Thurston Spine in Teichmüller Space

• Thurston spine is a subset of Teichmüller space defined by hyperbolic surfaces whose systoles fill the surface, leading to a natural polygonal decomposition.
• It is stratified by analytic equalities and inequalities that produce a semi-analytic, equivariant spine supporting controlled deformation retractions.
• Asymptotically, its codimension grows slowly with genus, offering deep insights into hyperbolic geometry, systolic flows, and mapping class group topology.

The Thurston spine of Teichmüller space is the subset

$$\mathcal{P}_g=\{x\in \mathcal{T}_g \mid \text{the systoles of }S_x\text{ fill the surface}\},$$

where $\mathcal{T}_g$ is the Teichmüller space of marked closed hyperbolic surfaces of genus $g\ge 2$, and a set of curves fills if its union cuts the surface into polygons. In this sense, $\mathcal{P}_g$ records those hyperbolic structures for which the shortest closed geodesics already determine a polygonal decomposition of the surface. The object sits at the intersection of hyperbolic geometry, mapping class group topology, and the nonsmooth Morse theory of the systole function, and it has become a central test case for understanding low-dimensional skeleta of Teichmüller space (Mathieu, 2023, Irmer, 2022).

1. Definition and basic geometric structure

For a closed, connected, orientable surface of genus $g\ge 2$, Teichmüller space $\mathcal{T}_g$ is a smooth manifold of dimension $6g-6$. A systole on a hyperbolic surface is a closed geodesic of minimal length. The defining condition for the Thurston spine is therefore not merely multiplicity of shortest curves, but the stronger requirement that the entire systole set be filling, equivalently that the complement of the union of systoles be a union of polygons (Mathieu, 2023).

A convenient stratified description is obtained by fixing a finite set $C$ of simple closed curves and considering

$$\mathrm{Sys}(C)=\{x\in \mathcal{T}_g\mid \text{the set of systoles at }x\text{ is exactly }C\}.$$

Then $\mathcal{P}_g$ is the union of the strata $\mathcal{T}_g$0 over all filling $\mathcal{T}_g$1. Each such stratum is specified by analytic equalities expressing equality of the corresponding length functions and inequalities excluding other curves from becoming equally short. This makes $\mathcal{T}_g$2 a semi-analytic subset of $\mathcal{T}_g$3, and it admits a triangulation; consequently its dimension is well defined by

$\mathcal{T}_g$4

This formulation is fundamental in the modern dimension theory of the spine (Mathieu, 2023, Irmer, 2022).

The same papers emphasize that the polygonal decomposition condition is intrinsic to the hyperbolic structure rather than to a chosen combinatorial model. In particular, the spine is $\mathcal{T}_g$5-invariant, where $\mathcal{T}_g$6 denotes the mapping class group, because it is defined entirely in terms of marked length data.

2. Equivariant topology and the retraction problem

Thurston proposed $\mathcal{T}_g$7 as a mapping class group-equivariant spine for Teichmüller space. The classical idea is that if a given systole set does not fill the surface, then there exists a direction in Teichmüller space along which the lengths of all those curves increase simultaneously. This makes non-filling configurations dynamically unstable and suggests a flow pushing points toward the filling locus. A neighborhood $\mathcal{T}_g$8 is defined by requiring the curves whose lengths lie within $\mathcal{T}_g$9 of the shortest length to fill; flowing by a suitably chosen vector field is intended to move points into such neighborhoods (Irmer, 2022).

Later work showed that one step in the original argument requires modification. An explicit genus $g\ge 2$0 example demonstrates that once a point enters $g\ge 2$1, it need not remain there under the naive flow. The repair uses tubular neighborhoods adapted to the stratified structure of $g\ge 2$2, rather than the raw $g\ge 2$3-neighborhoods originally envisioned. On that basis, a mapping class group-equivariant deformation retraction of the Thurston spine onto a CW-complex of dimension $g\ge 2$4 is constructed, aligning the resulting complex with the virtual cohomological dimension of $g\ge 2$5 (Irmer, 2022).

This point is often a source of confusion. The statement about a deformation retraction of $g\ge 2$6 onto a $g\ge 2$7-dimensional CW-complex does not imply that $g\ge 2$8 itself has dimension $g\ge 2$9. Indeed, later dimension estimates show that $\mathcal{P}_g$0 can be much larger. Related work on other $\mathcal{P}_g$1-equivariant spines, notably “well-rounded” deformation retracts to positive-codimension subspaces, highlights both the appeal of Thurston’s proposal and the technical difficulties in converting systolic flows into fully controlled global retractions (Ji, 2013). In still later literature, the status of the full conjectural retraction $\mathcal{P}_g$2 is described as open in general, even as explicit subfamilies are understood (Gao et al., 29 Sep 2025).

