An extension of Rourke’s proof that Ω₃=0 to nonorientable manifolds

Ancel, Fredric D.; Guilbault, Craig R.

doi:10.1090/S0002-9939-1992-1092913-0

An extension of Rourke’s proof that $\Omega _ 3=0$ to nonorientable manifolds
HTML articles powered by AMS MathViewer

by Fredric D. Ancel and Craig R. Guilbault PDF

Proc. Amer. Math. Soc. 115 (1992), 283-291 Request permission

Abstract:

A classical result in manifold theory states that every closed $3$-manifold bounds a compact $4$-manifold. In 1985 C. Rourke discovered a strikingly short and elementary proof of the orientable case of this theorem $({\Omega _3} = 0)$. In this note we show that Rourke’s approach can be extended to include nonorientable $3$-manifolds.

References

W. B. R. Lickorish, A representation of orientable combinatorial $3$-manifolds, Ann. of Math. (2) 76 (1962), 531–540. MR 151948, DOI 10.2307/1970373
W. B. R. Lickorish, Homeomorphisms of non-orientable two-manifolds, Proc. Cambridge Philos. Soc. 59 (1963), 307–317. MR 145498, DOI 10.1017/s0305004100036926

A

-dimensional manifold is the boundary of a

-dimensional manifold

81

Colin Rourke, A new proof that $\Omega _3$ is zero, J. London Math. Soc. (2) 31 (1985), no. 2, 373–376. MR 809959, DOI 10.1112/jlms/s2-31.2.373
René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86 (French). MR 61823, DOI 10.1007/BF02566923
Andrew H. Wallace, Modifications and cobounding manifolds, Canadian J. Math. 12 (1960), 503–528. MR 125588, DOI 10.4153/CJM-1960-045-7