Classical and new plumbed homology spheres bounding contractible manifolds

Abstract

A classical question in low-dimensional topology asks which homology 3-spheres bound contractible 4-manifolds. In this paper, we address this question for plumbed 3-manifolds and we present two new infinite families. Also, we reprove most of the classical works of Akbulut and Kirby, Fukuhara, Fickle, Maruyama, Martin, and Casson and Harer around the nineteen-eighties. This attempt continues and varies the recovery procedure of classical results started in our previous work. In addition, our approach provides a unification of classical and new results in a fairly simple way since we just use one folklore method.

On homology planes and contractible 4‐manifolds

“…Analyzing the Zaidenberg's construction, we see that Ramanujam manifolds studied by the second author in [58] are also homology planes. ) .…”

“…The generalization of the Ramanujam surface shown in Figure 1 appeared in [58], and the second author proved that 𝜕𝑊(𝑛) bound Poénaru manifolds with one 0-handle, two 1-handles, and two 2-handles. During the course of this paper, we also investigate that 𝑊(𝑛) originate from the homology planes for every 𝑛 ⩾ 1, see Proposition 2.5.…”

“…Since we want to control the number of handles of the 4-manifold precisely, we prefer to work with ribbon knots. Lemma 3.1 (Lemma 2.1, [58]). Let 𝑌 be the 3-manifold obtained by 0-surgery on a ribbon knot in 𝑆 3 with the fusion number 𝑛 ⩾ 1.…”

“…It was found in [6] and played a key role in the proof of their main theorem. Later, it was also effectively used by the second author [58,59].…”

“…Then the boundaries 𝜕𝑊(𝑛) are Kirby-Ramanujam homology spheres.Proof. By[58], we already know that the homology spheres 𝜕𝑊(𝑛) bound Poénaru manifolds. Also, their dual graph corresponds to the numerical data 𝑠 = 3 and 𝑀 =…”