On signatures associated with ramified coverings and embedding problems

J. Wood; Emery Thomas

Annales de l'institut Fourier (1973)

  • Volume: 23, Issue: 2, page 229-235
  • ISSN: 0373-0956

Abstract

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Given a cohomology class ξ ∈ H 2 ( M ; Z ) there is a smooth submanifold K ⊂ M Poincaré dual to ξ . A special class of such embeddings is characterized by topological properties which hold for nonsingular algebraic hypersurfaces in C P 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 .

How to cite

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Wood, J., and Thomas, Emery. "On signatures associated with ramified coverings and embedding problems." Annales de l'institut Fourier 23.2 (1973): 229-235. <http://eudml.org/doc/74127>.

@article{Wood1973,
abstract = {Given 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 $d=2$.},
author = {Wood, J., Thomas, Emery},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {2},
pages = {229-235},
publisher = {Association des Annales de l'Institut Fourier},
title = {On signatures associated with ramified coverings and embedding problems},
url = {http://eudml.org/doc/74127},
volume = {23},
year = {1973},
}

TY - JOUR
AU - Wood, J.
AU - Thomas, Emery
TI - On signatures associated with ramified coverings and embedding problems
JO - Annales de l'institut Fourier
PY - 1973
PB - Association des Annales de l'Institut Fourier
VL - 23
IS - 2
SP - 229
EP - 235
AB - Given 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 $d=2$.
LA - eng
UR - http://eudml.org/doc/74127
ER -

References

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  1. [1] M. ATIYAH and I. SINGER, The index of elliptic operators: III, Annals of Math. 87 (1968), 546-604. Zbl0164.24301MR38 #5245
  2. [2] F. HIRZEBRUCH, Topological methods in algebraic geometry, 3rd ed., New York, 1966. Zbl0138.42001MR34 #2573
  3. [3] F. HIRZEBRUCH, The signature of ramified coverings, Papers in honor of Kodiara, 253-265, Princeton, 1969. Zbl0208.51802MR41 #2707
  4. [4] W. HSIANG and R. SZCZARBA, On embedding surfaces in 4-manifolds, Proc. Symp. Pure Math. XXII. Zbl0234.57009
  5. [5] K. JÄNICH and E. OSSA, On the signature of an involution, Topology 8 (1969), 27-30. Zbl0184.27302MR38 #6613
  6. [6] P. JUPP, Classification of certain 6-manifolds, (to appear). Zbl0249.57005
  7. [7] M. KATO and Y. MATSUMOTO, Simply connected surgery of submanifolds in codimension two, I, (to appear). Zbl0238.57018
  8. [8] M. KERVAIRE and J. MILNOR, On 2-spheres in 4-manifolds, P.N.A.S. 47 (1961) 1651-1657. Zbl0107.40303MR24 #A2968
  9. [9] W. MASSEY, Proof of a conjecture of Whitney, Pacific J. Math. 31 (1969) 143-156. Zbl0198.56701MR40 #3570
  10. [10] V. ROKHLIN, Two dimensional submanifolds of four dimensional manifolds, Functional Analysis and its Applications, 5 (1971), 39-48. Zbl0268.57019MR45 #7733
  11. [11] C.T.C. WALL, Classification problems in differential topology. V. On certain 6-manifolds, Invent. Math. 2 (1966), 355-374. Zbl0149.20601
  12. [12] E. THOMAS and J. WOOD, On manifolds representing homology classes in codimension 2, (to appear). Zbl0283.57018

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