Signatures of singular branched covers. (English) Zbl 0793.57013

We consider branched cyclic \(d\)-fold covers of a closed smooth oriented manifold \(M\) along a collection of codimension-2 smooth oriented submanifolds \(\{K_ i\}\) which are in general position, where the branching index of each submanifold is \(d\). It is shown that these are rational homology manifolds. A formula for the eigenspace signatures of the cover in terms of the Hirzebruch \(L\)-class of \(M\) and 2-dimensional cohomology classes dual to the \(K_ i\) is given. Many singular algebraic varieties arise in this manner. As an application the signature of a \(d\)- fold branched cyclic cover of \(\mathbb{C} \text{P} (2k)\) along a collection of \(d\) hyperplanes in general position is calculated to be \(1+{d-1 \choose k+1}\). A formula for the stable tangent bundle of a branched cover along a single submanifold is also given.


57R20 Characteristic classes and numbers in differential topology
14F45 Topological properties in algebraic geometry
57M12 Low-dimensional topology of special (e.g., branched) coverings
Full Text: DOI EuDML


[1] [A-S] Atiyah, M.F., Singer, I.M.: The index of elliptic operators III. Ann. Math. II Ser.87, 546-604 (1968) · Zbl 0164.24301
[2] [C-F] Conner, P., Floyd, E.: Differentiable periodic maps: (Ergeb. Math. Vol. 33) G?ttingen Heidelberg: Springer Berlin, 1964
[3] [C-S1] Cappell S., Shaneson J.: Stratifiable maps and topological invariants. J. Am. Math. Soc.4, 521-551 (1991) · Zbl 0746.32016
[4] [C-S2] Cappell S, Shaneson J.: Singular spaces, characteristic classes, and intersection homology. Ann. Math. II Ser.134 (1991) · Zbl 0759.55002
[5] [F] Fox, R.H.: Covering spaces with singularities. In: Fox, R.H., Spencer, D.C., Tucker, A.W. (eds.) Algebraic geometry and topology, a symposium in honor of S. Lefschetz, pp. 243-257, Princeton: Princeton Univ. Press, 1957
[6] [G1] Gilmer, P.M.: Configuration of surfaces in 4-manifolds. PhD Thesis, University of California at Berkeley, 1978
[7] [G2] Gilmer, P.M.: Configuration of surfaces in 4-manifolds. Trans. Am. Math. Soc.264, 353-380 (1981) · Zbl 0509.57023
[8] [G3] Gilmer, P.M.: Topological proof of theG-signature theorem forG finite. Pac. J. Math 97, 105-114 (1981)
[9] [G-L] Gilmer, P., Livingston, C.: Casson-gordon invariants and link concordance. To appear in Topology · Zbl 0797.57001
[10] [G-M] Goresky, M., MacPherson, R.: Stratified morse theory (Ergebnisse 3 Folge, 14) Berlin Heidelberg New York: Springer 1988
[11] [Ha] Hattori, A.: Genera of ramified coverings. Math. Ann.195, 208-226 (1972) · Zbl 0226.57017
[12] [H1] Hirzebruch, F.: Topological methods in algebraic geometry, 3rd ed., New York: Springer 1966 · Zbl 0138.42001
[13] [H2] Hirzebruch, F.: The signatures of ramified coverings. In: Global Analysis. Papers in honor of Kodaira, pp. 253-265. Princeton: Princeton University Press 1969
[14] [H3] Hirzebruch, F.: Arrangements of lines and Algebraic surfaces. In Artin, M., Tate, G. (eds) Arithmetic and geometry papers dedicated to I.R. Shafarevich on the occasion of his sixthieth birthday Vol. II Geometry, pp 113-140 (Progress in Mathematics 36) Boston Basel Stuttgard: Birk?user, 1983
[15] [H-Z] Hirzebruch, F., Zagier, D.: The Atiyah-Singer theorem and elementary number theory. Math. Lect. Ser., 3. Boston: Publish or Perish, 1974 · Zbl 0288.10001
[16] [H-S] Hsiang, W., Szczarba, R.: On embedding surfaces in 4-manifolds, Proc. Symp. in Pure Math. XXII. Algebraic Topology, 97-103 (1971)
[17] [I1] Itenberg, V.I.: Medium-dimensional homologies of a submanifold of codimension two. Funct. Anal. Appl.8, 121-126 (1974) · Zbl 0322.57026
[18] [I2] Itenberg, V.I.: Medium-dimensional homologies of a submanifold of codimension two II. Zap. Naucn. Sem. Leningrad Otel. Mat. Inst. Stelkov (LOMI)56, 182-185 (1976) (in Russian with English summary) · Zbl 0363.57028
[19] [M] Milnor J.: Characteristic classes (Annals of Math. Studies No. 76) Princeton: Princeton University Press, 1974 · Zbl 0298.57008
[20] [R] Rokhlin, V.A.: Two-dimensional submanifolds of four-manifolds. Funct. Anal. Appl.5, 39-48 (1971) · Zbl 0268.57019
[21] [S] Spanier, E.: Algebraic topology. New York: McGraw Hill 1966 · Zbl 0145.43303
[22] [T-W1] Thomas, E., Wood, J.: On signatures associated with ramified covering and embedding problems: Ann. Inst. Fourier23, 225-235 (1973)
[23] [T-W2] Thomas, E., Wood, J.: On manifolds representing homology in codimension-2. Invent. Math.25, 63-89 (1974) · Zbl 0283.57018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.