Embedding homology equivalent 3-manifolds in 4-space

Embedding Seifert Fibred 3-Manifolds and Sol3 -Manifolds in 4-Space

“…We shall also show that precisely three Sol 3 -manifolds embed in S 4 . The strategy of [4] also succeeds in these cases.…”

“…Those which do not embed in S 4 admit no such locally¯at embedding in any homology 4-sphere. (Hence, by Lemma 1 of [4], neither can any 3-manifold homology equivalent to one of the latter embed in any homology 4-sphere. )…”

“…It is an open question whether there are any 3-manifolds with ®nite fundamental group which are not S 3 -manifolds.Here we shall determine strong constraints on the generalized Euler invariants of Seifert bundles over non-orientable base orbifolds which may embed in S 4 . Using this result we show that there are precisely thirteen geometric 3-manifolds with elementary amenable fundamental group which embed in S 4 and so may serve as the model P for the strategy of [4]. Exactly ten Seifert ®bred 3-manifolds of this kind (in other words, admitting one of the geometries S 3 , S 2´E 1 , E 3 or Nil 3 ) embed in S 4 .…”

“…Exactly ten Seifert ®bred 3-manifolds of this kind (in other words, admitting one of the geometries S 3 , S 2´E 1 , E 3 or Nil 3 ) embed in S 4 . It was shown in [4] that this strategy may be carried through successfully in eight of these cases, that is, every Zr-homology equivalent 3-manifold also embeds. In the remaining two cases r I Ã (the binary icosahedral group) and r Q (the quaternion group of order 8) the surgery obstruction group is large.…”

“…We shall use`embed' to mean`embed as a topologically locally¯at submanifold' and otherwise add the quali®cation`smoothly' where appropriate. In [4] it was shown that if a closed orientable 3-manifold P embeds in S 4 , and if M is Zr-homology equivalent to P then M also embeds in S 4 , provided that r p 1 P is elementary amenable, every automorphism of r can be realized by a self-homeomorphism of P and a surgery obstruction vanishes. The hypothesis that r be elementary amenable is needed in order to apply 4-dimensional topological surgery.…”