Free maps in critical dimension on πm
Abstract
The critical dimension for free maps on m-manifolds is qm=m(m+3)/2. We show that Freeβ(πm,βqm)β β for every mβ₯ 1. The main tool is a product construction: given free maps in critical dimension on πa and on πb, together with a cross-free map on πaΓ πb, whose mixed Hessian is invertible everywhere, we obtain a free map in critical dimension on πa+b, the osculating matrix of the product being block-triangular. Cross-free maps are additive in each argument, and in the torus setting none exists when one factor is one-dimensional. We construct explicit cross-free maps on π2Γ π2 and on π3Γπ4 using quaternionic multiplication. Combined with the low-dimensional cases mβ€ 5, constructed explicitly by the author in a previous publication, these yield the result by induction.
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BibTeX
@misc{leo2026free,
title = {{Free maps in critical dimension on $\mathbbT^m$}},
author = {Roberto De Leo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23172},
eprint = {2609.23172},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23172},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Free maps in critical dimension on π^m
AU - Roberto De Leo
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23172
UR - https://arxiv.org/abs/2609.23172
AB - The critical dimension for free maps on m-manifolds is q_m=m(m+3)/2. We show that Free^β(π^m,β^q_m)β β
for every mβ₯ 1. The main tool is a product construction: given free maps in critical dimension on π^a and on π^b, together with a cross-free map on π^aΓ π^b, whose mixed Hessian is invertible everywhere, we obtain a free map in critical dimension on π^a+b, the osculating matrix of the product being block-triangular. Cross-free maps are additive in each argument, and in the torus setting none exists when one factor is one-dimensional. We construct explicit cross-free maps on π^2Γ π^2 and on π^3Γπ^4 using quaternionic multiplication. Combined with the low-dimensional cases mβ€ 5, constructed explicitly by the author in a previous publication, these yield the result by induction.
ER -