Stable Nested Cobordism Groups

• Stable nested cobordism groups are a generalization of classical cobordism, capturing layered submanifolds with prescribed normal data.
• They employ nested Pontryagin–Thom constructions and filtration techniques to link stratified manifolds with stable homotopy invariants.
• Wall's splitting theorem and module structures reveal deep algebraic properties and pave the way for tackling open questions in manifold theory.

Stable nested cobordism groups constitute a central object in the study of manifolds equipped with a hierarchy of embedded submanifolds, generalizing classical cobordism theory via both topological and singularity-theoretic perspectives. Their algebraic structure encodes interactions between embedded strata with specified normal data, and, in the context of singular maps, naturally produces filtrations whose successive quotients connect with stable homotopy theory. This article presents a comprehensive account of stable nested cobordism groups, their classification, foundational constructions, explicit computations, and structural decompositions, synthesizing the framework for both nested manifolds and cooriented codimension-one Morin maps (Szűcs, 2011, Blanco, 20 Dec 2025).

1. Definitions: Nested Manifolds and Cobordism Classes

A once-nested $(\theta',\theta)$–submanifold of a closed $m$–manifold $M$ comprises a pair

$K' \subseteq K \subseteq M$

where $K$ is a closed $k_1$-dimensional submanifold equipped with a stable normal bundle classified via a fibration $\theta\colon B\to BO(m-k_1)$, and $K'\subset K$ is a closed $k_2$-submanifold with normal data encoded by $\theta'\colon B'\to BO(k_1-k_2)$. A nested cobordism between such pairs consists of

$W' \subseteq W \subseteq M\times[0,1]$

with $\partial W' = K'\sqcup \widetilde K'$, $\partial W = K\sqcup \widetilde K$, and compatible lifts of normal structures.

Let $\NCob^{(\theta',\theta)}(M)$ denote the set of cobordism classes of such pairs in $M$. For suitable dimension and codimension hypotheses, this set acquires a natural abelian group structure under disjoint union (Blanco, 20 Dec 2025).

In the context of singular maps, a cooriented codimension-one Morin map $f\colon M^n\to\mathbb{R}^{n+1}$ is characterized by only $A_k$-type singularities ($\Sigma^{1_k}$, $k\ge 0$), with $\tau\subset\{A_0,A_1,\dots,A_r\}$ denoting the allowed singularities. The cobordism groups $\text{Cob}_\tau(n)$ of $\tau$-maps form abelian groups and allow a filtration indexed by singularity type (Szűcs, 2011).

2. Pontryagin–Thom Construction and Stable Realizations

There exists a nested Pontryagin–Thom correspondence generalizing the classical construction: for suitable normal data and codimension bounds,

$\NCob^{(\theta',\theta)}(M) \cong [M, Th(\theta'^{*}\gamma_{k_1-k_2})_+ \wedge Th(\theta^*\gamma_{m-k_1})]$

where $Th(\theta^\ast\gamma_{d})$ denotes the Thom space for the universal bundle of rank $d$ induced by $\theta$ (or $\theta'$), and $(-)_+$ means adjoining a basepoint (Blanco, 20 Dec 2025). For stable normal structures $\Theta = \{B(n)\to BO(n)\}_{n\ge0}$, the stable nested cobordism group is given by

$$\Omega_{k_1}^{(\theta',\Theta)} \cong \pi_{k_1}\left(Th(\theta'^*\gamma_{k_1-k_2})_+\wedge Th\Theta\right)$$

where $Th\Theta$ is the Thom spectrum built from $\Theta$.

In the singularity context, the classifying spaces for cobordisms of $\tau$-maps fit into a “key-bundle” fibration, with a long exact sequence in homotopy: $$\dots \to \pi^{n+1}(X_{\tau'}) \to \pi^{n+1}(X_\tau) \to \pi^{n+1}(B_r) \xrightarrow{\partial} \pi^n(X_{\tau'}) \to \dots$$ where $X_\tau$ is the classifying spectrum of $\tau$-maps and $B_r=Th(E_r)$ is the Thom space of the universal normal bundle for the top singular stratum (Szűcs, 2011).

3. Explicit Computation and Filtration for Codimension-One Morin Maps

For $\tau$ corresponding to cooriented codimension-one Morin maps, $\text{Cob}_{1,2r+1}(n)$ admits a filtration and an essentially graded decomposition. Denote $r$ the highest Morin singularity allowed:

• Fold maps ($\tau = \{A_0,A_1\}$): The cobordism group splits as

$$\text{Cob}_{1,0}(n) \cong [\pi^s_n]_{(odd)} \oplus \ker\left( \kappa : \pi^s_{n-1}(\mathbb{R}P^\infty) \to \pi^s_{n-1} \right)$$

where $\kappa$ is the Kahn–Priddy homomorphism, and $[\pi^s_n]_{(odd)}$ denotes the odd-torsion part of the stable homotopy group of spheres (Szűcs, 2011).

