ANNALES DE L'INSTITUT FOURIER

On signatures associated with ramified coverings and embedding problems
Annales de l'Institut Fourier, Volume 23 (1973) no. 2, pp. 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 $d=2$.

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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. https://aif.centre-mersenne.org/articles/10.5802/aif.470/

[1] M. Atiyah and I. Singer, The index of elliptic operators: III, Annals of Math. 87 (1968), 546-604. | MR | Zbl

[2] F. Hirzebruch, Topological methods in algebraic geometry, 3rd ed., New York, 1966. | MR | Zbl

[3] F. Hirzebruch, The signature of ramified coverings, Papers in honor of Kodiara, 253-265, Princeton, 1969. | MR | Zbl

[4] W. Hsiang and R. Szczarba, On embedding surfaces in 4-manifolds, Proc. Symp. Pure Math. XXII. | Zbl

[5] K. Jänich and E. Ossa, On the signature of an involution, Topology 8 (1969), 27-30. | MR | Zbl

[6] P. Jupp, Classification of certain 6-manifolds, (to appear). | Zbl

[7] M. Kato and Y. Matsumoto, Simply connected surgery of submanifolds in codimension two, I, (to appear). | Zbl

[8] M. Kervaire and J. Milnor, On 2-spheres in 4-manifolds, P.N.A.S. 47 (1961) 1651-1657. | MR | Zbl

[9] W. Massey, Proof of a conjecture of Whitney, Pacific J. Math. 31 (1969) 143-156. | MR | Zbl

[10] V. Rokhlin, Two dimensional submanifolds of four dimensional manifolds, Functional Analysis and its Applications, 5 (1971), 39-48. | MR | Zbl

[11] C.T.C. Wall, Classification problems in differential topology. V. On certain 6-manifolds, Invent. Math. 2 (1966), 355-374. | Zbl

[12] E. Thomas and J. Wood, On manifolds representing homology classes in codimension 2, (to appear). | Zbl

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