Non-Splitting Lemma

• Non-Splitting Lemma is a collection of results stating that certain mathematical objects, from dynamical systems to set theory, cannot be decomposed into simpler parts, highlighting intrinsic complexity.
• In dynamical systems and algebra, the lemma uses techniques like dominated splitting and functorial obstructions to prevent naive decompositions, ensuring that pathological or trivial structures are avoided.
• Its applications extend to topology, forcing in set theory, and metric geometry, where the absence of splitting sequences or extensions confirms rigidity and underpins structural invariants.

The Non-Splitting Lemma encompasses a set of results across dynamical systems, topology, homological algebra, continuum theory, and logic which assert, under precise technical hypotheses, obstructions to decomposing objects—be they dynamical, topological, algebraic, or set-theoretic—into simpler "split" pieces. These results often pinpoint structural, spectral, or algebraic features that preclude a naive splitting and thus reveal fundamental rigidity or complexity in the objects of paper.

1. Non-Splitting Lemma in Dynamical Systems

Within the theory of differentiable dynamical systems, the Non-Splitting Lemma typically declares that certain dynamically robust sets must admit dominated splittings, or else pathological behavior (such as the emergence of sinks or sources or degenerate eigenvalue configurations) is inevitable.

• The main theorem in (Yang, 2010) establishes: any $C^1$-stably weakly shadowing transitive set $\Lambda$ is either a sink, a source, or admits a dominated splitting. The dichotomy is as follows:
  • Dominated splitting exists: The tangent bundle over $\Lambda$ splits into invariant subbundles $E \oplus F$ satisfying domination

  $$\|Df^n|_{E(x)}\| \cdot \|Df^{-n}|_{F(f^n(x))}\| \leq C\lambda^n, \quad \forall x \in \Lambda, ~ n \in \mathbb{N}$$

  which implies a weak hyperbolic structure. - Non-splitting ("no dominated splitting") alternative: There would exist periodic orbits with eigenvalues of equal modulus, leading to almost sinks or sources—a scenario ruled out by the stably weakly shadowing condition. - The proof synthesizes Pugh’s Closing Lemma, approximation by periodic orbits, and perturbative techniques such as Franks’ Lemma to demonstrate that robust shadowing is incompatible with the absence of a dominated splitting unless the dynamics are trivial.

• Relatedly, (Zhang, 2010) shows that a global dominated splitting precludes minimality of a diffeomorphism—meaning nontrivial recurrence is obstructed by the existence of the splitting. Techniques based on Mañé's argument and Liao's Selecting Lemma construct nonrecurrent or periodic points, demonstrating that the system cannot spread every orbit densely.

2. Non-Splitting Phenomena in Topology and Homological Algebra

"Non-splitting" in topology often refers to the impossibility of dividing a complex space into two or more simpler subspaces with restrictive properties (e.g., vanishing or finite homology). In algebra, it signals that certain short exact sequences do not split functorially, reflecting deep algebraic obstructions.

• (Ancel et al., 2018) proves that a generalized dunce hat cannot be decomposed into two proper subpolyhedra with finite first homology groups; any such decomposition must involve at least one piece of infinite complexity. The proof invokes Mayer–Vietoris arguments and handles the special singularity structure of the attaching map.

  • This result undermines classical approaches to proving splittability for contractible 4-manifolds such as the Mazur manifold, indicating that certain 'wild' spines cannot be split into collapsible subcomplexes.
• In homological algebra, (Mikhailov, 2012) analyzes extensions of polynomial functors using cross-effects and polynomial $\mathbb{Z}$-modules to show that sequences like

$$0 \to A_3(A) \to H_3(A) \to \Lambda^2(A) \otimes \mathbb{Z}/2 \to 0$$

do not split functorially, since there is no nontrivial natural transformation between functors of different degrees. This obstruction arises from the vanishing of Hom-groups between certain functor categories.

3. Non-Splitting in Topological Continuum Theory

In continuum theory and dynamics, the Non-Splitting Lemma can be framed as the non-existence of sequences or decompositions that force a space to be non-arc-like.

• (Greenwood et al., 2020) gives a precise characterization for continuous surjections $f: [0,1] \to [0,1]$:
  • The inverse limit $\varprojlim f$ is an arc if and only if no splitting sequence exists—i.e., there does not exist an infinite collection of proper subintervals $S_n \subset T_n$ with $f(S_n)=f(T_n)$, so that $f$ does not "fold" intervals together persistently.
  • The presence of splitting sequences signals a non-arc continuum (possibly pseudo-arcs or more complicated spaces), while their absence implies topological simplicity.

