Contact topology and the structure of 5-manifolds with

Hansjörg Geiges; Charles B. Thomas

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 4, page 1167-1188
  • ISSN: 0373-0956

Abstract

top
We prove a structure theorem for closed, orientable 5-manifolds with fundamental group and second Stiefel-Whitney class equal to zero on . This structure theorem is then used to construct contact structures on such manifolds by applying contact surgery to fake projective spaces and certain -quotients of .

How to cite

top

Geiges, Hansjörg, and Thomas, Charles B.. "Contact topology and the structure of 5-manifolds with $\pi _1={\mathbb {Z}}_2$." Annales de l'institut Fourier 48.4 (1998): 1167-1188. <http://eudml.org/doc/75313>.

@article{Geiges1998,
abstract = {We prove a structure theorem for closed, orientable 5-manifolds $M$ with fundamental group $\pi _1(M)=\{\Bbb Z\}_2$ and second Stiefel-Whitney class equal to zero on $H_2(M)$. This structure theorem is then used to construct contact structures on such manifolds by applying contact surgery to fake projective spaces and certain $\{\Bbb Z\}_2$-quotients of $S^2\times S^3$.},
author = {Geiges, Hansjörg, Thomas, Charles B.},
journal = {Annales de l'institut Fourier},
keywords = {surgery; contact structures; contact surgery; involutions; Brieskorn manifolds; characteristic submanifold; pin cobordism},
language = {eng},
number = {4},
pages = {1167-1188},
publisher = {Association des Annales de l'Institut Fourier},
title = {Contact topology and the structure of 5-manifolds with $\pi _1=\{\mathbb \{Z\}\}_2$},
url = {http://eudml.org/doc/75313},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Geiges, Hansjörg
AU - Thomas, Charles B.
TI - Contact topology and the structure of 5-manifolds with $\pi _1={\mathbb {Z}}_2$
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 4
SP - 1167
EP - 1188
AB - We prove a structure theorem for closed, orientable 5-manifolds $M$ with fundamental group $\pi _1(M)={\Bbb Z}_2$ and second Stiefel-Whitney class equal to zero on $H_2(M)$. This structure theorem is then used to construct contact structures on such manifolds by applying contact surgery to fake projective spaces and certain ${\Bbb Z}_2$-quotients of $S^2\times S^3$.
LA - eng
KW - surgery; contact structures; contact surgery; involutions; Brieskorn manifolds; characteristic submanifold; pin cobordism
UR - http://eudml.org/doc/75313
ER -

References

top
  1. [1] M.F. ATIYAH and R. BOTT, A Lefschetz fixed point formula for elliptic complexes : II, Applications, Ann. of Math. (2), 88 (1968), 451-491. Zbl0167.21703MR38 #731
  2. [2] K.S. BROWN, Cohomology of Groups, Springer, 1982. Zbl0584.20036MR83k:20002
  3. [3] Y. ELIASHBERG, Topological characterization of Stein manifolds of dimension &gt; 2, Internat. J. Math., 1 (1990), 29-46. Zbl0699.58002MR91k:32012
  4. [4] H. GEIGES, Contact structures on 1-connected 5-manifolds, Mathematika, 38 (1991), 303-311. Zbl0724.57017MR93e:57042
  5. [5] H. GEIGES, Constructions of contact manifolds, Math. Proc. Cambridge Philos. Soc., 121 (1997), 455-464. Zbl0882.57007MR98f:53027
  6. [6] H. GEIGES, Applications of contact surgery, Topology, 36 (1997), 1193-1220. Zbl0912.57019MR98d:57044
  7. [7] C.H. GIFFEN, Smooth homotopy projective spaces, Bull. Amer. Math. Soc., 75 (1969), 509-513. Zbl0195.53302MR39 #964
  8. [8] F. HIRZEBRUCH and K.H. MAYER, O(n)-Mannigfaltigkeiten, exotische Sphären und Singularitäten, Lecture Notes in Math., 57, Springer, 1968. Zbl0177.26401MR37 #4825
  9. [9] H. HOPF, Fundamentalgruppe und zweite Bettische Gruppe, Comment. Math. Helv., 14 (1941/1942), 257-309. Zbl0027.09503MR3,316eJFM68.0503.01
  10. [10] R.C. KIRBY and L.C. SIEBENMANN, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Ann. of Math. Studies, 88, Princeton University Press, 1977. Zbl0361.57004MR58 #31082
  11. [11] R.C. KIRBY and L.R. TAYLOR, Pin structures on low-dimensional manifolds, in : Geometry of Low-Dimensional Manifolds 2, (S.K. Donaldson and C.B. Thomas, eds.), London Math. Soc. Lecture Note Ser., 151, Cambridge University Press (1990), 177-242. Zbl0754.57020MR94b:57031
  12. [12] A.A. KOSINSKI, Differential Manifolds, Academic Press, 1993. Zbl0767.57001MR95b:57001
  13. [13] S. LÓPEZ DE MEDRANO, Involutions on Manifolds, Ergeb. Math. Grenzgeb., 59, Springer, 1971. Zbl0214.22501MR45 #7747
  14. [14] R. LUTZ and C. MECKERT, Structures de contact sur certaines sphères exotiques, C.R. Acad. Sci. Paris, Sér. I, Math., 282 (1976), 591-593. Zbl0326.53044MR53 #1471
  15. [15] J. MARTINET, Formes de contact sur les variétés de dimension 3, in : Proc. Liverpool Singularities Sympos. II (C.T.C. Wall, ed.), Lecture Notes in Math., 209, Springer (1971), 142-163. Zbl0215.23003MR50 #3263
  16. [16] C.P. ROURKE and B.J. SANDERSON, Introduction to Piecewise-Linear Topology, Ergeb. Math. Grenzgeb., 69, Springer (1972). Zbl0254.57010MR50 #3236
  17. [17] C.B. THOMAS, Contact structures on (n-1)-connected (2n+1)-manifolds, Banach Center Publ., 18 (1986), 255-270. Zbl0642.57014MR89b:53074
  18. [18] C.T.C. WALL, Surgery of non-simply-connected manifolds, Ann. of Math. (2), 84 (1966), 217-276. Zbl0149.20602MR35 #3692
  19. [19] A. WEINSTEIN, Contact surgery and symplectic handlebodies, Hokkaido Math. J., 20 (1991), 241-251. Zbl0737.57012MR92g:53028

NotesEmbed ?

top

You must be logged in to post comments.