Embedding Seifert manifolds in 𝑆⁴

Abstract

Using an obstruction based on Donaldson's theorem on the intersection forms of definite 4-manifolds, we determine which connected sums of lens spaces smoothly embed in S 4 . We also find constraints on the Seifert invariants of Seifert 3-manifolds which embed in S 4 when either the base orbifold is non-orientable or the first Betti number is odd. In addition, we construct some new embeddings and use these, along with the d and μ invariants, to examine the question of when the double branched cover of a 3 or 4 strand pretzel link embeds.

Embedding 3‐manifolds in spin 4‐manifolds

“…On the constructive side, Casson and Harer [6] gave several infinite families of Brieskorn homology spheres that smoothly embed in S 4 (see [5]). More general classes of 3-manifolds that smoothly embed in S 4 include those that arise as cyclic branched covers of doubly slice knots (see [8,18,25]) and homology spheres obtained by surgery on ribbon links [23]. For some specific classes of 3-manifolds it is known exactly which ones smoothly embed in S 4 , for example, circle bundles over closed surfaces [7] and connected sums of lens spaces [8].…”

“…More general classes of 3-manifolds that smoothly embed in S 4 include those that arise as cyclic branched covers of doubly slice knots (see [8,18,25]) and homology spheres obtained by surgery on ribbon links [23]. For some specific classes of 3-manifolds it is known exactly which ones smoothly embed in S 4 , for example, circle bundles over closed surfaces [7] and connected sums of lens spaces [8]. Budney and Burton [5] have examined this question from the perspective of the 11-tetrahedron census of triangulated 3-manifolds.…”

Distinguishing topologically and smoothly doubly slice knots

“…Let d(M, s) denote the correction term associated to the pair (M, s). The main tool in this paper is the following theorem, which also appears in [4,10] in one form or another. Theorem 2.2.…”

“…In this subsection, we will present a sufficient condition for a knot K to be doubly slice that applies when K is obtained by a certain type of infection. We remark that Donald [4] gives a different sufficient condition: one which involves systems of ribbon bands for K. Our criterion will make use of some well-known facts about topologically locally flat surfaces in 4-manifolds that result from the work of Freedman and Quinn [6,7]. (2) Let κ be a topologically locally flat 2-knot in S 4 with π 1 (S 4 − κ) ∼ = Z.…”

“…Then, Y = L(5, 1), and equation (1) Note that implicit in the presentation matrix is the affine identification Spin c (L(n, 1)# L(n, −1)) ∼ = Z 5 ⊕ Z 5 given by [s i , s j ] ∼ (i, j). For example, the correction terms vanish on all elements of the subgroups generated by (1,1) and (1,4)…”