Chern-Dold Character

• Chern-Dold character is a canonical transformation from generalized cohomology theories to ordinary cohomology that links abstract invariants with explicit cocycle data.
• It refines classical constructs by encoding additive and multiplicative structures through well-behaved cocycle categories and 2-monoidal operations.
• Its explicit formulations in complex cobordism and non-abelian settings demonstrate significant applications in differential geometry and theoretical physics.

The Chern-Dold character is a canonical transformation from generalized cohomology theories to ordinary cohomology, providing a crucial link between abstract homotopy-theoretic invariants and explicit cocycle-level data. It generalizes the classical Chern character in topological K-theory to a broad class of cohomology theories, encoding the passage from abstract spectra to differential and singular cocycle representatives. The construction and refinements of the Chern-Dold character serve as foundational tools for both algebraic topology and differential geometry, facilitating the articulation of additive and multiplicative structures at the level of cocycles and their applications in geometry and theoretical physics.

1. Formal Definition and Construction

Let $E^*$ be a (reduced) generalized cohomology theory represented by a spectrum $$\{E_n, \varepsilon_n : E_n \wedge S^1 \xrightarrow{\simeq} \Omega E_{n+1}\}_{n \in \mathbb{Z}}$$. The Chern-Dold character is defined as the unique natural transformation

$$\operatorname{ch}: E^n(X) = [X, E_n] \longrightarrow H^n(X; V)$$

where $V^* = E^*(S^0) \otimes_{\mathbb{Z}} \mathbb{R}$, the graded real vector space of coefficients. The image $H^n(X;V) = \prod_{i+j=n} H^i(X;V^j)$ is modeled strictly by singular cocycles $$Z^n(X;V) = \{\omega \in C^n(X;V)\mid \delta\omega=0\}$$ (Upmeier, 2014, Buchstaber et al., 2020, Fiorenza et al., 2020).

The Chern-Dold character is uniquely determined by its normalization on universal classes; for the universal element $\mathrm{id}_{E_n} \in E^n(E_n)$,

$$\operatorname{ch}(\mathrm{id}_{E_n}) = [\iota_n] \in H^n(E_n; V)$$

where $\iota_n$ is any choice of fundamental cocycle. For any continuous map $f: X \to E_n$,

$\operatorname{ch}([f]) = [f^*(\iota_n)]$

establishing a bridge between the abstract homotopy-theoretic formulation and explicit cocycle representatives (Upmeier, 2014).

2. Cocycle Categories and 2-Monoidal Structures

The Chern-Dold character induces a well-behaved interplay between the homotopy-theoretic and chain-level representations of cohomology theories through the language of cocycle categories:

• The homotopy-theoretic cocycles comprise the groupoid $\mathcal{C}_E(X) = \Pi_1(E_n^X)$, whose objects are pointed continuous maps $f: X \to E_n$, and morphisms are based homotopies.
• The singular cocycle category $\mathcal{C}_{H_V}(X) = Z^n(X; V)$ is a strict groupoid of closed singular cochains, with morphisms realized by cochain homotopies $h \in C^{n-1}(X; V)$ satisfying $\delta h = \omega_1 - \omega_0$.

The spectrum structure on $\mathcal{C}_E(X)$ provides two loop-sum operations $\varobar, \varominus$ that, together with an interchange isomorphism, make it a 2-monoidal category. Meanwhile, $\mathcal{C}_{H_V}(X)$ is a strict 2-monoidal category under pointwise addition of cocycles (Upmeier, 2014).

A refined Chern-Dold transformation $\Phi_X = \operatorname{ch}_X$ exists as a 2-monoidal functor: $\Phi_X: (\mathcal{C}_E(X), \varobar, \varominus) \to (\mathcal{C}_{H_V}(X), +)$ which sends the loop-sum operation on the homotopy side to strict addition of singular cocycles. The induced map at the isomorphism class level recovers the classical Chern-Dold character (Upmeier, 2014).

3. Explicit Cocycle Formulas and Differential Cohomology

Given a coherent system of fundamental cocycles $\{\iota_n\}_{n \in \mathbb{Z}}$, the refined Chern-Dold functor operates explicitly as follows:

• On objects $f: X \to E_n$, $\Phi_X(f) = f^*(\iota_n)$.
• On morphisms (homotopies) $H: f \simeq g$, the morphism in $\mathcal{C}_{H_V}(X)$ is realized by the slant-product $H^*(\iota_n)/[\Delta^1]$ with boundary $$\delta(H^*(\iota_n)/[\Delta^1]) = g^*(\iota_n) - f^*(\iota_n)$$.

