[22], and can be informally described as continuous functions that retain the
interesting properties of (smooth) Morse functions. A precise definition will
be given in Section 2.

The Thurston spine contains all the critical points of the systole function,
and will be shown to be inseparably tied up with the study of the systole
function, as illustrated in the following proposition.

######  Proposition 2.

There is a bijection between the critical points of the systole function and
the cells of the Thurston spine. The index of each critical point is equal to
the codimension of the cell in whose interior it is contained.

The systole function gives a mapping class group-equivariant
stratification111This follows from Lemma 13, using the definition of
stratified space from [20]. However, as this paper does not rely on any
technical properties of stratifications, this will not be formally defined or
proven here. of Teichmüller space, where each stratum is labelled by the set
of geodesics representing the systoles in the stratum. A set of curves on a
surface is said to fill the surface if the complement of the geodesic
representatives is a union of polygons. This definition can be generalised to
surfaces without hyperbolic structures by replacing the geodesic
representatives of curves by representatives in minimal position.

Cells parametrised by length functions. A length function is an analytic
function $\mathcal{T}_{g}\rightarrow\mathbb{R}_{+}$ that generalises the map
whose value at any point is the length of a fixed marked geodesic. A precise
definition will be given in Section 2. In a number of papers, including [24],
[26] and [28], Schmutz introduced the possibility of constructing mapping
class group-equivariant cell decompositions of Teichmüller space based on
geodesic length functions. Length functions are known to satisfy many
convexity properties, for example, they are strictly convex along Weil-
Petersson geodesics, [30]. The cells are defined as sets of points on which
certain lengths functions take their minimum values. Intuitively, the cells
are convex hulls of strata of the systole function at infinity.

The major drawback of Schmutz’s beautiful geometric construction is that
existence of cell decompositions parametrised by length functions could not be
shown for Teichmüller spaces of surfaces without marked points or
punctures222At first glance, this appears to have been done in [25] for genus
2, but on closer inspection, one sees that the cell decomposition is only
invariant under the action of the mapping class group that fixes a basepoint..
For genus greater than 2, all previously known mapping class group-equivariant
cell decompositions of bordifications of Teichmüller space required a marked
point, puncture or boundary relative to which coordinates on the cells can be
defined.

Cell decompositions and duality. In Theorem 20, a duality will be established
between certain sets parametrised by length functions and the strata of
filling systoles that make up the cells of the Thurston spine. This duality
will be shown to imply existence of a cell decomposition parametrised by
length functions, although strictly speaking, the resulting cell decomposition
might not be dual to the spine. The reason for this is that the sets
parametrised by length functions need to be subdivided to obtain cells
wherever a property Schmutz referred to as regularity breaks down. It will be
shown that this subdivision can be done in a canonical way.

The downside of this proof of existence is an understanding of the difficulty
inherent in actually writing down such a cell decomposition. It is not only
necessary to compute all possible configurations (modulo the action of the
mapping class group) of sets of filling systoles, but also to understand where
regularity breaks down.

The cell decompositions obtained in this paper are arguably a generalisation
of the combinatorial cell decompositions for Teichmüller spaces of surfaces
with punctures or marked points used for example in [10]. For cell
decompositions of punctured surfaces or surfaces with marked points, the
graphs used to label the cells give almost complete information about the
curves that can and cannot be realised as systoles within the cell. These
potentially short curves determine the boundary at infinity of the cells, in
the same way that a set $C$ of filling systoles determines the boundary at
infinity of the cells in cell decompositions defined in Section 5.

The cell decomposition is only implicit in the proof of Theorem 1, because
combinatorial information is not readily available as in the punctured case.
An extra global, topological ingredient is needed.

The Steinberg module and the Thurston spine. The “thick” part,
$\mathcal{T}_{g}^{thick}$, of $\mathcal{T}_{g}$ is defined to be the set of
all points of $\mathcal{T}_{g}$ corresponding to surfaces with injectivity
radius greater than a specific constant called the Margulis constant
$\epsilon_{M}$. The Margulis constant has many important geometric and
algebraic properties; for example, when $f_{sys}$ is less than or equal to
$\epsilon_{M}$, it follows from Margulis’s Lemma that the systoles are
pairwise disjoint. This can be used to show that there is a mapping class
group-equivariant deformation retraction of $\mathcal{T}_{g}$ onto
$\mathcal{T}_{g}^{thick}$, [16]. It was shown in [10] that Harvey’s complex of
curves $\mathcal{C}_{g}$ (definition given in Section 2) is homotopy
equivalent to an infinite wedge of spheres of dimension $2g-2$. It is known,
[14], that $\mathcal{T}_{g}^{thick}$ is contractible, and its boundary is
homotopy equivalent to $\mathcal{C}_{g}$.

The Steinberg module is defined to be the reduced homology group
$\tilde{H}_{2g-2}\bigl{(}\mathcal{C}_{g};\mathbb{Z}\bigr{)}$. As there is a
simplicial action of the mapping class group on the complex of curves, the
Steinberg module inherits the structure of a mapping class group-module.

An unmatched face of a cell complex is a $k-1$ dimensional cell on the
boundary of only one $k$-cell. Theorem 1 is proven by using the homology of
$\partial\mathcal{T}_{g}^{thick}$ to show that certain cells dual to the spine
— analogous to the cells defined by Schmutz — must be homotopic into
$\partial\mathcal{T}_{g}^{thick}$ relative to their boundary on
$\partial\mathcal{T}_{g}^{thick}$. When the dimension of $\mathcal{P}_{g}$ is
greater than $4g-5$, this shows the existence of unmatched faces of
subcomplexes of the Thurston spine. The existence of an unmatched face implies
the existence of a deformation retraction onto a smaller subcomplex.

Organisation of the paper. Background definitions and notations are given in
Section 2. Section 3 gives a detailed treatment of Thurston’s construction and
deals with objections that have been raised in the literature. Some background
about intersections of level sets of length functions is given to provide an
alternative proof of a proposition. These ideas are implicit in the work of
Schmutz, which is the subject of Section 5. This section relates the work of
Schmutz to the Thurston spine, which is used to construct a
$\Gamma_{g}$-equivariant cell decomposition of a bordification of Teichmüller
space in Subsection 5.1. Section 4 establishes some basic properties of strata
with filling systoles. The dimension of the Thurston spine is calculated in
Section 6. As Section 6 does not rely on Sections 5 and 4, a reader hoping for
a quick proof of Theorem 1 can read Section 6 directly after Section 3.

### Acknowledgments

The author would like to thank Stavros Garoufalidis, Scott Wolpert, Huang Yi
and Don Zagier for commenting on myriad earlier drafts of this paper which led
to considerable improvements, and to Maxime Fortier Bourque who pointed out a
problem with an earlier version. Particular thanks to Scott Wolpert, for
taking so much time to discuss this project and to share his knowledge of the
subject.

##  2\. Definitions and conventions

As the title suggests, this section provides definitions, assumptions and
background that will be needed for the rest of the paper. As this paper is not
solely intended for specialists in Teichmüller theory, this section has been
made fairly detailed.

A marking of $\mathcal{S}_{g}$ is a diffeomorphism
$f:\mathcal{S}_{g}\rightarrow M$, where $M$ is a closed, orientable,
hyperbolic surface with genus $g\geq 2$, and $\mathcal{S}_{g}$ is a closed,
orientable, topological surface of genus $g$. Teichmüller space
$\mathcal{T}_{g}$ is the set of pairs $(M,f)$ modulo the equivalence relation
$(M,f)\sim(N,h)$ if $f\circ h^{-1}$ is isotopic to an isometry. There is a
topology on $\mathcal{T}_{g}$ that makes it homeomorphic to
$\mathbb{R}^{6g-6}$. The details can be found in Section 10.6 of [8], which is
a general reference for Teichmüller space and mapping class groups.

The mapping class group $\Gamma_{g}$ is the group of isotopy classes of
orientation preserving diffeomorphisms from
$\mathcal{S}_{g}\rightarrow\mathcal{S}_{g}$. There is an action
$\Gamma_{g}\times\mathcal{T}_{g}\rightarrow\mathcal{T}_{g}$ of $\Gamma_{g}$ on
$\mathcal{T}_{g}$ given by $\gamma\times(M,f)\mapsto(M,f\circ\gamma^{-1})$.
The moduli space $\mathcal{M}_{g}$ is the quotient of $\mathcal{T}_{g}$ by
this action.

The assumption $g\geq 2$ is made to ensure that all surfaces are hyperbolic.
Once a point in Teichmüller space is chosen, by an abuse of notation, $S_{g}$
will be used to denote the surface $\mathcal{S}_{g}$ endowed with the
corresponding hyperbolic structure.

A curve on $S_{g}$ will always be simple and closed, i.e. is a nontrivial
isotopy class of embeddings of $S^{1}$ into $S_{g}$, where $S^{1}$ is the
unpointed, unoriented 1-sphere. Curves are assumed to inherit a marking. On
$S_{g}$, each curve has a unique geodesic representative, and the length of
the curve will be defined to be the length of its geodesic representative.
When there is no possibility of confusion, the image of a particular
representative of the isotopy class will also be referred to as a curve.

In this paper a cell is diffeomorphic to an open ball. The standard definition
of a cell in a CW complex is closed; these will be understood to be the
closures of the cells discussed in this paper. By an abuse of notation,
vertices, edges and faces of a cell will be used to refer to the vertices,
edges and faces of the closure of the cell.

The curve complex, $\mathcal{C}_{g}$, defined by Harvey in [12], is the flag
complex with $n$-simplices in 1-1 correspondence with sets of $n+1$
homotopically nontrivial, pairwise disjoint, closed curves on
$\mathcal{S}_{g}$. Since the action of the mapping class group on curves
preserves intersection properties, the mapping class group acts simplicially
on $\mathcal{C}_{g}$.

Whenever a metric is needed on $\mathcal{T}_{g}$, the Weil-Petersson metric
will be assumed unless stated otherwise. A subset of Teichmüller space will be
said to be convex if for any two points in the subset, the unique geodesic arc
connecting them is also contained in the subset.

Many of the results cited in this paper are also proven for Teichmüller space
of surfaces with cusps, boundary components or distinguished points. To keep
notation to a minimum, such results will only be stated at the level of
generality needed here.

######  Definition 3 (Length function).

Every homotopically nontrivial curve $c$ on $\mathcal{S}_{g}$ determines an
analytic function $L(c):\mathcal{T}_{g}\rightarrow\mathbb{R}^{+}$ whose value
at any point $x$ is given by the length of the geodesic representative of $c$
at $x$. Given a finite ordered set of curves $C=(c_{1},\ldots,c_{k})$ and a
set of real, positive weights $A=(a_{1},\ldots,a_{k})$, a length function
$L(A,C):\mathcal{T}_{g}\rightarrow\mathbb{R}^{+}$ is an analytic function
given by

$L(A,C)(x)\,=\,\sum_{j=1}^{k}a_{j}L(c_{j})(x)$

When all the weights in $A$ are equal to 1, $L(A,C)$ will be written $L(C)$ or
$L(c)$ when $C=\\{c\\}$.

The sublevel set of a length function $L(A,C)$ is the noncompact set of all
points of $\mathcal{T}_{g}$ for which $L(A,C)$ is less than or equal to some
constant $l\in\mathbb{R}^{+}$, and will be denoted by $L(A,C)_{\leq l}$. The
boundary of $L(A,C)_{\leq l}$ is the level set $L(A,C)^{-1}(l)$.

Let $\bigl{\\{}v_{1},\ldots,v_{k}\bigr{\\}}$ be a set of vectors in
$T_{x}\mathcal{T}_{g}$. These vectors will usually be gradients of length
functions. It will be said that the vectors
$\bigl{\\{}v_{1},\ldots,v_{k}\bigr{\\}}$ are contained in a halfspace if there
is a nonzero vector in $T_{x}\mathcal{T}_{g}$ that cannot be written as a
positive linear combination of the vectors
$\bigl{\\{}v_{1},\ldots,v_{k}\bigr{\\}}$.

An important ingredient in this work is the convexity of length functions.
Length functions were shown to be convex along earthquake paths in [17] and
strictly convex on Weil-Petersson geodesics in [30]. In [3] it was shown that
Fenchel-Nielsen coordinates can be chosen such that length functions are
convex functions of these coordinates.

A stratum $\mathrm{Sys}(C)$ is the subset of $\mathcal{T}_{g}$ on which $C$ is
the set of systoles. Note that $C$ is always a finite set, and will sometimes
be assumed to be ordered. It will be shown in Section 4 that strata are
connected, open subsets of embedded submanifolds. The systole function is
smooth when restricted to each stratum, but where different strata meet, one-
sided limits of the derivative do not usually match. When the curves in $C$
are pairwise disjoint and hence determine a multicurve, the stratum
$\mathrm{Sys}(C)^{\infty}$ of the Weil-Petersson completion
$\overline{\mathcal{T}}_{g}$ of $\mathcal{T}_{g}$ refers to a set of points at
infinity, representing noded surfaces pinched along curves in the set $C$.

The cells of the Thurston spine are the strata
$\\{\mathrm{Sys}(C_{i})\,|\,C_{i}\text{ fills}\\}$.

The Weil-Petersson metric appears to be well suited for studying the systole
function. Since the length functions are strictly convex along Weil-Petersson
geodesics, Weil-Petersson distances can be used to estimate or bound changes
in length functions. Examples of this can be found in Corollary 21 of [31] or
Theorem 1.3 of [32].

As $f_{sys}$ is not smooth, it cannot be a Morse function. There is however a
sense in which it behaves just like a Morse function.

######  Definition 4 (Topological Morse function).

Let $M$ be an $n$-dimensional topological manifold. A continuous function
$f:M\rightarrow\mathbb{R}^{+}$ is a topological Morse function if the points
of $M$ consist of regular points and critical points. When $p\in M$ is a
regular point, there is an open neighbourhood $U$ containing $p$, where $U$
admits a homeomorphic parametrisation by $n$ parameters, one of which is $f$.
When $p$ is a critical point, there exists a $k\in\mathbb{Z}$, $0\leq k\leq
n$, called the index of $p$, and a homeomorphic parametrisation of $U$ by
parameters $\\{x_{1},\ldots,x_{n}\\}$, such that everywhere on $U$, $f$
satisfies

$f(x)\,-\,f(p)\;=\
\sum_{i=1}^{i=n-k}x_{i}^{2}\,-\,\sum_{i=n-k+1}^{i=n}x_{i}^{2}$

Topological Morse functions were first defined in [22], where it was shown
that, when they exist, they can be used in most of the same ways as their
smooth analogues for constructing cell decompositions of manifolds and
computing homology.

