Exotic smooth structures on complex projective spaces
Abstract
We study smooth structures on the topological manifold underlying complex projective space. We conjecture that every complex projective space ℂℙn with n>3 admits an exotic smooth structure. We prove two results providing strong evidence for this conjecture. First, the set of complex dimensions n for which ℂℙn admits an exotic smooth structure has natural density one. Second, we explicitly identify $266$ congruence classes modulo $288$ such that ℂℙn admits an exotic smooth structure whenever n>3 lies in one of these classes. These classes cover more than 92% of complex dimensions, improving the previously known lower bound of approximately 36%. We prove these results by first relating the concordance group of ℂℙn to suitable Atiyah--Hirzebruch filtration subgroups of its stable cohomotopy group, and then applying computations in chromatic homotopy theory at height $2$.
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BibTeX
@misc{hou2026exotic,
title = {{Exotic smooth structures on complex projective spaces}},
author = {Yueshi Hou and Hana Jia Kong and Zhouli Xu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08406},
eprint = {2611.08406},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08406},
}
RIS
TY - UNPB
TI - Exotic smooth structures on complex projective spaces
AU - Yueshi Hou
AU - Hana Jia Kong
AU - Zhouli Xu
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08406
UR - https://arxiv.org/abs/2611.08406
AB - We study smooth structures on the topological manifold underlying complex projective space. We conjecture that every complex projective space ℂℙ^n with n>3 admits an exotic smooth structure. We prove two results providing strong evidence for this conjecture. First, the set of complex dimensions n for which ℂℙ^n admits an exotic smooth structure has natural density one. Second, we explicitly identify $266$ congruence classes modulo $288$ such that ℂℙ^n admits an exotic smooth structure whenever n>3 lies in one of these classes. These classes cover more than 92% of complex dimensions, improving the previously known lower bound of approximately 36%. We prove these results by first relating the concordance group of ℂℙ^n to suitable Atiyah--Hirzebruch filtration subgroups of its stable cohomotopy group, and then applying computations in chromatic homotopy theory at height $2$.
ER -