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

Basu, Somnath; Prasad, Sachchidanand

doi:10.2140/agt.2023.23.4185

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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

DOI: 10.2140/agt.2023.23.4185

Abstract

Associated to every closed, embedded submanifold $N$ in a connected Riemannian manifold $M$ , there is the distance function $d_{N}$ 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 $M ∕ Cu (N)$ . 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 $U (p) \times U (q)$ and a geometric deformation of $GL (n, ℝ)$ to $O (n, ℝ)$ which is different from the Gram–Schmidt retraction.

Keywords

cut locus, distance function, Morse–Bott function, Thom space

Mathematical Subject Classification

Primary: 53B21, 53C22, 55P10

Secondary: 32B20, 57R19, 58C05

References

Publication

Received: 4 June 2021

Revised: 15 February 2023

Accepted: 16 December 2021

Published: 23 November 2023

Authors

Somnath Basu
Department of Mathematics and Statistics Indian Institute of Science Education and Research Kolkata India

Sachchidanand Prasad
Department of Mathematics and Statistics Indian Institute of Science Education and Research Kolkata India

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Volume 23, issue 9 (2023)