## Osaka Journal of Mathematics

### Cobordisme des surfaces plongées dans $S^4$

#### Abstract

We show that a closed connected surface embedded in $S^{4} = \partial B^{5}$ bounds a handlebody of dimension 3 embedded in $B^{5}$ if and only if the Euler number of its normal bundle vanishes. Using this characterization, we show that two closed connected surfaces embedded in $S^{4}$ are cobordant if and only if they are abstractly diffeomorphic to each other and the Euler numbers of their normal bundles coincide. As an application, we show that a given Heegaard decomposition of a 3-manifold can be realized in $S^{5}$. We also give a new proof of Rohlin's theorem on embeddings of 3-manifolds into $\mathbf{R}^{5}$.

#### Article information

Source
Osaka J. Math., Volume 42, Number 4 (2005), 751-765.

Dates
First available in Project Euclid: 21 July 2006

https://projecteuclid.org/euclid.ojm/1153494550

Mathematical Reviews number (MathSciNet)
MR2195992

#### Citation

Blanlœeil, Vincent; Saeki, Osamu. Cobordisme des surfaces plongées dans $S^4$. Osaka J. Math. 42 (2005), no. 4, 751--765. https://projecteuclid.org/euclid.ojm/1153494550