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 ξH 2 (M;Z) there is a smooth submanifold KM Poincaré dual to ξ. A special class of such embeddings is characterized by topological properties which hold for nonsingular algebraic hypersurfaces in CP n . This note summarizes some results on the question: how does the divisibility of ξ 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 ξH 2 (M;Z), il existe une sous-variété KM duale à ξ dans le sens de Poincaré. Il existe un ensemble de tels plongements qui est caractérisé par des propriétés topologiques, que les hypersurfaces algébriques de CP n vérifient. Cet exposé résume quelques résultats sur la question : comment la divisibilité de ξ 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},
     address = {28 - Luisant},
     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/

[1] M. Atiyah and I. Singer, The index of elliptic operators: III, Annals of Math. 87 (1968), 546-604. | MR 38 #5245 | Zbl 0164.24301

[2] F. Hirzebruch, Topological methods in algebraic geometry, 3rd ed., New York, 1966. | MR 34 #2573 | Zbl 0138.42001

[3] F. Hirzebruch, The signature of ramified coverings, Papers in honor of Kodiara, 253-265, Princeton, 1969. | MR 41 #2707 | Zbl 0208.51802

[4] W. Hsiang and R. Szczarba, On embedding surfaces in 4-manifolds, Proc. Symp. Pure Math. XXII. | Zbl 0234.57009

[5] K. Jänich and E. Ossa, On the signature of an involution, Topology 8 (1969), 27-30. | MR 38 #6613 | Zbl 0184.27302

[6] P. Jupp, Classification of certain 6-manifolds, (to appear). | Zbl 0249.57005

[7] M. Kato and Y. Matsumoto, Simply connected surgery of submanifolds in codimension two, I, (to appear). | Zbl 0238.57018

[8] M. Kervaire and J. Milnor, On 2-spheres in 4-manifolds, P.N.A.S. 47 (1961) 1651-1657. | MR 24 #A2968 | Zbl 0107.40303

[9] W. Massey, Proof of a conjecture of Whitney, Pacific J. Math. 31 (1969) 143-156. | MR 40 #3570 | Zbl 0198.56701

[10] V. Rokhlin, Two dimensional submanifolds of four dimensional manifolds, Functional Analysis and its Applications, 5 (1971), 39-48. | MR 45 #7733 | Zbl 0268.57019

[11] C.T.C. Wall, Classification problems in differential topology. V. On certain 6-manifolds, Invent. Math. 2 (1966), 355-374. | Zbl 0149.20601

[12] E. Thomas and J. Wood, On manifolds representing homology classes in codimension 2, (to appear). | Zbl 0283.57018