Finite-Distance Cohomology

• Finite-distance cohomology is a framework capturing approximate topological invariants by allowing controlled finite-rank errors in algebraic and geometric settings.
• It employs rank metrics on module maps and constructs cohomological groups for hyperplane complements, where cocycles satisfy vanishing residue conditions at infinity.
• This approach bridges algebra, combinatorics, and physics, offering practical tools for analyzing bounded domains, ε-representations, and canonical forms in positive geometries.

Cohomology at finite distance refers to a suite of algebraic and geometric constructions designed to capture "approximate" or "finite-distance" topological invariants of spaces or algebraic objects, where traditional notions are refined or relaxed to account for equivalence only up to bounded error (as measured by rank or residue). Two primary approaches have been developed: an algebraic theory via rank metrics on module maps, leading to approximate cohomology, and a topological/homological theory for the complements of affine hyperplane arrangements, where cohomology is supported on classes with prescribed vanishing at infinity. These frameworks are closely tied to the theory of ε-representations and positive geometries.

1. Rank-Distance and Approximate Cohomology

Given a field $k$, an infinite-dimensional $k$-vector space $L$, and $End:=End_k(L)$ the endomorphism algebra, the rank function $$r(m)=\operatorname{rank}_k(m)\in\mathbb{N}\cup\{0,\infty\}$$ for $m\in End$ induces a pseudo-metric $d(m_1,m_2):=r(m_1-m_2)$. This metric determines when two operators, or more generally two cochains $A,B:G^n\to End$, are at finite distance: $d(A,B):=\sup_{g\in G^n} r(A(g)-B(g))<\infty$. The notion of "approximate" in this theory designates functions or cochains for which cocycle or coboundary relations hold up to finite-rank errors, as formalized via filtrations on $End$ by rank. Given a filtration $F$ on an abelian group $M$ (e.g., $M_i:=\{m\in End\,|\,r(m)<i\}$), one defines approximate $n$-cocycles $Z^n_F(G,M)$ as those cochains whose coboundary images lie in some $M_i$ and specifies the subgroup of approximate coboundaries $B^n_F(G,M)$. The resulting approximate cohomology is $H^n_F(G,M):=Z^n_F(G,M)/B^n_F(G,M)$ (Kazhdan et al., 2017).

2. Finite-Distance Cohomology for Hyperplane Arrangements

Let $A = \bigcup_{i=1}^m L_i \subset \mathbb{C}^n$ be an essential affine hyperplane arrangement and $M = \mathbb{C}^n \setminus A$ its complement. The finite-distance cohomology is defined topologically as the relative cohomology at finite distance:

$$H^k_{<\infty}(M):= H^k_c(\mathbb{C}^n, Rj_*\mathbb{Q}_M) \cong H_{2n-k}(\mathbb{C}^n, A),$$

where $j:M\hookrightarrow \mathbb{C}^n$ is the inclusion. Thus, $H^*_{<\infty}(M)$ is Poincaré dual to the relative homology of $(\mathbb{C}^n, A)$ (Pfister, 31 Jan 2026). Equivalently, these groups capture cocycles supported "away from infinity," i.e., whose residues at infinity vanish—a property that makes them canonical for describing bounded domains in real settings and "positive geometries."

3. Structural and Algebraic Descriptions

In the hyperplane arrangement context, the Orlik–Solomon algebra $A_\bullet(A)=\bigwedge^\bullet \mathbb{Q}^m / I(A)$ encodes the combinatorics of the intersection lattice. The boundary operator $\partial$ on $A_k(A)$ has kernel in degree $n$ given by

$$H^n_{<\infty}(M) \simeq \ker(\partial:A_n(A)\rightarrow A_{n-1}(A)),$$

while cohomology vanishes in other degrees. These classes correspond to logarithmic $n$-forms with vanishing residues along all divisors at infinity, i.e., differential forms regular at infinity in the appropriate partial compactification (Pfister, 31 Jan 2026). Under the identification $e_i \mapsto d\log f_i$, the kernel of $\partial$ matches the kernel of the global residue-at-infinity maps.

