Relative differential cohomology and generalized Cheeger-Simons characters. (English) Zbl 1484.55003
Differential cohomology theories are refinements of cohomology theories on smooth manifolds which enrich them by differential form information. The first examples are differential ordinary cohomology (in the versions of smooth Deligne cohomology or Cheeger-Simons characters) and differential/smooth \(K\)-theory. Later, the concept has been axiomized and intensely studied.
The long paper at hand is devoted to the situation where one deals with manifolds with boundary. Also this has been discussed in previous work. The goal of the paper is a very systematic and complete treatment. This is defined on the category of smooth maps between smooth (finite dimensional) manifolds, possibly with boundary.
In Section 2, the axioms for a differential extension of an arbitrary generalized cohomology in the relative setup are formulated (following the expected pattern). This includes the concept of “integration over the \(S^1\)-fiber” (for the product \(M\times S^1\)), proven to be of fundamental importance in [U. Bunke and T. Schick, J. Topol. 3, No. 1, 110–156 (2010; Zbl 1252.55002)].
Section 3 then derives a number of long exact sequences for an axiomatic differential cohomology theory, in particular the pair sequence. As expected from a differential cohomology theory (which is not homotopy invariant), there is no long exact sequence only in differential cohomology, but the pair sequence for the underlying cohomology theory is spliced together with the pair sequence for the \(R/Z\)-version of that theory, with intermediate terms differential cohomology: \[ h_{R/Z}^{*-1}(A) \to \hat h^*(f) \to\hat h^*(X)\to \hat h^*(A)\to h^{*+1}(f) \] for \(f\colon A\to X\).
The next chapter proves existence and uniqueness of relative differential cohomology refinements, established under the same conditions as in the absolute case. As a special example, a cycle model based on generalized Cheeger-Simons characters is developed and its properties are discussed.
Furthermore, integration over the fiber is studied in great detail for differential cohomology theories.
The long paper is carefully written with full details of constructions and proofs.
The long paper at hand is devoted to the situation where one deals with manifolds with boundary. Also this has been discussed in previous work. The goal of the paper is a very systematic and complete treatment. This is defined on the category of smooth maps between smooth (finite dimensional) manifolds, possibly with boundary.
In Section 2, the axioms for a differential extension of an arbitrary generalized cohomology in the relative setup are formulated (following the expected pattern). This includes the concept of “integration over the \(S^1\)-fiber” (for the product \(M\times S^1\)), proven to be of fundamental importance in [U. Bunke and T. Schick, J. Topol. 3, No. 1, 110–156 (2010; Zbl 1252.55002)].
Section 3 then derives a number of long exact sequences for an axiomatic differential cohomology theory, in particular the pair sequence. As expected from a differential cohomology theory (which is not homotopy invariant), there is no long exact sequence only in differential cohomology, but the pair sequence for the underlying cohomology theory is spliced together with the pair sequence for the \(R/Z\)-version of that theory, with intermediate terms differential cohomology: \[ h_{R/Z}^{*-1}(A) \to \hat h^*(f) \to\hat h^*(X)\to \hat h^*(A)\to h^{*+1}(f) \] for \(f\colon A\to X\).
The next chapter proves existence and uniqueness of relative differential cohomology refinements, established under the same conditions as in the absolute case. As a special example, a cycle model based on generalized Cheeger-Simons characters is developed and its properties are discussed.
Furthermore, integration over the fiber is studied in great detail for differential cohomology theories.
The long paper is carefully written with full details of constructions and proofs.
Reviewer: Thomas Schick (Göttingen)
MSC:
| 55N20 | Generalized (extraordinary) homology and cohomology theories in algebraic topology |
| 57R19 | Algebraic topology on manifolds and differential topology |
