Homology 4-spheres with distinct k-invariants. (English) Zbl 0617.57008

Suciu, Alexander

Topology Appl. 25, 103-110 (1987).

Let M be a smooth, closed orientable 4-manifold and consider the following two homotopy type invariants of M: (i) \(\pi_ 2(M)\), regarded as a \({\mathbb{Z}}[\pi_ 1(M)]\)-module and (ii) the k-invariant \(k(M)\in H^ 3(\pi_ 1(M);\pi_ 2(M))\), i.e. the first obstruction to a retraction of \(K(\pi_ 1(M),1)\) onto M [see, e.g., S. Eilenberg and S. MacLane, Trans. Am. Math. Soc. 65, 49-99 (1949; Zbl 0034.111)]. The purpose of the paper under review is to show that, given any natural number n, there exist n homology 4-spheres with isomorphic \(\pi_ 1\) and \(\pi_ 2\) (as \({\mathbb{Z}}\pi_ 1\)-modules), but with distinct k- invariants. These examples have equivariantly isometric intersection forms on \(\pi_ 2\) and isometric intersection forms on \(H_ 2\) (compatible with the Hurewicz homomorphism \(\rho\) : \(\pi\) \({}_ 2\to H_ 2)\) but are not homotopy equivalent. (This answers another problem from the Kirby list (Problem N4.53) [R. Kirby, Four-manifold theory, Contemp. Math. 35, 513-528 (1984; Zbl 0558.57004)].) The paper concludes with a discussion of some interesting questions in this area.

Reviewer: D.Repovš

MSC:

57N13	Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
55S45	Postnikov systems, \(k\)-invariants
57Q45	Knots and links in high dimensions (PL-topology) (MSC2010)

Keywords:

spinning; knots in \(S^ 4\); action of fundamental group on second homotopy group; homology 4-spheres; distinct k-invariants; intersection forms

Citations:

Zbl 0034.111; Zbl 0558.57004

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