Stratified Manifolds (s-manifolds) Overview

• Stratified manifolds are singular spaces decomposed into smooth manifold strata with controlled local models such as conical, tubular, or corner-type structures.
• They extend classical differential topology by incorporating tangential data, sheaf invariants, and bordism theories to analyze and resolve singularities.
• Applications include nonsmooth analysis, low-dimensional topology, and symplectic geometry, facilitating advances in moduli spaces and h-principle formulations.

Stratified manifolds, often abbreviated s-manifolds in one recent formulation, are singular spaces decomposed into strata that are smooth manifolds, with the strata organized by frontier relations and typically controlled by local conical, tubular, or corner-type models. In current usage the term spans several closely related categories, including conically smooth stratified spaces, Thom–Mather abstractly stratified spaces, PL stratified pseudomanifolds, and newer categories of s-manifolds and s-manifolds with corners. What unifies these frameworks is the requirement that singularities be structured rather than arbitrary, so that one can still define tangential data, sheaf-theoretic invariants, fibre products, bordism theories, and, in suitable settings, fundamental classes and symplectic structures (Ayala et al., 2014, Mj et al., 2022, Joyce, 29 Jul 2025).

1. Foundational definitions and local models

A common starting point is a decomposition of a Hausdorff space into locally closed manifold pieces satisfying a frontier condition. In one formulation, a decomposed space $(X,\mathcal S)$ consists of a second countable metric space together with a locally finite collection of pairwise disjoint locally closed subsets $S_i$, each a smooth topological manifold, such that $S_i\cap \overline{S_j}\neq\emptyset$ if and only if $S_i\subset \overline{S_j}$. Thom–Mather theory refines this by adding tube systems $(T_i,p_i)$ around each stratum, with compatibility and submersion conditions on overlaps, yielding an abstractly stratified space. A key structural result is that every point in a stratum of dimension $n$ has a neighborhood isomorphic to

$\mathbb{R}^n\times cL,$

where $cL$ is the open cone on a lower-dimensional abstractly stratified space $L$ (Mj et al., 2022).

The conically smooth approach makes the local model itself the basic object. Its $C^0$ basics are spaces of the form $S_i$0, and a $S_i$1 stratified space is precisely a space for which such basics form a basis for the topology. A conically smooth atlas then requires transition maps between these basics to be conically smooth in an inductive sense, with a conical derivative controlling behavior along the cone locus. This framework is designed to support a genuine differential topology on singular spaces, including inverse function, tubular neighborhood, and isotopy extension theorems (Ayala et al., 2014).

A PL version appears in the theory of stratified pseudomanifolds. There the local model near a point in an $S_i$2-dimensional stratum is again $S_i$3, but $S_i$4 is a compact PL stratified pseudomanifold of dimension $S_i$5, and the filtration records how the singular part sits inside the ambient space. This setting is especially suited to bordism and intersection-homological questions (Friedman, 2013).

A more recent definition isolates only codimension $S_i$6 and $S_i$7 control. An s-manifold $S_i$8 of dimension $S_i$9 is a locally compact, Hausdorff, second countable space stratified by locally closed smooth manifolds of dimension at most $S_i\cap \overline{S_j}\neq\emptyset$0, locally embeddable in $S_i\cap \overline{S_j}\neq\emptyset$1, with closure relations constrained so that only strata of dimension $S_i\cap \overline{S_j}\neq\emptyset$2 or $S_i\cap \overline{S_j}\neq\emptyset$3 are forced to interact with lower-dimensional strata. In that framework the top stratum $S_i\cap \overline{S_j}\neq\emptyset$4 is a smooth $S_i\cap \overline{S_j}\neq\emptyset$5-manifold, the codimension-one stratum $S_i\cap \overline{S_j}\neq\emptyset$6 is a smooth $S_i\cap \overline{S_j}\neq\emptyset$7-manifold, and the remaining singular locus $S_i\cap \overline{S_j}\neq\emptyset$8 is closed (Joyce, 29 Jul 2025).

2. Smooth structure, tangential data, and stratified bundles

Once local models are fixed, the next issue is how to define smooth maps and tangent-type data. In the conically smooth theory, morphisms are required to be compatible with the cone structure, not merely smooth on each stratum. The tangent bundle of an ordinary manifold is replaced by a tangent classifier

$S_i\cap \overline{S_j}\neq\emptyset$9

or equivalently by the enter-path $S_i\subset \overline{S_j}$0-category $S_i\subset \overline{S_j}$1. Tangential structures are then encoded as lifts of this classifier to a right fibration $S_i\subset \overline{S_j}$2, and a $S_i\subset \overline{S_j}$3-manifold is a stratified space equipped with such a lift (Ayala et al., 2014).

