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PreprintCC BY-NC-SA 4.0arXiv

Homeomorphisms of surfaces in $4$-manifolds

  • Anthony Conway
  • Daniel Kasprowski
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Abstract

This paper establishes necessary and sufficient conditions for locally flat knotted surfaces in simply-connected $4$-manifolds to be equivalent. For surfaces with knot group ℤd, we extend results of Lee-Wilczynski from spheres to surfaces of arbitrary genus; the surfaces are permitted to be nonorientable and have boundary. We prove that most projective planes with knot group ℤ2 and the same Euler number are determined by the equivariant intersection form of their exterior. We also prove that knots with prime power determinants bound at most one Moebius band in D4 with knot group ℤ2 and a given Euler number. Cancellation results lead to new criteria for homologous discs to be equivalent rel. boundary. Finally, we determine the topological extendable mapping class group of knotted surfaces with abelian knot group.

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Cite

BibTeXRIS
BibTeX
@misc{conway2026homeomorphisms,
  title = {{Homeomorphisms of surfaces in $4$-manifolds}},
  author = {Anthony Conway and Daniel Kasprowski},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01545},
  eprint = {2609.01545},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01545},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Homeomorphisms of surfaces in $4$-manifolds
AU  - Anthony Conway
AU  - Daniel Kasprowski
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.01545
UR  - https://arxiv.org/abs/2609.01545
AB  - This paper establishes necessary and sufficient conditions for locally flat knotted surfaces in simply-connected $4$-manifolds to be equivalent. For surfaces with knot group ℤ_d, we extend results of Lee-Wilczynski from spheres to surfaces of arbitrary genus; the surfaces are permitted to be nonorientable and have boundary. We prove that most projective planes with knot group ℤ_2 and the same Euler number are determined by the equivariant intersection form of their exterior. We also prove that knots with prime power determinants bound at most one Moebius band in D^4 with knot group ℤ_2 and a given Euler number. Cancellation results lead to new criteria for homologous discs to be equivalent rel. boundary. Finally, we determine the topological extendable mapping class group of knotted surfaces with abelian knot group.
ER  -