On signatures associated with ramified coverings and embedding problems
Annales de l'Institut Fourier, Volume 23 (1973) no. 2, p. 229-235

Given a cohomology class $\xi \in {H}^{2}\left(M;Z\right)$ there is a smooth submanifold $K\subset M$ Poincaré dual to $\xi$. A special class of such embeddings is characterized by topological properties which hold for nonsingular algebraic hypersurfaces in $\mathbf{C}{P}_{n}$. This note summarizes some results on the question: how does the divisibility of $\xi$ restrict the dual submanifolds $K$ in this class ? A formula for signatures associated with a $d$-fold ramified cover of $M$ branched along $K$ is given and a proof is included in case $d=2$.

Étant donné une classe de cohomologie $\xi \in {H}^{2}\left(M;Z\right)$, il existe une sous-variété $K\subset M$ duale à $\xi$ dans le sens de Poincaré. Il existe un ensemble de tels plongements qui est caractérisé par des propriétés topologiques, que les hypersurfaces algébriques de $\mathbf{C}{P}_{n}$ vérifient. Cet exposé résume quelques résultats sur la question : comment la divisibilité de $\xi$ limite-t-elle les sous-variétés duales, $K$, dans cet ensemble ? Et nous donnons une formule pour la signature associée à un revêtement d’ordre $d$ sur $M$, ramifiée sur $K$ ; nous le démontrons dans le cas où $d=2$.

@article{AIF_1973__23_2_229_0,
author = {Wood, J. and Thomas, Emery},
title = {On signatures associated with ramified coverings and embedding problems},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Durand},
volume = {23},
number = {2},
year = {1973},
pages = {229-235},
doi = {10.5802/aif.470},
zbl = {0262.57012},
mrnumber = {49 \#3964},
language = {en},
url = {http://www.numdam.org/item/AIF_1973__23_2_229_0}
}

Wood, J.; Thomas, Emery. On signatures associated with ramified coverings and embedding problems. Annales de l'Institut Fourier, Volume 23 (1973) no. 2, pp. 229-235. doi : 10.5802/aif.470. http://www.numdam.org/item/AIF_1973__23_2_229_0/

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