3. Stratification, local differential topology, and the systole function

The local geometry of the Thurston spine is governed by the systole function

$\mathcal{P}_g$3

where $\mathcal{P}_g$4 is the length of the geodesic representative of $\mathcal{P}_g$5 at $\mathcal{P}_g$6. Although $\mathcal{P}_g$7 is only piecewise smooth, it behaves as a topological Morse function. Its critical points lie in $\mathcal{P}_g$8, and the spine can be viewed as the locus where the nonsmooth Morse theory of $\mathcal{P}_g$9 becomes geometrically meaningful (Irmer, 2024).

A key local model is the equal-length locus

$g\ge 2$0

For minimal filling sets $g\ge 2$1, if $g\ge 2$2 is nonempty, it is a connected, embedded submanifold of $g\ge 2$3; moreover, the relevant length restrictions are strictly convex and attain a unique minimum. These loci provide the smooth pieces from which the spine is assembled. They refine the rougher stratification by $g\ge 2$4 and make it possible to discuss tangent cones, local embeddedness, and the failure of global smoothness in a controlled way (Irmer, 2024).

Another decisive notion is the cone of increase at a point, namely the cone of tangent directions along which all systolic lengths increase to first order. The local Morse–Smale analogue proved for the spine states that if this cone is full at a point of $g\ge 2$5, then it intersects the tangent cone of $g\ge 2$6 nontrivially. In particular, one can move inside the spine while increasing all systolic lengths. Top-dimensional strata are always balanced in this sense, whereas unbalanced loci are adjacent to higher-dimensional balanced strata. This gives the Thurston spine a structure analogous to a Morse–Smale complex for a smooth Morse function, even though the underlying function is only piecewise smooth (Irmer, 2024).

The differential-topological viewpoint also clarifies how singularities arise. At a non-smooth point, the tangent cone of the spine can be a union of cones coming from several intersecting strata, and these intersections reflect degeneracies in the systole set rather than pathologies of Teichmüller space itself (Irmer, 2022).

4. Dimension, codimension, and asymptotic behavior

A central modern question concerns the actual dimension of $g\ge 2$7 inside the $g\ge 2$8-dimensional manifold $g\ge 2$9. Topological and group-cohomological considerations give an a priori upper bound $\mathcal{T}_g$0 for the codimension. At the opposite end, for genus $\mathcal{T}_g$1 it is known that

$\mathcal{T}_g$2

which matches the virtual cohomological dimension of $\mathcal{T}_g$3 (Mathieu, 2023).

Subsequent work showed that the spine is often much larger than one might expect from cohomological dimension alone. One paper proves that for every $\mathcal{T}_g$4, there exists some $\mathcal{T}_g$5 such that the set of closed hyperbolic surfaces of genus $\mathcal{T}_g$6 whose systoles fill has dimension at least $\mathcal{T}_g$7. Since $\mathcal{T}_g$8, this implies that the filling-systole locus itself can exceed the virtual cohomological dimension, so $\mathcal{T}_g$9 is not generally a minimal-dimensional spine in the literal sense of its own dimension (Bourque, 2022, Irmer, 2022).

An even stronger asymptotic phenomenon is now known. Fortier Bourque proved

$6g-6$0

and conjectured that

$6g-6$1

That conjecture has been resolved affirmatively: there exists an infinite set $6g-6$2 of genera such that

$6g-6$3

Thus, along an infinite sequence of genera, the codimension grows much more slowly than $6g-6$4, so the spine occupies an asymptotically very large portion of Teichmüller space (Mathieu, 2023).

Quantity	Statement	Source
Low genus	$6g-6$5	(Mathieu, 2023)
vcd	$6g-6$6	(Irmer, 2022)
Large-dimension lower bound	For every $6g-6$7, some $6g-6$8 has $6g-6$9	(Bourque, 2022)
Asymptotic codimension upper bound	For infinite $C$0, $C$1	(Mathieu, 2023)

These results dispel a common misconception. The fact that the Thurston spine deformation retracts onto a $C$2-dimensional CW-complex does not mean that $C$3 itself has that dimension. Rather, $C$4 can be substantially larger while still admitting further $C$5-equivariant collapse.

5. Explicit constructions and the transversality machinery

Two construction paradigms dominate the current quantitative theory. The first uses right-angled polygons and large-girth combinatorial maps. For parameters $C$6 and $C$7, one considers a right-angled hyperbolic $C$8-gon $C$9 with alternating side lengths $$\mathrm{Sys}(C)=\{x\in \mathcal{T}_g\mid \text{the set of systoles at }x\text{ is exactly }C\}.$$0 and $$\mathrm{Sys}(C)=\{x\in \mathcal{T}_g\mid \text{the set of systoles at }x\text{ is exactly }C\}.$$1, determined by

$$\mathrm{Sys}(C)=\{x\in \mathcal{T}_g\mid \text{the set of systoles at }x\text{ is exactly }C\}.$$2

