Knot concordance and homology cobordism

We consider the question: "If the zero-framed surgeries on two oriented knots in the 3-sphere are integral homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?" We show that this question has a negative answer in the smooth category, even for topologically slice knots. To show this we first prove that the zero-framed surgery on K is Z-homology cobordant to the zero-framed surgery on many of its winding number one satellites P(K). Then we prove that in many cases the tau and s-invariants of K and P(K) differ. Consequently neither tau nor s is an invariant of the smooth homology cobordism class of the zero-framed surgery. We also show, that a natural rational version of this question has a negative answer in both the topological and smooth categories, by proving similar results for K and its (p,1)-cables.Comment: 15 pages, 8 figure

Filtering smooth concordance classes of topologically slice knots

We propose and analyze a structure with which to organize the difference between a knot in the 3-sphere bounding a topologically embedded 2-disk in the 4-ball and it bounding a smoothly embedded disk. The n-solvable filtration of the topological knot concordance group, due to Cochran-Orr-Teichner, may be complete in the sense that any knot in the intersection of its terms may well be topologically slice. However, the natural extension of this filtration to what is called the n-solvable filtration of the smooth knot concordance group, is unsatisfactory because any topologically slice knot lies in every term of the filtration. To ameliorate this we investigate a new filtration, {B_n}, that is simultaneously a refinement of the n-solvable filtration and a generalization of notions of positivity studied by Gompf and Cochran. We show that each B_n/B_{n+1} has infinite rank. But our primary interest is in the induced filtration, {T_n}, on the subgroup, T, of knots that are topologically slice. We prove that T/T_0 is large, detected by gauge-theoretic invariants and the tau, s, and epsilon-invariants; while the non-triviliality of T_0/T_1 can be detected by certain d-invariants. All of these concordance obstructions vanish for knots in T_1. Nonetheless, going beyond this, our main result is that T_1/T_2 has positive rank. Moreover under a "weak homotopy-ribbon" condition, we show that each T_n/T_{n+1} has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity.Comment: 41 pages, slightly revised introduction, minor corrections and up-dated references, this is the final version to appear in Geometry and Topolog

In-Cave and Surface Geophysics to Detect a “Lost” River in the Upper Levels of the Mammoth Cave System

In early 1960, explorers accessed a significant underground river through a crawlspace beneath a ledge in Swinnerton Avenue southeast of the Duck-Under. However, later expeditions failed to find this crawlspace. Instead, the level of sediment in the passage is now generally at or above the rock ledge, leaving no openings to lower level passages other than the Duck-Under itself. Apparently recent organic material (leaves, twigs, etc.) observed in passages just below the Duck-Under may be related to open channel fl ow from storm events which could theoretically provide local sediment transport. Therefore we have used in-cave spontaneous potential (SP), ground penetrating radar (GPR), and acoustic profiling, as well as surface mise-a-la-masse resistivity profiling, in an attempt to locate the river itself rather than the missing crawlway. Incave dye studies and additional geophysical profiling are needed to work out the detailed 3-D hydraulics of this region of the cave system