We show that a closed connected surface embedded in S4 = ∂B5 bounds a handlebody of dimension 3 embedded in B5 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 S5. We also give a new proof of Rohlin's theorem on embeddings of 3-manifolds into R5.
|Number of pages||15|
|Journal||Osaka Journal of Mathematics|
|Publication status||Published - Dec 1 2005|
All Science Journal Classification (ASJC) codes