Gluing copies of these polygons according to a map $$\mathrm{Sys}(C)=\{x\in \mathcal{T}_g\mid \text{the set of systoles at }x\text{ is exactly }C\}.$$3 and then doubling across the boundary produces closed surfaces $$\mathrm{Sys}(C)=\{x\in \mathcal{T}_g\mid \text{the set of systoles at }x\text{ is exactly }C\}.$$4. For a distinguished value $$\mathrm{Sys}(C)=\{x\in \mathcal{T}_g\mid \text{the set of systoles at }x\text{ is exactly }C\}.$$5 satisfying $$\mathrm{Sys}(C)=\{x\in \mathcal{T}_g\mid \text{the set of systoles at }x\text{ is exactly }C\}.$$6, the blue and red curves arising from the gluing are precisely the systoles, and they fill the surface. The dimension calculation of the corresponding filling-systole locus is

$$\mathrm{Sys}(C)=\{x\in \mathcal{T}_g\mid \text{the set of systoles at }x\text{ is exactly }C\}.$$7

which yields the lower bound $$\mathrm{Sys}(C)=\{x\in \mathcal{T}_g\mid \text{the set of systoles at }x\text{ is exactly }C\}.$$8 for suitable large $$\mathrm{Sys}(C)=\{x\in \mathcal{T}_g\mid \text{the set of systoles at }x\text{ is exactly }C\}.$$9 and odd $\mathcal{P}_g$0 (Bourque, 2022).

The second paradigm, used in the asymptotic codimension theorem, constructs for infinitely many genera $\mathcal{P}_g$1 a surface $\mathcal{P}_g$2 with a standard right-angled hexagonal tessellation $\mathcal{P}_g$3 such that

$\mathcal{P}_g$4

Among the tessellation curves, those of index $\mathcal{P}_g$5 form a filling subset $\mathcal{P}_g$6 of size asymptotic to about two thirds of the total number of systoles. The locus where the curves in $\mathcal{P}_g$7 are exactly the systoles is then expected to have codimension $\mathcal{P}_g$8, because it is defined by $\mathcal{P}_g$9 independent equal-length conditions together with inequalities preventing other curves from tying for the minimum (Mathieu, 2023).

Making that expectation rigorous requires a nontrivial transversality argument. Along suitable Sanki paths, the differentials

$\mathcal{T}_g$00

are shown to be linearly independent except at finitely many exceptional parameter values. A dual family of length combinations is introduced; for a blue curve $\mathcal{T}_g$01,

$\mathcal{T}_g$02

where $\mathcal{T}_g$03 are associated red curves and $\mathcal{T}_g$04 is a geodesic determined by a pair of pants. Using the Weil–Petersson Poisson structure, the matrix

$\mathcal{T}_g$05

is proved to approach the identity in a degenerating regime. Combined with asymptotic analysis of Wolpert’s formula for Poisson brackets of length functions, this yields the required linear independence and hence the codimension formula at the constructed point (Mathieu, 2023).

These constructions show that the large dimension of the spine is not an abstract existence phenomenon. It is produced by highly structured hyperbolic surfaces whose systolic geometry is controlled by explicit tessellations and by precise differential-topological analysis of equal-length loci.

6. Special subspaces, essential loops, and distinct usages of the term

A recent line of work studies the Thurston spine on lower-dimensional invariant slices of $\mathcal{T}_g$06. In one such construction, a Teichmüller curve $\mathcal{T}_g$07 is obtained from genus $\mathcal{T}_g$08 surfaces with an order-$\mathcal{T}_g$09 rotational symmetry. The quotient is a spherical orbifold with four cone points of order $\mathcal{T}_g$10, so $\mathcal{T}_g$11 is the image of $\mathcal{T}_g$12 under a totally geodesic embedding. Within this slice, the systoles belong to three families $\mathcal{T}_g$13, and the intersection $\mathcal{T}_g$14 is described explicitly by the equalities $\mathcal{T}_g$15 and $\mathcal{T}_g$16. The resulting set is a trivalent tree, with a unique vertex where

$\mathcal{T}_g$17

Moreover, $\mathcal{T}_g$18 is an equivariant deformation retract of $\mathcal{T}_g$19, and the quotient produces explicit essential loops in the spine, including both reducible and pseudo-Anosov classes (Gao et al., 29 Sep 2025).

The term “Thurston spine” also appears in a distinct dynamical context. For Thurston maps $\mathcal{T}_g$20 with four postcritical points, one paper uses the phrase for a combinatorial graph formed by preimages of core arcs of an obstruction curve in the square-pillow model. In that setting the spine is tied to the pull-back dynamics on curves, the slope map on $\mathcal{T}_g$21, and a “blowing up arcs” surgery that eliminates obstructions without creating new ones, thereby producing rational maps with controlled curve dynamics (Bonk et al., 2021). This usage is not the Teichmüller-space spine $\mathcal{T}_g$22, even though both objects serve as skeletal structures organizing geometric or dynamical behavior.

Taken together, these developments place the Thurston spine at a junction of several active theories. In Teichmüller geometry it is the filling-systole locus, stratified by equal-length conditions, governed locally by topological Morse theory, and globally large in dimension. In special subspaces it can become explicitly computable, as in the trivalent-tree intersection with a Teichmüller curve. And across adjacent fields, the same terminology persists for related skeletal constructions, which underscores both the influence and the breadth of Thurston’s original viewpoint.