Flat bundles, von Neumann algebras and $K$-theory with $\R/\Z$-coefficients

Abstract

Let M be a closed manifold and α ∶ π1(M ) → Un a representation. We give a purely K-theoretic description of the associated element [α] in the K-theory of M with R Z-coefficients. To that end, it is convenient to describe the R Z-K-theory as a relative K-theory with respect to the inclusion of C in a finite von Neumann algebra B. We use the following fact: there is, associated with α, a finite von Neumann algebra B together with a flat bundle E → M with fibers B, such that Eα ⊗ E is canonically isomorphic with C n ⊗ E , where Eα denotes the flat bundle with fiber C n associated with α. We also discuss the spectral flow and rho type description of the pairing of the class [α] with the K-homology class of an elliptic selfadjoint (pseudo)-differential operator D of order 1.

R/Z-valued index theory via geometric K-homology

“…This completes the proof that the slant product K 1 Basu (Y × X; R/Z) × K 1 (X) → K 0 (Y ; R/Z) is well-defined.…”

“…K-homology with coefficients In this example, we discuss K-homology with coefficients in certain abelian groups. An introduction to K-theory and K-homology with coefficients in the abelian groups of interest here can be found in [13,Section 23.15] (also see [1,3,4]); geometric K-homology with coefficients in Z/kZ is the topic of [15,16]. Given an abelian group, G, and a finite CW-complex, X, we denote the K-theory of X with coefficients in G by K * (X; G) and the K-homology of X with coefficients in G by K * (X; G).…”

“…(2) F is the vector bundle defined by ( Ē ⊗f * (V 1 )) on W ×{0} and ( Ē ⊗f * (V 2 )) on W × {1}; (3) ᾱ is the identity on the fibers at W × {0} and ϕ on the fibers at W × {1}; Finally, for vector bundle modification, let (V 1 , V 2 , ϕ) be a cocycle in K 1 Basu (Y × X; R/Z), (M, E, f ) be a Baum-Douglas cycle and V be an even rank spin c -vector bundle over M . Then, the well-definedness of the slant product follows since…”

“…We will only discuss the odd pairing in detail. A cocycle in K 1 Lott (X; R/Z) is given by ((V 1 , ∇ 1 ), (V 2 , ∇ 2 ), ω) where V 1 and V 2 are complex Hermitian vector bundles, ∇ 1 and ∇ 2 are Hermitian connections on V 1 and V 2 respectively, and ω ∈ Ω odd (M )/im(d) such that dω = ch(∇ 1 ) − ch(∇ 2 ).…”

“…The reader should also note that the starting point for the construction considered here was [22,Remark 6.12]. Recently (see [1]) Antonini, Azzali, and Skandalis have considered an operator algebraic approach to R/Z-valued index theory.…”