With cedram.org English français Home Overview Search for an article Submit a paper Informations for the authors Subscription Limited access - RUCHE Table of contents for this issue | Previous article J. Wood; Emery ThomasOn signatures associated with ramified coverings and embedding problemsAnnales de l'institut Fourier, 23 no. 2 (1973), p. 229-235, doi: 10.5802/aif.470 Article PDF | Reviews MR 49 #3964 | Zbl 0262.57012 Résumé - AbstractGiven a cohomology class $\xi \in H^2(M;Z)$ 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 ${\bf C}P_n$. 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 $d=2$. Bibliography[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. Article |  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 © Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310