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Unitary S1-Manifolds with Four Fixed Points and Kosniowski's Conjecture

  • Zhi Lü
  • Jun Ma
  • Qiangbo Tan
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Abstract

We study unitary S1-manifolds with four isolated fixed points in relation to Kosniowski's conjectural inequality 4χ(M)>dim M. For every integer m≥3, we show that there exists a unitary S1-manifold Mm18 with exactly four isolated fixed points, and these manifolds are pairwise distinct and nonzero in ordinary complex bordism. Since χ(Mm18)=4, they give an infinite family of counterexamples to 4χ(M)>dim M. By gluing two blocks formed by G2/SU(3)≅ S6 and the unit disk D(V) of 12- dimensional real S1-representation V along their equivariantly identified free boundaries, we explicitly construct such a manifold M183. We also prove that the four-point case is null-bordant in real dimension 16 and in every real dimension greater than 18; hence 18 is the smallest dimension in which such a counterexample occurs.

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Cite

BibTeXRIS
BibTeX
@misc{lu2026unitary,
  title = {{Unitary $S^1$-Manifolds with Four Fixed Points and Kosniowski's Conjecture}},
  author = {Zhi Lü and Jun Ma and Qiangbo Tan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23855},
  eprint = {2610.23855},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23855},
}
RIS
TY  - UNPB
TI  - Unitary S^1-Manifolds with Four Fixed Points and Kosniowski's Conjecture
AU  - Zhi Lü
AU  - Jun Ma
AU  - Qiangbo Tan
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23855
UR  - https://arxiv.org/abs/2610.23855
AB  - We study unitary S^1-manifolds with four isolated fixed points in relation to Kosniowski's conjectural inequality 4χ(M)>dim M. For every integer m≥3, we show that there exists a unitary S^1-manifold M_m^18 with exactly four isolated fixed points, and these manifolds are pairwise distinct and nonzero in ordinary complex bordism. Since χ(M_m^18)=4, they give an infinite family of counterexamples to 4χ(M)>dim M. By gluing two blocks formed by G_2/SU(3)≅ S^6 and the unit disk D(V) of 12- dimensional real S^1-representation V along their equivariantly identified free boundaries, we explicitly construct such a manifold M^18_3. We also prove that the four-point case is null-bordant in real dimension 16 and in every real dimension greater than 18; hence 18 is the smallest dimension in which such a counterexample occurs.
ER  -