Co-dimension two spinal open book embeddings of 3-manifolds

Abstract

In this paper, we show that every spinal open book decomposition of a closed oriented [Formula: see text]-manifold [Formula: see text] spinal open book embeds into a certain spinal open book decomposition of [Formula: see text], the connected sum of [Formula: see text] copies of the twisted [Formula: see text]-bundle over [Formula: see text], where [Formula: see text] depends on the spinal open book decomposition of [Formula: see text]. We also discuss spinal open book embeddings of a huge class of spinal open books of closed oriented [Formula: see text]-manifolds into the trivial spinal open book of the [Formula: see text]-sphere [Formula: see text]. Finally, we show that given a closed oriented [Formula: see text]-manifold [Formula: see text], there exists a spinal open book for [Formula: see text] such that [Formula: see text] spinal open book embeds into the trivial spinal open book of [Formula: see text] In particular, this gives another proof of Hirsch’s theorem which states that every closed orientable [Formula: see text]-manifold embeds in [Formula: see text]