Kervaire's Characterization Theorem

Let G be a group, then there exists a piecewise linear knot K^(n-2) in S^n for n>=5 with G=pi_1(S^n-K) iff G satisfies

1. G is finitely presentable,

2. The Abelianization of G is infinite cyclic,

3. The normal closure of some single element is all of G,

4. H_2(G)=0; the second homology of the group is trivial.

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Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 350-351, 1976.

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Kervaire's Characterization Theorem

Cite this as:

Weisstein, Eric W. "Kervaire's Characterization Theorem." From MathWorld--A Wolfram Web Resource.

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