Stratified Manifolds with Corners

• Stratified manifolds with corners are geometric structures that blend smooth manifolds with explicit depth stratifications and local corner models.
• They employ local polyhedral, monoidal, and conical models to capture complex singularities and non-simplicial incidence combinatorics.
• Their resolution techniques use iterative blow-ups and boundary fibration structures to convert singular spaces into smooth manifolds with corners.

“Stratified manifolds with corners” names a family of geometric frameworks in which cornered smooth spaces and singular stratified spaces are related by explicit local models, depth decompositions, and resolution procedures. In the most elementary sense, a manifold with corners already carries a canonical decomposition by boundary codimension. In a stronger sense, a compact Thom–Mather stratified space can be resolved to a manifold with corners with an iterated fibration structure, and the original stratified space can be recovered by collapsing the fibres of those boundary fibrations (Albin, 2016). Later developments broaden ordinary corners to manifolds with generalized corners modeled on toric monoids (Joyce, 2015), to b-complex manifolds with generalized corners (Argüz et al., 24 Apr 2026), and to smooth atlas stratified spaces whose resolutions are manifolds with fibered corners and which coincide with Thom–Mather stratified spaces (Albin et al., 20 May 2025).

1. Corner stratifications and local models

A manifold with corners is locally modeled on

$[0,\infty)^k\times \mathbb{R}^{n-k},$

or equivalently on open subsets of convex polytopes in the locally polyhedral setting (Albin, 2016, Jakob, 12 Jun 2026). This local model immediately produces a codimension filtration: interior points have codimension $0$, boundary hypersurface points have codimension $1$, and higher corners occur where several boundary coordinates vanish simultaneously. In the locally polyhedral formulation, if $P(x)$ is the smallest face of a polytope containing $x$, then

$\operatorname{ind}_P(x)=n-\dim P(x),$

and for a compact locally polyhedral manifold $M$,

$\partial_iM=\{x\in M:\operatorname{ind}_M(x)=i\}$

is an $(n-i)$-dimensional submanifold (Jakob, 12 Jun 2026). In the generalized-corners setting, the analogous notation is

$S^k(X)=\{x\in X:\mathrm{depth}_X x=k\},$

and each $0$0 is a smooth manifold without boundary of dimension $0$1 (Argüz et al., 24 Apr 2026).

Three recurrent local formalisms organize the subject.

Framework	Local model	Stratification datum
Manifolds with corners	$0$2	codimension/depth of vanishing boundary coordinates
Manifolds with generalized corners	$0$3	faces of a weakly toric monoid $0$4
Smooth atlas stratified spaces	$0$5	poset-indexed conical strata

For smooth atlas stratified spaces, the local model is conical rather than orthant-like. A point in stratum $0$6 has a neighborhood modeled on

$0$7

where $0$8 is a compact lower-depth stratified space (Albin et al., 20 May 2025). This shifts the emphasis from “how many boundary coordinates vanish” to “what link sits transverse to the stratum.”

This suggests a useful distinction. Ordinary manifolds with corners are already stratified by depth, but only in the simplicial, orthant-type sense. More general theories replace orthants by cones on links or by monoidal models, thereby retaining a stratification while allowing non-simplicial corner combinatorics (Joyce, 2015, Albin et al., 20 May 2025).

2. Resolution of singular stratified spaces to manifolds with corners

A central paradigm is that singular stratified spaces can be replaced by smooth manifolds with corners that remember the singularities through boundary fibrations. For compact Thom–Mather stratified spaces, iterative radial blow-up of the deepest strata produces a manifold with corners $0$9 and a blow-down map

$1$0

such that $1$1 restricts to a diffeomorphism from the interior $1$2 onto the regular part $1$3, and each singular stratum $1$4 corresponds to a boundary hypersurface $1$5 (Albin, 2016). The boundary hypersurface $1$6 is itself the total space of a fibration

$1$7

where $1$8 is the resolution of the closure of the stratum and $1$9 is the resolution of the link.

This is formalized by the notion of a manifold with corners with an iterated fibration structure. Such a structure is a triple $P(x)$0, where $P(x)$1 is a manifold with corners, $P(x)$2 is a collection of collective boundary hypersurfaces, and $P(x)$3 assigns to each $P(x)$4 a fibration

$P(x)$5

subject to compatibility conditions over corners. If $P(x)$6 and $P(x)$7 intersect and $P(x)$8, then $P(x)$9 has a collective boundary hypersurface $x$0 fibering over $x$1, and the resulting square commutes (Albin, 2016). Corners therefore record incidence among strata: if $x$2, then the corresponding boundary hypersurfaces satisfy

$x$3

The structural statement is Theorem 6.3 of Albin’s survey: iteratively blowing up the deepest stratum of a compact Thom–Mather stratified space produces a manifold with corners with an iterated fibration structure, and conversely collapsing the fibres of the boundary fibrations in the correct order reconstructs a Thom–Mather stratified space (Albin, 2016). In this sense, a stratified space and its resolved manifold with corners encode the same geometry in singular and desingularized forms.

