Download 2-knots and their groups by Jonathan A. Hillman PDF

By Jonathan A. Hillman

To assault convinced difficulties in four-dimensional knot idea the writer attracts on numerous ideas, concentrating on knots in S^T4, whose primary teams include abelian general subgroups. Their type comprises the main geometrically attractive and most sensible understood examples. furthermore, it's attainable to use fresh paintings in algebraic tips on how to those difficulties. New paintings in 4-dimensional topology is utilized in later chapters to the matter of classifying 2-knots.

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Group is simply connected and has Euler characteristic 1, and so is contractible. Therefore if the Whitehead conjecture is true the subcomplex CCP) must also be aspherical, and so G is free or has finite geome tric dimension 2. (In particular, this is so if G is a I-relator group Ccf. [Ly 1950, Go 1981)), or is locally indicable [Ho 19821 or if it has no nontrivial superperfect normal subgroup [Oy 1987)). Thus Theorem 6 and the Whitehead conjecture together imply that a 3-knot group has finite geometric dimension 2 if and only if it has deficiency 1, in which case it is a 2-knot group.

Homology stably free monomorphically module Q[G ]-module is K(O,l) of Q~E. 3]. Since only 0 or 2. & which in nonis a induces a rational homology equivalence the Euler characteristics of Nand K(O,l) are equal. As these are sequence 0 is also C. - O. Therefore the the Euler characteristics of C. D. - stably E. , and as 0 is exact, the Euler characteristic of E. ) rank 0 and so must in fact be 0, by Kaplansky's Lemma. ) and f is a homotopy equivalence. the 0 has = 0 49 Localization and Aspbericity Suppose Corollary tbat G is a PDt-group over Q and tbat tbe cobo- mology ring H-(N;Q) is generated by Hl(N;Q).

3]. Since only 0 or 2. & which in nonis a induces a rational homology equivalence the Euler characteristics of Nand K(O,l) are equal. As these are sequence 0 is also C. - O. Therefore the the Euler characteristics of C. D. - stably E. , and as 0 is exact, the Euler characteristic of E. ) rank 0 and so must in fact be 0, by Kaplansky's Lemma. ) and f is a homotopy equivalence. the 0 has = 0 49 Localization and Aspbericity Suppose Corollary tbat G is a PDt-group over Q and tbat tbe cobo- mology ring H-(N;Q) is generated by Hl(N;Q).

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