In the algebraic context, for abelian $G$ and module $M$ with trivial $G$-action, $H^1_F(G,M)$ is interpretable as the group of genuine homomorphisms modulo those of finite rank:

$$H^1_F(G, M)=\operatorname{Hom}(G,M)/\operatorname{Hom}_f(G,M),$$

where $\operatorname{Hom}_f$ denotes the subgroup of finite-rank homomorphisms (Kazhdan et al., 2017).

4. Finiteness Theorems and Approximation Properties

A central question is whether near-homomorphisms (additive up to uniformly bounded finite-rank error) can be approximated by genuine homomorphisms to within uniformly bounded error. For $G=V$ a countable $k$-vector space over a finite field $k=\mathbb{F}_p$, $d < p-1$, and $M^d$ the space of degree-$d$ homogeneous polynomials, for any map $A:V\to M^d$ satisfying $r(A(v+v')-A(v)-A(v'))\leq r$, there exists a homomorphism $X:V\to M^d$ with $r(A(v)-X(v))\leq R(r,k,d)$ for all $v$, uniquely determined up to finite-rank. This result employs Gowers norms and inverse theorems in the finite-dimensional case, and compactness/diagonal arguments for the infinite-dimensional extension (Kazhdan et al., 2017).

A precise version for $G=\mathbb{Z}$ and $k$ of characteristic $\neq 2$ is: if $r(A(m+n)-A(m)-A(n))\leq 1$, then $R(\mathbb{Z},1,k)\leq 2$, so $r(A(n)-nX)\leq 2$ for some $X$. In contrast, in low characteristic (e.g., $k=\mathbb{F}_2$ and quadratic forms), counterexamples demonstrate the failure of bounded-rank approximation, necessitating further generalizations.

5. Connections to Residues, Compactification, and Positive Geometry

For affine arrangements, finite-distance cohomology classes correspond to logarithmic forms with vanishing residues along exceptional divisors at infinity. The relevant partial compactification is constructed by adding the hyperplane at infinity and blowing up only those strata entirely at infinity, ensuring the boundary divisor $Y = X \setminus \mathbb{C}^n$ captures the geometry of faces "at infinity" but not those inside affine space. This yields an identification

$$H^n_{<\infty}(M) \simeq \ker\left( \bigoplus_{i\in I} \operatorname{Res}_{Y_i^\circ}: H^n(\mathbb{C}^n \setminus A)\to \bigoplus_{i\in I} H^{n-1}(Y_i^\circ) \right),$$

with $Y_i$ components of the boundary divisor (Pfister, 31 Jan 2026).

These cohomology classes coincide with the canonical forms of positive geometry in the sense of Arkani-Hamed, Bai, and Lam: each chamber (relative cycle) in $H_n(\mathbb{C}^n,A)$ is paired, via the canonical mapping, with a unique logarithmic form in $H^n_{<\infty}(M)$. Explicitly, such forms are sums over non-broken-circuit sets and constructed from basic logarithmic forms.

6. Illustrative Examples and Physical Applications

A five-line arrangement in $\mathbb{C}^2$,

$A=\{x=0,\, x-1=0,\, y=0,\, y-1=0,\, x-y=0 \},$

has $H^2_{<\infty}(M)\cong \ker(\partial)$ of dimension $2$, matching the number of bounded regions in the real locus. In a physical context, the two-site cosmological correlator corresponds to an arrangement of three hyperplanes in $\mathbb{C}^2$ with $H^2_{<\infty}=1$ and the explicit canonical form recovers the known integrand via the residue method (Pfister, 31 Jan 2026).

7. Interpretation, Scope, and Obstructions

The algebraic "approximate cohomology" theory generalizes the concept of $\epsilon$-representations, measuring "defects" of near-morphisms not correctable by finite-rank adjustment. For hyperplane complements, cohomology at finite distance provides a bridge between combinatorial, topological, and geometric frameworks, and directly realizes canonical forms with precise normalization and vanishing properties at infinity. Counterexamples in low characteristic show the necessity of further structural refinements (such as the introduction of nonclassical obstructions) for broad generality in bounded-rank representation theory (Kazhdan et al., 2017).