Knots, Singular Embeddings, and Monodromy

  • Markus BanaglEmail author
  • Sylvain E. Cappell
  • Julius L. Shaneson
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 1)


The Goresky-MacPherson L-class of a PL pseudomanifold piecewise-linearly embedded in a PL manifold in a possibly nonlocally flat way, can be computed in terms of the Hirzebruch-Thom L-class of the manifold and twisted L-classes associated to the singularities of the embedding, as was shown by Cappell and Shaneson. These formulae are refined here by analyzing the twisted classes. We treat the case of Blanchfield local systems that extend into the singularities as well as cases where they do not extend. In the latter situation, we consider fibered embeddings of strata and 4-dimensional singular sets, using work of Banagl. Rho-invariants enter the picture.


Local System Real Vector Space Cyclic Cover Constant Sheaf Seifert Manifold 
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  1. [APS75]
    Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry II. Math. Proc. Camb. Philos. Soc. 78, 405–432 (1975) zbMATHCrossRefMathSciNetGoogle Scholar
  2. [Ban06]
    Banagl, M.: Computing twisted signatures and L-classes of non-Witt spaces. Proc. Lond. Math. Soc. (3) 92, 428–470 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  3. [Ban07]
    Banagl, M.: Topological Invariants of Stratified Spaces. Springer, Berlin (2007) zbMATHGoogle Scholar
  4. [Ban08]
    Banagl, M.: The signature of partially defined local coefficient systems. J. Knot Theory Ramif. 17(12), 1455–1481 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  5. [BCS03]
    Banagl, M., Cappell, S.E., Shaneson, J.L.: Computing twisted signatures and L-classes of stratified spaces. Math. Ann. 326(3), 589–623 (2003) zbMATHMathSciNetGoogle Scholar
  6. [CS91]
    Cappell, S.E., Shaneson, J.L.: Singular spaces, characteristic classes, and intersection homology. Ann. Math. 134, 325–374 (1991) CrossRefMathSciNetGoogle Scholar
  7. [Kea73]
    Kearton, C.: Classification of simple knots by Blanchfield duality. Bull. Am. Math. Soc. 79(5), 952–955 (1973) zbMATHCrossRefMathSciNetGoogle Scholar
  8. [Ker65]
    Kervaire, M.A.: Les nœuds de dimensions supérieures. Bull. Soc. Math. France 93, 225–271 (1965) zbMATHMathSciNetGoogle Scholar
  9. [Lev65]
    Levine, J.: Unknotting spheres in codimension two. Topology 4, 9–16 (1965) zbMATHCrossRefMathSciNetGoogle Scholar
  10. [Lev70]
    Levine, J.: An algebraic classification of some knots of codimension two. Commun. Math. Helv. 45, 185–198 (1970) zbMATHCrossRefGoogle Scholar
  11. [Lus71]
    Lusztig, G.: Novikov’s higher signature and families of elliptic operators. J. Differ. Geom. 7, 229–256 (1971) MathSciNetGoogle Scholar
  12. [Mey72]
    Meyer, W.: Die Signatur von lokalen Koeffizientensystemen und Faserbündeln. Bonner Math. Schr. 53 (1972) Google Scholar
  13. [Tro73]
    Trotter, H.F.: On S-equivalence of Seifert matrices. Invent. Math. 20, 173–207 (1973) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Markus Banagl
    • 1
    Email author
  • Sylvain E. Cappell
    • 2
  • Julius L. Shaneson
    • 3
  1. 1.Mathematisches InstitutUniversität HeidelbergHeidelbergGermany
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  3. 3.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA

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