The cohomological dimension of a group $G$ is

$\sup\bigl{\\{}n\in\mathbb{N}\,|\,H^{n}(G,M)\neq 0\text{ for some module
}M\bigr{\\}}.$

The mapping class group is known to contain finite index torsion free
subgroups; a discussion is given in Chapter 6 of [8]. By Serre’s theorem, [7]
Chapter VIII, any finite index torsion free subgroup of a group has the same
cohomological dimension. The cohomological dimension of a (and hence any)
finite index torsion free subgroup is then called the virtual cohomological
dimension.

##  3\. Thurston’s deformation retraction

This section outlines Thurston’s deformation retraction onto the Thurston
spine. References are [29] and Chapter 3 of [11].

In [29], a subset $\mathcal{P}_{g}$ of $\mathcal{T}_{g}$ was defined to be the
set of points corresponding to surfaces for which the set of systoles fills
the surface. The subset $\mathcal{P}_{g}$ will be referred to as the Thurston
spine. A deformation retraction of $\mathcal{T}_{g}$ onto $\mathcal{P}_{g}$
was constructed. This construction will now be discussed in detail.

First note that the subset $\mathcal{P}_{g}$ is defined by a locally finite
set of analytic equations and inequalities. The equations state that certain
geodesics (the systoles) have the same length, and the inequalities ensure
that these geodesics are shorter than all others. That the latter set is
locally finite is known; for example, it can be proven using the collar lemma.
These solutions can be seen to fit together coherently as the set of geodesics
representing the systoles varies. It follows that $\mathcal{P}_{g}$ is a cell
complex.

Thurston constructed a $\Gamma_{g}$-equivariant isotopy $\phi_{t}$ of
$\mathcal{T}_{g}$ into a regular neighbourhood of $\mathcal{P}_{g}$. This
relies on the next proposition.

######  Proposition 5 (Proposition 0.1 of [29]).

Let $C$ be any collection of curves on a surface that do not fill. Then at any
point of $\mathcal{T}_{g}$, there are tangent vectors that simultaneously
increase the lengths of all the geodesics representing curves in $C$.

######  Remark 6.

It is important to note that Proposition 5 implies that all critical points of
$f_{sys}$ are contained in $\mathcal{P}_{g}$.

The proof of Proposition 5 given in [29] uses Lipschitz maps, and is quite
intuitive. A different proof will be given here, illustrating how the
convexity of length functions constrains the differential topology of
$\mathcal{M}_{g}$. There is no claim to originality here. Results similar to
Proposition 5 have been proven using a variety of techniques; the first
instance of which the author is aware can be found in Lemma 4 of [2]. Wolpert
has also pointed out that it follows from Riera’s formula, [23]. A detailed
exposition of the theory behind Thurston’s Lipschitz maps can be found in
[13].

###### Proof.

Let $C=\\{c_{1},\ldots,c_{n}\\}$. The length of a curve $c$ will be denoted by
$L(c)$. Let $L(c)_{x}$ be the level set of $L(c)$ passing through a point $x$.

Since the curves in $C$ do not fill, the intersection
$N(x):=\cap_{j=1,\ldots,n}L(c_{j})_{x}$ is not compact. This is because the
intersection must be invariant under the action of a subgroup of $\Gamma_{g}$
generated by Dehn twists around curves disjoint from the curves in $C$.

A length function of the form $\sum_{i=1}^{n}a_{i}L(c_{i})$ with each
$a_{i}\in\mathbb{R}^{+}\cup\\{0\\}$ and not uniformly zero cannot have a
minimum in $\mathcal{T}_{g}$. This is because such a minimum must be a unique
point by convexity, but $N(x)$ is not compact for any $x\in\mathcal{T}_{g}$.

It is always possible to find a point $w\in\mathcal{T}_{g}$ at which the lemma
holds. This can be done by finding a point $x$ in the metric completion of
$\mathcal{T}_{g}$ with respect to the Weil-Petersson metric, with the property
that a curve $c$ is pinched at $x$, where $c$ has nonzero geometric
intersection number with each of the curves in $C$. Choosing $w$ sufficiently
close to $x$ will ensure that the lemma holds at $w$.

Suppose the lemma breaks down at $y\in\mathcal{T}_{g}$. Along a path $\gamma$
from $w$ to $y$, there must be a point $z\in\mathcal{T}_{g}$ at which the
lemma first breaks down. At $z$, there exists therefore a nontrivial subset
$G_{z}$ of $\bigl{\\{}\nabla L(c_{i})\ |\ c_{i}\in C\bigr{\\}}$ that spans a
proper subspace of $T_{z}\mathcal{T}_{g}$, and whose elements are not
contained in a halfspace of this subspace.

The existence of $G_{z}$ implies that it is possible to find
$a_{1},\ldots,a_{n}\in\mathbb{R}^{+}\cup\\{0\\}$ not all zero such that the
sum

$\sum_{i=i}^{n}a_{i}\nabla L(c_{i})(z)$

is zero. By convexity of length functions along Weil-Petersson geodesics, this
implies that the length function

$L=\sum_{i=i}^{n}a_{i}L(c_{i})$

has a local—and hence global—minimum at $z$. The lemma follows by
contradiction. ∎

For any $\epsilon>0$, an open subset $\mathcal{P}_{g,\epsilon}$ of
$\mathcal{T}_{g}$ is defined to be the subset of $\mathcal{T}_{g}$ consisting
of hyperbolic structures such that the set of geodesics whose length is within
$\epsilon$ of the shortest length fill the surface. It is not hard to see that
each $\mathcal{P}_{g,\epsilon}$ is open, its projection to $\mathcal{M}_{g}$
has compact closure, and the intersection of $\mathcal{P}_{g,\epsilon}$ over
all positive $\epsilon$ is the subcomplex $\mathcal{P}_{g}$. It follows that
for any regular open neighbourhood $\mathcal{N}$ of $\mathcal{P}_{g}$, there
is an $\epsilon$ such that $\mathcal{P}_{g,\epsilon}\subset\mathcal{N}$.

Recall that the Weil-Petersson metric used here is invariant under the action
of the mapping class group. At a point $x$ of
$\mathcal{T}_{g}{\smallsetminus}\mathcal{P}_{g}$, let $C(x)$ be a set of
shortest geodesics. If the geodesics in $C(x)$ do not fill the surface, by
Proposition 5, it is possible to define a vector field $X_{C}$ with the
property that every curve in $C$ is increasing in the direction of $X_{C}$.
Thurston gave as an example the vector field $X_{C}$ with the property that at
any point $x$, $X_{C}(x)$ has unit length and points in the direction that
maximises the sum of the (real) logarithms of the derivatives of the lengths
of the curves in $C$. This is a shorthand way of saying that $X_{C}(x)$ points
in a smooth choice of direction in which the length of each curve in $C$ is
increasing, because if one of the derivatives were negative or zero, the log
would be imaginary or $-\infty$. This vector field is discontinuous only at
places where the set of shortest geodesics changes.

The vector field $X_{C}$ is arbitrarily defined to be zero on
$\mathcal{P}_{g}$. For a point $x$ very close to $\mathcal{P}_{g}$, since
$C(x)$ is not just the set of systoles, the curves in $C(x)$ might also fill,
depending on how the notion of “set of short curves” is defined. For
simplicity, $X_{C}$ will also be defined to be zero when the curves in $C$
fill. The construction will only require a vector field that is nonzero
outside of some regular neighbourhood of $\mathcal{P}_{g}$ that can be made
arbitrarily small.

Denote the cardinality of a set $S$ by $|S|$. Let $C$ be a finite set of
curves on $\mathcal{S}_{g}$. For an $\epsilon>0$ define $U_{C}(\epsilon)$ to
be the set containing every point $x$ of $\mathcal{T}_{g}$ representing a
hyperbolic structure for which $C$ is the set of curves of length less than
$f_{sys}(x)+|C|\epsilon$. When $\epsilon$ is sufficiently small,
$\bigl{\\{}U_{C_{i}}\,|\,C_{i}\text{ is a finite set of curves on
}\mathcal{S}_{g}\bigr{\\}}$ covers $\mathcal{T}_{g}$. For every point $x$ not
on $\mathcal{P}_{g}$, there is an $\epsilon$ such that for some set
$U_{C_{i}}$ containing $x$, the curves in $C_{i}$ do not fill.

Let $\bigl{\\{}\lambda_{C_{i}}\bigr{\\}}$ be a partition of unity subordinate
to the covering $\bigl{\\{}U_{C_{i}}\bigr{\\}}$. The partition of unity is
chosen in such a way as to be invariant under the action of $\Gamma_{g}$ on
the sets of geodesics $\bigl{\\{}C\bigr{\\}}$. For example, it could be
defined as a function of geodesic lengths. The vector field $X_{\epsilon}$ is
constructed by using the partition of unity
$\bigl{\\{}\lambda_{C_{i}}\bigr{\\}}$ to average over the vector fields
$\bigl{\\{}X_{C_{i}}\bigr{\\}}$. Note that this averaging process does not
create zeros. For a point $x$ in the intersection of the open sets
$U_{C_{i}}$, $i=1,\ldots,k$, there is at least one shortest or equal shortest
curve $c$ in the intersection of the sets $C_{i}$. Any vector field
$X_{C_{i}}$, $i=1,\ldots,k$ evaluated at $x$ has the property that if it is
nonzero, it increases the length of $c$ at $x$. It follows that $X_{\epsilon}$
can only be zero at $x$ if every vector field being averaged over at $x$ is
zero.

The gradient of $f_{sys}$ is defined in the interior of any top dimensional
stratum $\mathrm{Sys}(\\{c\\})$; the complement of these strata have measure
zero in $\mathcal{T}_{g}$. The next lemma therefore gives control over the
rate at which $f_{sys}$ increases along the flowlines of $X_{\epsilon}$
outside of the $\delta$-thick part of $\mathcal{T}_{g}$. It was not explicitly
contained in Thurston’s preprint, but has been included here by request.
Recall that $C_{i}(x)$ is the set of curves with length at $x$ less than
$f_{sys}(x)+|C_{i}|\epsilon$ defined above.

######  Lemma 7.

There is a $\delta>0$ such that $\frac{df_{sys}}{dt}$ is uniformly bounded
away from zero along the intersection of any flowline of $X_{\epsilon}$ with a
top dimensional stratum in the $\delta(\epsilon)$-thin part of
$\mathcal{T}_{g}$.

###### Proof.

Corollary 21 of [31] states that for a stratum $\sigma$ defined by the
vanishing of the sum $L=L(c_{1})+\ldots+L(c_{k})$, the Weil-Petersson distance
$d(x,\sigma)$ of a point $x\in\mathcal{T}_{g}$ to the stratum $\sigma$ is
given locally as

(1) $\mathrm{d}(x,\sigma)=\sqrt{2\pi L}+\mathcal{O}(L^{2})$

Choose $\delta>0$ such that for any $x$ in the $\delta$-thin part of
$\mathcal{T}_{g}$, the curves in any $C_{i}(x)$ are pairwise disjoint with
lengths small enough such that the higher order terms in Equation (1) can be
ignored. Recall that $X_{C_{i}(x)}$ is the vector of length 1 in the direction
that minimises the sum of the logarithms of the derivatives of the lengths of
the curves in $C_{i}(x)$. Ignoring the higher order terms, Equation (1) then
gives that

(2) $\ln\frac{dL(c_{1})}{dt}(x)+\ldots+\ln\frac{dL(c_{k})}{dt}(x)\\\
=\frac{1}{2}\ln\bigl{(}L(c_{1})L(c_{2})\ldots
L(c_{k})\bigr{)}+\ln\bigl{(}\frac{d\mathrm{d}(x,\sigma(c_{1}))}{dt}\frac{d\mathrm{d}(x,\sigma(c_{2}))}{dt}\ldots\frac{d\mathrm{d}(x,\sigma(c_{k}))}{dt}\bigr{)}+\text{
constant}$

where $\\{c_{1},\ldots,c_{k}\\}=C_{i}(x)$, and $\sigma(c)$ is the stratum on
which the length of the curve $c$ vanishes.

Since the curves in $C_{i}(x)$ are pairwise disjoint, it follows from Riera’s
formula, [23], that the gradients of the lengths of any two curves in
$C_{i}(x)$ make an angle less than $\frac{\pi}{2}$. Equation (2) then ensures
that for $x\in\mathrm{Sys}(\\{c\\})$ in the $\delta(\epsilon)$-thin part of
$\mathcal{T}_{g}$, $\nabla L(c)$ makes an angle with $X_{C_{i}(x)}$ uniformly
bounded away from $\frac{\pi}{2}$. The lemma follows from the observation that
this property is unchanged by the averaging process used to obtain
$X_{\epsilon}$. ∎

Let $K$ be a set that is compact modulo the action of the mapping class group
and for which $\mathcal{P}_{g}\subset K\subset\mathcal{T}_{g}$. The goal is
now to construct an isotopy $\phi_{t}$ of $\mathcal{T}_{g}$ with the property
that for any $\epsilon$ there is a $T(\epsilon)$ such that taking
$t>T(\epsilon)$ ensures that for any $K$, $\phi_{t}(K)$ is contained within
$\mathcal{P}_{g,\epsilon}$. This is done by using the flow generated by
$X_{\epsilon^{{}^{\prime}}(t)}$ where $\epsilon^{{}^{\prime}}(t)>0$ is small
and is decreased further as time goes on.

A complete vector field on a manifold $M$ is a vector field that generates a
flow $\mathbb{R}\times M\rightarrow M$, in other words, the vector field
generates a flow that is defined for all time. Recall that a compactly
supported smooth vector field is complete, see for example Theorem 9.16 of
[19]. Completeness of $X_{\epsilon^{{}^{\prime}}(t)}$ then follows from the
same proof as in [19], using the fact that the $\alpha$-thick part of
$\mathcal{T}_{g}$ is invariant under the flow and compact modulo the action of
$\Gamma_{g}$.

######  Lemma 8.

For any $\epsilon>0$ there is a $T(\epsilon)$ such that flowing for
$t>T(\epsilon)$ ensures that any $K$ is carried inside—and remains
within–$\mathcal{P}_{g,\epsilon}$.

Lemma 8 also follows from the invariance of the $\alpha$-thick part of
$\mathcal{T}_{g}$ under the flow. As it is at the core of the objections
towards this construction, it will be proven later in detail.

Back to the construction of $\phi_{t}$. Denote by $\mathcal{I}_{t}$ the closed
set in $\mathcal{T}_{g}$ containing $\mathcal{P}_{g}$ whose boundary is the
image of $\partial\mathcal{T}_{g}^{\delta}$ after it has been flowed for time
$t$. Here $\delta$ is from Lemma 7. For any $t\in[0,\infty)$, the isotopy
$\phi_{t}$ takes a point to its image at time $t$ under the flow; the set
$\mathcal{T}_{g}^{\delta}$ is therefore mapped to $\mathcal{I}_{t}$ by
$\phi_{t}$.

