• DOI:10.1007/BF01394055
• Corpus ID: 122833780

A surgery sequence in dimension four; the relations with knot concordance

@article{Freedman1982ASS,
  title={A surgery sequence in dimension four; the relations with knot concordance},
  author={Michael H. Freedman},
  journal={Inventiones mathematicae},
  year={1982},
  volume={68},
  pages={195-226},
  url={https://api.semanticscholar.org/CorpusID:122833780}
}

• M. Freedman
• Published 1 June 1982
• Mathematics
• Inventiones mathematicae

We present a systematic treatment of the classification problem for compact smooth 4-manifolds M. It is modeled on the surgery exact sequence, the central theorem in the classification of n-manifolds n>5. The price for the extension to dimension=4 is a hole in M where a (homotopy) 1-skeleton should be. There is no homotopy theoretic or surgical obstruction to complet- ing M with a wedge of circles so that the completion has the topology of a compact smooth manifold. This point-set problem is… 

We present a systematic treatment of the classification problem for compact smooth 4-manifolds M. It is modeled on the surgery exact sequence, the central theorem in the classification of n-manifolds n>5. The price for the extension to dimension=4 is a hole in M where a (homotopy) 1-skeleton should be. There is no homotopy theoretic or surgical obstruction to complet- ing M with a wedge of circles so that the completion has the topology of a compact smooth manifold. This point-set problem is all that stands between 4- manifolds and the tranquility that prevails in higher dimensions. When M is simply connected only a point is missing from the model. The applications of this are discussed in (F) and (FQ 1). The general theory has application to knot and link theory. In particular, knots with Alexander polynomial= 1 are characterized geometrically as knots admitting a certain