Locally Smooth Cohomology

• Locally smooth cohomology is a framework that refines classical cohomology by requiring cochains to be smooth only locally, thus bridging smooth and discrete models.
• It utilizes bicomplexes and spectral sequences, comparing models such as Alexander–Spanier, singular, and Lie group cohomology through canonical isomorphisms.
• Its applications span infinite-dimensional groups, differential generalized cohomology, and geometric constructions like Chern–Simons cocycles and Gelfand–Fuks classes.

A locally smooth cohomology theory systematically refines classical cohomology by combining the properties of smooth and local constructions. It arises in the context of group cohomology, Lie algebra cohomology, differential graded algebra, and generalized cohomology on smooth manifolds. The essential feature is that cochains—functions or differential forms used to define cocycles—need only be smooth in a neighborhood of the diagonal (or the identity) rather than globally. This local smoothness condition provides a bridge between continuous/smooth and discrete models, and underlies comparisons and equivalences with more classical cohomological invariants such as singular or Alexander–Spanier cohomology, and supports geometric constructions such as Chern–Simons cocycles and Gelfand–Fuks classes. Locally smooth cohomology is implemented by various bicomplexes and spectral sequences, plays a central role for infinite-dimensional groups (e.g., diffeomorphism groups), and underpins new developments in differential generalized cohomology.

1. Core Definitions and Construction

Locally smooth cochains are functions that are required to be smooth only on a neighborhood of a reference “diagonal” in the domain, such as the identity element in $G^{n+1}$ for Lie groups or the small diagonal in $X^{n+1}$ for manifolds. The principal complexes and models are:

• Abstract Local Cochains: For an open cover $\mathcal{U} = \{U_i\}_{i\in I}$ of a topological space $X$, cochains $$C^n_{\operatorname{loc},\mathcal{U}}(X; V) = \{ f: \mathcal{U}[n] \to V \}$$, with $\mathcal{U}[n] = \bigcup_{i\in I} U_i^{n+1}$, and differential $\delta$ given by the Alexander–Spanier formula. No smoothness is required.
• Continuous and Smooth Local Cochains: $C^n_{\operatorname{loc},\mathcal{U},c}(X; V)$ for continuous cochains, or the further subcomplex for smoothness in the context of smooth manifolds.
• Lie Group Case: For a Lie group $G$ with left-invariant covering neighborhoods $U$, cochains $C^n_{\germ,s}(G; V)$ are germs of smooth functions defined on $(g_0U) \times \cdots \times (g_nU)$, required to be smooth near the diagonal in $G^{n+1}$ (Fuchssteiner, 2011, Wagemann et al., 2011).
• Bar Complex for Locally Smooth Cochains: For a (possibly infinite-dimensional) Lie group $G$ and smooth $G$-module $A$, set

$$C^n_{\operatorname{loc}}(G, A) = \{ f : G^n \to A \mid \exists \, \text{open } U \ni e,\, f|_U \text{ smooth} \}$$

with group cochain differential as in the standard bar complex (Wagemann et al., 2011).

The cohomology groups obtained from these complexes—denoted $H^*_{\operatorname{loc},\mathcal{U}}(X;V)$, $H^*_{\operatorname{germ},s}(G;V)$, or $H^*_{\operatorname{loc}}(G,A)$—are the locally smooth (or continuous) cohomology groups in the respective contexts.

2. Structural Properties, Isomorphism Theorems, and Comparison Results

The central structural result is the existence of canonical isomorphisms between locally smooth cohomology and other cohomology theories under certain contractibility and covering assumptions:

• Continuity–Abstraction Comparison: If $V$ is loop contractible and $\mathcal{U}$ is a cover by cozero sets of a (generalized) partition of unity, then

$$H^*(C^*_{\operatorname{loc},\mathcal{U},c}(X; V)) \cong H^*(C^*_{\operatorname{loc},\mathcal{U}}(X; V))$$

((Fuchssteiner, 2011), Theorem 2.1).

• Locally Smooth Cohomology and Alexander–Spanier/Singular Cohomology: For locally contractible Lie groups $G$ and loop contractible $V$,

$$H^*(C^*_{\operatorname{germ},s}(G;V)) \cong H^*_{AS}(G;V) \cong H^*_{\mathrm{sing}}(G;V)$$

((Fuchssteiner, 2011), Theorem 2.2; also (Wagemann et al., 2011)).

• Cocycle Model Equivalences: For suitable $G$ and $A$, the following models are canonically isomorphic:

$$H^n_{\operatorname{loc}}(G,A) \cong H^n_{\operatorname{cont}}(G,A) \cong H^n_{\operatorname{Segal–Mitchison}}(G,A) \cong H^n_{\operatorname{simp}}(G,A)$$

((Wagemann et al., 2011), §3). For contractible coefficients, all these models coincide.

These isomorphisms rely on explicit constructions: bicomplexes combining Čech and Alexander–Spanier differentials, row contractions via partitions of unity, and homotopy operators exploiting loop contractibility (Fuchssteiner, 2011).

