Jonathan Hillman (University of Sydney)
Friday 11th August, 12.05-12.55pm, Carslaw 373Slice knots and deficiency conditionsIt 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:S^{n}->S^{n+2} is a slice knot if it bounds an (n+1)-disc knot Delta:D^{n+1}->D^{n+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(D^{n+3}\Delta) into D^{n+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.) |