The sheaf-theoretic extension of differential topology requires more than stratumwise smoothness. A stratified continuous sheaf assigns data not just to open subsets of $S_i\subset \overline{S_j}$4, but to open subsets of closures of strata, together with restriction morphisms across incidences $S_i\subset \overline{S_j}$5. The new object governing extension across singular strata is the homotopy fiber sheaf $S_i\subset \overline{S_j}$6, which measures the gap between intrinsic and extrinsic restrictions. In this language, stratumwise flexibility together with flexibility of these homotopy fiber sheaves yields the parametric h-principle on stratified spaces (Mj et al., 2022).

A complementary linear theory is provided by stratified vector bundles. A stratified vector bundle $S_i\subset \overline{S_j}$7 is a stratified morphism such that for each stratum $S_i\subset \overline{S_j}$8, the preimage $S_i\subset \overline{S_j}$9 is itself a stratum of $(T_i,p_i)$0, and the restriction $(T_i,p_i)$1 is an ordinary smooth vector bundle. Scalar multiplication

$(T_i,p_i)$2

must be a stratified morphism. In the Whitney $(T_i,p_i)$3 case, the tangent object $(T_i,p_i)$4 becomes a stratified vector bundle, and there is an alternate characterization by regular actions of the multiplicative monoid $(T_i,p_i)$5: every smooth stratified regular monoid action determines a unique differentiable stratified vector bundle structure with base $(T_i,p_i)$6 (Ross, 2023).

3. Major constructions and model examples

Several important geometric sources produce stratified manifolds canonically. One such source is nonsmooth analysis. A continuous selection of smooth functions $(T_i,p_i)$7 on a smooth manifold can determine strata

$(T_i,p_i)$8

and under the conditions $(T_i,p_i)$9 and affine independence of active gradients, each $n$0 is an embedded smooth submanifold of codimension $n$1. The boundary relation

$n$2

gives a stratification in the sense of filtered spaces with frontier condition. For $n$3 on a closed connected $n$4-manifold, the resulting seven strata—three $n$5-dimensional sectors, three $n$6-dimensional pairwise intersections, and one $n$7-dimensional triple intersection—refine Gay–Kirby trisections and support a stratified handle decomposition (Horvat, 2022).

A second family is low-dimensional and combinatorial. A closed $n$8-stratifold is a compact connected $n$9-dimensional cell complex with a branch $\mathbb{R}^n\times cL,$0-skeleton $\mathbb{R}^n\times cL,$1 consisting of circles, such that $\mathbb{R}^n\times cL,$2 is a $\mathbb{R}^n\times cL,$3-manifold. Near a branch circle, several sheets meet with monodromy described by a permutation. These spaces are encoded by bicolored labeled graphs, and they occur as spines of precisely those closed $\mathbb{R}^n\times cL,$4-manifolds that are connected sums of lens spaces, $\mathbb{R}^n\times cL,$5-bundles over $\mathbb{R}^n\times cL,$6, and $\mathbb{R}^n\times cL,$7 (Gómez-Larrañaga et al., 2017).

Proper group actions and proper Lie groupoids provide a third large source. Orbit type and Morita type decompositions give canonical Whitney $\mathbb{R}^n\times cL,$8 stratifications on both the object manifold and the orbit space. If $\mathbb{R}^n\times cL,$9 is a $cL$0-equivariant vector bundle for a proper action, the fiberwise invariant part

$cL$1

and its quotient $cL$2 become differentiable stratified vector bundles. Singular foliations admit analogous tangent constructions when the foliation is compatible with a stratification of the ambient manifold (Ross, 2023).

4. Resolution, corners, and product constructions

A recurrent theme is that stratified spaces are often easier to analyze after passage to a manifold with corners. In the conically smooth theory, the unzipping construction produces an $cL$3-manifold with corners $cL$4 and a natural map $cL$5 that resolves each conical singularity into a boundary face. The faces map to the strata as bundles of links, and this functorial resolution underlies the tubular neighborhood theorem and the open handlebody decomposition theorem (Ayala et al., 2014).

Manifolds with fibered corners refine this idea. A manifold with fibered corners carries, for each boundary hypersurface $cL$6, a fibration $cL$7 whose base models a stratum and whose fiber models the corresponding link. In this framework every smoothly stratified space can be resolved by iteratively desingularizing cone bundles to cylinders, and conversely an interior maximal manifold with fibered corners collapses back to a stratified space. The ordered product

$cL$8

is then defined by a controlled blow-up of $cL$9, and for resolutions of stratified spaces it resolves the product stratification on $L$0. The link of a product stratum is expressed by join constructions, and the ordered product is adapted to wedge and quasi-fibered boundary metrics, including QAC and QALE metrics (Kottke et al., 2022).

