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

Abstract

We determine strong constraints on the generalized Euler invariants of Seifert bundles over non‐orientable base orbifolds which may embed as topologically locally flat submanifolds of S4. In particular, a circle bundle over a non‐orientable surface F embeds if and only if it embeds as the boundary of a regular neighbourhood of an embedding of F in S4, and we show that precisely thirteen geometric 3‐manifolds with elementary amenable fundamental groups embed. With the exception of the Poincaré homology sphere, each member of the latter class may be obtained by 0‐framed surgery on a link which is the union of two slice links, and so embeds smoothly in S4. 1991 Mathematics Subject Classification: 57N13.

Embedding 3-manifolds with circle actions

“…(The analogous implication in the case of nonorientable base orbifolds is not reversible. See [3]. )…”

“…As a consequence we were able to settle there the question of embeddability for manifolds having one of the geometries S 3 , E 3 , Nil 3 , S 2 × E 1 or Sol 3 . When the Seifert data is "skew-symmetric" (i.e., is a set of complementary pairs) and all cone point orders are odd, the corresponding Seifert manifold embeds smoothly [3]. Such a manifold has generalized Euler invariant ε = 0 and so is geometric of type…”

“…If, moreover, the α i s are all odd, a fibre sum construction based on embeddings of punctured lens spaces shows that M (g; S) embeds smoothly in S 4 , for all g ≥ 0. (See Lemma 3.1 of [3].) Here we shall show that when ε S = 0 and g = 0, skew-symmetry is a necessary condition for M (0; S) to embed in a homology 4-sphere.…”

“…(The relevant papers known to us are [2,3,[5][6][7][8][9][10][11].) In particular, it is not yet known which Seifert fibred 3-manifolds embed, although in many other respects this is a well-understood class of spaces, with natural parametrizations in terms of Seifert data.…”

“…These admit no compatible S 1 -action. However in [3] we used the Z/2Z-index theorem to constrain the Euler invariants for such 3-manifolds. As a consequence we were able to settle there the question of embeddability for manifolds having one of the geometries S 3 , E 3 , Nil 3 , S 2 × E 1 or Sol 3 .…”