[Paper Review] Differential cohomology

Ulrich Bunke|arXiv (Cornell University)|Aug 20, 2012

Homotopy and Cohomology in Algebraic Topology7 references4 citations

TL;DR

This paper presents a comprehensive, homotopy-theoretic framework for differential cohomology, unifying secondary characteristic classes and differential refinements of generalized cohomology theories. It establishes a stable $∞$-categorical approach to differential extensions, including products and Umkehr maps, with explicit constructions for $K$-theory and bordism, and provides a systematic treatment of integration, Chern-Simons invariants, and differential refinements via smooth Deligne cohomology and Cheeger-Simons characters.

ABSTRACT

These course note first provide an introduction to secondary characteristic classes and differential cohomology. They continue with a presentation of a stable homotopy theoretic approach to the theory of differential extensions of generalized cohomology theories including products and Umkehr maps.

Motivation & Objective

• To develop a systematic, stable homotopy-theoretic foundation for differential cohomology that generalizes classical constructions.
• To unify secondary characteristic classes and differential refinements of generalized cohomology theories under a single framework.
• To provide explicit calculational tools and examples for differential $K$-theory and complex bordism, including integration and products.
• To establish a differential refinement of the Chern-Weyl homomorphism via Cheeger-Simons differential characters and smooth Deligne cohomology.
• To introduce a full theory of differential extensions with Umkehr maps, multiplicative structures, and geometric realization via $∞$-categories.

Proposed method

• Uses $∞$-categories and stable homotopy theory to formalize differential extensions of generalized cohomology theories.
• Applies sheaf-theoretic methods to define smooth Deligne cohomology as a model for differential integral cohomology.
• Constructs differential refinements of characteristic classes via the Cheeger-Simons homomorphism, refining Chern-Weil theory.
• Introduces differential function spectra to model differential cohomology groups with compatible products and integration maps.
• Employs partition-of-unity techniques and local trivializations to construct global connections and curvature forms.
• Develops a theory of differential Thom classes and orientation in the differential setting, extending topological Thom isomorphisms.

Experimental results

Research questions

• RQ1 How can secondary characteristic classes be systematically refined into differential cohomology classes using curvature and connection data?
• RQ2 What is the stable homotopy-theoretic structure underlying differential extensions of generalized cohomology theories, including products and Umkehr maps?
• RQ3 How do differential $K$-theory and differential bordism theory arise as differential refinements of their topological counterparts?
• RQ4 What is the role of smooth Deligne cohomology in realizing differential refinements of integral characteristic classes?
• RQ5 How can integration and index theorems be formulated in the differential cohomological setting, particularly for $χ$-classes and Adams operations?

Key findings

• The Cheeger-Simons differential character construction provides a canonical differential refinement of the Chern-Weyl homomorphism for complex vector bundles.
• Differential $K$-theory is realized as a differential extension of topological $K$-theory with a well-defined product structure and integration map.
• The differential refinement of complex bordism theory is constructed with a compatible multiplicative structure and Thom isomorphism.
• The differential $e$-invariants for $σ \in \pi_3(\mathbf{S})$ and $\eta \in \pi_1(\mathbf{S})$ are computed as elements in $\mathbb{C}/\mathbb{Z}$, yielding $\hat{e}(\hat{\eta}) = \frac{1}{2} \mod \mathbb{Z}$ and $\hat{e}(\hat{\sigma}) = \frac{1}{24} \mod \mathbb{Z}$.
• The differential ${\mathbf{MU}}$-index theorem is formulated and proven in the context of differential cohomology, extending the classical index theorem to the differential setting.