3. Generalized corners and monoidal stratification

Joyce’s category $x$4 enlarges ordinary manifolds with corners by replacing orthant charts with monoidal models

$x$5

where $x$6 is a weakly toric monoid (Joyce, 2015). Ordinary corners are recovered from

$x$7

The advantage is that the local corner combinatorics are no longer forced to be simplicial.

For $x$8, the support

$x$9

is a face of $\operatorname{ind}_P(x)=n-\dim P(x),$0, and $\operatorname{ind}_P(x)=n-\dim P(x),$1 decomposes as

$\operatorname{ind}_P(x)=n-\dim P(x),$2

Depth is given by

$\operatorname{ind}_P(x)=n-\dim P(x),$3

hence

$\operatorname{ind}_P(x)=n-\dim P(x),$4

Each stratum is a smooth manifold without boundary of the expected dimension (Joyce, 2015).

The model

$\operatorname{ind}_P(x)=n-\dim P(x),$5

illustrates what generalized corners add. It is $\operatorname{ind}_P(x)=n-\dim P(x),$6 for a toric monoid $\operatorname{ind}_P(x)=n-\dim P(x),$7, and at its vertex four 2-dimensional boundary faces and four 1-dimensional edges meet. That incidence pattern cannot occur in an ordinary 3-manifold with corners, where a vertex locally looks like $\operatorname{ind}_P(x)=n-\dim P(x),$8 (Joyce, 2015). This suggests that generalized corners capture non-simplicial corner stratifications that ordinary corner charts exclude.

The same paper shows that $\operatorname{ind}_P(x)=n-\dim P(x),$9-corners continue to exist, but for $M$0 they no longer admit the simplicial description familiar from ordinary corners. For ordinary corners, $M$1 is controlled by unordered $M$2-tuples of local boundary components; for generalized corners this fails in higher codimension (Joyce, 2015). The resulting geometry is still cornered and stratified, but its incidence combinatorics are toric rather than simplicial.

4. Smooth atlas stratified spaces and manifolds with fibered corners

Smooth atlas stratified spaces were introduced to provide a category of smoothly stratified spaces closed under cartesian products and therefore suitable for defining fiber bundles of stratified spaces (Albin et al., 20 May 2025). The local charts are conical,

$M$3

and smoothness of transition maps is defined by requiring a smooth stratified lift

$M$4

before collapsing the cone variable (Albin et al., 20 May 2025). The paper proves that this class coincides with Thom–Mather stratified spaces and is closed under products.

The resolution of a smooth atlas stratified space is a manifold with fibered corners. Here a boundary stratification is a poset stratification of a manifold with corners $M$5 such that each stratum $M$6 is either the interior or a collective boundary hypersurface, and each $M$7 carries a bundle

$M$8

compatible with the incidence of boundary hypersurfaces (Albin et al., 20 May 2025). The blow-up procedure is iterative: one replaces a minimal singular stratum by a new boundary hypersurface $M$9 equipped with a bundle over $\partial_iM=\{x\in M:\operatorname{ind}_M(x)=i\}$0, then repeats.

The full resolution theorem states that for every smooth atlas stratified space $\partial_iM=\{x\in M:\operatorname{ind}_M(x)=i\}$1, the resolution $\partial_iM=\{x\in M:\operatorname{ind}_M(x)=i\}$2 is a manifold with fibered corners, with blowdown

$\partial_iM=\{x\in M:\operatorname{ind}_M(x)=i\}$3

and

$\partial_iM=\{x\in M:\operatorname{ind}_M(x)=i\}$4

If $\partial_iM=\{x\in M:\operatorname{ind}_M(x)=i\}$5 is the link of $\partial_iM=\{x\in M:\operatorname{ind}_M(x)=i\}$6, then the fibre of the boundary fibration over $\partial_iM=\{x\in M:\operatorname{ind}_M(x)=i\}$7 is $\partial_iM=\{x\in M:\operatorname{ind}_M(x)=i\}$8 (Albin et al., 20 May 2025). In the opposite direction, every manifold with fibered corners arises as the resolution of some smooth atlas stratified space, and smooth stratified maps are exactly those maps that lift to smooth fibered-corners maps between the resolutions.

A distinctive feature of the smooth-atlas approach is product compatibility. If $\partial_iM=\{x\in M:\operatorname{ind}_M(x)=i\}$9 and $(n-i)$0 are smooth atlas stratified spaces, then $(n-i)$1 is again smoothly stratified, locally through

$(n-i)$2

with $(n-i)$3 the join (Albin et al., 20 May 2025). This gives a workable notion of fiber bundle of stratified spaces, something the paper presents as a substantial practical advantage.

