[Paper Review] Foliated manifolds, algebraic K-theory, and a secondary invariant

Ulrich Bunke|arXiv (Cornell University)|Jul 23, 2015

Homotopy and Cohomology in Algebraic Topology13 references3 citations

TL;DR

This paper introduces a ${\mathbb{C}}/{\mathbb{Z}}$-valued secondary invariant for odd-dimensional, stably framed foliated manifolds equipped with a partially flat vector bundle, defined via differential $K$-theory or spectral invariants. The key result establishes a formula linking this invariant to a regulator map from algebraic $K$-theory of smooth functions to connective $K$-theory with ${\mathbb{C}}/{\mathbb{Z}}$ coefficients, unifying classical invariants like the Adams $e$-invariant and Godbillon-Vey class under a single framework.

ABSTRACT

We introduce a $\mathbb{C}/\mathbb{Z}$-valued invariant of a foliated manifold with a stable framing and with a partially flat vector bundle. This invariant can be expressed in terms of integration in differential $K$-theory, or alternatively, in terms of $η$-invariants of Dirac operators and local correction terms. Initially, the construction of the element in $\mathbb{C}/\mathbb{Z}$ involves additional choices. But if the codimension of the foliation is sufficiently small, then this element is independent of these choices and therefore an invariant of the data listed above. We show that the invariant comprises various classical invariants like Adams' $e$-invariant, the $ρ$-invariant of twisted Dirac operators, or the Godbillon-Vey invariant from foliation theory. Using methods from differential cohomology theory we construct a regulator map from the algebraic $K$-theory of smooth functions on a manifold to its connective $K$-theory with $\mathbb{C}/\mathbb{Z}$ coefficients. Our main result is a formula for the invariant in terms of this regulator and integration in algebraic and topological $K$-theory.

Motivation & Objective

• To define a new ${\mathbb{C}}/{\mathbb{Z}}$-valued invariant for foliated manifolds with stable framing and partially flat bundles, independent of geometric choices under codimension constraints.
• To establish a deep connection between this invariant and algebraic $K$-theory of smooth functions on the base space of the foliation.
• To unify classical invariants such as the Adams $e$-invariant, $\rho$-invariant, and Godbillon-Vey invariant within a single cohomological framework.
• To construct a regulator map from algebraic $K$-theory to connective $K$-theory with ${\mathbb{C}}/{\mathbb{Z}}$ coefficients, and show its compatibility with integration in differential $K$-theory.

Proposed method

• The invariant is constructed using integration in differential complex $K$-theory $\widehat{KU}^*$, which provides a geometric realization of the invariant via curvature and connection data.
• An alternative expression is given in terms of $\eta$-invariants of twisted Dirac operators and correction terms involving transgressed characteristic forms.
• The paper defines a regulator map $\mathrm{reg}_X: K_p(C^\infty(X)) \to \mathbf{ku}{\mathbb{C}}/{\mathbb{Z}}^{-p-1}(X)$, linking algebraic $K$-theory of smooth functions to topological $K$-theory with ${\mathbb{C}}/{\mathbb{Z}}$ coefficients.
• A Riemann-Roch-type diagram is used to relate algebraic and topological $K$-theory, with differential refinements, to prove the main formula.
• The construction is shown to be independent of auxiliary choices (metric, connection extensions) precisely when $2\mathrm{codim}(\mathcal{F}) < \dim(M)$.
• The invariant is interpreted as the composition of the regulator map and integration in connective $K$-theory, yielding a class in ${\mathbb{C}}/{\mathbb{Z}}$.

Experimental results

Research questions

• RQ1 How can a secondary invariant be defined for foliated manifolds with stable framing and partially flat bundles, independent of geometric choices?
• RQ2 What is the precise relationship between this invariant and algebraic $K$-theory of smooth functions on the base space of the foliation?
• RQ3 How does this invariant unify classical invariants such as the Adams $e$-invariant, $\rho$-invariant, and Godbillon-Vey class?
• RQ4 Can the regulator map from algebraic $K$-theory to connective $K$-theory with ${\mathbb{C}}/{\mathbb{Z}}$ coefficients be constructed and used to express the invariant?
• RQ5 Under what topological conditions does the invariant become independent of auxiliary geometric data?

Key findings

• The invariant $\rho(M,\mathcal{F},\nabla^I,s) \in \mathbb{C}/\mathbb{Z}$ is well-defined and independent of geometric choices precisely when $2\mathrm{codim}(\mathcal{F}) < \dim(M)$.
• The invariant admits two equivalent descriptions: one via integration in differential $K$-theory, and another via $\eta$-invariants and transgressed characteristic forms.
• The main result expresses the invariant as $\rho(M,\mathcal{F},\nabla^I,s) = \pi_!^o(\mathrm{reg}_X(f^{o_s}_!( [V,\nabla^I]^{\mathrm{alg}} )))$, linking it to algebraic $K$-theory and a regulator map.
• The invariant subsumes the Adams $e$-invariant, the $\rho$-invariant for flat bundles, and the Godbillon-Vey invariant as special cases.
• The regulator map $\mathrm{reg}_X$ is constructed as a natural transformation from algebraic $K$-theory of $C^\infty(X)$ to connective $K$-theory with ${\mathbb{C}}/{\mathbb{Z}}$ coefficients.
• In the case of a foliation of the form $(P \times X, T_{\mathbb{C}}P \boxplus \{0\})$, the invariant is shown to factor through the algebraic $K$-theory of $C^\infty(X)$, with the codimension condition ensuring well-definedness.