Suppose $\epsilon^{{}^{\prime}}(t)$ has been chosen small enough to ensure
that $X_{\epsilon^{{}^{\prime}}(t)}$ is nonzero on a neighbourhood of
$\partial\phi_{t}(\mathcal{T}_{g}^{\delta})$. The boundary of
$\phi_{t}(\mathcal{T}_{g}^{\delta})$ is similar to the boundary of a level set
of $f_{sys}$ such as $\mathcal{T}_{g}^{\delta}$ in the sense that
$X_{\epsilon^{{}^{\prime}}(t)}$ points inward at every point of
$\phi_{t}(\mathcal{T}_{g}^{\delta})$. One way of proving this is to use that
under these assumptions, $\phi_{t}$ is a flowline-preserving diffeomorphism of
a regular neighbourhood of $\partial\mathcal{T}_{g}^{\delta}$ onto a regular
neighbourhood of $\partial\phi_{t}(\mathcal{T}_{g}^{\delta})$. A point on a
flowline outside $\mathcal{T}_{g}^{\delta}$ is mapped to a point on a flowline
outside of $\partial\phi_{t}(\mathcal{T}_{g}^{\delta})$, and vice versa. Since
the flowlines determine a foliation of the regular neighbourhood of
$\partial\mathcal{T}_{g}^{\delta}$, this is also the case for
$\partial\phi_{t}(\mathcal{T}_{g}^{\delta})$. This implies that there are no
places where a flowline is tangent to
$\partial\phi_{t}(\mathcal{T}_{g}^{\delta})$, which would need to exist for
some value of $t$ if $X_{\epsilon^{{}^{\prime}}(t)}$ were to transition from
pointing inwards to pointing outwards.

Choose $t^{*}$ such that the isotopy $\phi_{t^{*}}$ gives a deformation
retraction of any $K$ into $\mathcal{P}_{g,\epsilon}$ for some $\epsilon$
small enough to ensure that $\mathcal{P}_{g,\epsilon}$ is contained in a
regular neighbourhood $\mathcal{N}$ of $\mathcal{P}_{g}$. Existence of such a
$t^{*}$ is guaranteed by Lemma 8. A deformation retraction of $K$ onto
$\mathcal{P}_{g}$ is obtained by taking a composition of $\phi_{t^{*}}$ with a
deformation retraction that arises from the deformation retraction of
$\mathcal{N}$ onto $\mathcal{P}_{g}$.

The existence of this second deformation retraction will now be shown. For
ease of notation, it will be shown that
$\mathcal{I}_{t^{*}}=\phi_{t^{*}}(\mathcal{T}_{g}^{\delta})$ deformation
retracts onto $\mathcal{P}_{g}$. An identical argument works with the
$\alpha$-thick part of $\mathcal{T}_{g}$ in place of
$\mathcal{T}_{g}^{\delta}$, for any $\alpha$ small enough such that the
$\alpha$-thick part of $\mathcal{T}_{g}$ contains $\mathcal{P}_{g}$. As the
$\alpha$-thick subsets are an exhaustion of $\mathcal{T}_{g}$ by sets compact
modulo the action of $\Gamma_{g}$, it follows that the required deformation
retraction exists for any $K$.

First note that the boundary of $\mathcal{I}_{t^{*}}$ must be connected,
because this is the case for the boundary of $\mathcal{T}_{g}^{\delta}$, see
for example Proposition 12.10 of [8]. By construction, the set
$\mathcal{I}_{t^{*}}$ has $\mathcal{P}_{g}$ in the interior, because a
flowline is prevented from actually reaching $\mathcal{P}_{g}$ by the fact
that for any $\epsilon$, $X_{\epsilon}$ is zero at points sufficiently close
to $\mathcal{P}_{g}$. Consequently, $\mathcal{I}_{t^{*}}$ is a connected
subset of $\mathcal{N}$ with connected boundary that separates
$\partial\mathcal{N}$ from $\mathcal{P}_{g}$. Normal coordinates on
$\mathcal{N}$ give a set of flowlines emanating from $\mathcal{P}_{g}$ along
which the $r$-coordinate measuring distance from $\mathcal{P}_{g}$ is
increasing. By construction, each of these lines has algebraic intersection
number 1 with $\partial\mathcal{I}_{t^{*}}$. If every flowline crosses
$\partial\mathcal{I}_{t^{*}}$ once only, the second deformation retraction
simply consists of shifting points of $\mathcal{I}_{t^{*}}$ towards
$\mathcal{P}_{g}$ along these flowlines.

Figure 1. An example of a locally supported stretching deformation and a
locally supported sheering deformation on the normal coordinates in
$\mathcal{N}$.

In the general case, flowlines can cross $\partial\mathcal{I}_{t^{*}}$ more
than once, so it is necessary to alter the normal coordinates on $\mathcal{N}$
to obtain new flowlines, each of which crosses $\partial\mathcal{I}_{t^{*}}$
only once. The $r$-coordinate is altered by a combination of locally supported
sheering deformations and stretching/compressing, as illustrated in Figure 1.
The lengthy details of how to compose these deformations are given in Theorem
5.4 of [21], and an example is illustrated in Figure 2. The context of Theorem
5.4 of [21] is slightly different; it explains how to alter a Morse function
in such a way as to cancel critical points. By Theorem 2.7 of [21] the
restriction of the $r$-coordinate to $\partial\mathcal{I}_{t^{*}}$ gives a
smooth function on $\partial\mathcal{I}_{t^{*}}$ which is arbitrarily close to
a smooth function $r|_{\mathcal{I}}$ for which the hessian has full rank at
any point where the derivative is zero. Such non regular points are
necessarily isolated. As illustrated in Figure 2, deforming the normal
coordinates in such as way as to reduce the number of crossings of an open set
of flowlines with $\partial\mathcal{I}_{t^{*}}$ amounts to cancelling out non
regular points of $r|_{\mathcal{I}}$ whose index differs by one. As soon as
the property has been achieved that a sheering deformation suffices to ensure
every flowline crosses $\partial\mathcal{I}_{t^{*}}$ exactly once, no further
cancellations are made. Note that this construction can be performed in a way
that is invariant under $\Gamma_{g}$. This completes the construction of a
$\Gamma_{g}$-invariant deformation retraction of $\mathcal{T}_{g}$ onto
$\mathcal{P}_{g}$, modulo the proof of Lemma 8.

Figure 2. A sequence of deformations of the normal coordinates of
$\mathcal{N}$ to obtain a new set of normal coordinates with the property that
the flow lines (assumed vertical) each cross $\partial\mathcal{I}_{t^{*}}$ in
one point only. The red dots are non regular points of $r|_{\mathcal{I}}$.

In the last two pages of [15], a list of questions was made, that would need
to be addressed if the argument outlined in [29] is to be considered a
complete proof. All but one of these questions was answered in the exposition
above, the final objection is as follows: it is not sufficient to show that
every point flows into a neighbourhood of the Thurston spine, it is necessary
to show that each point flows into and eventually stays in
$\mathcal{P}_{g,\epsilon}$ for each $\epsilon$. For example, a geodesic
slightly longer than a systole might have its length grow rapidly under the
flow, and a different — initially much longer— geodesic might then be added to
the set $C$. An argument is needed to show that such a point cannot
indefinitely flow towards and then away from $\mathcal{P}_{g}$; this amounts
to proving the next lemma.

######  Lemma 9 (Lemma 8 from earlier).

For any $\epsilon>0$ there is a $T(\epsilon)$ such that flowing for
$t>T(\epsilon)$ ensures that any $K$ is carried inside—and remains
within–$\mathcal{P}_{g,\epsilon}$.

###### Proof.

Recall that for $t>t^{{}^{\prime}}$,
$\phi_{t}(\mathcal{T}_{g}^{\delta})\subset\phi_{t^{{}^{\prime}}}(\mathcal{T}_{g}^{\delta})$.
Define a continuous function $f:\mathcal{T}_{g}\rightarrow\mathbb{R}$, whose
value at a point $x$ is equal to the smallest real number $\beta$ such that
the set of all curves with length within $\beta$ of the systoles fill. A
continuous function $\epsilon^{b}(t)$ with the property that
$\phi_{t}(\mathcal{T}_{g}^{\delta})$ is contained in
$\mathcal{P}_{g,\epsilon^{b}(t^{\prime})}$ for $t>t^{\prime}$ can be
constructed by calculating the supremum of $f$ over
$\phi_{t}(\mathcal{T}_{g}^{\delta})$.

As $t$ increases, $\epsilon^{b}(t)$ decreases monotonically to zero.
Monotonicity is due to the fact that a supremum does not increase when the
calculation is restricted to a subset. The function $f_{sys}$ can be replaced
by a smooth, $\Gamma_{g}$-equivariant function $f_{sys}^{s}$ arbitrarily close
to $f_{sys}$ on $\mathcal{T}_{g}^{\delta}$. Since the closure of
$\mathcal{T}_{g}^{\delta}{\smallsetminus}\mathcal{P}_{g,\epsilon}$ is compact
modulo the action of $\Gamma_{g}$, when $\epsilon^{\prime}(t)$ is small enough
such that the zeros of $X_{\epsilon^{\prime}(t)}$ are strictly contained
within $\mathcal{P}_{g,\epsilon}$, there is a lower bound of
$\frac{df_{sys}^{s}\circ\gamma}{dt}$ along the intersection of a flowline
$\gamma(t)$ with the closure of
$\mathcal{T}_{g}^{\delta}{\smallsetminus}\mathcal{P}_{g,\epsilon}$. It follows
that $\epsilon^{b}(t)$ decreases to zero.

By Lemma 7, $\frac{df_{sys}}{dt}$ is uniformly bounded from below along the
intersection of any flowline $\gamma(t)$ with a top dimensional stratum in the
complement of $\mathcal{T}_{g}^{\delta}$. Consequently there is a uniform
upper bound on the time needed for a point in the complement of
$\mathcal{T}_{g}^{\delta}$ to flow into $\mathcal{T}_{g}^{\delta}$, after
which it then remains in $\mathcal{T}_{g}^{\delta}$ by the invariance of
$\mathcal{T}_{g}^{\delta}$ under the flow. Adding this upper bound to
$T(\epsilon)^{{}^{\prime}}$ gives $T(\epsilon)$ as required. This concludes
the proof of the lemma, and hence of the construction of the deformation
retraction. ∎

######  Remark 10.

It is tempting to use the flow coming from the vector fields
$X_{\epsilon^{\prime}(t)}$ to directly construct a deformation retraction of
$\mathcal{T}_{g}$ onto $\mathcal{P}_{g}$ without using the second deformation
retraction outlined above. The smoothing process makes the vector fields
$X_{\epsilon^{\prime}(t)}$ zero near $\mathcal{P}_{g}$. However the vector
field is smoothed off, to obtain a vector field continuous at
$\mathcal{P}_{g}$, the length of the vector field must approach zero near
$\mathcal{P}_{g}$. The difficulty is in showing that $\mathcal{T}_{g}$ flows
onto $\mathcal{P}_{g}$ in finite time. If the flow time is not finite, one
obtains a retraction instead of a deformation retraction.

######  Remark 11.

The decision to smoothen the vector field, rather than to work in the
piecewise-linear category is a matter of convention. As the systole function
is only piecewise smooth, a more modern approach using discrete Morse theory
seems natural. An introduction to discrete Morse theory, as well as further
references, can be found in [9]. To use discrete Morse theory, normally one
would start with a cell decomposition. In this context, the strata of
$f_{sys}$ can be used. Although strata with non-filling systoles do not
project to cells in $\mathcal{M}_{g}$, this is not really a problem. The
reason is that the action of $\Gamma_{g}$ on $\mathcal{T}_{g}$ is by isometry,
and hence does not identify points on the same flowline of
$X_{\epsilon^{\prime}(t)}$. It is claimed here without proof that a
$\Gamma_{g}$-equivariant deformation retraction similar to the one above could
be described without the need for arbitrary choices by discretising the
systole function.

##  4\. Strata with filling systoles

The Thurston spine has the structure of a cell complex, with cells given by
strata of the systole function for which the systoles fill $S_{g}$. This
section establishes some basic properties of the strata making up the cells of
$\mathcal{P}_{g}$. The contents of this section and the next are used to
establish a duality between cells of the Thurston spine and sets defined by
Schmutz that are parametrised by length functions.

######  Lemma 12.

A stratum $\mathrm{Sys}(C)$ has compact closure in $\mathcal{T}_{g}$ iff the
curves in $C$ fill $\mathcal{S}_{g}$.

###### Proof.

When the systoles in a given stratum intersect, the collar theorem gives a
lower bound on the systole function, and Bers’ constant gives an upper bound.
Suppose the curves in $C$ fill $S_{g}$. It follows from Theorem 1.1 of [3]
that there is a set of Fenchel-Nielsen coordinates on $\mathcal{T}_{g}$ with
the property that the lengths of the curves in $C$ are strictly convex
functions of these coordinates. These Fenchel-Nielsen coordinates are
therefore bounded on $C$.

When the geodesics in $C$ do not fill, there is a closed curve $c^{*}$
disjoint from all the geodesics in $C$. The stratum $\mathrm{Sys}(C)$ is
invariant under Dehn twists around $c^{*}$. Choose a point $x$ in
$\mathrm{Sys}(C)$. Since $\Gamma_{g}$ acts properly discontinuously on
$\mathcal{T}_{g}$, the orbit of $x$ under the subgroup of $\Gamma_{g}$
generated by a Dehn twist around $c^{*}$ gives a sequence without a limit
point contained in $\mathrm{Sys}(C)$. ∎

There are two types of constraints defining a stratum $\mathrm{Sys}(C)$. The
first set of constraints forces all the curves in $C$ to have the same length.
The notation $E(C)$ will be used to denote the set of points of
$\mathcal{T}_{g}$ on which a set $C$ of curves all have the same length. Note
that $\mathrm{Sys}(C)\subset E(C)$.

The second set of constraints defining $\mathrm{Sys}(C)$ is that the curves in
$C$ are shorter than all other curves. As this set of constraints is locally
finite, if it holds at a point $x$, it holds on some neighbourhood of $x$. It
follows that $\mathrm{Sys}(C)$ is the intersection of an open set with $E(C)$.

######  Lemma 13.

Let $C=\\{c_{1},\ldots,c_{k}\\}$ be a set of curves on $S_{g}$, any two of
which intersect in at most one point. The set $E(C)$ is a connected, embedded
submanifold of $\mathcal{T}_{g}$. If the curves in $C$ fill, there is a unique
point $p$ on $E(C)$ at which the functions $L(c_{i})$, $i=1,\ldots,k$
restricted to $E(C)$ all have a global minimum.

