Abstract
Infra-solvmanifolds are a certain class of aspherical manifolds which generalize both flat manifolds and almost flat manifolds (i.e., infra-nilmanifolds). Every 4-dimensional infra-solvmanifold is diffeomorphic to a geometric \(4\)-manifold with geometry of solvable Lie type. There were questions about whether or not all 4-dimensional infra-solvmanifolds bound. We answer this affirmatively. On each infra-solvmanifold \(M\) admitting \(\text {Nil}^{3} \times {\mathbb R},\,\text {Nil}^{4},\,\text {Sol}^{3} \times {\mathbb R}\), or \(\text {Sol}_1^{4}\) geometry, an isometric involution with 2-dimensional fixed set is constructed. The Stiefel–Whitney number \(\omega _1^4(M)\) vanishes by a result of R.E. Stong and from this it follows that all Stiefel–Whitney numbers vanish.
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The result of this paper is one part of the author’s Ph.D. thesis. The author would like to thank his adviser, Kyung Bai Lee, for his guidance and Jim Davis for very helpful comments.
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Thuong, S.V. All 4-dimensional infra-solvmanifolds are boundaries. Geom Dedicata 176, 315–328 (2015). https://doi.org/10.1007/s10711-014-9970-6
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DOI: https://doi.org/10.1007/s10711-014-9970-6