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Factorisation of equivariant spectral triples in unbounded KK-theory

Journal Article


Abstract


  • We provide sufficient conditions to factorise an equivariant spectral triple as a Kasparov product of unbounded classes constructed from the group action on the algebra and from the fixed point spectral triple. We show that if factorisation occurs, then the equivariant index of the spectral triple vanishes. Our results are for the action of compact abelian Lie groups, and we demonstrate them with examples from manifolds and θ-deformations. In particular, we show that equivariant Dirac-type spectral triples on the total space of a torus principal bundle always factorise. Combining this with our index result yields a special case of the Atiyah-Hirzebruch theorem. We also present an example that shows what goes wrong in the absence of our sufficient conditions (and how we get around it for this example).

Publication Date


  • 2019

Citation


  • Forsyth, I. & Rennie, A. (2019). Factorisation of equivariant spectral triples in unbounded KK-theory. Journal of the Australian Mathematical Society, 107 (2), 145-180.

Scopus Eid


  • 2-s2.0-85072116757

Number Of Pages


  • 35

Start Page


  • 145

End Page


  • 180

Volume


  • 107

Issue


  • 2

Place Of Publication


  • United Kingdom

Abstract


  • We provide sufficient conditions to factorise an equivariant spectral triple as a Kasparov product of unbounded classes constructed from the group action on the algebra and from the fixed point spectral triple. We show that if factorisation occurs, then the equivariant index of the spectral triple vanishes. Our results are for the action of compact abelian Lie groups, and we demonstrate them with examples from manifolds and θ-deformations. In particular, we show that equivariant Dirac-type spectral triples on the total space of a torus principal bundle always factorise. Combining this with our index result yields a special case of the Atiyah-Hirzebruch theorem. We also present an example that shows what goes wrong in the absence of our sufficient conditions (and how we get around it for this example).

Publication Date


  • 2019

Citation


  • Forsyth, I. & Rennie, A. (2019). Factorisation of equivariant spectral triples in unbounded KK-theory. Journal of the Australian Mathematical Society, 107 (2), 145-180.

Scopus Eid


  • 2-s2.0-85072116757

Number Of Pages


  • 35

Start Page


  • 145

End Page


  • 180

Volume


  • 107

Issue


  • 2

Place Of Publication


  • United Kingdom