In the Hopkins-Singer model, differential $E$-cocycles on a smooth manifold $M$ are given by triples $(c, \omega, h)$, where $c: M \to E_n$, $\omega \in \Omega^n(M; V)$ is a closed form, and $h \in C^{n-1}(M; V)$ satisfies $\delta h = \omega - c^*(\iota_n)$. The refined Chern-Dold character enables the lift of additive structures to the differential cocycle level, yielding a strictly associative and commutative group law (Upmeier, 2014, Fiorenza et al., 2020): $$(c_1, \omega_1, h_1) + (c_2, \omega_2, h_2) = (\alpha_n(c_1, c_2), \omega_1 + \omega_2, h_1 + h_2 + (c_1, c_2)^*A_n)$$ where $\alpha_n$ is the loop-sum, and $A_n$ the cochain measuring non-additivity of the fundamental cocycle.

4. The Chern-Dold Character in Complex Cobordism

In complex cobordism (MU-theory), the Chern-Dold character,

$$\chi^y: MU^*(X) \to H^*(X; \Omega_* \otimes \mathbb{Q}),$$

is the unique multiplicative, natural transformation coinciding with the tautological embedding on a point (Buchstaber et al., 2020). For $\mathbb{C}P^\infty$, the value $\chi^y(u)$ (with $u$ the first MU-Chern class) is the universal formal group exponential $B(z)$, admitting an expansion with coefficients given by cobordism classes of theta divisors: $$B(z) = z + \sum_{n \geq 1} \frac{[\Theta_n]}{(n+1)!} z^{n+1}$$ where $[\Theta_n] \in MU_{2n}(pt) = \Omega_{2n}$ (Buchstaber et al., 2020).

Operations by the Landweber–Novikov algebra $S_*$ act on the coefficients of the Chern-Dold character, and the structure of these operations and their geometric correspondence (e.g., to Chern numbers and theta divisors) is explicitly described. The quantization functor (mapping to $MU^*(X) \otimes S^*$) and the dequantization via the cycle-class map factor the Chern-Dold character as $\chi^y = p^* \circ q^*$, revealing deep algebraic structure (Buchstaber et al., 2020).

5. Generalizations: Twisted and Non-Abelian Character Maps

The Chern-Dold character admits further generalization to (twisted, differential) non-abelian cohomology. Here, the character targets non-abelian de Rham cohomology, encoded in the space of flat $L_\infty$-algebra-valued differential forms (Chevalley–Eilenberg algebras) (Fiorenza et al., 2020). Given a homotopy type $A$, the non-abelian character map is

$$\operatorname{ch}_A : H_t(X;A) \longrightarrow H_{\mathrm{dR},t}^{A}(X)$$

constructed via rationalization and the non-abelian de Rham theorem. This framework recovers and unifies the classical Chern-Dold character, the Chern–Weil homomorphism (for $A=BG$), and the Cheeger–Simons character (for $A = B^{n+1}U(1)$), and extends to twisted K-theory, higher K-theories, and 4-cohomotopy relevant in M-theory:

Example	Source Space $A$	Character Map Target
K-theory	$KU$	$H^{2*}(X; \mathbb{R})$
Twisted K-theory	$Z \times BU \sslash B^2 U(1)$	$H^\bullet_{d - H_3}(X; \mathbb{R})$
4–Cohomotopy (M-theory)	$S^4 \sslash Sp(2)$, $\mathbb{C}P^3 \sslash Sp(2)$	Bianchi-identity-constrained forms

This formalism incorporates twisting, differential refinement (moduli of connections), and produces a unifying framework for character maps across abelian and non-abelian, twisted and differential settings (Fiorenza et al., 2020).

6. Applications and Geometric Representatives

Geometric realization of the Chern-Dold character's coefficients is prominent in the context of complex cobordisms. Smooth theta divisors in principally polarized abelian varieties provide algebraic representatives for the coefficients $B_{2n}$ in the Chern-Dold expansion. Real-analytic representatives are constructed via Weierstrass $\sigma$- and $\zeta$-functions, offering explicit smooth manifolds for given cobordism classes. Actions of the Landweber–Novikov algebra on these geometric representatives mirror the structural properties of the character, and the framework provides precise answers to classical questions such as the Milnor–Hirzebruch problem on the existence of algebraic varieties with prescribed Chern numbers (Buchstaber et al., 2020).

7. Structural and Unifying Significance

The Chern-Dold character, in both its classical and refined (including twisted and non-abelian) forms, constitutes a central apparatus for translating between generalized cohomological invariants and explicit geometric or differential data. Its structural properties—uniqueness, functoriality, compatibility with additive/multiplicative structures, and adaptability to twisted and differential settings—have established it as the canonical character map in modern homotopy theory, differential cohomology, and higher gauge theory (Upmeier, 2014, Buchstaber et al., 2020, Fiorenza et al., 2020). The unification achieved by the non-abelian character map provides a comprehensive categorical context for both classical invariants and those of contemporary physical theories such as M-theory, integrating all forms of character maps under a single formalism.