Persistent Manifold Theory

• Persistent Manifold Theory is a framework that defines how topological invariants and geometric structures persist under perturbation and filtration.
• It unifies classical dynamical systems, such as normally hyperbolic invariant manifolds, with modern sheaf-theoretic methods and topological data analysis.
• The approach enables efficient computation of stable invariants like barcodes and characteristic classes, with applications in geometry, machine learning, and imaging.

Persistent manifold theory is the body of results, constructions, and algorithms that describe how manifold- and fiber-bundle-valued topological invariants, geometric structures, and invariant submanifolds persist under perturbation and filtration. The subject integrates classical dynamical systems—specifically, normally hyperbolic invariant manifolds under perturbation—with modern topological data analysis (TDA), sheaf-theoretic and spectral constructions over manifolds, and the persistent extension of characteristic classes. Recent advances subsume classical persistence (sublevel/barcode invariants), extend the setting to maps or filtrations over higher-dimensional, possibly noncompact manifolds, and yield robust, computable, and stable invariants for use in geometry, analysis, and machine learning.

1. Persistence of Invariant Manifolds in Dynamical Systems

The classical context for persistence of manifolds arises in geometric dynamical systems. For a $C^1$ diffeomorphism $f: M \to M$ with a compact, $C^1$ invariant submanifold $N \subset M$, normal hyperbolicity is defined via a continuous, $Df$-invariant splitting of the tangent bundle: $TM|_N = E^s \oplus TN \oplus E^u$ with contraction/expansion rates in $E^s$ and $E^u$ dominating tangent dynamics in $TN$. The persistence theorem (Hirsch–Pugh–Shub, Fenichel, Berger–Bounemoura) asserts that if $f$ is normally hyperbolic at $N$, then for all $C^1$-small perturbations $g$ there exists a unique $C^1$ invariant submanifold $N_g$ diffeomorphic and close to $N$ which persists as a normally hyperbolic manifold for $g$ (Berger et al., 2011).

In a noncompact setting, normal hyperbolicity and its persistence remain valid if the ambient manifold has $k$-th order bounded geometry: uniform injectivity radius, uniform bounds on curvature and its derivatives. Under these conditions, uniform tubular neighborhood and smooth approximation theorems permit global control and extend persistence results to noncompact NHIMs with explicit $C^{k, \alpha}$ regularity determined by the spectral gap (Eldering, 2012, Eldering, 2012).

Methods include graph transform (invariant graph fixed point construction in Lipschitz or $C^{k, \alpha}$ Banach spaces), cone-field geometries, and uniform estimates on derivatives and curvature, all of which generalize to Banach bundles and parameterized settings.

2. Topological Persistence via Sheaf and Local System Frameworks

Modern persistent manifold theory generalizes persistent homology to mappings and filtrations over arbitrary base manifolds. Given a continuous map $f: X \to M$ where $M$ is an oriented $n$-manifold and $X$ a fibered space, one constructs two key objects:

• The Leray homology cosheaf $\mathcal{F}_k(U) = H_k(f^{-1}(U); \Bbbk)$ for open $U \subset M$.
• The Leray relative homology sheaf $\mathcal{G}_k(U) = H_{k+n}(X, X-f^{-1}(U); \Bbbk)$.

These are functorial on the poset of open sets and are constructible with respect to any suitable stratification. The cap product with the pullback of the orientation class $[M]$ yields a natural bisheaf map $\Phi_k: \mathcal{G}_k \to \mathcal{F}_k$. Processing through canonically defined "epification" and "monofication" truncations produces an isobisheaf, whose image defines a persistent local system $L^k$—a local system on each path-connected open $U \subset M$.

The main invariants are $\rho_k(f, U) = \mathrm{rank}(L^k(U))$, which generalize the rank invariant ("barcode") of classical persistence to arbitrary base manifolds, all homological degrees, and with local system structure rather than mere vector-space stalks. Two fundamental theorems govern this structure:

• $\mathrm{rank}\;H_k(f^{-1}(U)) \ge \rho_k(f, U)$ (Lower bound),
• $$|\rho_k(f, U) - \rho_k(f', U)| \le \sup_{x \in X} d_M(f(x), f'(x))$$ (Stability under perturbation) (MacPherson et al., 2018).

This framework recovers ordinary persistence for $M = \R$, circle-valued persistence for $M = S^1$, and general multiparameter persistence for $M = \R^n$. The persistent local system acts as a robust, locally constant, and computable invariant of maps into manifolds.

3. Categorical and Characteristic-Class Extensions

Persistence has been extended beyond homology to the category of vector bundles, classifying maps, and characteristic classes, notably in the computation of persistent Stiefel–Whitney classes. For a point-cloud approximation $P \subset \R^D$ of an $n$-manifold $M$ and an associated Čech or alpha filtration $\{X^r\}$, one constructs a persistent vector bundle whose fibers are homotopically equivalent to the tangent bundle of $M$; the sequence of classifying maps $X^r \to \mathrm{Gr}_n(\R^\infty)$ induces a coherent filtration of characteristic classes.

