Weights of finite cyclic actions on definite $4$-manifolds
Abstract
Let X be a closed, smooth, orientable, positive definite $4$-manifold with H1(X ; ℤ) = 0. Let p be a prime and suppose that G = ℤp acts smoothly on X (if p=2 we also require an assumption on how G acts on H2(X ; ℤ)). Using equivariant Yang--Mills theory, Hambleton--Lee and Hambleton--Tanase proved (in the simply-connected case) that the fixed point set and tangential isotropy representations coincide with that of an equivariant connected sum of copies of ℂℙ2 on which G acts linearly. We give a new proof of this result using equivariant Seiberg--Witten theory. Furthermore we also determine the weights of all equivariant line bundles on X, showing that these likewise coincide with that of an equivariant connected sum of linear actions on ℂℙ2.
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BibTeX
@misc{baraglia2026weights,
title = {{Weights of finite cyclic actions on definite $4$-manifolds}},
author = {David Baraglia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14328},
eprint = {2609.14328},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14328},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Weights of finite cyclic actions on definite $4$-manifolds
AU - David Baraglia
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14328
UR - https://arxiv.org/abs/2609.14328
AB - Let X be a closed, smooth, orientable, positive definite $4$-manifold with H_1(X ; ℤ) = 0. Let p be a prime and suppose that G = ℤ_p acts smoothly on X (if p=2 we also require an assumption on how G acts on H^2(X ; ℤ)). Using equivariant Yang--Mills theory, Hambleton--Lee and Hambleton--Tanase proved (in the simply-connected case) that the fixed point set and tangential isotropy representations coincide with that of an equivariant connected sum of copies of ℂℙ^2 on which G acts linearly. We give a new proof of this result using equivariant Seiberg--Witten theory. Furthermore we also determine the weights of all equivariant line bundles on X, showing that these likewise coincide with that of an equivariant connected sum of linear actions on ℂℙ^2.
ER -