Embedding 3-manifolds with circle actions

Abstract

Abstract. Constraints on the Seifert invariants of orientable 3-manifolds M which admit fixed-point free S 1 -actions and embed in R 4 are given. In particular, the generalized Euler invariant of the orbit fibration is determined up to sign by the base orbifold B unless H 1 (M ; Z) is torsion free, in which case it can take at most one nonzero value (up to sign). An H 2 × E 1 -manifold M with base orbifold B = S 2 (α 1 , . . . , α r ) where all cone point orders are odd embeds in R 4 if and only if its Seifert data S is skew-symmetric.The question of which closed 3-manifolds M (other than homology spheres) embed in R 4 has received surprisingly little attention. (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. It was shown earlier that if M embeds in R 4 , then it must be orientable and the torsion subgroup T (M ) of H 1 (M ; Z) must be a direct double: T (M ) ∼ = U ⊕ U for some finite abelian group U [8]. Moreover, the linking pairing M on T (M ) must be hyperbolic [13]. Most of the known constructions give smooth embeddings, but these conditions must also hold if M embeds as a TOP locally flat submanifold.The first two sections establish our notation and summarize some basic facts about compact 3-manifolds in R 4 . In §3 we shall observe that T (M ) being a direct double imposes strong constraints on the Seifert data of orientable 3-manifolds which admit fixed-point free S 1 -actions and which embed in R 4 . The present simple argument does not work for orientable Seifert fibred 3-manifolds with nonorientable base orbifolds. 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 . 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 E 3 , S 2 × E

Smoothly embedding Seifert fibered spaces in 𝑆⁴

“…. , p k q k , − p k q k ) [Don15, Theorem 1.3], see also [Hil09]. Donald also obtained similar results when the base surface is non-orientable [Don15, Theorem 1.2] and further results in the non-orientable case can be found in [CH98].…”

“…Over the years many different techniques and obstructions have been developed to address the question. For example, Hantzsche [Han38] proved that if Y embeds in S 4 then the torsion part of H 1 (Y ) must split as a direct double, that is, tor H 1 (Y ) ∼ = G ⊕ G for some abelian group G. There have also been applications of topological obstructions based on linking forms [Hil09], Casson-Gordon signatures [GL83] and the G-index theorem [CH98], as well as smooth obstructions based on Rokhlin's theorem, the Neumann-Siebenman invariant, Furuta's 10/8 theorem, Donaldson's theorem and the Ozsváth-Szabó d-invariants, see e.g. [BB12] and [Don15].…”

Seifert hypersurfaces of 2-knots and Chern-Simons functional

“…On the other hand, it is still difficult to solve the following problem: for a given connected 4-manifold X and c ∈ H 3 (X; Z), which closed connected 3-manifold Y is embedded into X with [Y ] = c ∈ H 3 (X; Z) ? This problem has been studied in several situations ( [19], [17], [22], [18], [10]). We give a partial answer to this problem under the assumptions that Y is an oriented homology 3-sphere and X is a closed negative definite 4-manifold.…”

Embedding Seifert manifolds in 𝑆⁴

“…For a kcomponent link there are 2 k−1 spin structures on the double branched cover and this is a square precisely when k is odd. When L is a pretzel link, Y is a Seifert manifold with base S 2 and it follows from, for example, [11,Theorem 3.1], that b 1 (Y ) ≤ 1.…”

“…We also consider orientable base surfaces. An interesting special case, considered by Hillman [11], occurs when e(Y ) = 0. These are the only examples where b 1 (Y ) is odd.…”