Stefan Friedl

3-Manifold Groups

We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

A SURVEY OF TWISTED ALEXANDER POLYNOMIALS

We give a short introduction to the theory of twisted Alexander polynomials of a 3-manifold associated to a representation of its fundamental group. We summarize their formal properties and we explain their relationship to twisted Reidemeister torsion. We then give a survey of the many applications of twisted invariants to the study of topological problems. We conclude with a short summary of the theory of higher order Alexander polynomials.

Eta invariants as sliceness obstructions and their relation to Casson–Gordon invariants

We give a useful classification of the metabelian unitary repre- sentations of π1(MK), where MK is the result of zero-surgery along a knot K ⊂ S 3 . We show that certain eta invariants associated to metabelian representations π1(MK) → U(k) vanish for slice knots and that even more eta invariants vanish for ribbon knots and doubly slice knots. We show that our vanishing results contain the Casson-Gordon sliceness obstruction. In many cases eta invariants can be easily computed for satellite knots. We use this to study the relation between the eta invariant sliceness obstruction, the eta-invariant ribbonness obstruction, and the L 2 -eta invariant sliceness obstruction recently introduced by Cochran, Orr and Teichner. In partic- ular we give an example of a knot which has zero eta invariant and zero metabelian L 2 -eta invariant sliceness obstruction but which is not ribbon. AMS Classification 57M25, 57M27; 57Q45, 57Q60

Twisted Alexander Polynomials of Hyperbolic Knots

We study a twisted Alexander polynomial naturally associated to a hyperbolic knot in an integer homology 3-sphere via a lift of the holonomy representation to . It is an unambiguous symmetric Laurent polynomial whose coefficients lie in the trace field of the knot. It contains information about genus, fibering, and chirality, and moreover, is powerful enough to sometimes detect mutation. We calculated this invariant numerically for all 313 209 hyperbolic knots in S 3 with at most 15 crossings, and found that in all cases it gave a sharp bound on the genus of the knot and determined both fibering and chirality. We also study how such twisted Alexander polynomials vary as one moves around in an irreducible component X 0 of the -character variety of the knot group. We show how to understand all of these polynomials at once in terms of a polynomial whose coefficients lie in the function field of X 0. We use this to help explain some of the patterns observed for knots in S 3, and explore a potential relationship between this universal polynomial and the Culler–Shalen theory of surfaces associated to ideal points.

The decategorification of sutured Floer homology

We define a torsion invariant T for every balanced sutured manifold (M, γ), and show that it agrees with the Euler characteristic of sutured Floer homology (SFH). The invariant T is easily computed using Fox calculus. With the help of T, we prove that if (M, γ) is complementary to a Seifert surface of an alternating knot, then SFH(M, γ) is either 0 or ℤ in every Spin c structure. The torsion invariant T can also be used to show that a sutured manifold is not disc decomposable, and to distinguish between Seifert surfaces.The support of SFH gives rise to a norm z on H 2 (M, ∂ M; ℝ). The invariant T gives a lower bound on the norm z, which in turn is at most the sutured Thurston norm x s . For closed 3-manifolds, it is well known that Floer homology determines the Thurston norm, but we show that z<x s can happen in general. Finally, we compute T for several wide classes of sutured manifolds.

A vanishing theorem for twisted Alexander polynomials with applications to symplectic 4-manifolds

In this paper we show that given any 3-manifold N and any non-fibered class in H^1(N;Z) there exists a representation such that the corresponding twisted Alexander polynomial is zero. This is obtained by extending earlier work of the authors, together with results of Agol and Wise on separability of 3-manifold groups. This result allows us to completely classify symplectic 4-manifolds with a free circle action, and to determine their symplectic cones.

TWISTED ALEXANDER NORMS GIVE LOWER BOUNDS ON THE THURSTON NORM

We introduce twisted Alexander norms of a compact connected orientable 3-manifold with first Betti number greater than one, generalizing norms of McMullen and Turaev. We show that twisted Alexander norms give lower bounds on the Thurston norm of a 3-manifold. Using these we completely determine the Thurston norm of many 3-manifolds which are not determined by norms of McMullen and Turaev.

Twisted Alexander polynomials and symplectic structures

Let <i>N</i> be a closed, oriented 3-manifold. A folklore conjecture states that <i>S</i>¹ × <i>N</i> admits a symplectic structure only if <i>N</i> admits a fibration over the circle. The purpose of this paper is to provide evidence to this conjecture studying suitable twisted Alexander polynomials of <i>N</i>, and showing that their behavior is the same as of those of fibered 3-manifolds. In particular, we will obtain new obstructions to the existence of symplectic structures and to the existence of symplectic forms representing certain cohomology classes of <i>S</i>¹ × <i>N</i>. As an application of these results we will show that <i>S</i>¹ × <i>N</i>(<i>P</i>) does not admit a symplectic structure, where <i>N</i>(<i>P</i>) is the 0-surgery along the pretzel knot <i>P</i> = (5, -3, 5), answering a question of Peter Kronheimer.

New constructions of slice links

We use techniques of Freedman and Teichner to prove that, under certain circumstances, the multi-infection of a slice link is again slice (not necessarily smoothly slice). We provide a general context for proving links are slice that includes many of the previously known results.

The cobordism group of homology cylinders

Garoufalidis and Levine introduced the homology cobordism group of homology cylinders over a surface. This group can be regarded as an enlargement of the mapping class group. Using torsion invariants, we show that the abelianization of this group is infinitely generated provided that the first Betti number of the surface is positive. In particular, this shows that the group is not perfect. This answers questions of Garoufalidis and Levine, and Goda and Sakasai. Furthermore, we show that the abelianization of the group has infinite rank for the case that the surface has more than one boundary component. These results also hold for the homology cylinder analogue of the Torelli group.