REVIEW OF: "Kawauchi Akio, Splitting a 4-manifold with infinite cyclic fundamental group, revised, J. Knot Theory Ramifications 22, No. 14, Article ID 1350081, 9 p. (2013)". [DE062730205]

In [Osaka J. Math. 31(3), 489-495 (1994; Zbl 0849.57018 )], the author stated that every closed connected orientable 4-manifold M with infinite cyclic fundamental group is TOP-split, i.e. it is homeomorphic to the connected sum (S1 × S3)#M1, M1 being a closed simply connected 4-manifold. However, in [Manuscr. Math. 93(4), 435-442 (1997; Zbl 0890.57034)], Hambleton and Teichner obtained a counterexample to the above general statement. In the paper under review, the author makes a revision and proves that TOP-splittability holds under the additional hypothesis that a finite covering of M is TOP-split. In particular, the original statement turns out to be true in the case of indefinite intersection form, as well as for any smooth spin 4-manifold (with infinite cyclic fundamental group). The proof of the revised statement makes use of notions developed in [Knots in Hellas 98, Ser. Knots Everything. 24 (World Scientific Publishing), 208-228 (2000; Zbl 0969.57020)] and [Atti Semin. Mat. Fis. Univ. Modena 48(2), 405-424 (2000; Zbl 1028.57019)], together with the key result - proved in [Osaka J. Math. 31(3), 489-495 (1994; Zbl 0849.57018 )] - that every closed connected orientable 4-manifold M with infinite cyclic fundamental group is homology cobordant to (S1 × S3)#M1. Consequences about surface-knots in S4 are also considered (see [J. Knot Theory Ramifications 4(2), 213-224 (1995; Zbl 0844.57020)]).