3. Spectral Sequences, Filtrations, and Differential Generalized Cohomology

Locally smooth cohomology admits rich bicomplex and spectral sequence structures, essential for effective calculations and for connections with more general frameworks:

• Čech–Alexander–Spanier Double Complex: Embeds local (and Čech) cochain complexes into a total bicomplex $\check C^p(\mathcal{U},A^q)$, robustly controlling their interplay and enabling explicit comparison via row contractions (Fuchssteiner, 2011).
• AHSS-type Spectral Sequence in Smooth Generalized Cohomology: For any smooth generalized cohomology theory $\mathcal{E}^\ast$ represented by a smooth spectrum, and a manifold $M$ covered by a good cover $\mathcal{U}$, there is a spectral sequence with

$$E_1^{p,q} = \check{C}^p(\mathcal{U}; E^q(*)), \qquad E_2^{p,q} = H^p(M; \underline{E^q(*)})$$

converging to $\mathcal{E}^{p+q}(M)$ or its differential refinement in the smooth/differential context (Grady et al., 2016).

• Differential Deligne and K-theory: In differential cohomology, the relevant differentials capture the obstruction to forms having integral periods and encode secondary invariants arising from smooth structures, as in

$$d_n: \Omega^{n}_{\mathrm{cl}}(M) \to H^n(M; U(1)), \qquad d_n(\omega) = \exp\left(\int_{\Delta^n}\omega\right)$$

((Grady et al., 2016), §3).

Thus, locally smooth cochains underpin the passage from purely algebraic/topological cohomology to geometric invariants detecting smooth and torsion information.

4. Applications: Lie Groups, Diffeomorphism Groups, and Vector Field Cohomology

Locally smooth cohomology is particularly effective in the infinite-dimensional or diffeomorphism group regime and for geometric representation theory:

• Diffeomorphism Groups and Configured Cohomology: For Fréchet–Lie groups such as $\mathrm{Diff}_{\mathrm{vol}}(S^3)$, locally smooth cohomology supports explicit construction of $\mathbb{R}/\mathbb{Z}$ or $\mathbb{Q}/\mathbb{Z}$-valued Chern–Simons type 3-cocycles detecting torsion in $H_3$ (Nosaka, 12 Jan 2026). This is enabled by "well-configured" complexes and geometric fillings which are only locally (not globally) smooth yet yield group cohomology classes not captured by smooth or discrete cohomology alone.
• String Group Cocycle and Higher Group Extensions: In compact, simply connected Lie groups, the canonical String class in locally smooth $U(1)$-valued cohomology can be constructed explicitly using smooth fillings of simplices in neighborhoods of the identity, representing the universal central extension in degree three (Wagemann et al., 2011).
• Vector Field Cohomology: For the Lie algebra $\Vect(M)$ of smooth vector fields, the locally smooth (“diagonal” or “local”) complex encodes Gelfand–Fuks/Losik–Guillemin–Fuks classes as local jet functionals, with

$H^k_{\mathrm{loc}}(\Vect(M)) \cong H^{d+k}_{\mathrm{dR}}(Fr^\C_M\times^{U(d)}X(d))$

where $d = \dim M$, $Fr^\C_M$ is the frame bundle, and $X(d)$ is a finite-type skeleton. This produces the universal characteristic classes for modules and anomalies in field theory, and allows explicit realization of Virasoro-type cocycles via descent formulas (Williams, 2024).

5. Local-to-Global Analysis and Factorization

Locally smooth cohomology admits fine local-to-global and factorization structures, crucial for both explicit computation and modern approaches:

• Diagonal/Support-Controlled Complexes: The “diagonal” subcomplexes $C^\bullet_\triangle$ in Lie algebra cohomology consist of functionals vanishing unless the arguments have common support, thus localizing the cohomology near the diagonal and enabling cosheaf/factorization algebra strategies (Miaskiwskyi, 2022).
• $k$-Good Covers and Bott–Segal Double Complex: For any $k$-good cover $\mathcal{U}$ on a manifold, the Bott–Segal double complex computes global cohomology from local diagonal pieces, yielding spectral sequences whose $E_2$-pages are expressed in terms of relative homology $(M^r, M^r_{r-1})$ and the cohomology of the local model $W_n$ (formal vector fields on $\mathbb{R}^n$) (Miaskiwskyi, 2022). The diagonal filtration stabilizes, so that for large enough $k$, the inclusion map is a quasi-isomorphism and global cohomology is recovered.

6. Conceptual Role, Generalizations, and Research Directions

Locally smooth cohomology unifies several approaches to smooth, continuous, and local geometric invariants and enables systematic comparison and computation:

• Unified $\delta$-Functor Perspective: All main locally smooth, smooth, Segal–Mitchison, and simplicial sheaf cohomology functors define equivalent $\delta$-functors for smooth groups and modules, providing canonical comparison maps and transfer of results across models (Wagemann et al., 2011).
• Refinement and Interpolation: Locally smooth cohomology interpolates between the smooth and discrete settings; configured and well-configured variants make possible explicit construction of secondary and torsion invariants in geometric group theory (Nosaka, 12 Jan 2026).
• Foundational Role in Differential Cohomology: The theory provides the local and smooth models essential for spectral sequence constructions in differential generalized cohomology (Grady et al., 2016), undergirds descent formulas and jet-theoretic cocycle constructions (Williams, 2024), and informs local-to-global approaches for factorization algebras and higher category methods in modern geometry (Miaskiwskyi, 2022).

Locally smooth cohomology thus offers a powerful, flexible, and geometrically meaningful framework bridging classical and modern invariants of smooth spaces, groups, and Lie algebras, and serves as a backbone for subsequent theories in differential, higher, and factorization cohomology.