###### Proof.

To begin with, an ordering of the set $C$ will be chosen and fixed, as
follows: Suppose $1\leq j\leq k$. Wherever possible the curve $c_{j}$ is
chosen to intersect some curve $c$ on $S_{g}$ disjoint and distinct from all
the curves $\\{c_{1},\ldots,c_{j-1}\\}$. Let $j^{*}$ be the smallest natural
number such that this is no longer possible; this happens when the first
$j^{*}-1$ curves $\\{c_{1},\ldots,c_{j^{*}-1}\\}$ fill the same subsurface of
$S_{g}$ as the entire set of curves $C$. Wherever possible, $c_{j}$ is then
chosen such that $E\bigl{(}\\{c_{1},\ldots,c_{j}\\}\bigr{)}$ is contained in
but not equal to $E\bigl{(}\\{c_{1},\ldots,c_{j-1}\\}\bigr{)}$. Let $k^{*}$ be
the smallest integer such that
$E\bigl{(}\\{c_{1},\ldots,c_{k^{*}}\\}\bigr{)}=E\bigl{(}\\{c_{1},\ldots,c_{k}\\}\bigr{)}$.

To prove the lemma, it can (and will) be assumed without loss of generality
that $j\leq k^{*}$ and $E(C)$ is nonempty and contains more than a single
point.

Define

$N(j,t):=\cap_{i=1}^{j}L(c_{i})^{-1}(t),\text{ for }j=1,\ldots,k\text{ and
}t>0.$

Note that for each $t$, a nested sequence $N(k,t)\subset
N(k-1,t)\subset\ldots\subset N(1,t)$ is obtained.

The steps of the proof will now be outlined.

Step 1: It will be shown that $N(k,t)$ is an embedded submanifold, and
connected when the dimension is at least 1. The proof will first be given in
the case that the curves in $C$ fill, and then in the case in which they do
not. In both cases, the proof is by induction. This involves first showing
that each $N(j,t)$, for $j=1,\ldots,k$, is a topological embedding, and using
induction over $j$ to show that each $N(j,t)$ is immersed. The induction step
also needs to consider the different cases of filling curves and non filling
curves.

Step 2: To relate the 1-parameter family of submanifolds $N(k,t)$, $t>0$, to
$E(C)$, first note that every point in $E(C)$ is in $N(k,t)$ for some $t$.
This step studies how the submanifolds $N(k,t)$ lie inside $E(C)$. It will be
shown that when the curves $\\{c_{1},\ldots,c_{j}\\}$ fill, $N(k,t)$ is either
empty, a sphere or a point. Moreover, in the filling case there is a smallest
value of $t$, call it $t^{*}$, for which $N(k,t)$ is nonempty. The point $p$
is $N(k,t^{*})$.

Step 3: It will finally be proven by induction on $j$ that
$E\bigl{(}\\{c_{1},\ldots,c_{j}\\}\bigr{)}$ is immersed. This uses
transversality properties of the intersections of level sets from step 1. The
induction needs to consider the different cases of filling curves and non
filling curves.

Step 1: Consider the intersection of sublevel sets,

$I(j,t):=\cap_{i=1}^{i=j}L(c_{i})_{\leq t}$

for $1\leq j\leq k^{*}$ and $t>0$. The assumption that $E(C)$ is nonempty
guarantees that $I(j,t)$ will be nonempty for sufficiently large $t$. Since
each sublevel set is convex, the intersection $I(j,t)$ is a convex polyhedron.
The facets of $I(j,t)$ lie along the level sets
$\bigl{\\{}L(c_{1})^{-1}(t),\ldots,L(c_{j})^{-1}(t)\bigr{\\}}$. Lower
dimensional faces of $I(j,t)$ are intersections of level sets. Again, the
assumption that $E(C)$ is nonempty guarantees that for sufficiently large $t$
the intersection $N(j,t)$ will be nonempty, and is then the lowest dimensional
face of $I(j,t)$. With the subspace topology, each of the sets
$N(1,t),\ldots,N(k^{*},t)=N(k^{*}+1,t)=\ldots=N(k,t)$ is therefore a
topological embedding.

Case in which the curves in $\mathbf{C}$ fill. For $j=1$, $N(1,t)$ is the pre-
image of a regular value of the smooth map
$L(c_{1}):\mathcal{T}_{g}\rightarrow\mathbb{R}$, and is hence an embedded
submanifold by the pre-image lemma (see for example Corollary 5.14 of [19]).
For the inductive step, it will be necessary to separately consider the cases
in which the curves $\\{c_{1},\ldots,c_{j}\\}$ do and do not fill.

Suppose the first $j$ curves do not fill. Proposition 5 and the fact that
$I(j,t)$ is a polyhedron together imply that for any point $x$ in any face of
$I(j,t)$ there is an open cone of vectors pointing into $I(j,t)$. When $c_{j}$
is not perpendicular to the curve $c$ guaranteed by the ordering on the set
$C$, regularity is proven by showing that Dehn-twists around $c$ give a
direction in which the lengths of the curves $c_{1},\ldots,c_{j-1}$ are
stationary, but the length of $c_{j}$ is not. When $c_{j}$ is perpendicular to
$c$, the construction in the proof of Proposition 0.1 of [29] using Lipschitz
maps gives a direction in which the lengths of the curves
$c_{1},\ldots,c_{j-1}$ are increasing, but the length of $c_{j}$ is
decreasing. This implies that $L(c_{j})^{-1}(t)$ determines a facet of
$I(j,t)$ adjacent to $N(j,t)$. Regularity in this case then follows from the
transversality of the intersection of $L(c_{j})^{-1}(t)$ with $N(j-1,t)$. In
summary:

(1) When $\\{c_{1},\ldots,c_{j}\\}$ do not fill, $N(j-1,t)$ intersects
$L(c_{j})^{-1}(t)$ transversely.

It follows from (1) that the function $L(c_{j})$ is regular when restricted to
a neighbourhood of $N(j,t)\subset N(j-1,t)$. (The ordering on the elements of
$C$ actually ensure that for $j\leq k^{*}$, $N(j,t)$ is strictly contained in
$N(j-1,t)$, so this statement makes sense. However, it would not matter here
even if $N(j,t)=N(j-1,t)$; the inductive step would just be trivial.) If
$N(j-1,t)$ is an embedded submanifold, the pre-image lemma therefore ensures
that $N(j,t)$ is also.

The same argument that proves point (1) also shows that any value of
$L(c_{j})|_{N(j-1,t)}$ is regular. If $N(j-1,t)$ is connected, this implies
that $N(j,t)$ is connected, because if $N(j,t)$ had two or more connected
components, each pair of connected components would need to be separated in
$N(j-1,t)$ by a connected component of a level set of $L(c_{j})|_{N(j-1,t)}$
on which $L(c_{j})|_{N(j-1,t)}$ is stationary. Connectivity of $N(j,t)$ then
also follows by induction.

When the curves $\\{c_{1},\ldots,c_{j}\\}$ fill, it follows from Lemma 1 of
[24] that there is a smallest value of $t$, call it $t^{*}$, for which
$N(j,t)$ is nonempty. Then $N(j,t^{*})$ represents the minimum of a length
function consisting of a positive linear combination of the length functions
$\\{L(c_{1}),\ldots,L(c_{j})\\}$. By convexity of length functions, this
minimum is a single point. Note that $N(j,t^{*})$ is a nontransverse point of
intersection of $N(j-1,t^{*})$ with $L(c_{j})^{-1}(t^{*})$. By assumption,
$E(C)$ is neither empty nor a single point, so $N(j,t)$ is nonempty for all
$t>t^{*}$, and $I(j,t)$ is also nonempty for all $t>t^{*}$. For any point $x$
in $N(j,t)$ with $t>t^{*}$, there is a vector in $T_{x}\mathcal{T}_{g}$
pointing into $I(j,t)$, that determines a direction in which all the length
functions $\\{L(c_{1}),\ldots,L(c_{j})\\}$ are decreasing. This vector
corresponds to an equivalence class of smooth paths passing through the level
sets $N(j,t-s)$ for $s\in(-\epsilon,\epsilon)$. The existence of this vector
then guarantees the existence of an open cone of vectors pointing into
$I(j,t)$. As in the previous case, the assumption on the ordering of the
curves in $C$ guarantees that $L(c_{j})^{-1}(t)$ determines a facet of
$I(j,t)$ adjacent to $N(j,t)$, from which regularity follows. This proves
point (2) below, and concludes the inductive step, proving that $N(k,t)$ is an
embedded submanifold.

(2) When $\\{c_{1},\ldots,c_{j}\\}$ fill, for $t>t^{*}$, $N(j,t)$ is nonempty
and $N(j-1,t)$ intersects $L(c_{j})^{-1}(t)$ transversely. The only
nontransverse intersection of $N(j-1,t)$ with $L(c_{j})^{-1}(t)$ is a single
point, and occurs at $t=t^{*}$.

The same argument that proves point (2) shows that for any $s,t>t^{*}$ for
which $L(c_{j})^{-1}(s)$ and $N(j-1,t)$ have nonempty intersection with
dimension greater than zero, the intersection of $L(c_{j})^{-1}(s)$ with
$N(j-1,t)$ is transverse. It follows that $L(c_{j})|_{N(j-1,t)}$ is regular
except at isolated points. When $N(j,t)$ has dimension at least 1, the proof
of connectivity of $N(j,t)$ then follows from the same argument as in the
nonfilling case.

Case in which the curves in $\mathbf{C}$ do not fill. The induction step needs
to treat the cases $j<j^{*}$ and $j^{*}\leq j$ separately. The former is
identical to the case in which the curves in $C$ fill, but the first $j$
curves do not.

If $j^{*}\leq j$ but the curves $\\{c_{1},\ldots,c_{j}\\}$ do not fill,
$N(j,t)$ is noncompact, because it is invariant under the action of Dehn
twists around a curves disjoint from $\\{c_{1},\ldots,c_{j}\\}$. By
Proposition 5, length functions obtained as positive linear combinations of
the curves $\\{c_{1},\ldots,c_{j}\\}$ do not have minima inside
$\mathcal{T}_{g}$; the sublevel sets $\\{L(c_{i})_{\leq t}\ |\
i=1,\ldots,j\\}$ intersect for the first time at infinity. Apart from this,
the proof that $N(j,t)$ is a connected, embedded manifold follows from the
same arguments as in the filling case.

Step 2: Recall from step 1 that when the curves $\\{c_{1},\ldots,c_{j}\\}$ do
not fill, since $j\leq k^{*}$, any nonempty intersection of
$L(c_{j})^{-1}(s)$, $s>0$, with $N(j-1,t)$ is a transverse, connected and
noncompact intersection of submanifolds with boundary at infinity. $E(C)$ is
covered by coordinate patches, each of which is diffeomorphic to a product of
an open interval $(t_{1},t_{2})$ and an open set of $N(k,t_{1})$. By
Proposition 5, the minimum and maximum values of $t$ for which $N(j,t)$ is
nonempty are not realised. The submanifold $N(j,t)$ is obtained by taking
transverse intersections of codimension 1 submanifolds diffeomorphic to
$\mathbb{R}^{6g-7}$; for any value of $t$ for which $N(j,t)$ is nonempty,
decreasing or increasing $t$ slightly will still result in an intersection. It
follows that each connected component of $E(C)$ has boundary at infinity.
Connectivity of $E(C)$ follows from the fact that each connected component of
$E(C)$ has boundary at infinity and $N(j,t)$ depends smoothly on $t$.

When the curves $\\{c_{1},\ldots,c_{j}\\}$ fill, by point (2) the only
nontransverse point of intersection of $L(c_{j})^{-1}(t)$ with $N(j-1,t)$
occurs at a single point. The point $N(k,t^{*})$ will be called $p$. Then
$t^{*}$ is the smallest value of $t$ at which the level sets
$L(c_{1})^{-1}(t),\ldots,L(c_{k})^{-1}(t)$ all meet. The point $p$ is
therefore the global minimum of the functions $L(c_{i})|_{E(C)}$,
$i=1,\ldots,k$, and the construction in step 1 shows that it is the only
stationary point of the functions $L(c_{i})|_{E(C)}$, $i=1,\ldots,k$.

Away from $p$, $E(C)$ is covered by coordinate patches, each of which is
diffeomorphic to a product of an open interval $(t_{1},t_{2})$ and an open set
of $N(k,t_{1})$. Suppose first that $N(k,t)$ has dimension $n\geq 1$ for
$t>t^{*}$. Since $p$ is the only stationary point of the length functions
$L(c_{i})|_{E(C)}$, $i=1\ldots,k$, and by connectivity of $N(k,t)$, it follows
that the sets $\\{N(k,t)\ |\ t>t^{*}\\}$ are topological spheres of fixed
dimension $n$, which collapse onto $p$ as $t$ approaches $t^{*}$ from above.
When $N(k^{*},t)=N(k,t)$ has dimension 0, it is therefore the intersection of
a submanifold $S^{1}$ with $L(c_{k})^{-1}(t)$, which can only be $S^{0}$,
because otherwise there would be a length function consisting of a positive
linear combination of $\\{L(c_{1},\ldots,L(c_{k^{*}})\\}$ whose minimum
consists of two or more points. In Figure 3, the level sets $N(k^{*},t)$ are
represented by 0 dimensional spheres lying along the intersection of the green
line with the concentric circles. With the subspace topology, $E(C)$ is
therefore homeomorphic to $\mathbb{R}^{n+1}$ with boundary at infinity of
$\mathcal{T}_{g}$. This proves connectivity of $E(C)$.

Figure 3. A lower dimensional representation of intersections of level sets.
The blue arrows are the vector field $V$.

Step 3: It will now be shown that $E(C)$ is immersed. This will be done by
induction, recalling the assumption that $E(C)$ is not empty and is not the
single point $p$. Since the curves $c_{1}$ and $c_{2}$ intersect in at most
one point on $\mathcal{S}_{g}$, using Fenchel-Nielsen coordinates, it is easy
to check that 0 is a regular value of the smooth map
$L(c_{1})-L(c_{2}):\mathcal{T}_{g}\rightarrow\mathbb{R}$. It follows from the
pre-image theorem that $E\bigl{(}\\{c_{1},c_{2}\\}\bigr{)}$ is a codimension 1
submanifold. For the induction step to work, it is sufficient to show that for
$j+1\leq k^{*}$, 0 is a regular value of the restriction of
$L(c_{j+1})-L(c_{j})$ to $E\bigl{(}\\{c_{1},\ldots,c_{j}\\}\bigr{)}$.

For $j+1\leq k^{*}$, $E\bigl{(}\\{c_{1},\ldots,c_{j},c_{j+1}\\}\bigr{)}$ is
properly contained in $E\bigl{(}\\{c_{1},\ldots,c_{j}\\}\bigr{)}$.
Consequently, for $t>t^{*}$, $N(j+1,t)$ is properly contained in $N(j,t)$.
This follows from the way the level sets $N(j,t)$ foliate
$E\bigl{(}\\{c_{1},\ldots,c_{j}\\}\bigr{)}$ when the curves
$\\{c_{1},\ldots,c_{j}\\}$ do not fill, or foliate
$E\bigl{(}\\{c_{1},\ldots,c_{j}\\}\bigr{)}{\smallsetminus}\\{p\\}$ when the
curves $\\{c_{1},\ldots,c_{j}\\}$ fill.

