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Odd-dimensional lens-space Yamabe counterexamples
Abstract
Let n≥5 be odd and let L3qn=Sn/C3q be the standard lens space, where q>3 is prime. We prove that for all sufficiently large q, σ(L3qn) > (3q)−2/nY(Sn)=Y(L3qn,[grd]). Here [grd] denotes the quotient-round conformal class. Combining spin bordism with the surgery theorem of Ammann--Dahl--Humbert, we establish σ(L3qn)≥ min {q−2/nY(Sn),Λn}, where Λn>0 depends only on n. For sufficiently large q, this exceeds the quotient-round value, showing that the quotient-round conformal class need not maximize the smooth Yamabe invariant in any odd dimension n≥5.
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BibTeX
@misc{loren2026odddimensional,
title = {{Odd-dimensional lens-space Yamabe counterexamples}},
author = {Jakob Loren},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18662},
eprint = {2610.18662},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18662},
}
RIS
TY - UNPB
TI - Odd-dimensional lens-space Yamabe counterexamples
AU - Jakob Loren
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18662
UR - https://arxiv.org/abs/2610.18662
AB - Let n≥5 be odd and let L_3q^n=S^n/C_3q be the standard lens space, where q>3 is prime. We prove that for all sufficiently large q, σ(L_3q^n) > (3q)^−2/nY(S^n)=Y(L_3q^n,[g_rd]). Here [g_rd] denotes the quotient-round conformal class. Combining spin bordism with the surgery theorem of Ammann--Dahl--Humbert, we establish σ(L_3q^n)≥ min {q^−2/nY(S^n),Λ_n}, where Λ_n>0 depends only on n. For sufficiently large q, this exceeds the quotient-round value, showing that the quotient-round conformal class need not maximize the smooth Yamabe invariant in any odd dimension n≥5.
ER -