Toward an exotic S^2× S^2 and an exotic ℂℙ^2#ℂℙ̅^2

Abstract

We study a specified Luttinger filling in the construction of Lidman and Piccirillo whose simple connectivity would yield exotic S²× S² and ℂℙ²#ℂℙ̅². An earlier version of this manuscript claimed simple connectivity from incorrectly derived group relations; that claim is withdrawn. We derive the complete fundamental-group presentation and express the group as a one-relator quotient of an iterated HNN extension containing the genus-two surface group. The filled group is trivial exactly when the final relation kills a specified fiber commutator. We also determine its universal central extension with cyclic kernel, construct geometric realizations of it, and extend the boundary involution over the filling. The final filling problem remains unresolved.

Cite

BibTeX RIS

BibTeX

@misc{wuebben2026toward,
  title = {{Toward an exotic $S^2\times S^2$ and an exotic $\mathbbCP^2\#\overline\mathbbCP^2$}},
  author = {Bernd Johannes Wuebben},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.17267},
  eprint = {2608.17267},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.17267},
}

RIS

TY  - UNPB
TI  - Toward an exotic S^2× S^2 and an exotic ℂℙ^2#ℂℙ̅^2
AU  - Bernd Johannes Wuebben
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2608.17267
UR  - https://arxiv.org/abs/2608.17267
AB  - We study a specified Luttinger filling in the construction of Lidman and Piccirillo whose simple connectivity would yield exotic S^2× S^2 and ℂℙ^2#ℂℙ̅^2. An earlier version of this manuscript claimed simple connectivity from incorrectly derived group relations; that claim is withdrawn. We derive the complete fundamental-group presentation and express the group as a one-relator quotient of an iterated HNN extension containing the genus-two surface group. The filled group is trivial exactly when the final relation kills a specified fiber commutator. We also determine its universal central extension with cyclic kernel, construct geometric realizations of it, and extend the boundary involution over the filling. The final filling problem remains unresolved.
ER  -