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arxiv: 2601.14545 · v2 · submitted 2026-01-20 · 🧮 math.GN

On {Gamma}-embeddings and partial actions of function spaces

Luis A. Mart\'inez-S\'anchez , H\'ector Pinedo , Jos\'e L. Vilca-Rodr\'iguez This is my paper

Pith reviewed 2026-05-16 12:26 UTC · model grok-4.3

classification 🧮 math.GN
keywords Gamma-embeddingspartial actionstopological groupsfunction spacescompact-open topologyinverse semigroupsglobalizationsenveloping spaces
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The pith

Every topological space admits a Γ-embedding into a space of continuous functions from a compact domain, inducing partial actions there.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Γ-embeddings, a class of topological embeddings defined using the inverse semigroup of homeomorphisms between open subsets. It proves that for any topological space Y and any compact space X, Y admits a Γ-embedding into C(X, Y) equipped with the compact-open topology. This embedding allows any partial action of a topological group G on Y to induce a corresponding partial action on the function space. The work then explores how these induced actions relate to the original ones through their globalizations and enveloping spaces.

Core claim

Every topological space Y admits a Γ-embedding into C(X, Y) with the compact-open topology when X is compact. Consequently, any partial action θ of a topological group G on Y induces a partial action ˆθ on C(X, Y). The paper examines the relationships between these actions, their globalizations, and enveloping spaces.

What carries the argument

Γ-embeddings constructed from the inverse semigroup of homeomorphisms between open subsets of the space.

If this is right

  • Any partial action on Y extends naturally to one on C(X, Y).
  • The globalizations of the induced action correspond to those of the original action.
  • Relationships between the enveloping spaces of the two actions can be established.
  • This construction applies to arbitrary topological spaces Y.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This lifting might preserve dynamical properties like minimality or transitivity in the partial actions.
  • Applications could include extending results on group actions from base spaces to their function spaces in topology.
  • One could investigate whether the induced action on C(X, Y) has simpler globalizations for specific choices of X and Y.

Load-bearing premise

The inverse semigroup of homeomorphisms between open subsets defines a class of Γ-embeddings that works for arbitrary topological spaces Y and compact X.

What would settle it

Finding a topological space Y and compact X where no Γ-embedding from Y to C(X,Y) exists, or a partial action on Y that fails to induce a valid partial action on the function space.

read the original abstract

This paper deals with the extension of partial actions of topological groups on topological spaces. Within this framework, we introduce a class of topological embeddings defined via the inverse semigroup of homeomorphisms between open subsets of a topological space. We describe several embeddings of this type, referred to as $\Gamma$- embeddings, and we place particular emphasis on one of them. In particular, we prove that every topological space $Y$ admits a $\Gamma$-embedding into the space of continuous functions $C(X, Y )$, equipped with the compact-open topology, where $X$ is a compact space. Consequently, any partial action $\theta$ of a topological group $G$ on $ Y$ naturally induces a partial action $\hat\theta$ on $C(X, Y ).$ Throughout the paper, we investigate various relationships between these actions, as well as between their corresponding globalizations and enveloping spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper introduces Γ-embeddings, topological embeddings defined via the inverse semigroup of partial homeomorphisms between open subsets of a space. It proves that every topological space Y admits a Γ-embedding into C(X, Y) equipped with the compact-open topology, where X is a suitable compact space. Consequently, any partial action θ of a topological group G on Y induces a partial action ˆθ on C(X, Y). The manuscript investigates relationships between these actions, their globalizations, and enveloping spaces.

Significance. If the central construction holds, the result supplies a systematic, direct method for extending partial actions from arbitrary spaces Y to function spaces C(X, Y) while preserving the inverse-semigroup structure. This is a concrete, verifiable embedding (via constant functions or evaluations) that works without additional separation axioms on Y and is a strength of the paper; it could facilitate further study of globalizations in topological dynamics.

minor comments (3)
  1. [§2] §2 (Definition of Γ-embedding): the inverse-semigroup construction is central; explicitly verify that the embedding map is continuous and open onto its image with respect to the compact-open topology on C(X, Y) for arbitrary Y.
  2. [Theorem 3.1] Theorem 3.1 (main embedding result): state the precise choice of compact X (e.g., whether X = [0,1] or a singleton suffices) and confirm that the induced ˆθ is indeed a partial action of G on C(X, Y).
  3. [§4] §4 (relationships with globalizations): clarify whether the enveloping space of ˆθ is the function space over the enveloping space of θ, or provide a counter-example if not.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript, recognition of the strength of the central Γ-embedding construction, and recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines Γ-embeddings directly in terms of the inverse semigroup of partial homeomorphisms between open subsets of a space. It then proves by explicit construction that every topological space Y admits a Γ-embedding into C(X,Y) equipped with the compact-open topology for suitable compact X, and that this embedding intertwines partial actions of topological groups. No step reduces a claimed prediction or result to a fitted parameter, self-citation chain, or definitional tautology; the central theorem is a direct verification that the constructed map satisfies the semigroup-based definition of Γ-embedding and preserves the partial action structure. The derivation is therefore self-contained against the paper's own definitions and constructions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the new definition of Γ-embeddings and standard topological constructions.

axioms (1)
  • standard math Standard axioms of topological spaces, continuous functions, and inverse semigroups
    Relies on basic definitions from general topology.
invented entities (1)
  • Γ-embedding no independent evidence
    purpose: Class of topological embeddings defined via inverse semigroup of homeomorphisms
    Newly introduced construction in the paper.

pith-pipeline@v0.9.0 · 6693 in / 965 out tokens · 64840 ms · 2026-05-16T12:26:13.674775+00:00 · methodology

discussion (0)

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Reference graph

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