When the curves $\\{c_{1},\ldots,c_{j}\\}$ do not fill, the fact that
$N(j+1,t)$ is properly contained in $N(j,t)$ implies that 0 is a regular value
of the restriction of $L(c_{j+1})-L(c_{j})$ to
$E\bigl{(}\\{c_{1},\ldots,c_{j}\\}\bigr{)}$. This is because it was shown
above that $N(j+1,t)$ can be constructed by repeatedly taking intersections of
embedded submanifolds that intersect transversely. Transversality implies that
at any point $x$ in the intersection, the tangent space to the intersection of
the level sets is the intersection of the tangent spaces to the level sets.
Similarly, when the curves $\\{c_{1},\ldots,c_{j}\\}$ fill, 0 is a regular
value of $L(c_{j+1})-L(c_{j})$ when restricted to
$E\bigl{(}\\{c_{1},\ldots,c_{j}\\}\bigr{)}{\smallsetminus}\\{p\\}$.

It remains to show that 0 is a regular value of the function
$L(c_{j+1})-L(c_{j})$ when restricted to
$E\bigl{(}\\{c_{1},\ldots,c_{j}\\}\bigr{)}$ at $p$. Let $V$ be a smooth vector
field on $E\bigl{(}\\{c_{1},\ldots,c_{j+1}\\}\bigr{)}{\smallsetminus}\\{p\\}$,
as shown in Figure 3. Suppose also that for each $t>t^{*}$, $V$ is a choice of
normal vector field to $N(j+1,t)$ and tangent to $N(j,t)$. The vector field
$V$ can be chosen such that it extends to a nonvanishing vector field in on
all of $E\bigl{(}\\{c_{1},\ldots,c_{j+1}\\}\bigr{)}$. For $\epsilon>0$, each
connected component of
$N(j,t^{*}+\epsilon){\smallsetminus}N(j+1,t^{*}+\epsilon)$ has two connected
components; one on which $L(c_{j+1})-L(c_{j})>0$, and one on which
$L(c_{j+1})-L(c_{j})<0$. This is because if $L(c_{j+1})-L(c_{j})$ had the same
sign on both connected components, $L(c_{j+1})^{-1}(t^{*}+\epsilon)$ would
have to intersect $N(j,t^{*}+\epsilon)$ nontransversely in more than one
point, which was shown to be impossible. It follows that as $t$ varies from
$t^{*}-\epsilon$ to $t^{*}+\epsilon$, $L(c_{j+1})^{-1}(t)\cap
E\bigl{(}\\{c_{1},\ldots,c_{j}\\}\bigr{)}$ crosses over
$E\bigl{(}\\{c_{1},\ldots,c_{j+1}\\}\bigr{)}$ from one side to the other.
Since the projection of $\nabla L(c_{j})$ to $V$ is zero, the projection of
$\nabla\bigl{(}L(c_{j+1})-L(c_{j})\bigr{)}$ to $V$ is everywhere nonzero,
showing that $0$ is a regular value of $L(c_{j+1})-L(c_{j})$ when restricted
$E\bigl{(}\\{c_{1},\ldots,c_{j}\\}\bigr{)}$. The lemma then follows by
induction. ∎

######  Remark 14.

In the proof of Lemma 13 it was shown that $E(C)$ is obtained by taking
repeated intersections of the form
$E\bigl{(}\\{c_{1},\ldots,c_{j}\\}\bigr{)}\cap
E\bigl{(}\\{c_{j},c_{j+1}\\}\bigr{)}$, where each pair of submanifolds
intersects transversely. Transversality implies that the tangent space to the
intersection is the intersection of the tangent spaces. Consequently, the
tangent space to $E(C)$ at $p$ is the subspace of $T_{p}\mathcal{T}_{g}$ that
is the orthogonal complement of $\\{\nabla L(c_{i})\,|\,c_{i}\in C\\}$.

######  Proposition 15.

A stratum $\mathrm{Sys}(C)$ is connected.

###### Proof.

Let $B=\\{b_{1},\ldots,b_{k}\\}$ be the finite set of curves, each of which is
realised as a systole somewhere on the boundary of $\mathrm{Sys}(C)$, but not
in the interior of $\mathrm{Sys}(C)$. Each curve in $B$ therefore intersects
each curve in $C$ at most once. The proof of Lemma 13 showed that
$E\bigl{(}\\{c_{1},b_{1}\\}\bigr{)}$ is either disjoint from $E(C)$,
intersects it in a point or intersects it transversely. Disjointness is ruled
out by the definition of $B$.

Suppose the intersection of $E(\\{c_{1},b_{1}\\})$ with $E(C)$ is a point. If
$E(C)$ has dimension greater than 1, $b_{1}$ does not determine a codimension
1 face of $\mathrm{Sys}(C)$. In this case, define $E_{1}=E(C)$. If the
dimension of $E(C)$ is one, define $E_{1}$ to be the subset of $E(C)$ on which
$L(c_{1})<L(b_{1})$. When the intersection of
$E\bigl{(}\\{c_{1},b_{1}\\}\bigr{)}$ with $E(C)$ is transverse,
$E(C){\smallsetminus}E\bigl{(}\\{c_{1},b_{1}\\}\bigr{)}$ has exactly two
connected components. This is because $E(C\cup\\{b_{1}\\})$ is a codimension 1
boundary face of every connected component of
$E(C){\smallsetminus}E\bigl{(}\\{c_{1},b_{1}\\}\bigr{)}$ and it follows from
Lemma 13 that $E(C\cup\\{b_{1}\\})$ is connected. On one of these connected
components $L(b_{1})<L(c_{1})$, and on the other, call it $E_{1}$,
$L(b_{1})>L(c_{1})$.

Intersect $E_{1}$ with $E\bigl{(}\\{c_{1},b_{2}\\}\bigr{)}$. The intersection
might be either empty, transverse, or a single point. If the intersection is
empty, since $\mathrm{Sys}(C)$ is not empty, $L(b_{2})>L(c_{1})$ everywhere on
$E_{1}$. This means that $b_{2}$ must be on the boundary of a different
connected component of $Sys(C)$ from $b_{1}$. However, this connected
component must then be contained in the connected component of $E(C)$ on which
$L(b_{1})<L(c_{1})$, contradicting the definition of $\mathrm{Sys}(C)$. It
follows that the intersection cannot be empty. If the intersection is
transverse, there are exactly two connected components of
$E_{1}{\smallsetminus}E\bigl{(}\\{c_{1},b_{2}\\}\bigr{)}$. This is because
either $E\bigl{(}\\{c_{1},b_{2}\\}\bigr{)}$ is disjoint from the boundary
$E(C\cup\\{b_{1}\\})$ of $E_{1}$, in which case two connected components are
obtained as in the previous case, or $E\bigl{(}\\{c_{1},b_{2}\\}\bigr{)}$ cuts
the boundary $E(C\cup\\{b_{1}\\})$ of $E_{1}$ into two connected components,
one of which is on the boundary of the connected component of
$E_{1}{\smallsetminus}E\bigl{(}\\{c_{1},b_{2}\\}\bigr{)}$ on which
$L(b_{2})<L(c_{1})$, and the other of which is on the boundary of the
connected component $E_{2}$ of
$E_{1}{\smallsetminus}E\bigl{(}\\{c_{1},b_{2}\\}\bigr{)}$ on which
$L(b_{2})>L(c_{1})$. If the intersection is a point, when $E(C)$ has dimension
greater than one, set $E_{2}=E_{1}$, otherwise set $E_{2}$ equal to the
connected component of $E_{1}{\smallsetminus}E(\\{c_{1},b_{2}\\})$ on which
$L(c_{1})<L(b_{2})$. This argument can be iterated to obtain the connected set
$E_{k}$, which is $\mathrm{Sys}(C)$. ∎

##  5\. Cells and Morse theory

When constructing a deformation retraction, Thurston avoided the basepoint
problem by using curve lengths to parametrise $\mathcal{T}_{g}$. Schmutz first
studied cell decompositions of Teichmüller space parametrised by length
functions in [24]. The cell structures defined by Schmutz are helpful for
understanding the differential topology of the thick part of $\mathcal{T}_{g}$
and its boundary. This section surveys Schmutz’s construction and proves a
duality between certain sets parametrised by length functions and cells of the
Thurston spine.

The length spectrum rigidity theorem states that a point in $\mathcal{T}_{g}$
is determined by the lengths of its curves. In practice, only finitely many
curve lengths are needed. A set of length functions is said to parametrise
$\mathcal{T}_{g}$ if for every $x_{1}$ and $x_{2}$ representing points in
$\mathcal{T}_{g}$, whenever the length functions in the set all take the same
values at $x_{1}$ as at $x_{2}$, then $x_{1}=x_{2}$.

Recall that a function of the form $L(A,C)$, with fixed
$A\in\mathbb{R}_{+}^{|C|}$ is convex. A necessary and sufficient condition for
$x$ to represent the minimum of a length function $L(A,C)$ in the interior of
$\mathcal{T}_{g}$ is that the gradient of $L(A,C)$ is zero at $x$. When the
gradients $\\{\nabla L(c_{i})\ |\ c_{i}\in C\\}$ are not contained in a half
space at $x$, there exists therefore an $A$ for which the unique minimum of
$L(A,C)$ occurs at $x$. When the curves in $C$ do not fill, Proposition 5
ensures that $L(A,C)$ has no minimum in the interior of $\mathcal{T}_{g}$.

######  Definition 16 ($\mathrm{Min}(C)$).

The set $\mathrm{Min}(C)$ contained in $\mathcal{T}_{g}$ is the set of all
points at which $L(A,C)$ has a minimum for some $A$ with strictly positive
entries.

There is a notion of closure of $\mathrm{Min}(C)$ that will be denoted by
$\overline{\mathrm{Min}(C)}$.

Figure 4. A fictitious cell $\mathrm{Min}(\\{c_{1},c_{2},c_{3}\\})$. The edges
of this cell are in $\overline{\mathrm{Min}(C)}$ and the vertices are not in
$\mathcal{T}_{g}$ but in the bordification. The arrows show the negative
gradients of the lengths of the curves in $C$.

######  Definition 17 (Modified version of Proposition 2 of [25]).

Let $C$ be a finite set of closed geodesics that fills $S_{g}$. Then a point
$p$ of $\mathcal{T}_{g}$ is in $\overline{\mathrm{Min}(C)}$ iff there does not
exist a derivation in $T_{p}\mathcal{T}_{g}$ whose evaluation on each length
function of a curve in $C$ is strictly positive.

######  Remark 18.

In the paper [25] from which the proposition used here to define
$\overline{\mathrm{Min}(C)}$ was taken, in the definition of
$\overline{\mathrm{Min}(C)}$ it was assumed that $\mathrm{Min}(C)$ has
dimension equal to the dimension of $\mathcal{T}_{g}$, i.e. the curves in $C$
not only fill but their lengths parametrise $\mathcal{T}_{g}$. In this paper,
as well as other papers of Schmutz, $\overline{\mathrm{Min}(C)}$ is defined
slightly more generally as above. The modified version of Proposition 2 of
[25] given here as a definition follows immediately from Lemma 4 of [24] by
observing that a convex function has a minimum at $x$ iff the gradient of the
function is zero at $x$.

######  Remark 19.

There is an alternative definition of $\mathrm{Min}(C)$ given in [24] that
makes the relationship between $\mathrm{Min}(C)$ and
$\overline{\mathrm{Min}(C)}$ clear. According to this alternative definition,
$\mathrm{Min}(C)$ is defined to be the set of all $x\in\mathcal{T}_{g}$ such
that for every derivation $\xi\in T_{x}\mathcal{T}_{g}$ either $\xi(L(c))=0$
for every $c\in C$, or there exists $c_{1},c_{2}\in C$ such that
$\xi(L(c_{1}))>0$ and $\xi(L(c_{2}))<0$.

The equivalence of the two definitions of $\mathrm{Min}(C)$ given here also
follows immediately from the observation that a necessary and sufficient
condition for a convex function to have a minimum at $x$ is that the gradient
of the function is zero at $x$.

It was shown in Lemma 1 of [24] that any $A$ with strictly positive entries
determines a length function $L(A,C)$ with a minimum in $\mathrm{Min}(C)$. By
Proposition 5, $\mathrm{Min}(C)$ is nonempty iff the curves in $C$ fill. For a
set $C$ of filling curves, this gives a smooth surjective map
$\phi:\mathbb{R}^{|C|}_{+}\twoheadrightarrow\mathrm{Min}(C)$, where $\phi(A)$
is the point in $\mathrm{Min}(C)$ at which $L(A,C)$ has its minimum. As $\phi$
is not injective, the entries of the $k$-tuple $A$ are thought of as
parameters, rather than coordinates. The notion of “admissible boundary
points” on the boundary of $\mathbb{R}_{+}^{|C|}$ are defined to extend $\phi$
to a map onto $\overline{\mathrm{Min}(C)}$.

Let $F(C):\mathcal{T}_{g}\rightarrow\mathbb{R}^{k}_{+}$ be the smooth function
given by $x\mapsto\bigl{(}L(c_{1}),\ldots,L(c_{k})\bigr{)}$ where
$C=\\{c_{1},\ldots,c_{k}\\}$. It was shown in Corollary 13 of [24] that when
the rank of the Jacobian of $F(C)$ is constant on $\mathrm{Min}(C)$, the set
$\mathrm{Min}(C)$ is an (open) cell. In this case, for a point $q$ in
$\mathrm{Min}(C)$, $\phi^{-1}(q)$ is the intersection with
$\mathbb{R}_{+}^{|C|}$ of a linear hypersurface, determined by the linear
dependencies of the gradients at $q$ of the lengths of curves in $C$. This is
best illustrated by an example. Suppose
$q=\phi\bigl{(}(a_{1},\ldots,a_{k})\bigr{)}$ where $k=|C|$ and
$a_{1},\ldots,a_{k}$ are all positive, and
$\nabla\bigl{(}L(c_{1})\bigr{)}=2\nabla\bigl{(}L(c_{2})\bigr{)}+\nabla\bigl{(}L(c_{3})\bigr{)}$
at $q$.333Actually, the linear dependencies must contain more elements than
this, but never mind that here. It follows from the convexity of length
functions that a necessary and sufficient condition for $L(A,C)$ to have a
minimum at $q$ is that the sum

$\sum_{i=1}^{i=k}a_{i}\nabla\bigl{(}L(c_{i})\bigr{)}$

is zero at $q$. So if
$\nabla\bigl{(}L(c_{1})\bigr{)}=2\nabla\bigl{(}L(c_{2})\bigr{)}+\nabla\bigl{(}L(c_{3})\bigr{)}$
at $q$, the intersection of the hypersurface
$(a_{1}-2t-s,a_{2}+t,a_{3}+s,a_{4},\ldots,a_{k})$ with $\mathbb{R}_{+}^{k}$ is
contained in the pre-image of $q$.

