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Shake-slice knots and smooth contractible 4-manifolds

Published online by Cambridge University Press:  24 October 2008

Steven Boyer
Steven Boyer
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge
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Let V be a smooth, 1-connected 4-manifold. It has long been known that there may be 2-dimensional homology classes in V which are not represented by smoothly embedded 2-spheres, owing to the failure of the Whitney lemma in this dimension. One of the simplest examples of this failure may be described as follows. Let K be a smooth knot in S3 and r an integer. By V(K, r) we mean the 4-manifold obtained by attaching a 2-handle to B4 along an r-framed tubular neighbourhood of K. Evidently each V(K, r) is homotopy equivalent to S2 and we say that K is an r-shake-slice knot if there is a 2-sphere, smoothly embedded in V(K, r), which represents a generator of the second homology group. For example, if K is a slice knot then it is r-shake-slice for each integer r. On the other hand, it is known that arf (K) = 0 whenever K is r-shake-slice and so in particular a trefoil is never such a knot.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 98 , Issue 1 , July 1985 , pp. 93 - 106
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

REFERENCES

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