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Representing fine shape of local compacta by homotopy classes of ordinary maps

Vladislav Zemlyanoy

TL;DR

This paper advances fine shape theory by constructing, for every local compactum $X$, a metrizable space $|X|$ that represents fine shape morphisms from any locally compact metrizable space $Y$ via a bijection $[Y,X]_{fSh} \cong [Y,|X|]$, functorial in $Y$. The construction hinges on embedding into absolute retracts, using approaching maps, and exploiting exhaustion functions to extend Cathey’s compact-case framework to noncompact settings. Core contributions include the establishment of $|X|$ as a universal classifying space for fine shape from locally compact domains, with $|X|$ unique up to homotopy equivalence and a canonical fine shape inverse encoded by a distinguished embedding $X\hookrightarrow|X|$ that is a fine shape equivalence (FDR-embedding). Additional results address the structure of $|X|$, its universal properties, and the potential for broader categorification, while also linking fine shape invariants to Čech cohomology and Steenrod-Sitnikov homology via map excision. Overall, the work provides a concrete, homotopy-theoretic model for fine shape morphisms into local compacta, enabling practical computation and conceptual clarity in the fine shape category.

Abstract

Fine shape, as defined by Melikhov, is an extension of the strong shape category of compacta (compact metrizable topological spaces) to all metrizable spaces, notable for being compatible with both Čech cohomology and Steenrod-Sitnikov homology. In this work we study fine shape of local compacta (locally compact separable metrizable spaces), and construct, for every local compactum $X$, a space $|X|$ unique up to a homotopy equivalence and such that fine shape classes from any locally compact metrizable space $Y$ to $X$ bijectively correspond to homotopy classes of ordinary maps from $Y$ to $|X|$. This correspondence is (contravariatly) functorial in $Y$, thus giving a representation of $Y$-dependent contravariant functor for a fixed $X$; the universal class corresponding to the identity map of $X$ is the homotopy class of a specific embedding of $X$ into $|X|$ that is a fine shape equivalence.

Representing fine shape of local compacta by homotopy classes of ordinary maps

TL;DR

This paper advances fine shape theory by constructing, for every local compactum , a metrizable space that represents fine shape morphisms from any locally compact metrizable space via a bijection , functorial in . The construction hinges on embedding into absolute retracts, using approaching maps, and exploiting exhaustion functions to extend Cathey’s compact-case framework to noncompact settings. Core contributions include the establishment of as a universal classifying space for fine shape from locally compact domains, with unique up to homotopy equivalence and a canonical fine shape inverse encoded by a distinguished embedding that is a fine shape equivalence (FDR-embedding). Additional results address the structure of , its universal properties, and the potential for broader categorification, while also linking fine shape invariants to Čech cohomology and Steenrod-Sitnikov homology via map excision. Overall, the work provides a concrete, homotopy-theoretic model for fine shape morphisms into local compacta, enabling practical computation and conceptual clarity in the fine shape category.

Abstract

Fine shape, as defined by Melikhov, is an extension of the strong shape category of compacta (compact metrizable topological spaces) to all metrizable spaces, notable for being compatible with both Čech cohomology and Steenrod-Sitnikov homology. In this work we study fine shape of local compacta (locally compact separable metrizable spaces), and construct, for every local compactum , a space unique up to a homotopy equivalence and such that fine shape classes from any locally compact metrizable space to bijectively correspond to homotopy classes of ordinary maps from to . This correspondence is (contravariatly) functorial in , thus giving a representation of -dependent contravariant functor for a fixed ; the universal class corresponding to the identity map of is the homotopy class of a specific embedding of into that is a fine shape equivalence.
Paper Structure (10 sections, 18 theorems, 3 equations)

This paper contains 10 sections, 18 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spaces
  4. Approaching maps
  5. Absolute retracts
  6. Fine shape
  7. FDR-embeddings and fine shape cylinders
  8. Representing fine shape from arbitrary metrizable spaces into a compactum
  9. Exhaustion functions
  10. Representing fine shape from locally compact spaces into a local compactum

Key Result

Proposition 1.2

(Lemma extend_map_to_fSh) Let $f\colon X \to Y$ be a continuous map between metrizable spaces $X$ and $Y$, and let the those be embedded as closed homotopy negligible subsets in absolute retracts $M$ and $N$ respectively. Then (1)There eixsts an $X-Y$-approaching map $\phi\colon M\setminus X \to N\s

Theorems & Definitions (53)

  • Definition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 43 more