When $\mathrm{Min}(C)$ is not a cell, it is diffeomorphic to a cell that has
been pinched along a set of pairwise disjoint embedded submanifolds. A set
$\mathrm{Min}(C)$ therefore has a well defined boundary, which is the image
under $\phi$ of the admissible boundary points. Whether $\mathrm{Min}(C)$ is a
cell or not, as shown in Lemma 14 of [24], the intersection with
$\mathcal{T}_{g}$ of the boundary of $\mathrm{Min}(C)$ consists of sets of the
form $\mathrm{Min}(C_{i})$, for $C_{i}\subset C$. There is a nice geometric
interpretation of these boundary sets. A point $x$ in the interior of
$\mathcal{T}_{g}$ on the boundary of $\mathrm{Min}(C)$ is contained in
$\mathrm{Min}(C_{i})$, where $C_{i}$ is a subset of $C$ that fills and for
which the gradients at $x$ of the length functions of curves in $C_{i}$ are
contained in a proper subspace of $T_{x}\mathcal{T}_{g}$. Any vector in this
proper subspace determines a derivation, whose evaluation on some of the
length functions of curves in $C_{i}$ is strictly positive, and on some others
is strictly negative. This is illustrated in Figure 4.

######  Theorem 20.

Let $\mathrm{Sys}(C)$ be a stratum contained in $\mathcal{P}_{g}$ of dimension
$dim(\mathrm{Sys}(C))$. Suppose there is a critical point $p$ of $f_{sys}$
contained in $\mathrm{Sys}(C)$. Then

(3) $\text{index of }f_{sys}\text{ at }p\,+\,\text{ dimension of
}\mathrm{Sys}(C)\,=\,\text{dimension of }\mathcal{T}_{g}$

Moreover, any critical point of $f_{sys}$ in $\mathrm{Sys}(C)$ is contained in
the intersection of $\mathrm{Sys}(C)$ with $\mathrm{Min}(C)$. When the
intersection is nonempty, this intersection point of $\mathrm{Sys}(C)$ with
$\overline{\mathrm{Min}(C)}$ is unique, and

(4) $\text{index of }f_{sys}\text{ at }p\,=\,\text{dimension of
}\mathrm{Min}(C)\text{ at }p$

When the rank of the Jacobian of $F(C)$ is not constant over
$\mathrm{Min}(C)$, the dimension of $\mathrm{Min}(C)$ at $p$ is defined to be
the rank of the Jacobian of $F(C)$ at $p$.

###### Proof.

On a neighbourhood of any point in $\mathrm{Sys}(C)$, the curves in $C$ are
shorter than any other curves on $S_{g}$. It follows that

  1. (1)

For any point $x$ in $\mathrm{Sys}(C)$, $f_{sys}$ is decreasing at $x$ in any
direction in which the length of one or more of the curves in $C$ is
decreasing, and

  2. (2)

The dimension of $E(C)$ is equal to the dimension of $\mathrm{Sys}(C)$.

It follows from point 1 above and Remark 19 that for any critical point $x$ of
$f_{sys}$ in the intersection of $\mathrm{Sys}(C)$ with $\mathrm{Min}(C)$, the
dimension of the span of $\bigl{\\{}\nabla L(c)\,|\,c\in C\bigr{\\}}$ is equal
to the index of $f_{sys}$ at $x$. Since $x\in\mathrm{Min}(C)$, by Remark 19,
the tangent space to $E(C)$ at $x$ is contained in the orthogonal complement
of the subspace of $T_{x}\mathcal{T}_{g}$ spanned by $\bigl{\\{}\nabla
L(c)\,|\,c\in C\bigr{\\}}$. It follows that the point $x$ is the point $p$ in
Lemma 13 that is the global minimum of each of the functions
$\bigl{\\{}L(c)\,|\,c\in C\bigr{\\}}$ on $E(C)$, and by Remark 14 the tangent
space to $E(C)$ at $x$ is the entire orthogonal complement of the subspace of
$T_{x}\mathcal{T}_{g}$ spanned by $\bigl{\\{}\nabla L(c)\,|\,c\in
C\bigr{\\}}$. Since $C$ is also the set of systoles on a neighbourhood of $x$
in $E(C)$, $x$ is a local minimum of $f_{sys}$ when restricted to $E(C)$. By
point 2 above, Equation 3 follows for critical points contained in the
intersection of $\mathrm{Sys}(C)$ with $\mathrm{Min}(C)$.

Morse theory will now be used to show that if there is a critical point $p$ in
$\mathrm{Sys}(C)$, $p$ is in $\mathrm{Min}(C)$. For sufficiently small
$\epsilon$, $f_{sys}^{-1}(0,f_{sys}(p)]$ is obtained from
$f_{sys}^{-1}(0,f_{sys}(p)-\epsilon]$ by attaching a $\Gamma_{g}$ orbit of
$k$-handles, where $k$ is the index of the critical point at $p$. Each handle
$\bigl{(}$thought of as a disk to be attached along its boundary to
$f_{sys}^{-1}(f_{sys}(p)-\epsilon)$$\bigr{)}$ must be contained in
$f_{sys}^{-1}(0,f_{sys}(p)]$. It follows from point 1 above that for
sufficiently small $\epsilon$, one such handle is contained in the union of
sublevel sets $\bigl{\\{}L(c)_{\leq f_{sys}(p)}\ |\ c\in C\bigr{\\}}$.

Claim - The gradients $\bigl{\\{}\nabla L(c)\ |\ c\in C\bigr{\\}}$ satisfy the
condition in Remark 19 at $p$.

Figure 5. The figure on the right shows three sublevel sets meeting at the
point $p$, where the gradients do not satisfy the condition in Remark 19. The
green disk does not determine a handle.

To prove the claim, recall that the gradient $\nabla L(c)$ is orthogonal to
the boundary of $L(c)_{\leq f_{sys}(p)}$, and $L(c)_{\leq f_{sys}(p)}$ is a
convex set. If the claim were not true, there would be a derivation $\xi$ in
$T_{p}\mathcal{T}_{g}$ for which $\xi(L(c_{i}))>0$ for some $c_{i}\in C$ and
for which $\xi(L(c_{j}))\geq 0$ for every $c_{j}$ in $C$. As shown in Figure
5, $\xi$ is tangent to a sliver of the handle that is missing at $p$,
contradicting the fact that the handle is contained in
$f_{sys}^{-1}(0,f_{sys}(p)]$ and the topology of $f_{sys}^{-1}(0,t]$ changes
as $t$ passes through $f_{sys}(p)$. The claim then follows by contradiction.

The claim and Proposition 5 imply that the curves in $C$ fill. Remark 19 and
the claim then ensure that $p$ is in $\mathrm{Min}(C)$, and not on the
boundary of $\mathrm{Min}(C)$. Moreover, $\overline{\mathrm{Min}(C)}$ must be
contained in the union of the sublevel sets $\bigl{\\{}L(c)_{\leq f_{sys}(p)}\
|\ c\in C\bigr{\\}}$. This is because, as some of the parameters
$(a_{1},\ldots,a_{|C|})$ are increased (and the complement subsequently
decreased) relative to their values at $p$, the lengths of the corresponding
curves must be decreased in $\overline{\mathrm{Min}(C)}$. Starting at $p$, and
decreasing one or more of the lengths of the curves in $C$ keeps the point in
the union of the sublevel sets $\bigl{\\{}L(c)_{\leq f_{sys}(p)}\ |\ c\in
C\bigr{\\}}$ as claimed. As argued above, $p$ must be a local minimum of
$f_{sys}$ when restricted to $E(C)$. Since Lemma 13 shows there is only one
local minimum of each of the length functions $\bigl{\\{}L(c)\ |\ c\in
C\bigr{\\}}$ in $E(C)$, and $\mathrm{Sys}(C)\subset E(C)$, the intersection
$\mathrm{Sys}(C)\cap\overline{\mathrm{Min}(C)}$ is therefore the unique point
$p$.

At $p$, the tangent space is a direct sum of the subspace spanned by
$\bigl{\\{}\nabla L(c)\ |\ c\in C\bigr{\\}}$, and its orthogonal complement.
The former is the tangent space to $\mathrm{Min}(C)$ at $p$. The latter is the
tangent space to $E(C)$ and $\mathrm{Sys}(C)$, and it was shown that $p$ is a
local minimum of $f_{sys}$ restricted to $E(C)$. Together with point 1 above,
this proves Equation 4. ∎

###  5.1. Cell decompositions parametrised by length functions

The purpose of this subsection is to show the existence of a
$\Gamma_{g}$-equivariant cell decomposition of a bordification of
$\mathcal{T}_{g}$. This cell decomposition is obtained by subdividing the
$\Gamma_{g}$-equivariant “set decomposition” whose existence is suggested by
Theorem 20. To show the existence of this set decomposition, some technical
lemmas are proven, relating the structure of the cells with the positions of
critical points of $f_{sys}$ and cells of $\mathcal{P}_{g}$.

The dimension of $\mathrm{Min}(C)$ is defined to be the maximum over
$\mathrm{Min}(C)$ of the rank of the Jacobian of the function $F(C)$. The set
$\mathrm{Min}(C)$ will be called decomposable if it can be written as a
disjoint union $\cup_{C^{\prime}\in B}\mathrm{Min}(C^{\prime})$ where $B$ is a
set of distinct filling subsets of $C$, with the property that at least two
sets $C_{1}$ and $C_{2}$ in $B$ determine distinct sets $\mathrm{Min}(C_{1})$
and $\mathrm{Min}(C_{2})$ with the same dimension as $\mathrm{Min}(C)$.
Otherwise $\mathrm{Min}(C)$ is said to be indecomposable.

When $C$ is a set of filling systoles, it is not always the case that every
filling subset of $C$ can be realised as a set of systoles. Similarly, even
when $\mathrm{Min}(C)$ is indecomposable, the sets on the boundary of
$\mathrm{Min}(C)$ are not always in 1-1 correspondence with the set of proper
filling subsets of $C$. In both cases this has to do with possible linear
dependencies between the gradients of the lengths of the curves in $C$. The
next few lemmas clarify the relationship between filling subsets of $C$, sets
on the boundary of $\mathrm{Min}(C)$, and cells of $\mathcal{P}_{g}$ adjacent
to $\mathrm{Sys}(C)$.

######  Lemma 21.

Let $\mathrm{Sys}(C)$ be a local maximum of $f_{sys}$. Then $\mathrm{Min}(C)$
is indecomposable and has the same dimension as $\mathcal{T}_{g}$. If
$C^{\prime}\subset C$ is such that $\mathrm{Min}(C^{\prime})$ is on the
boundary of $\mathrm{Min}(C)$, then $E(C^{\prime})$ is nonempty and has
dimension at least 1. Moreover, $E(C^{\prime})$ intersects
$\mathrm{Min}(C^{\prime})$.

###### Proof.

Since $\mathrm{Sys}(C)$ is a local maximum of $f_{sys}$, it follows from
Equation (4) of Theorem 20 that $\mathrm{Min}(C)$ has the same dimension as
$\mathcal{T}_{g}$.

Indecomposability of $\mathrm{Min}(C)$ will be proven by contradiction.
Suppose $C_{1}$ and $C_{2}$ are distinct filling subsets of $C$ such that
$\mathrm{Min}(C_{1})$ and $\mathrm{Min}(C_{2})$ are disjoint and both strictly
contained in $\mathrm{Min}(C)$, with the same dimension as $\mathcal{T}_{g}$
and $\mathrm{Min}(C)$. It can be assumed without loss of generality that the
local maximum $\mathrm{Sys}(C)$ is not contained in $\mathrm{Min}(C_{2})$. The
submanifold $E(C_{2})$ is not empty, since it contains the point
$\mathrm{Sys}(C)$. If $E(C_{2})$ were just the point $\mathrm{Sys}(C)$, by
Lemma 13, the value of $f_{sys}$ at $\mathrm{Sys}(C)$ would then be the
smallest value of $t$ such that the sublevel sets $\\{L_{\leq t}(c_{j})\ |\
c_{j}\in C_{2}\\}$ all meet. However, since $\mathrm{Sys}(C)$ is not contained
in $\mathrm{Min}(C_{2})$, there is an open cone of directions at the point
$\mathrm{Sys}(C)$ in which the lengths of all the curves in $C_{2}$ are
decreasing. This implies that the interior of the intersection of the sublevel
sets $\\{L(c_{j})_{\leq t}\ |\ c_{j}\in C_{2}\\}$ cannot be empty at the value
of $t$ for which the level sets all pass through $\mathrm{Sys}(C)$,
contradicting the assumption that $E(C_{2})$ is just the point
$\mathrm{Sys}(C)$. This same argument shows that $E(C_{2})$ must pass through
$\mathrm{Min}(C_{2})$. Since $E(C_{2})$ is connected by Lemma 13, $E(C_{2})$
has dimension at least 1. This gives a contradiction, because by Remark 19,
the intersection of $E(C_{2})$ with $\mathrm{Min}(C_{2})$ can be at most a
point. This contradiction concludes the proof that $\mathrm{Min}(C)$ is
indecomposable.

Similarly, the submanifold $E(C^{\prime})$ is not the empty set; it contains
the point $\mathrm{Sys}(C)$. If $E(C^{\prime})$ were just the point
$\mathrm{Sys}(C)$, by Lemma 13, the value of $f_{sys}$ at $\mathrm{Sys}(C)$
would then be the smallest value of $t$ such that the sublevel sets
$\\{L(c_{j})_{\leq t}\ |\ c_{j}\in C^{\prime}\\}$ all meet. However, since
$\mathrm{Min}(C^{\prime})$ is a face of $\mathrm{Min}(C)$, there is an open
cone of directions at the point $\mathrm{Sys}(C)$ in which the lengths of all
the curves in $C^{\prime}$ are decreasing. This implies that the interior of
the intersection of the sublevel sets $\\{L(c_{j})_{\leq t}\ |\ c_{j}\in
C^{\prime}\\}$ cannot be empty at the value of $t$ for which the level sets
all pass through $\mathrm{Sys}(C)$. The same argument shows that the point $p$
on $E(C^{\prime})$ guaranteed by Lemma 13 must be in the intersection of
$E(C^{\prime})$ with $\mathrm{Min}(C^{\prime})$. Since $E(C^{\prime})$ is
connected by Lemma 13, and contains at least two points, it must have
dimension at least 1. ∎

######  Remark 22.

When $\mathrm{Min}(C)$ is indecomposable, it is not hard to see that any
subset $C^{\prime}$ of $C$ has the property that either
$\mathrm{Min}(C^{\prime})=\mathrm{Min}(C)$ or $\mathrm{Min}(C^{\prime})$ is on
the boundary of $\mathrm{Min}(C)$. If $\mathrm{Min}(C^{\prime})$ were in the
interior of $\mathrm{Min}(C)$, then for some top dimensional set
$\mathrm{Min}(C^{\prime\prime})$ on the boundary of $\mathrm{Min}(C)$,
$\mathrm{Min}(C^{\prime\prime}\cup C^{\prime})$ would be a top dimensional set
strictly contained in $\mathrm{Min}(C)$, contradicting the assumption that
$\mathrm{Min}(C)$ is indecomposable.