The recent theory of s-manifolds with corners internalizes corner structure directly at the stratified level. Instead of deriving corners from charts on $L$1, it specifies spaces $L$2 of $L$3-corners with proper maps $L$4, together with relation spaces $L$5 encoding how corners sit over the same base point. At each $L$6, the finite set $L$7 is required to form a finite lattice under the relation induced by $L$8, and simplicial corners are those locally isomorphic to the coordinatewise poset on $L$9. The induced depth pieces $C^0$0 recover the familiar filtration by interior, boundary, and higher corners (Joyce, 29 Jul 2025).

5. Homology, sheaves, and bordism theories

Stratified manifolds support a substantial homological formalism. For conically smooth stratified spaces, continuous space-valued sheaves are completely determined by their values on basics: $C^0$1 For a fixed stratified space $C^0$2, constructible sheaves are equivalently presheaves on the enter-path category $C^0$3, and $C^0$4 is weakly homotopy equivalent to $C^0$5. The same formalism characterizes $C^0$6-excisive invariants by tangential data on basics (Ayala et al., 2014).

Factorization homology extends this local-to-global principle from manifolds to stratified spaces with tangential structure. For a $C^0$7-algebra $C^0$8, the value on a $C^0$9-manifold is computed as a colimit over embedded basic pieces,

$S_i$00

and homology theories on $S_i$01-manifolds are characterized by Eilenberg–Steenrod-type axioms based on collar-gluings and sequential unions. The paper explicitly lists intersection homology, compactly supported stratified mapping spaces, and Hochschild homology with coefficients as examples of such theories, and proves a stratified form of nonabelian Poincaré duality (Ayala et al., 2014).

Bordism theory exhibits a different kind of invariance. For classes of PL stratified pseudomanifolds determined by local link properties, the stratified and unstratified bordism groups coincide, and the same holds for the associated bordism homology theories. This covers Witt spaces, IP spaces, locally orientable Witt spaces, $S_i$02-duality spaces, and locally square-free spaces. In that setting the stratification is indispensable for formulating the local conditions, but once the link data are fixed it does not create additional bordism invariants (Friedman, 2013).

A recent s-manifold framework is explicitly designed to recover fundamental classes despite codimension-$S_i$03 singularities: an oriented s-manifold $S_i$04 of dimension $S_i$05 has a fundamental class

$S_i$06

and transverse fibre products exist in the category of s-manifolds (Joyce, 29 Jul 2025).

6. Symplectic and differential-topological structures, applications, and terminology

Stratified manifolds also support genuinely geometric structures. A symplectic stratified space is an abstractly stratified space equipped with a smooth $S_i$07-form $S_i$08 whose restriction $S_i$09 to each stratum is symplectic. Because $S_i$10 need not be globally closed on the singular space, the relevant cohomology class is defined using the compression map $S_i$11: integrality means that $S_i$12 lies in the image of $S_i$13. Under this condition, a compact symplectic stratified space admits an embedding $S_i$14 such that $S_i$15 on each stratum, extending the Gromov–Tischler theorem from manifolds to stratified spaces. The same paper also relates this geometric notion to the Poisson-symplectic theory of Sjamaar–Lerman (Mj et al., 2022).

On the differential-topological side, the h-principle has been extended to stratified spaces in both the sheaf-theoretic and jet-theoretic senses. The extension requires stratified continuous sheaves, smooth stratified bundles, and the flexibility of homotopy fiber sheaves across strata. Within this framework one obtains a parametric h-principle, a stratified holonomic approximation theorem, and a stratified analog of the Smale–Hirsch immersion theorem (Mj et al., 2022).

These tools are being positioned for symplectic applications. One recent program states that s-manifolds are designed for applications in Symplectic Geometry, with the specific hope that, after suitable perturbations, the moduli spaces $S_i$16 of $S_i$17-holomorphic curves used for Gromov–Witten invariants, Lagrangian Floer cohomology, and Fukaya categories can be made into s-manifolds or s-manifolds with corners, so that their fundamental classes can be used directly (Joyce, 29 Jul 2025).

A persistent source of confusion is terminology. In other parts of differential geometry, “$S_i$18-manifold” and “indefinite S-manifold” refer to smooth manifolds with $S_i$19-structures, transversely Kähler foliations, or $S_i$20-Einstein conditions; those notions are explicitly unrelated to stratified spaces in the sense discussed above (Raźny, 2022, Brunetti, 2012). In the stratified literature, by contrast, s-manifolds are singular spaces with manifold strata, frontier relations, and controlled local models.