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arXiv · 2611.08406

Exotic smooth structures on complex projective spaces

Yueshi Hou·Hana Jia Kong·Zhouli Xu
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Abstract

We study smooth structures on the topological manifold underlying complex projective space. We conjecture that every complex projective space $\mathbb{CP}^n$ 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 $\mathbb{CP}^n$ admits an exotic smooth structure has natural density one. Second, we explicitly identify $266$ congruence classes modulo $288$ such that $\mathbb{CP}^n$ 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 $\mathbb{CP}^n$ 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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Yueshi Hou, Hana Jia Kong, Zhouli Xu. 2026-11-05. Exotic smooth structures on complex projective spaces. https://arxiv.org/abs/2611.08406

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