
Speaker: 
Ezio Vasselli, Roma

Title: 
\(KK\)theory for \(C^*\)precosheaves and holonomyequivariance

Abstract: 
Let \(X\) be a space with a good base \(\Delta\) ordered under inclusion.
Then any \(C^*\)precosheaf over \(\Delta\) defines a \(C^*\)dynamical system for the fundamental group
\(\pi_1(X)\),
called the holonomy \(C^*\)system.
We define Kasparov cycles for \(C^*\)precosheaves \(A,B\) and describe them in terms of
holonomyequivariant
cycles carrying an additional filtration structure. This leads to the notion of holonomyequivariant
\(KK\)theory,
that we denote by \(KK^\Delta(A,B)\).
When the \(C^*\)precosheaves are those defined by ideal structures of \(C^*\)algebras, \(KK^\Delta\)
can be regarded as a twist of KasparovKirchberg \(KK_X\)theory, where \(X\) is the space acting on the
Jacobson spectra:
if \(\pi_1(X)\) is trivial, then there is a natural transformation \(KK_X \to KK^\Delta\).
Joint works with Giuseppe Ruzzi.