Via the persistent Wu formula and persistent cohomology operations (cup product, Steenrod squares), one computes at each filtration step the Wu classes $v_{(k,n)}$ as persistent cohomology classes solved by $v_{(k,n)} \smile x = \mathrm{Sq}^k(x)$ (for all $x \in H^{n-k}(X^r)$), and assembles the persistent Stiefel–Whitney class $w_{(k,n)}$ by $w_{(k,n)} = \sum_{i+j=k} \mathrm{Sq}^i(v_{(j,n)})$.

If the filtration $X^r$ deformation retracts to $M$ (guaranteed by sufficient sampling), these persistent classes coincide with the classical topological invariants of the manifold. This approach extends characteristic-class invariants to point-cloud filtrations and demonstrates discriminatory power beyond Betti numbers, for both synthetic complexes (e.g., exotic 4-manifolds) and real data (e.g., image patches, molecular conformation spaces) (Gang, 20 Mar 2025).

4. Morse Theory, Signed Distance, and Persistent Homology of Shapes

Persistent manifold theory subsumes Morse-theoretic approaches to the persistent homology of manifold-embedded functions, particularly those arising from signed distance fields to smooth surfaces. For a smooth compact surface $S \subset \R^3$, the signed distance function $d$ to $S$ is not generally $C^2$, but can be analyzed as a "Min-type" function comprising minima over smooth branches, allowing the application of topological Morse theory and transversality arguments.

On generic surfaces, only finitely many non-degenerate critical points of $d$ occur (proved via a transversality argument on embeddings), and each produces a handle attachment in a finite filtration by sublevel sets $V^a = \{x: d(x) \le a\}$. The persistent homology modules thus decompose into a finite sum of interval modules, yielding barcodes that quantify multiscale geometric and topological features (Song et al., 2023). This pipeline is robust under perturbations and is applicable to real data for quantifying vascular, porous, and phase-field structure via shape barcodes.

5. Persistent Homology in Floer-Novikov and Multiparameter Settings

Persistent manifold theory further encompasses generalized chain-level and multiparameter frameworks. In the Floer-Novikov context, chain complexes parametrized by Novikov fields (with valuations tracking action or energy) admit filtered barcodes via non-Archimedean singular value decomposition. These barcodes completely determine the filtered chain homotopy type, encode invariants such as boundary depth and torsion exponents, and are stable under $C^0$ perturbations, yielding stability theorems analogous to classical bottleneck stability (Usher et al., 2015).

Multiparameter persistence modules arise from one-parameter families of functions $\tilde f: I \times X \to \R$ on a compact manifold $X$. The three-parameter module $$H_j F(a,b,c) = H_j(f^{-1}([a,b] \times (-\infty, c]); k)$$ captures the evolution of homology under time and value, is stable with respect to supremum perturbations, and may decompose into indecomposable modules of arbitrarily large rank. Cobordism and Cerf theory describe the handle attachments and critical transitions, providing a geometric underpinning for the module's structure (Bubenik et al., 2021).

6. Computational and Algorithmic Aspects

Key computational frameworks include:

• Graph and cone-field fixed point methods for invariant manifold computation under perturbation (Berger et al., 2011, Eldering, 2012).
• Sheaf-theoretic methods for explicit calculation of persistent local systems on stratified or triangulated manifolds (MacPherson et al., 2018).
• Polynomial-time algorithms for computing optimal persistent cycles in levelset-zigzag persistence, especially in (weak) pseudomanifold complexes; dual-graph min-cut formulations yield globally optimal cycle sequences for all four interval types arising in PL levelset barcodes (Dey et al., 2021).
• Discretized de Rham–Hodge Laplacian approaches: persistent Hodge Laplacians computed on Eulerian grids provide joint topological-spectral signatures (track both Betti numbers and geometric Laplace eigenvalues), are numerically stable under grid refinement and Morse function perturbation, and support supervised/unsupervised learning from manifold data (Su et al., 1 Aug 2024).
• Intrinsic persistent homology via density-based Fermat distance, providing robustness to outliers and independence from ambient embeddings. The sample Fermat distance converges (in Gromov–Hausdorff sense) to the intrinsic manifold metric, with the persistence diagram converging exponentially in probability to that of the manifold. This approach is computationally tractable, with provable convergence guarantees, and underpins robust manifold learning from data (Fernández et al., 2020).

7. Broader Significance and Applications

Persistent manifold theory unifies geometric, topological, analytic, and computational techniques for quantifying invariant structure and descriptors on manifolds under perturbation or filtration. Applications are diverse:

• Dynamical systems: persistence of stable/unstable invariant manifolds and their robustness to perturbation, even under loss of rate conditions or non-invertible maps (Capinski et al., 2018).
• Topological data analysis: extraction of multiscale, robust, and discriminative invariants from point cloud or volumetric data, including characteristic classes and Laplacian spectra.
• Learning and inference on manifolds: representation learning, classification, and regression tasks utilizing persistent topological and geometric features, with theoretical guarantees of invariance and stability (Su et al., 1 Aug 2024).
• Symplectic and Morse–Floer theory: stability and computation of analytical invariants (spectral invariants, boundary depth), with barcodes fully capturing the filtered chain types (Usher et al., 2015).
• Computational geometry and imaging: geometric shape quantification, texture and morphometry of biological or synthetic surfaces, with rigorous barcode-based quantifiers (Song et al., 2023).

In all contexts, theory emphasizes the interplay between topological invariance, analytic stability, computational feasibility, and geometric specificity, underpinning both fundamental mathematics and manifold-structured data science.