Morse-Bott inequalities on Lefschetz complexes

@misc{nishikawa2026morsebott,
  title = {{Morse-Bott inequalities on Lefschetz complexes}},
  author = {Yuto Nishikawa},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33329},
  eprint = {2609.33329},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33329},
  note = {CC BY-NC-ND 4.0},
}

TY  - UNPB
TI  - Morse-Bott inequalities on Lefschetz complexes
AU  - Yuto Nishikawa
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33329
UR  - https://arxiv.org/abs/2609.33329
AB  - We develop a discrete Morse-Bott theory for Lefschetz complexes with real-valued incidence functions and finitely many cells in each dimension. Our main result is a reduction procedure, based on four elementary operations called Moves, for computing the remainder series R_t associated with the Morse-Bott inequality. These Moves preserve R_t, and the procedure recursively constructs, using finitely many Moves in each dimension, a disjoint union of two-cell elementary blocks with vanishing Poincaré series. The resulting decomposition computes the coefficients of R_t by counting the blocks in the corresponding dimensions. The same procedure also computes the Betti numbers of the original complex by counting the cells removed as isolated cells. As consequences, we obtain the nonnegativity of the coefficients of R_t and the Morse-Bott inequality for Lefschetz complexes. The method applies in particular to CW complexes, where the nonnegativity is obtained by identifying each coefficient of R_t with the number of elementary blocks in the corresponding dimension.
ER  -