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Irreducible 3-manifolds that cannot be obtained by 0-surgery on a knot. (English) Zbl 1429.57010

It is a well-known fact that any closed, orientable 3-manifold can be obtained as a surgery of a link in \(S^3\). It is possible that this link has only one component; i.e., it is a knot. The authors cite a conjectured algebraic characterization (due to M. Aschenbrenner et al. [3-manifold groups. Zürich: European Mathematical Society (EMS) (2015; Zbl 1326.57001)]) of the irreducible 3-manifolds \(M\) that are obtained through surgery on a knot in \(S^3\) based on the number of normal generators of \(\pi_1(M)\) and on \(H_1(M)\). Specifically, it is known that the result of surgery on non-trivial knots in \(S^3\) produces an irreducible 3-manifold with \(b_1(M) = 1\) and requiring only one normal generator of \(\pi_1(M)\), but the conjecture is that the converse holds. The authors have produced two infinite families of two-component links offering counter-examples to this conjecture, among which are the first known examples of irreducible homology \(S^1\times S^2\)’s which are not the result of surgery on a knot in \(S^3\). This marks a major milestone in the surgery classification of 3-manifolds.
The authors expressly assume prior knowledge of knot Floer homology as the veracity of their counter-examples is based in the \(d\)-invariants. For those with relatively little experience with knot Floer homology, this is not a total barrier to understanding; but the results are certainly more understandable with this prerequisite. It also doesn’t hurt to have a working knowledge of 4-manifold theory since the authors make use of some plumbing arguments as well as the Rohlin and Arf invariants.
The arguments do get a bit technical in places (including some work by cases), but the authors do an excellent job of highlighting the details on which the readers should focus their attention. This is especially true of the early sections, where all results are linked back to the original conjecture and the main results that disprove it. They also identify places where proofs can be extended to larger families of manifolds, including L-spaces. Also noteworthy are the exceptionally detailed surgery diagrams with clear, easily followed intermediate steps.
This article is a great introduction to the practical use of the invariants of knot Floer homology and 4-manifold theory. Though somewhat technical, this article is accessible to early-career researchers and advanced graduate students.

MSC:

57K30 General topology of 3-manifolds
57K10 Knot theory
57K16 Finite-type and quantum invariants, topological quantum field theories (TQFT)
57M05 Fundamental group, presentations, free differential calculus
57R58 Floer homology
57R65 Surgery and handlebodies
57R90 Other types of cobordism

Citations:

Zbl 1326.57001
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References:

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