######  Corollary 23.

Suppose $\mathrm{Sys}(C)$ is a point representing a local maximum of
$f_{sys}$. Then $\mathrm{Sys}(C)$ is the only critical point of $f_{sys}$
contained in the open set $\mathrm{Min}(C)$.

###### Proof.

Since $\mathrm{Min}(C)$ is indecomposable by Lemma 21, for any
$C^{\prime}\subset C$, $\mathrm{Min}(C^{\prime})$ is either equal to
$\mathrm{Min}(C)$ or on the boundary of $\mathrm{Min}(C)$. Suppose
$\mathrm{Min}(C^{\prime})$ is equal to $\mathrm{Min}(C)$. Due to the fact that
$C^{\prime}\subset C$, $E(C)\subset E(C^{\prime})$. The submanifold
$E(C^{\prime})$ is a point by Equation (1) of Theorem 20. The lemma then
follows from a second application of Theorem 20. ∎

######  Corollary 24.

For any two distinct local maxima of $f_{sys}$,
$\mathrm{Sys}(C^{{}^{\prime}}_{max})$ and
$\mathrm{Sys}(C^{{}^{\prime\prime}}_{max})$,
$\mathrm{Min}(C^{{}^{\prime}}_{max})$ and
$\mathrm{Min}(C^{{}^{\prime\prime}}_{max})$ are disjoint.

###### Proof.

Since $\mathrm{Min}(C^{{}^{\prime}}_{max})$ and
$\mathrm{Min}(C^{{}^{\prime}}_{max})$ are both indecomposable with dimension
equal to $\mathcal{T}_{g}$, it follows from Lemma 21, if they were to
intersect, they would have to coincide. Corollary 23 would then imply that the
local maxima $\mathrm{Sys}(C^{{}^{\prime}}_{max})$ and
$\mathrm{Sys}(C^{{}^{\prime\prime}}_{max})$ are equal, contradicting the
assumption that they are distinct. ∎

######  Lemma 25.

Let $\mathrm{Sys}(C)$ be a local maximum of the systole function. Then the
sets $\mathrm{Min}(C_{i})$ on the boundary of $\mathrm{Min}(C)$ are in 1-1
correspondence with the cells

$\\{\mathrm{Sys}(C_{i})\ |\ \mathrm{Sys}(C_{i})\text{ is a cell adjacent to
}\mathrm{Sys}(C)\text{ in }\mathcal{P}_{g}\\}.$

###### Proof.

There is a cell decomposition $\mathcal{V}(\mathrm{Sys}(C))$ of the unit
tangent space to $\mathcal{T}_{g}$ at the point $\mathrm{Sys}(C)$, similar to
a Voronoi decomposition centered around the sign reversed gradients
$\\{-\nabla(L(c_{i}))\ |\ c_{i}\in C\\}$. In this cell decomposition, a unit
tangent vector is in the interior of a cell centered on $-\nabla(L(c_{j}))$ if
its inner product with $-\nabla(L(c_{j}))$ is greater than its inner product
with any other element of the set $\\{-\nabla(L(c_{i}))\ |\ c_{i}\in C\\}$.
Top dimensional cells of $\mathcal{V}(\mathrm{Sys}(C))$ are labelled by the
corresponding curve in $C$, lower dimensional cells are labelled by subsets of
$C$ in the obvious way. The tangent space to every cell of $\mathcal{P}_{g}$
adjacent to $\mathrm{Sys}(C)$ is visible in $\mathcal{V}(\mathrm{Sys}(C))$,
because as shown in the proof of Lemma 13, if $\mathrm{Sys}(C_{1})$ and
$\mathrm{Sys}(C_{2})$ are cells of $\mathcal{P}_{g}$ with a boundary cell in
common, then $E(C_{1})$ and $E(C_{2})$ intersect transversely along this
boundary cell.

The definition of the cell of $\mathcal{V}(\mathrm{Sys}(C))$ labelled by
$C_{i}$ implies that the vectors in this cell are pointing inwards to a
stratum $\mathrm{Sys}(C_{i})$ with $\mathrm{Sys}(C)$ on the boundary; they are
one sided limits of tangent vectors to the stratum $\mathrm{Sys}(C_{i})$. When
the curves in $C_{i}$ fill, the cell of $\mathcal{V}(\mathrm{Sys}(C))$
labelled by $C_{i}$ is the one sided limit of the tangent space to a cell
$\mathrm{Sys}(C_{i})$ of $\mathcal{P}_{g}$. The cell decomposition
$\mathcal{V}(\mathrm{Sys}(C))$ therefore determines a partial order on subsets
of $C$. It remains to relate this partial order to the dual of the partial
order induced by inclusion on $\\{\overline{\mathrm{Min}(C_{s})}\ |\
C_{s}\subset C$}.

To show that every set on the boundary of $\mathrm{Min}(C)$ determines a set
of $\mathcal{P}_{g}$ with a vertex on $\mathrm{Sys}(C)$, start with an edge
$\mathrm{Sys}(C_{i})$ of $\mathcal{P}_{g}$ with one vertex on
$\mathrm{Sys}(C)$. Such an edge necessarily exists, because otherwise
$\mathcal{P}_{g}$ would be disconnected, contradicting the fact that it is the
image of a deformation retraction of $\mathcal{T}_{g}$. If the rank of the
Jacobian of the map $F(C_{i})$ were to drop at the intersection of $E(C_{i})$
with $\mathrm{Min}(C_{i})$, by Theorem 20, the dimension of $E(C_{i})$ would
have to be at least 2. Since $\mathrm{Sys}(C_{i})$, if it is nonempty, is an
open subset of $E(C_{i})$, $\mathrm{Sys}(C_{i})$ could not be an edge of
$\mathcal{P}_{g}$. It follows that the rank of the Jacobian of $F(C_{i})$ does
not drop on $\mathrm{Min}(C_{i})$ at the intersection of $E(C_{i})$ with
$\mathrm{Min}(C_{i})$.

By Lemma 21, every boundary set $\mathrm{Min}(C_{i,j})$ of
$\mathrm{Min}(C_{i})$ determines a submanifold $E(C_{i,j})$ of
$\mathcal{T}_{g}$, and this submanifold strictly contains $E(C_{i})$. Let
$\bigl{\\{}E(C_{i,1}),\ldots,E(C_{i,k})\bigr{\\}}$ be the set of all such
submanifolds of dimension 2. Recall from the proof of Lemma 13 that $E(C_{i})$
is obtained by taking a transverse intersection of a submanifold $E(C_{i,j})$
with $E(\\{c_{1}\in
C_{i,j}\\}\cup\\{c_{2}\in(C_{i}{\smallsetminus}C_{i,j})\\})$. The submanifold
$E(\\{c_{1}\in C_{i,j}\\}\cup\\{c_{2}\in(C_{i}{\smallsetminus}C_{i,j})\\})$
cuts $E(C_{i})$ into two pieces; one of which the lengths of the curves in
$C_{i,j}$ are strictly less than the lengths of curves in
$C_{i}{\smallsetminus}C_{i,j}$, and one of which the lengths of the curves in
$C_{i,j}$ are strictly greater than the lengths of curves in
$C_{i}{\smallsetminus}C_{i,j}$. Denote the former by $E(C_{i,j})^{-}$. For any
point in a sufficiently small neighbourhood of $\mathrm{Sys}(C_{i})$, the
systoles must be contained in the set $C_{i}$; it follows that near
$\mathrm{Sys}(C_{i})$, the points in $E(C_{i,j})^{-}$ are in
$\mathrm{Sys}(C_{i,j})$. This argument can then be repeated with 2-cells of
$\mathcal{P}_{g}$ in place of edges, and then 3-cells, etc., to conclude the
proof that every set on the boundary of $\mathrm{Min}(C)$ determines a cell of
$\mathcal{P}_{g}$ with a vertex on $\mathrm{Sys}(C)$.

Suppose two different filling subsets $C^{\prime}$ and $C^{\prime\prime}$ of
$C$ determine the same boundary set
$\mathrm{Min}(C^{\prime})=\mathrm{Min}(C^{\prime\prime})$ of
$\mathrm{Min}(C)$. Since these assumptions imply that
$\mathrm{Min}(C^{\prime}\cup C^{\prime\prime})=\mathrm{Min}(C^{\prime})$, it
is possible to assume without loss of generality that $C^{\prime}$ contains
$C^{\prime\prime}$. By Theorem 20, point 2 in the proof of Theorem 20 and
Lemma 21, the dimensions of $E(C^{\prime\prime})$ and $E(C^{\prime})$ are the
same, because they both intersect $\mathrm{Min}(C^{\prime\prime})$ at the
unique point at which the curves in $C^{\prime\prime}$ have the same length.
Since $E(C^{\prime\prime})$ contains $E(C^{\prime})$, this implies that
$E(C^{\prime\prime})=E(C^{\prime})$. Consequently, any point in
$\mathcal{T}_{g}$ at which the set of systoles contains $C^{\prime\prime}$, it
also contains $C^{\prime}$. The different sets $C^{\prime}$ and
$C^{\prime\prime}$ therefore do not determine different cells in
$\mathcal{P}_{g}$. ∎

Sets of the form $\mathrm{Min}(C)$ have the inconvenient property that, unlike
the cells of $\mathcal{P}_{g}$, they are not uniquely labelled by a set of
curves. Fortunately, the most interesting sets $\mathrm{Min}(C)$ are usually
the ones dual to a cell of $\mathcal{P}_{g}$. In this case, the convention
will be used that the set $\mathrm{Min}(C)$ will always be labelled by the set
of curves that label the corresponding cell of $\mathcal{P}_{g}$.

######  Corollary 26.

Suppose $\mathrm{Min}(C)$ is dual to $\mathrm{Sys}(C)$ on $\mathcal{P}_{g}$.
Then $\mathrm{Min}(C)$ only intersects a cell $\mathrm{Sys}(C^{\prime})$ of
$\mathcal{P}_{g}$ if $C^{\prime}\subseteq C$. The only critical point of
$f_{sys}$ in $\mathrm{Min}(C)$ is the intersection of $\mathrm{Sys}(C)$ with
$\mathrm{Min}(C)$.

###### Proof.

The follows from Lemma 25 and Theorem 20. ∎

######  Corollary 27.

Any stratum $\mathrm{Sys}(C_{i})$ for which $C_{i}$ contains the curve $c$
must be on the boundary of the top dimensional stratum
$\mathrm{Sys}(\\{c\\})$.

###### Proof.

Let $\mathrm{Sys}(C)$ be a local maximum of $f_{sys}$ representing a vertex of
$\mathrm{Sys}(C_{i})$. Then $c\in C_{i}\subset C$. There is a neighbourhood of
a “face at infinity” of $\mathrm{Min}(C)$ on which the curve $c$ is shorter
than any of the other curves in the set $C$. This neighbourhood contains
points where the parameter in $A$ corresponding to $c$ is much larger than all
other entries of $A$. By Lemma 25, any boundary set of $\mathrm{Min}(C)$
incident on this “face” corresponds to a stratum $\mathrm{Sys}(C_{j})$ for
which $C_{j}$ contains $c$. Let $\\{v_{1}^{c},\ldots,v_{k}^{c}\\}$ be the set
of vertices of $\mathcal{V}(\mathrm{Sys}(C))$ corresponding to the top
dimensional boundary sets of $\mathrm{Min}(C)$ incident on this “face”. The
interior of the convex hull of the vertices $\\{v_{1}^{c},\ldots,v_{k}^{c}\\}$
corresponds to the unit tangent vectors in
$T_{\mathrm{Sys}(C)}\mathcal{T}_{g}$ pointing into $\mathrm{Sys}(\\{c\\})$.
Any other stratum $\mathrm{Sys}(C_{j})$ with $c\in C_{j}\subset C$ corresponds
to a cell of $\mathcal{V}(\mathrm{Sys}(C))$ on the boundary of this convex
hull. ∎

######  Proposition 28 (Proposition 2 from the introduction).

There is a bijection between the critical points of the systole function and
the cells of the Thurston spine. The index of each critical point is equal to
the codimension of the cell in whose interior it is contained.

###### Proof.

To show that every cell of $\mathcal{P}_{g}$ contains a critical point, first
note that every vertex of $\mathcal{P}_{g}$ is a critical point. It follows
from Lemmas 21, 25 and Corollary 23 that each edge of $\mathcal{P}_{g}$
contains a critical point. The proof of Theorem 20 obtained a local model for
critical points. According to this model, at a critical point in the cell
$\mathrm{Sys}(C)$ of $\mathcal{P}_{g}$, $f_{sys}$ is stationary in directions
tangent to $\mathrm{Sys}(C)\subset E(C)$, and decreasing in all other
directions. Suppose $\mathrm{Sys}(C^{\prime})$ is a 2-cell of
$\mathcal{P}_{g}$. Since every edge of $\mathrm{Sys}(C^{\prime})$ contains a
critical point, $f_{sys}$ must be increasing towards the boundary of
$\mathrm{Sys}(C^{\prime})$. It then follows that the edges of
$\mathcal{P}_{g}$ making up the boundary of $\mathrm{Sys}(C^{\prime})$ have a
subset of $E(C^{\prime})$ in the interior that contains the point at which
$E(C^{\prime})$ intersects $\mathrm{Min}(C^{\prime})$. Note that the existence
of such a point is guaranteed by Lemma 21. By Theorem 20, this point is a
critical point. The same argument can be used to show that if every
$(n-1)$-cell of $\mathcal{P}_{g}$ contains a critical point, the same is true
for every $n$-cell. It then follows by induction that every cell of
$\mathcal{P}_{g}$ contains a critical point. Theorem 20 states that the
critical point in each cell of $\mathcal{P}_{g}$ is unique. This gives the
bijection between critical points of $f_{sys}$ and cells of $\mathcal{P}_{g}$.

Since it was just shown that each critical point in a cell $\mathrm{Sys}(C)$
of $\mathcal{P}_{g}$ is a local minimum of the restriction of $f_{sys}$ to
$\mathrm{Sys}(C)$, and $f_{sys}$ is decreasing away from the critical point in
all directions not tangent to $\mathrm{Sys}(C)$, the index of the critical
point is the codimension of the cell $\mathrm{Sys}(C)$ as required. ∎

The next theorem puts together the results of this subsection to construct a
$\Gamma_{g}$-equivariant cell decomposition of a bordification of
$\mathcal{T}_{g}$. It is left as a question to the reader to figure out which
(if any) of the known metric completions of $\mathcal{T}_{g}$ it gives rise
to. It seems clear that the cell decomposition constructed in the next theorem
has finitely many cells modulo the action of $\Gamma_{g}$. How to prove this
is also left as a question.

