A Morse-Bott type complex for intersection homology and the Bismut-Zhang torsion

Ursula Ludwig
University of Duisburg-Essen, Germany

Date(s) : 17/12/2020   iCal
14 h 00 min - 16 h 30 min

The famous Morse-Thom-Smale complex on a smooth compact manifold M associated to a smooth real valued Morse function f is a complex generated by the critical points of the Morse function f and computing the singular (co-)homology of M. An important generalisation of this complex for smooth Morse-Bott functions is due to Austin and Braam. The Morse-Bott complex of Austin and Braam is generated by the de Rham complexes of all critical submanifolds.

The aim of this talk is to adapt the construction of Austin and Braam for a stratified pseudomanifold and intersection cohomology. The main idea is to replace the de Rham complex in the construction of Austin and Braam with the complex of liftable intersection forms, due to Brasselet, Hector and Saralegi.

While the above constructed « singular » Morse-Bott type complex may be of some interest in itself, our main motivation comes from the the comparison theorem of analytic and topological torsion, aka Cheeger-Müller theorem, and its generalisation to singular spaces, which in recent years has become a fruitful area of research.

The Morse-Bott complex can be used to define a torsion, which conjecturally could serve as a « topological counterpart » in a Cheeger-Müller theorem for wedge singularities.


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