• Cusp maps ($\tau = \{A_0,A_1,A_2\}$): Modulo the $2$- and $3$-primary subgroups,

$$\text{Cob}_{1,1}(n) \cong \pi^s_n \oplus \pi^s_{n-4} \mod C_{\{2,3\}}$$

where $C_{\{2,3\}}$ is the class of abelian groups whose only prime divisors are $2$ or $3$.

• Higher Morin maps ($\tau_{r}=\{A_0,\dots,A_{2r+1}\}$): Modulo torsion at primes $\le2r+1$,

$$\text{Cob}_{1,2r+1}(n) \cong \bigoplus_{i=0}^{r} \pi^s_{n-4i} \mod C_{p\le 2r+1}$$

with $C_{p\le 2r+1}$ denoting abelian groups whose only prime divisors are at most $2r+1$ (Szűcs, 2011).

The associated filtration of classifying spaces,

$$X_{\{A_0\}} \leftarrow X_{\{A_0,A_1\}} \leftarrow X_{\{A_0,A_1,A_2\}} \leftarrow \dots$$

gives, in each degree $n$, a filtration of cobordism groups whose graded pieces are

$F_i/F_{i+1} \cong \pi^s_{n-4i}$

up to orders divisible by prescribed primes (Szűcs, 2011).

4. Stable Splitting and Wall’s Theorem for Nested Cobordism

Wall’s splitting theorem provides a direct sum decomposition for stable nested cobordism groups: $$\Omega_{k_1}^{(\theta',\Theta)} \cong \Omega_{k_2}^{\theta'\times\Theta} \oplus \Omega_{k_1}^\Theta$$ where $\theta'\times\Theta$ is the product stable structure on $B'\times B(n)\to BO(d'+n)$, $d' = k_1-k_2$ (Blanco, 20 Dec 2025). This result is grounded in a split cofibre sequence of spectra: $$Th\Theta \to Th(\theta'^*\gamma_{d'})_+\wedge Th\Theta \to \Sigma Th\Theta$$ with the splitting arising from a homotopy-theoretic retraction.

In practical terms, the stable nested cobordism group for a once-nested manifold reduces to the direct sum of stable cobordism groups for the smaller (nested) and larger strata, reflecting the independence of the cobordism types in the stable regime.

5. Low-Dimensional Cases and Link-Cobordism Invariants

In the classical framed case $(\theta',\theta)\simeq(*,*)$, the unstable cobordism set

$\NCob^{(*,*)}(S^m) \cong [S^m, S^{m-k_1}\vee S^{m-k_2}]$

splits into summands linked to Whitehead products, describing higher interactions between the strata. Wang’s invariants $\Delta_\lambda$ provide quantitative measures for these components (Blanco, 20 Dec 2025). Specifically, there exist examples in dimension $m=2$, $k_1=1$, $k_2=0$ where both $S^1$ and $S^0$ are nullbordant, but the nesting is nontrivial due to a nonzero $[\iota,\iota']$-invariant.

Moreover, when the normal bundle of the ambient manifold admits one framed direction (i.e., $\theta$ factors through $BO(d-1)\to BO(d)$), the nested cobordism group is related to the cobordism of links formed by pushing off the smaller stratum along the framing, with identification via link cobordism groups and associated Thom spaces (Blanco, 20 Dec 2025).

6. Multiplicative Structures and Open Questions

Stable nested cobordism groups $\Omega_{*}^{(\theta',\Theta)}$ possess a module structure over the stable cobordism ring $\Omega_{*}^{\Theta}$, derived from the Cartesian product of nested manifolds with ordinary $\Theta$-manifolds (Blanco, 20 Dec 2025). However, the explicit computation of structure constants remains undeveloped. Open directions include:

• Explicit product formulas $$\Omega_{i}^{(\theta',\Theta)}\otimes \Omega_{j}^{(\theta',\Theta)}\to\Omega_{i+j}^{(\theta',\Theta)}$$ in light of Wall’s splitting.
• The behavior of Whitehead product summands under module actions.
• Extension of the stable homology theory of nested-cobordism categories, generalizing cobordism spectra for “flagged” or multistratified manifolds.

A plausible implication is that understanding these structures offers a path to a more general classification of multi-filtration phenomena in manifold and singularity theory, paralleling the role of stable homotopy in the study of smooth maps with prescribed singularities (Blanco, 20 Dec 2025, Szűcs, 2011).