Morse-Bott inequalities for manifolds with boundary

Ryuma Orita
Tokyo Journal of Mathematics, vol. 41 (2018), no. 1, pp. 113–130
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Overview

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Morse theory relates the topology of a manifold to the critical points of a smooth function. Morse–Bott theory allows these critical points to form submanifolds. On a manifold with boundary, the behavior of the function along the boundary must also enter this relationship.

This paper defines Morse–Bott functions on compact manifolds with boundary and proves inequalities relating their critical submanifolds to the Betti numbers of the manifold. The boundary need not be a level set of the function. The inequalities include contributions from the interior and from the restriction of the function to the boundary, and the proof uses an explicit perturbation to Morse functions while controlling the behavior near the boundary.

@article{orita2018morsebott,
  author        = {Orita, Ryuma},
  title         = {{Morse-Bott inequalities for manifolds with boundary}},
  journal       = {Tokyo Journal of Mathematics},
  volume        = {41},
  number        = {1},
  pages         = {113--130},
  year          = {2018},
  doi           = {10.3836/tjm/1502179256},
  eprint        = {1609.02381},
  archivePrefix = {arXiv}
}