Transfinitely iterated wild sets

Abstract: In this paper, we study homotopical analogues of the Cantor-Bendixson derivative. For each $n\geq 0$, the "$πn$-wild set" $\mathbf{w}_n(X)$ of a topological space $X$ is the subspace of $X$ consisting of the points at which there exists a shrinking sequence of essential based maps $S^n\to X$. Since the operator $\mathbf{w}_n$ permits iteration, every given space $X$ yields a descending transfinite sequence of nested subspaces ${\mathbf{w}_n^κ(X)}κ$ that stabilizes at some smallest ordinal $\mathbf{wrk}n(X)$ called the "$π_n$-wild rank" of $X$. We show that the entire transfinite sequence ${ho(\mathbf{w}_n^κ(X))}κ$ of homotopy types is a homotopy invariant of $X$ and that $\mathbf{wrk}_n(X)$ can be an arbitrary countable ordinal when $X$ is an $n$-dimensional Peano continuum. It remains open if there exists a continuum $X$ with uncountable $π_n$-wild rank. This difficulty motivates the parallel study a basepoint-free version $\mathbf{fwrk}_n(X)$, called the "free $π_n$-wild rank" of $X$. We show that for every continuum $X$, $\mathbf{fwrk}_n(X)$ is always countable and can be any countable ordinal.

• The paper presents a homotopical analogue of the Cantor-Bendixson derivative for Peano continua by defining the πₙ-wild set operator.
• It demonstrates that for every countable ordinal, there exists a continuum with the corresponding πₙ-wild rank achieved via controlled topological attachments and transfinitely iterated constructions.
• The methodology rigorously distinguishes homotopy invariants from classical invariants and opens avenues for applications in computational topology and motion planning.

Transfinitely Iterated Wild Sets: Homotopical Cantor-Bendixson Analogues

Introduction

The paper "Transfinitely iterated wild sets" (2604.14929) introduces and systematically develops the homotopical notion of "wild sets" in the context of topological spaces—particularly, Peano continua—by analogy to the classical Cantor-Bendixson derivative. For each integer $n \geq 0$, the $\pi_n$-wild set $w(X)$ is the subset of $X$ comprised of points with shrinking sequences of essential based maps $S^n \to X$, encapsulating local homotopy-theoretic complexity. The wild set operator $w$ admits transfinite iteration, yielding a descending sequence of subspaces $w^\kappa(X)$ indexed by ordinals and stabilizing at a minimal rank $wrk(X)$, called the $\pi_n$-wild rank. The paper rigorously proves that for Peano continua, wild ranks can achieve arbitrarily large countable ordinals, and explores related basepoint-free versions (free wild rank $fwrk(X)$), establishing countable bounds and providing precise constructions.

Definitions and Framework

The investigation is grounded in homotopy theory and set-theoretic topology, extending the concept of wild points where classical local connectivity fails (e.g., lack of semilocal simple connectivity at $\pi_n$0), but generalizing for all $\pi_n$1. For a space $\pi_n$2 and $\pi_n$3:

• A $\pi_n$4-wild point is one admitting a sequence of essential based maps $\pi_n$5 converging (compact-open topology) to the constant map at that point.
• The $\pi_n$6-wild set $\pi_n$7 consists of all such points.
• $\pi_n$8 may be iterated: $\pi_n$9, $w(X)$0, and so on, extended transfinitely by $w(X)$1 for limit ordinals.

The stabilization ordinal $w(X)$2 defines the $w(X)$3-wild rank, capturing the "depth" of homotopical wildness.

A parallel, basepoint-free version $w(X)$4 defines free wild points via sequences of essential unbased maps, with corresponding free wild rank $w(X)$5.

Main Results and Constructions

Homotopy Invariance and Rank Realizability

The paper establishes strong structural results:

• For each $w(X)$6, the sequence of homotopy types $w(X)$7 is a homotopy invariant (Thm. \ref{homotopyinvariance2}).
• For indeed any countable ordinal $w(X)$8, there exists a Peano continuum $w(X)$9 with $X$0-wild rank $X$1 and terminal wild set $X$2 (Thm. \ref{thm1}).
• The analogous result for free wild rank: For each countable ordinal $X$3, a continuum $X$4 with $X$5 exists (Thm. \ref{thm2}).

The constructions proceed via controlled topological attachings—shrinking wedge and point-attachment spaces—leveraging local contractibility and first-countability. Iterative, transfinite induction schemes using these spaces yield explicit Peano continua with prescribed wild ranks.

Technical Analysis of Iteration and Stabilization

Key results include:

• The wild set operator $X$6 is not necessarily closed, and rank stabilization intricately depends on basepointing and space topology.
• The countable nature of free wild rank for second countable spaces is proved by analysis of descending sequences of closed subspaces.
• Neither $X$7 nor $X$8 are homotopy invariants, but the sequence $X$9 is invariant under homotopy equivalences.
• In dimension $S^n \to X$0, $S^n \to X$1, aligning wildness with classical isolated/non-isolated point behavior.

Comparative and Contradictory Claims

The paper makes several strong technical assertions:

• Every countable ordinal is realized as both $S^n \to X$2-wild rank and free wild rank within compact, metrizable spaces of dimension $S^n \to X$3.
• The existence of uncountable wild rank (for any $S^n \to X$4) remains open and is suspected to be independent of $S^n \to X$5; the authors speculate its independence from ZFC.
• The difference $S^n \to X$6 can be arbitrarily large countable ordinals, demonstrated explicitly.
• Wild ranks do not align with Cantor-Bendixson rank, but share analogous recursive structure and limit behavior.

Implications and Directions

Practical Impact

These results elucidate the fine structure of homotopical complexity in fractal-like and locally wild spaces, providing a hierarchy of invariants beyond classical shape theory and homotopy groups. The techniques offer precise tools for motion planning (as shown in (Brazas et al., 27 Oct 2025)) and distinguishing spaces where traditional invariance (simple connectivity, local contractibility) fails.

Theoretical Perspectives

From a theoretical standpoint, transfinite iteration of wild set operators adds a layer to the already rich interplay of set-theoretic and algebraic topology. These wild ranks may reflect the depth of nontrivial topological and homotopy-theoretic phenomena, especially in infinite-dimensional, fractal, or non-locally contractible spaces. The homotopy-invariant sequence $S^n \to X$7 suggests new avenues for classification and possibly connections to descriptive set theory and large-cardinal behavior.

Future Developments

Potential directions include:

• Construction or nonexistence proofs for spaces with uncountable wild rank, possibly depending on set-theoretic axioms.
• Investigation of wild ranks in higher-dimensional continua and their relation to shape theory and higher homotopy group behavior.
• Application to computational topology and robotics, especially for motion planning in non-locally simply connected spaces.
• Extension to non-metrizable spaces and broader categories, connecting with homotopy limits and colimits.

Conclusion

The paper rigorously defines and explores transfinitely iterated wild sets, introducing homotopical analogues of the Cantor-Bendixson process for Peano continua and related spaces. It proves that every countable ordinal can be realized as a (free) wild rank, establishing strong existence and invariance results and drawing attention to unresolved questions concerning wildness of uncountable rank. The implications reach both practical applications and foundational topology, positioning wild ranks as nuanced invariants for locally complicated spaces and opening substantial avenues for future research in algebraic and geometric topology.