By Jonathan A. Hillman
To assault yes difficulties in four-dimensional knot idea the writer attracts on various innovations, concentrating on knots in S^T4, whose primary teams include abelian basic subgroups. Their type includes the main geometrically attractive and most sensible understood examples. in addition, it's attainable to use fresh paintings in algebraic how you can those difficulties. New paintings in 4-dimensional topology is utilized in later chapters to the matter of classifying 2-knots.
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Extra resources for 2-knots and their groups
O. & 0, considering between the Wang sequences of the projections of M onto K(Tr,l) we see that f 41M1 '" 0 and so f so the for all c in by duality map induced onto M and K(Tr',1) induces isomorphisms in (co)homology with coefficients F. & Proof Tr". Tben M(K) is aspberical. & O. By Theorem 8 the classifying map has nonzero degree and therefore induces a rational homology equivalence, so the corollary follows from Theorem 7. 0 As a contrast we have the following theorem. 50 Loca1ization and Asphericity Theorem suppose 9 Let K be a 2-knot wbose group Tr bas deficiency I, and tbat Tr' '" Tr".
Geometrically this corresponds to the knot having a "minimal" Seifert hypersurface with abelian fundamental group). Theorem E >:1 - ] 19861 [Yo Let H be a finitely generated abelian group and an isomorphism between two subgroups of H extension G = H- > (with presentation
The condition on the subgroup U could be replaced by the apparently more general condition that R [GIT) have a flat extension with the strong invariant basis number property [C), where some annihilator of the augmentation 42 Localization and Aspbericity module R becomes invertible (cf. [Hi 1980». But we know of no other construction of such extensions of group rings. an Note also tha t if a group has abelian normal subgroup which is either central or finitely generated then it has a torsion free abelian normal subgroup of the same rank.