4. Non-Splitting and Splitting in Set Theory and Logic

In logic and set theory, non-splitting refers to properties of families (e.g., splitting families) preserved under iterations of forcing, and to the inability to further "split" cardinal invariants.

• (Goldstern et al., 2020) demonstrates that splitting families added by a constructed forcing are preserved under finite-support symmetric iterations, leading to a model in which 15 cardinal characteristics of the continuum are pairwise different. The Non-Splitting Lemma ensures that the splitting family continues to split every infinite set in the extension, and that no further splitting occurs—guaranteeing control over the splitting number $\mathfrak{s}$ within the model.
  • The preservation involves posets defined via finite functions and the careful handling of automorphisms and history in the iteration.

5. Non-Splitting and Splitting Principles in Derived Categories and Singularity Theory

The "splitting principle" in derived categories—a duality to non-splitting lemmas—states that morphisms with left inverses in derived contexts are quasi-isomorphisms, i.e., splits are genuine and not accidental (Kovács, 2011). The various technical incarnations imply that the existence of a split forces good singularity properties (e.g., rational, Du Bois).

In singularity theory (Greuel et al., 22 Jul 2025), the splitting lemma guarantees that after coordinate changes, formal or analytic power series split into a nondegenerate quadratic form plus a residual, and obstructions to splitting (non-splitting) are tied to moduli or lack of finite determinacy.

6. Non-Splitting Lemma in Nonsmooth Morse Theory

In infinite-dimensional Morse theory and variational analysis (Lu, 2012), the Non-Splitting Lemma signals the limits of decomposing functionals near degenerate critical points when smoothness falls below $C^2$—functionals that are merely directionally differentiable admit a splitting into quadratic and reduced parts after suitable coordinate changes, yet non-splitting phenomena arise when further regularity fails.

7. Non-Splitting in Metric Geometry

In the theory of metric measure spaces (Gigli, 2013), the "non-splitting lemma" is manifest in the proof that infinitesimally Hilbertian $CD(0,N)$ spaces containing a line necessarily split off an $\mathbb{R}$ factor. Obstructions to splitting are absent due to symmetry properties of the Sobolev space and the gradient flow of the Busemann function, with the splitting encoded in the metric via

$$d_{X' \times \mathbb{R}}((x', t), (y', s)) = \sqrt{d'(x', y')^2 + |t-s|^2}$$

and the structure of the quotient space $X'$.

References and Technical Summary Table

Context	Non-Splitting Content	Key Formula / Criterion
Dynamical Systems	No dominated splitting $\implies$ pathological orbits excluded by robust properties	$$\|Df^n|_{E(x)}\| \cdot \|Df^{-n}|_{F(f^n(x))}\| \leq C\lambda^n$$
Algebraic Topology	No splitting of homological extension functors	$$0 \to A_3(A) \to H_3(A) \to \Lambda^2(A) \otimes \mathbb{Z}/2 \to 0$$
Topology (Dunce Hat)	Cannot split generalized dunce hat into finite $H_1$ pieces	$\nexists~A,B \subset D$ both proper, $H_1(A), H_1(B)$ finite
Set Theory / Forcing	Splitting families persist, no new infinite set unsplit	$$\mathfrak{s} = \min \{ |F|: F~\text{splitting family}\}$$
Continuum Theory	No splitting sequence $\implies$ inverse limit is an arc	$f(S_n) = f(T_n)$ forbidden on infinitely many $n$
Metric Geometry	Line $\implies$ space splits, non-splitting absent	$$X \cong X' \times \mathbb{R},~d((x', t),(y',s)) = \sqrt{d'(x',y')^2+|t-s|^2}$$

Conclusion

The Non-Splitting Lemma, in its many contexts, formalizes the rigidity encountered when object decomposition is precluded by geometric, algebraic, or combinatorial constraints. It asserts that obstructions—be they in dynamical recurrences, functorial extensions, topological complexity, or set-theoretic splitting—are intrinsic and immune to naive partitioning or splitting strategies. When splittings do exist, structural domination properties, normal forms, or isometric product structures often result, whereas persistent non-splitting reveals inherent complexity and often guides the identification of key invariants or phenomena in the underlying mathematical theory.