NCG Seminar

[Ludwik Dabrowski]

Walter van Suijlekom (Radboud University Nijmegen)

"Factorization of Dirac operators in unbounded KK-theory"

Tuesday 26 April, 9:00, SISSA room 5.

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Abstract: Unbounded KK-cycles are used in a geometric refinement
of Kasparov's bivariant K-theory, capturing for instance also notions
such as curvature. I will explain the general theory and introduce
a notion of curvature associated to the internal tensor product
of (suitable) unbounded KK-cycles. It is essentially given by the
difference between the square of the Kasparov product, and the Kasparov
product of the squares.

We illustrate our theory by several examples:
- Almost-commutative manifolds, generalizing the usual notion
of curvature on a vector bundle
- Riemannian submersions of (compact) Riemannian spin manifolds,
where we establish the factorization of Dirac operators in a vertical
and horizontal part, up to an explicit obstructing curvature term
- Toric deformations (eg the Connes-Landi sphere) factorized over
a commutative base
- Immersions of spheres into Euclidean space

Based on joint work with Jens Kaad, Bram Mesland and Adam Rennie.