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A connection between cut locus, Thom space and Morse–Bott functions

Somnath Basu and Sachchidanand Prasad

Algebraic & Geometric Topology 23 (2023) 4185–4233

Associated to every closed, embedded submanifold N in a connected Riemannian manifold M, there is the distance function dN which measures the distance of a point in M from N. We analyze the square of this function and show that it is Morse–Bott on the complement of the cut locus Cu (N) of N provided M is complete. Moreover, the gradient flow lines provide a deformation retraction of M Cu (N) to N. If M is a closed manifold, then we prove that the Thom space of the normal bundle of N is homeomorphic to MCu (N). We also discuss several interesting results which are either applications of these or related observations regarding the theory of cut locus. These results include, but are not limited to, a computation of the local homology of singular matrices, a classification of the homotopy type of the cut locus of a homology sphere inside a sphere, a deformation of the indefinite unitary group U(p,q) to U(p) × U(q) and a geometric deformation of GL (n, ) to O(n, ) which is different from the Gram–Schmidt retraction.

cut locus, distance function, Morse–Bott function, Thom space
Mathematical Subject Classification
Primary: 53B21, 53C22, 55P10
Secondary: 32B20, 57R19, 58C05
Received: 4 June 2021
Revised: 15 February 2023
Accepted: 16 December 2021
Published: 23 November 2023
Somnath Basu
Department of Mathematics and Statistics
Indian Institute of Science Education and Research
Sachchidanand Prasad
Department of Mathematics and Statistics
Indian Institute of Science Education and Research

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