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On the non-realizability of certain 3-manifolds by Dehn surgery

Published online by Cambridge University Press:  24 October 2008

Steven Boyer
Steven Boyer
Affiliation:
University of Toronto
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Extract

We construct examples of compact 3-manifolds which have finite cyclic first homology groups but which are not realizable by Dehn surgery on a knot in a homotopy 3-sphere. A necessary condition for a 3-manifold M to be the result of such a surgery is that π1( M) be normally generated by a single element (corresponding to a meridian of the knot), that is π1(M) should have weight 1. As we do not expect this to hold in general for the free product of non-trivial groups we are naturally led to consider connected sums. For instance, if G denotes the binary icosahedral group, it is known that ℤ* G has weight greater than 1 (see [2]). Thus, if H is the Poincaré sphere and M is any 3-manifold with first homology ℤ, M # H is not the result of surgery on a knot in a homotopy 3-sphere. Unfortunately this purely group-theoretic approach is too difficult in general owing to the intractability of the weight 1 condition (see § 9 of [4]). Thus we adopt a strategy involving an interplay between group theory and geometry. Our method is based on the following:

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 99 , Issue 1 , January 1986 , pp. 103 - 106
Copyright
Copyright © Cambridge Philosophical Society 1986

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References

REFERENCES

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