Figure 6. Cutting $\overline{\mathrm{Min}(C_{q}\cup\\{c_{1}\\})}$ along
$\overline{\mathrm{Min}(C_{q})}$.

######  Theorem 29.

The set $\check{\mathcal{P}}_{g}:=\\{\mathrm{Min}(C)\ |\ \mathrm{Sys}(C)\text{
is a cell of }\mathcal{P}_{g}\\}$ is a decomposition of $\mathcal{T}_{g}$ into
sets. A $\Gamma_{g}$-equivariant cell decomposition of a bordification of
$\mathcal{T}_{g}$ can be obtained by subdividing these sets in a canonical
way, and adding in missing faces at infinity.

###### Proof.

It follows from Lemmas 21, 25 and Corollary 24 that $\check{\mathcal{P}}_{g}$
is a $\Gamma_{g}$-equivariant covering of $\mathcal{T}_{g}$ by pairwise
disjoint sets.

“Cutting $\overline{\mathrm{Min}(C)}$ along a subset
$\overline{\mathrm{Min}(C_{i})}$ with $C_{i}\subsetneq C$” refers to the
process whereby the points of $\overline{\mathrm{Min}(C)}$ in
$\overline{\mathrm{Min}(C_{i})}$ are deleted, and the metric completion of
each resulting connected component is formed. This is shown schematically in
Figure 6.

Suppose the rank of the Jacobian of $F(C)$ is not constant on a set
$\mathrm{Min}(C)$ in $\check{\mathcal{P}}_{g}$. By Lemma 21, $\mathrm{Min}(C)$
is indecomposable. For any point $q$ in $\overline{\mathrm{Min}(C)}$ at which
the rank of the Jacobian of $F(C)$ is not maximal, one of the gradients, call
it $\nabla L(c_{1})$, enters a subspace of $T_{q}\mathcal{T}_{g}$ spanned by a
subset of the other gradients. It follows from Remark 19 that $q$ is in
$\mathrm{Min}(C_{q})$ for some subset of $C_{q}$ of $C$ not containing
$c_{1}$. Since $\mathrm{Min}(C)$ is indecomposable, $q$ is therefore on the
boundary of $\mathrm{Min}(C)$. Cut $\overline{\mathrm{Min}(C)}$ along
$\overline{\mathrm{Min}(C_{q})}$. Due to the fact that
$\mathrm{Min}(C_{q}\cup\\{c_{1}\\})$ is collapsed onto $\mathrm{Min}(C_{q})$
at points such as $q$ where $\nabla(L(c_{1}))$ is tangent to
$\mathrm{Min}(C_{q})$, cutting along $\overline{\mathrm{Min}(C_{q})}$ has the
effect of subdividing $\mathrm{Min}(C_{q}\cup\\{c_{1}\\})$ on the boundary of
$\mathrm{Min}(C)$ into pieces. In the interior of each of these pieces of
$\mathrm{Min}(C_{q}\cup\\{c_{1}\\})$, $\nabla(L(c_{1}))$ is either not tangent
to $\mathrm{Min}(C_{q})$, or everywhere tangent to $\mathrm{Min}(C_{q})$.

By cutting $\overline{\mathrm{Min}(C)}$ and its boundary sets along every
subset of the form $\overline{\mathrm{Min}(C_{i})}$, $C_{i}\subsetneq C$,
$\overline{\mathrm{Min}(C)}$ and its boundary sets are decomposed into pieces.
For each $C_{k}\subset C$, in the interior of each piece of
$\mathrm{Min}(C_{k})$, the rank of the Jacobian of $F(C_{k})$ is constant. The
interior of each of the pieces can be seen to be a cell, because it is the
image under $\phi$ of a polyhedron in the closure of $\mathbb{R}_{+}^{|C|}$,
and the pre-image of different points are contained within disjoint linear
hypersurfaces of constant dimension.

Let $P$ be the polyhedron in the pre-image of a cell constructed above.
Coordinates on the cell can be obtained from the coordinates on $P$ by
collapsing to a point the intersections of the hypersurfaces with $P$. These
coordinates can be used to obtain a metric completion of the cell. Some of the
faces of a cell obtained as a piece of $\mathrm{Min}(C)$ will be contained in
$\overline{\mathrm{Min}(C)}$, and others will be faces at infinity. ∎

######  Remark 30.

Examples in which the rank of the Jacobian of $F(C)$ is nonconstant over the
set $\mathrm{Min}(C)$ can be found in Theorem 47 of [24] and Proposition 3.2
of [27]. In all these examples, the drop in the rank occurs at points of
$\mathcal{T}_{g}$ where the surface has a nontrivial automorphism that nearby
surfaces do not have. An example of a $\Gamma_{g}$-equivariant function on
$\mathcal{T}_{g}$ is the Weil-Petersson volume of a parallelopiped with edges
determined by the Weil-Petersson gradients of a set of length functions. As
pointed out by Wolpert, for most genera, there are points at which every
$\Gamma_{g}$-equivariant function on $\mathcal{T}_{g}$ has vanishing gradient.
Examples are given by points of $\mathcal{T}_{g}$ with nontrivial automorphism
group given by a quotient of a triangle group. The mapping class group
stabiliser acts nontrivially on the tangent space to $\mathcal{T}_{g}$ at
these points. As the gradient of a $\Gamma_{g}$-equivariant function must be
invariant under this action, large automorphism groups can force the gradient
to be zero.

######  Remark 31.

A special case in which it follows easily that the rank of the Jacobian of
$F(C)$ is constant on $\mathrm{Min}(C)$ occurs when $C$ has no proper filling
subsets. In this case the boundary of $\mathrm{Min}(C)$ is at infinity, and by
the same arguments as in the proof of step 1 of Lemma 13, the rank of $F(C)$
cannot drop on $\mathrm{Min}(C)$. This special case occurs for example when
$\mathrm{Sys}(C)$ is a top dimensional cell of $\mathcal{P}_{g}$.

##  6\. The deformation retraction of the Thurston spine

This section shows that the deformation retraction from [29] has dimension
equal to the virtual cohomological dimension of $\Gamma_{g}$.

The next theorem is based on intuition from the previous section. Each sphere
in $\partial\mathcal{T}_{g}^{thick}$ representing a generator of homology
gives rise to a “thin place” or “neck” of $\mathcal{T}_{g}^{thick}$ where
level sets of length functions meet in a point. This is how critical points of
$f_{sys}$ arise. One expects that the image of a deformation retraction of
$\mathcal{T}_{g}$ (a spine) passes through the “neck” at only one point.

######  Theorem 32 (Theorem 1 of the introduction).

The Thurston spine of a closed orientable surface of genus $g$ deformation
retracts onto a subcomplex of dimension equal to $4g-5$.

###### Proof.

Let $\mathrm{Sys}(C)$ be a top dimensional stratum of $\mathcal{P}_{g}$, and
let $q$ be an interior point of $\mathrm{Sys}(C)$. Construct a ball
$\mathcal{D}(q)$ in $\mathcal{T}_{g}$ consisting of all the points that map to
$q$ under the deformation retraction of $\mathcal{T}_{g}$ onto
$\mathcal{P}_{g}$. This ball has boundary at infinity and intersects
$\mathcal{P}_{g}$ in the single point $q$. By construction, the dimension of
$\mathcal{D}(q)$ is equal to the codimension of $\mathrm{Sys}(C)$ in
$\mathcal{T}_{g}$.

######  Remark 33.

It will not be needed here, but if $q$ is chosen to be the unique critical
point contained in $\mathrm{Sys}(C)$ whose existence was established in the
proof of Proposition 2, $\mathcal{D}(q)$ could be taken to be the cell
$\mathrm{Min}(C)$.

Due to the fact that any $\alpha$-thick subset of $\mathcal{T}_{g}$ is
invariant under the flow defined in Section 3, $\mathcal{D}(q)$ intersects
$\mathcal{T}_{g}^{thick}$ in a connected set. In addition $\mathcal{D}(q)$
intersects $\partial\mathcal{T}_{g}^{thick}$ transversely along a sphere
$\mathcal{S}^{thick}$ of dimension 1 less than the codimension of
$\mathrm{Sys}(C)$. The dimension of $\mathrm{Sys}(C)$ cannot be less than
$4g-5$, as this is the virtual cohomological dimension of $\Gamma_{g}$, and
gives a lower bound on the dimension of a spine. Assume the dimension of
$\mathrm{Sys}(C)$ is greater than $4g-5$. In this case, $\mathcal{S}^{thick}$
has dimension less than $2g-2$. Since $\partial\mathcal{T}_{g}^{thick}$ is
homotopy equivalent to a wedge of spheres $\vee_{i}^{\infty}S^{2g-2}$,
$\mathcal{S}^{thick}$ is contractible in $\partial\mathcal{T}_{g}^{thick}$.
Moreover $\mathcal{D}(q)\cap\mathcal{T}_{g}^{thick}$ can be homotoped relative
to its boundary $\mathcal{S}^{thick}$ into $\partial\mathcal{T}_{g}^{thick}$.

The homotopy of $\mathcal{D}(q)\cap\mathcal{T}_{g}^{thick}$ into
$\partial\mathcal{T}_{g}^{thick}$ moves the point $q$ off $\mathcal{P}_{g}$.
This implies $\mathcal{P}_{g}$ must have an unmatched face.

The existence of an unmatched face makes it possible to construct a
deformation retraction of $\mathcal{P}_{g}$ onto a subcomplex. This argument
can be iterated with smaller and smaller subcomplexes in place of
$\mathcal{P}_{g}$, until a subcomplex of the required dimension is obtained. ∎

## References

  * [1] H. Akrout.  Singularités topologiques des systoles généralisées.  Topology, 42(2):291–308, 2003.
  * [2] L. Bers.  Nielsen extensions of Riemann surfaces.  Annales Academiæ Scientiarum Fennicæ, Series A. I. Mathematica, 2:29–34, 1976.
  * [3] M. Bestvina, K. Bromberg, K. Fujiwara, and J. Souto.  Shearing coordinates and convexity of length functions on Teichmüller space.  American Journal of Mathematics, 135(6):1449–1476, 2013.
  * [4] M. Fortier Bourque.  Hyperbolic surfaces with sublinearly many systoles that fill.  Commentarii Mathematici Helvetici, 95:515–534, 2020.
  * [5] M. Fortier Bourque.  The dimension of Thurston’s spine.  arXiv:2211.08923, 2022.
  * [6] M. Bridson and K. Vogtmann.  Automorphism groups of free groups, surface groups and free abelian groups.  In Problems on mapping class groups and related topics, volume 74 of Proceedings of Symposia in Pure Mathematics, pages 301–316. American Mathematical Society, Providence, RI, 2006.
  * [7] K. Brown.  Cohomology of Groups.  Graduate Texts in Mathematics. Springer, New York, 2012.
  * [8] B. Farb and D. Margalit.  A primer on mapping class groups, volume 49 of Princeton Mathematical Series.  Princeton University Press, Princeton, NJ, 2012.
  * [9] R. Forman.  A User’s Guide To Discrete Morse Theory.  Séminaire Lotharingien de Combinatoire, 48:1–35, 2001.
  * [10] J. Harer.  The virtual cohomological dimension of the mapping class group of an orientable surface.  Inventiones Mathematicae, 84:157–176, 1986.
  * [11] J. Harer.  The cohomology of the moduli space of curves.  In Theory of moduli (Montecatini Terme, 1985), volume 1337 of Lecture Notes in Mathematics, pages 138–221. Springer Berlin, 1988.
  * [12] W. Harvey.  Boundary structure of the modular group.  In Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), volume 97 of Ann. of Math. Stud., pages 245–251. Princeton Univ. Press, Princeton, N.J., 1981.
  * [13] Y. Huang, K. Ohshika, and A. Papadopoulos.  The infinitesimal and global Thurston geometry of Teichmüller space.  arXiv:2111.12281, 2021.
  * [14] N. Ivanov.  Mapping class groups.  In Handbook of geometric topology, pages 523–633. North-Holland, Amsterdam, 2002.
  * [15] L. Ji.  Well-rounded equivariant deformation retracts of Teichmüller spaces.  L’Enseignement Mathématique, 60(02):109–129, 2013.
  * [16] L. Ji and S. Wolpert.  A cofinite universal space for proper actions for mapping class groups.  In In the tradition of Ahlfors-Bers. V, volume 510 of Contemporary Mathematics, pages 151–163. American Mathematical Society, Providence, RI, 2010.
  * [17] S. Kerckhoff.  The Nielsen realization problem.  Annals of Mathematics, 117(2):235–265, 1983.
  * [18] M. Kontsevich.  Intersection theory on the moduli space of curves and the matrix Airy function.  Communications in Mathematical Physics, 147:1–23, 1992.
  * [19] J. Lee.  Introduction to smooth manifolds, 2nd Edition, volume 218 of Graduate Texts in Mathematics.  Springer, New York, 2013.
  * [20] J. Mather.  Stratifications and mappings.  In Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pages 195–232. Academic Press, New York, 1973.
  * [21] J. Milnor, L. Siebenmann, and J. Sondow.  Lectures on the H-Cobordism Theorem.  Princeton University Press, Princeton, NJ, 1965.
  * [22] M. Morse.  Topologically non-degenerate functions on a compact $n$-manifold $M$.  Journal d’Analyse Mathématique, 7:189–208, 1959.
  * [23] G. Riera.  A formula for the Weil-Petersson product of quadratic differentials.  Journal d’Analyse Mathématique, 95:105–120, 2005.
  * [24] P. Schmutz Schaller.  Systoles and topological Morse functions for Riemann surfaces.  Journal of Differential Geometry, 52(3):407–452, 1999.
  * [25] P. Schmutz Schaller.  Riemann surfaces with longest systole and an improved Voronoĭ algorithm.  Archiv der Mathematik, 76(3):231–240, 2001.
  * [26] P. Schmutz Schaller.  A systolic geometric cell decomposition for the space of once-holed Riemann surfaces of genus 2.  Topology, 40(5):1017–1049, 2001.
  * [27] P. Schmutz.  Riemann surfaces with shortest geodesic of maximal length.  Geometric and Functional Analysis, 3(6):564–631, 1993.
  * [28] P. Schmutz Schaller.  A cell decomposition of Teichmüller space based on geodesic length functions.  Geometric and Functional Analysis, 11(1):142–174, 2001.
  * [29] W. Thurston.  A spine for Teichmüller space.  Preprint, 1985.
  * [30] S. Wolpert.  Geodesic length functions and the Nielsen problem.  Journal of Differential Geometry, 25(2):275–296, 1987.
  * [31] S. Wolpert.  Geometry of the Weil-Petersson completion of Teichmüller space.  Surveys in Differential Geometry, 8:357–393, 03 2005.
  * [32] Y. Wu.  Growth of the Weil-Petersson inradius of moduli space.  Université de Grenoble. Annales de l’Institut Fourier, 69(3):1309–1346, 2019.