Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 35 min ago
PreprintarXiv licencearXiv

On Reeb spaces of non-proper smooth functions which are $1$-dimensional CW complexes

  • Naoki Kitazawa
Open at sourcePDFDOICite

Abstract

We discuss Reeb spaces of non-proper smooth functions. (Non-)proper functions are maps between topological spaces the preimages of compact sets by which are compact (resp. may not be compact). Their Reeb spaces are the spaces of connected components of their level sets and with the natural quotient topologies. In proper cases, explicit theory are developing since the 1950s. This is closely related to Morse(-Bott) function theory and in such cases we have graphs naturally. Recently, related general topological studies with combinatorial ones are actively developing, mainly due to Gelbukh and Saeki, and they are at most $1$-dimensional in tame cases. We extend a theorem by Saeki in the 2020s: the Reeb space of a smooth function on a closed manifold is naturally a graph if and only if its critical value set is finite. The author has been a pioneer of the non-proper case and previously investigated examples.

Related

  1. Adaptive Local Proper Orthogonal Decomposition via Graph Partitioning and Provable Error-Bounded Lossy Compression for Parametric Series in Finite-Dimensional Function Spaces
    Roberto Nuca, Matteo Parsani, Lisandro Dalcin
    arXivPreprintarXiv licenceFull textNumerical analysis & scientific computingarXiv
  2. Smooth maps onto regions in Euclidean spaces respecting height functions
    Naoki Kitazawa
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  3. Diagonalization of nonlinear functions in finite-dimensional spaces and the case of ℝ2
    Roger Arnau, Juan Carlos Cortés López, Álvaro González Cortés +1
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  4. Rigidity of symmetric holomorphic functions on infinite-dimensional sequence spaces
    David Feldman, Jon Bannon
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  5. Refined proper descent for left completed sheaves on locally compact Hausdorff spaces
    Li He
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{kitazawa2026reeb,
  title = {{On Reeb spaces of non-proper smooth functions which are $1$-dimensional CW complexes}},
  author = {Naoki Kitazawa},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.00053},
  eprint = {2609.00053},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00053},
}
RIS
TY  - UNPB
TI  - On Reeb spaces of non-proper smooth functions which are $1$-dimensional CW complexes
AU  - Naoki Kitazawa
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00053
UR  - https://arxiv.org/abs/2609.00053
AB  - We discuss Reeb spaces of non-proper smooth functions. (Non-)proper functions are maps between topological spaces the preimages of compact sets by which are compact (resp. may not be compact). Their Reeb spaces are the spaces of connected components of their level sets and with the natural quotient topologies. In proper cases, explicit theory are developing since the 1950s. This is closely related to Morse(-Bott) function theory and in such cases we have graphs naturally. Recently, related general topological studies with combinatorial ones are actively developing, mainly due to Gelbukh and Saeki, and they are at most $1$-dimensional in tame cases. We extend a theorem by Saeki in the 2020s: the Reeb space of a smooth function on a closed manifold is naturally a graph if and only if its critical value set is finite. The author has been a pioneer of the non-proper case and previously investigated examples.
ER  -