We calculate the intersection ring of three-dimensional graph manifolds with rational coefficients and give an algebraic characterization of these rings when the manifold’s underlying graph is a tree. We are able to use this characterization to show that the intersection ring obstructs arbitrary three-manifolds from being homology cobordant to certain graph manifolds.

Cochran defined the \(n\)th-order integral Alexander module of a knot in the three sphere as the first homology group of the knot’s (n+1)th-iterated abelian cover. The case \(n=0\) gives the classical Alexander module (and polynomial). After a localization, one can get a finitely presented module over a principal ideal domain, from which one can extract a higher-order Alexander polynomial. We present an algorithm to compute the first-order Alexander module for any knot. Included in this algorithm is a solution to the word problem in finitely presented \(\mathbb{Z}[\mathbb{Z}]\)-modules.

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We introduce a geometric invariant of knots in the three-sphere, called the first-order genus, that is derived from certain 2-complexes called gropes, and we show it is computable for many examples. In computing this invariant, we draw some interesting conclusions about the structure of a general Seifert surface for some knots.

We consider the Grope filtration of the classical knot concordance group that was introduced in a paper of Cochran, Orr and Teichner. Our main result is that successive quotients at each stage in this filtration have infinite rank. we also establish the analogous result for the Grope filtration of the concordance group of string links consisting of more than one component.

For certain classes of knots we define geometric invariants called higher-order genera. Each of these invariants is a refinement of the slice genus of a knot. We find lower bounds for the higher-order genera in terms of certain von Neumann \(\rho\)-invariants, which we call higher-order signatures. The higher-order genera offer a refinement of the Grope filtration of the knot concordance group.

It is known that knot Floer homology detects the genus and Alexander polynomial of a knot. We investigate whether knot Floer homology of \(K\) detects more structure of minimal genus Seifert surfaces for \(K\). We define an invariant of algebraically slice, genus one knots and provide examples to show that knot Floer homology does not detect this invariant. Finally, we remark that certain metabelian \(L^2\)-signatures bound this invariant from below.

We define families of invariants for elements of the mapping class group of \(S\), a compact orientable surface. Fix any characteristic subgroup \(H\) of \(\pi_1(S)\) and restrict to \(J(H)\), any subgroup of mapping classes that induce the identity modulo \(H\). To any unitary representation, \(r\) of \(\pi_1(S)/H\) we associate a higher-order \(\rho_r\)-invariant and a signature 2-cocycle \(\sigma_r\). These signature cocycles are shown to be generalizations of the Meyer cocycle. In particular each \(\rho_r\) is a quasimorphism and each \(\sigma_r\) is a bounded 2-cocycle on \(J(H)\). In one of the simplest non-trivial cases, by varying \(r\), we exhibit infinite families of linearly independent quasimorphisms and signature cocycles. We show that the \(\rho_r\) restrict to homomorphisms on certain interesting subgroups. Many of these invariants extend naturally to the full mapping class group and some extend to the monoid of homology cylinders based on \(S\).

We consider the question: "If the zero-framed surgeries on two oriented knots in \(S^3\) 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 \(\mathbb{Z}\)-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.

Recall that the n-solvable filtration of the smooth knot concordance group, suggested by work of Cochran-Orr-Teichner, has the property that any topologically slice knot lies in every term of the filtration. With the aim of studing topologically slice knots, 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. As is the case for the n-solvable filtration, 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-triviality 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.