[Paper Review] Algebraic topology of Polish spaces. II: Axiomatic homology

Sergey A. Melikhov|arXiv (Cornell University)|Aug 30, 2018

Advanced Topology and Set Theory4 references3 citations

TL;DR

This paper provides a complete axiomatic characterization of Steenrod–Sitnikov homology and Čech cohomology on closed pairs of Polish spaces by introducing a new Controlled Additivity Axiom, which generalizes Milnor’s additivity axioms. The key result is a uniqueness theorem showing that these homology and cohomology theories are uniquely determined by the Eilenberg–Steenrod axioms, the Wallace–Milnor map excision axiom, and the Controlled Additivity Axiom, with homology computable via a cellular (co)chain complex satisfying a lim-colim symmetry.

ABSTRACT

Milnor proved two uniqueness theorems for axiomatic (co)homology: one for pairs of compacta (1960) and another, in particular, for pairs of countable simplicial complexes (1961). We obtain their common generalization: the Eilenberg-Steenrod axioms along with Milnor's map excision axiom and a (non-obvious) common generalization of Milnor's two additivity axioms suffice to uniquely characterize (co)homology of closed pairs of Polish spaces (=separable complete metrizable spaces). The proof provides a combinatorial description of the (co)homology of a Polish space in terms of a cellular (co)chain complex satisfying a symmetry of the form $\lim ext{colim} = ext{colim} \lim$.

Motivation & Objective

• To provide a complete axiomatic characterization of Steenrod–Sitnikov homology and Čech cohomology on closed pairs of Polish spaces.
• To generalize Milnor’s two additivity axioms—valid for compacta and countable simplicial complexes—into a single, unified axiom applicable to Polish spaces.
• To establish that the Eilenberg–Steenrod axioms, together with the Wallace–Milnor map excision and the Controlled Additivity Axiom, uniquely determine the homology and cohomology theories on Polish spaces.
• To offer a computational framework by expressing the homology and cohomology of Polish spaces as (co)homology of a cellular (co)chain complex with a lim-colim symmetry.

Proposed method

• Introduce the Controlled Additivity Axiom: if a Polish space $M$ minus a closed subset $X$ decomposes into a disjoint union of compact sets $K_i$ with diameters shrinking toward $X$, then $H_n(M,X)$ is isomorphic to a subgroup of $\prod H_n(K_i)$, specifically the $\mathbb{K}$-direct sum for a certain ideal $\mathbb{K}$.
• Use the duality lemma from Part I to establish the symmetry $\lim\,\operatorname*{colim} = \operatorname*{colim}\,\lim$ in the construction of the cellular (co)chain complex.
• Construct a cellular (co)chain complex based on inverse systems of triangulated polyhedra $P_i$ with bonding maps $P_{i+1} \to P_i$, where each $P_i$ captures the intersection of $X$ with a neighborhood of shrinking diameter.
• Define the chain complex as the mapping telescope of the inverse system $\dots \to Q_{1K} \to Q_{0K}$, where $Q_{iK}$ is the finite subcomplex of $P_i$ intersecting the image of a compact $K \subset X$, and show this complex computes the correct homology.
• Show that the same complex arises both as $\varinjlim \varprojlim$ and $\varprojlim \varinjlim$, proving the lim-colim symmetry and justifying its use as a canonical object.
• Apply the map excision axiom to replace fibers of the universal resolution $E_\Delta(X) \to K_\Delta(X)$ with compact pairs $(P_{\alpha,[0,\infty]}, K_\alpha)$, enabling homology computation via telescopes of clusters.

Experimental results

Research questions

• RQ1 Can Milnor’s two additivity axioms—valid for compacta and countable simplicial complexes—be unified into a single axiom that applies to Polish spaces?
• RQ2 Does the Controlled Additivity Axiom, which governs countable decompositions of $M \setminus X$ with shrinking diameters, suffice to characterize Steenrod–Sitnikov homology and Čech cohomology on Polish spaces?
• RQ3 Is the homology of a Polish space computable via a cellular (co)chain complex that exhibits a symmetry between inverse and direct limits?
• RQ4 Can the uniqueness of homology and cohomology theories on Polish spaces be established using only the Eilenberg–Steenrod axioms, map excision, and the Controlled Additivity Axiom?
• RQ5 What is the geometric and algebraic structure of the lim-colim symmetric chain complex that computes the homology of Polish spaces?

Key findings

• The Controlled Additivity Axiom, which generalizes Milnor’s additivity axioms, is sufficient to uniquely characterize Steenrod–Sitnikov homology and Čech cohomology on closed pairs of Polish spaces.
• The homology and cohomology of a Polish space are isomorphic to the (co)homology of a cellular (co)chain complex built from inverse systems of triangulated polyhedra with shrinking neighborhoods.
• This (co)chain complex admits two equivalent descriptions: as $\varinjlim \varprojlim$ and as $\varprojlim \varinjlim$, establishing a nontrivial symmetry between limits and colimits.
• The $\mathbb{K}$-direct sum in the Controlled Additivity Axiom arises from a specific ideal $\mathbb{K}$ in the power set of $\mathbb{N}$, which encodes the geometric condition that the diameters of the $K_i$ tend to zero near $X$, and this ideal is derived from the filtrations introduced in Part I.
• The uniqueness theorem shows that any graded isomorphism $H(pt) \to H'(pt)$ extends uniquely to a natural equivalence between two homology theories satisfying the axioms, proving full axiomatic uniqueness.
• The construction allows direct computation of homology and cohomology of Polish spaces from the axioms, making the theory computationally accessible in a way that previous characterizations were not.