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arXiv · 2401.15412

Obstruction theory for the $\mathbb{Z}_2$-index of $4$-manifolds

Chahrazade Matmat·Christian Blanchet
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Abstract

We develop a complete obstruction theory for the $\mathbb{Z}_2$-index of a compact connected 4-dimensional manifold with free involution. This $\mathbb{Z}_2$-index, equal to the minimum integer $n$ for which there exists an equivariant map with target the $n$-sphere with antipodal involution, is computed in two steps using cohomology with twisted coefficients. The key ingredient is a spectral sequence computing twisted cohomology of the orbit space of a free involution on odd complex projective spaces. We illustrate the main results with various examples including computation of the secondary obstruction.

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Chahrazade Matmat, Christian Blanchet. 2024-08-25. Obstruction theory for the $\mathbb{Z}_2$-index of $4$-manifolds. https://arxiv.org/abs/2401.15412

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