5. Analytic, differential, and complex structures on resolved corners

Once a stratified space has been resolved to a manifold with corners carrying fibration data, one can formulate analysis in smooth terms. In Hodge theory on stratified spaces, the relevant cotangent object is the wedge cotangent bundle $(n-i)$4, locally spanned near a depth-one boundary fibration by

$(n-i)$5

and a rigid wedge metric has the form

$(n-i)$6

near the boundary (Albin, 2016). In higher depth this becomes a rigid iterated wedge metric, and the de Rham operator is analyzed on $(n-i)$7. Mezzoperversities then define ideal boundary conditions and yield Hodge and signature theories for Cheeger spaces (Albin, 2016).

For manifolds with fibred corners, the pseudodifferential side is governed by the groupoid of $(n-i)$8-calculus. The associated groupoid $(n-i)$9 is the holonomy groupoid of the singular foliation defined by vector fields tangent to the boundary fibrations, and the paper proves that the compactly supported pseudodifferential calculus on $S^k(X)=\{x\in X:\mathrm{depth}_X x=k\},$0 coincides with Mazzeo–Melrose’s small $S^k(X)=\{x\in X:\mathrm{depth}_X x=k\},$1-calculus (Guillaume, 2013). In the fibred-boundary case,

$S^k(X)=\{x\in X:\mathrm{depth}_X x=k\},$2

This places analysis on fibred-corner resolutions into the framework of singular foliations, Lie groupoids, and noncommutative geometry (Guillaume, 2013).

Other differential structures persist on manifolds with corners themselves. Moser’s theorem extends to compact manifolds with corners: if $S^k(X)=\{x\in X:\mathrm{depth}_X x=k\},$3 and $S^k(X)=\{x\in X:\mathrm{depth}_X x=k\},$4 are smooth positive densities with equal total volume, there exists a diffeomorphism $S^k(X)=\{x\in X:\mathrm{depth}_X x=k\},$5 such that

$S^k(X)=\{x\in X:\mathrm{depth}_X x=k\},$6

and $S^k(X)=\{x\in X:\mathrm{depth}_X x=k\},$7 can be chosen to be the identity on $S^k(X)=\{x\in X:\mathrm{depth}_X x=k\},$8 if and only if

$S^k(X)=\{x\in X:\mathrm{depth}_X x=k\},$9

(Bruveris et al., 2016). In a different direction, if $0$00 is a compact locally polyhedral smooth manifold, then $0$01 is a regular Lie group modeled on the stratified vector fields

$0$02

for all $0$03; manifolds with corners are a special case of this locally polyhedral setting (Jakob, 12 Jun 2026).

The complex-geometric variant replaces the ordinary tangent bundle by the b-tangent bundle. A b-complex structure is a complex structure

$0$04

satisfying a transversality condition along every corner stratum, and the paper proves a formal Newlander–Nirenberg theorem: near each depth-$0$05 stratum, $0$06 agrees to infinite order with a standard model on a manifold with generalized corners (Argüz et al., 24 Apr 2026). This suggests that corner strata can support logarithmic complex geometry compatible with the ambient stratification.

6. Applications, examples, and categorical variants

Several recent constructions treat the corner stratification as active geometric input rather than as a technical nuisance. In toric topology, a “nice manifold with corners” $0$07 is stratified by codimension, and the generalized moment-angle manifold $0$08 is built from the face data of $0$09. Its stable splitting is

$0$10

and the cohomology groups satisfy

$0$11

so the topology of the face stratification directly controls the invariants (Yu, 2020). In Morse theory, compactified trajectory spaces admit smooth manifold-with-corners structures whose $0$12-stratum is the space of $0$13-fold broken trajectories, with associative gluing maps built from the corner charts (Wehrheim, 2012).

Low-dimensional topology gives a different manifestation. “Cornered skein lasagna theory” extends skein lasagna theory to compact oriented smooth 4-manifolds with corners

$0$14

where the codimension-2 corner $0$15 carries marked-point data and gluing along a common codimension-1 face is expressed by tensor product of bimodules (Blackwell et al., 5 Dec 2025). Although that paper does not formulate a general theory of stratified spaces, it treats codimensions $0$16, $0$17, and $0$18 in a distinctly stratified manner.

Categorical and sheaf-theoretic variants show the same pattern. There is a fully faithful embedding of the category of manifolds with corners into the Cahiers topos that preserves open covers and transverse fibre products, and gluing manifolds with corners along a common face coincides with a pushout along an infinitesimally thickened face (Schlegel, 2015). In another direction, Poisson structures on manifolds with corners can be formulated sheaf-theoretically as morphisms

$0$19

satisfying Leibniz and Jacobi identities, where $0$20 is the sheaf of admissible smooth functions (Antonio-Vásquez, 2016).

Taken together, these constructions support a broad but coherent picture. The phrase “stratified manifolds with corners” may refer to manifolds with corners equipped with their canonical depth stratification, to desingularized models of Thom–Mather spaces via iterated or fibered corners, or to generalized-corner and b-geometric extensions. Across these variants, the recurring principle is stable: strata are encoded by corner faces, links appear as fibers of boundary fibrations, and singular geometry is converted into smooth corner geometry without discarding the incidence structure that made the space stratified in the first place.