By Jonathan A. Hillman

ISBN-10: 0521371732

ISBN-13: 9780521371735

ISBN-10: 0521378125

ISBN-13: 9780521378123

To assault definite difficulties in four-dimensional knot thought the writer attracts on quite a few thoughts, targeting knots in S^T4, whose primary teams include abelian basic subgroups. Their category includes the main geometrically beautiful and most sensible understood examples. furthermore, it's attainable to use contemporary paintings in algebraic 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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**Extra resources for 2-knots and their groups**

**Sample text**

Tbe 1 induces a rational bomology equivalence. f Proof As these conditions are clearly necessary, we need only show they are sufficient. , D. complexes of Since the and E. universal covers HS(O;Z[G)) = 0 for be the equivariant cellular chain and (K(G,l),ii) ii, K(O,l) s < 4, Poincare that duality respectively. ) trivial maps Z[G] = As H 2(ii;Q). 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.

Superperfect group and is the They observe G is a knot group then by the KUnneth theorem Hi(GXP;Z) = HlG;Z) for i " 2, and If moreover there is an c1ement P in GxP has centre ((G XP) = CG XCP. P such that the subgroup [p,PI generated by {(p,xl = pxp- 1 x- 1 :x in P} is the weight whole of P, and if g element for erally if P l' in Pi Pi) CG (i)CP 1(i). it shall then which is GXP, Pr is a weight element are r GXP 1 X therefore a for G, then gp 3-knot group. is a More gen- such superperfect groups (with such elements is .

Lemma, if X(M) = 0 O. Since HS then R [GIT) embeds in r, HÂ· embeds in HS == Homr(HS,n and so is also O. Therefore MT is R -acyclic. Conversely, if MT is acyclic then H = 0 so HS = 0 and hence X(M) = O. 0 The orientability of M is not essential assumption on T is satisfied if T is perfect (T is a torsion group and R = Q. = for this T) and R theorem. The = Z or if T (in particular it holds when T = 1). 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.

### 2-knots and their groups by Jonathan A. Hillman

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