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Abstract
We show that the main theorem of Morse theory holds for a large class of functions on singular spaces. The function must satisfy certain conditions extending the usual requirements on a manifold that Condition C holds and the gradient flow around the critical sets is wellbehaved, and the singular space must satisfy a local deformation retract condition. We then show that these conditions are satisfied when the function is the normsquare of a moment map on an affine variety, and that the homotopy equivalence from this theorem is equivariant with respect to the associated Hamiltonian group action. An important special case of these results is that the main theorem of Morse theory holds for the norm square of a moment map on the space of representations of a finite quiver with relations.
Original language  English 

Pages (fromto)  4730–4763 
Number of pages  34 
Journal  International Mathematics Research Notices 
Volume  2019 
Issue number  15 
DOIs  
Publication status  Published  18 Nov 2017 
Bibliographical note
© The Author 2017. Published by Oxford University Press.Profiles
Activities
 5 Invited talk

The Morse complex on singular spaces
Graeme Peter Desmond Wilkin (Invited speaker)
17 Sept 2022Activity: Talk or presentation › Invited talk

Morse Complex on Singular Spaces
Graeme Peter Desmond Wilkin (Invited speaker)
18 Mar 2022Activity: Talk or presentation › Invited talk

Algebraic and geometric classification of YangMillsHiggs flow lines
Graeme Peter Desmond Wilkin (Invited speaker)
13 May 2020Activity: Talk or presentation › Invited talk