Jonathan Hillman (University of Sydney)

Friday 11th August, 12.05-12.55pm, Carslaw 373

Slice knots and deficiency conditions

It is an open question whether there is a finitely presentable group G with deficiency def(G) strictly less than the module deficiency def_Z[G](g) of the augmentation ideal g as a Z[G]-module. This "deficiency gap" problem is related to various open problems in topology, through the interpretation of combinatorial group theory as the theory of 2-complexes, and through "general position" arguments which imply that k-complexes are homotopy equivalent to (2k+1)-manifolds (with boundary). We shall consider an example arising in high-dimensional knot theory.

An n-knot K:Sⁿ->Sⁿ⁺² is a slice knot if it bounds an (n+1)-disc knot Delta:Dⁿ⁺¹->Dⁿ⁺³. We shall show that if n > 1 and G is a knot group then def_Z[G](g)=1 is necessary and def(G)=1 is sufficient for there to be an n-knot with group G which bounds a slice Delta such that the inclusion of ð(Dⁿ⁺³\Delta) into Dⁿ⁺³\Delta is n-connected. (Every knot group is so realized with the inclusion (n-1)-connected, and only the trivial knot with the inclusion (n+1)-connected.)