Abstract: "Diese Arbeit wurde mit dem Preis der Johannes Gutenberg-Universität Mainz 1992 ausgezeichnet." Thesis (doctoral)--Johannes Gutenberg-Universität Mainz, 1992. Includes bibliographical references (p. 125-127).
Abstract: We give a definition of 6-connected covering groups String(n) → Spin(n) in terms of "local fermions" on the circle. These are certain very explicit von Neumann algebras, the easiest examples of hyper-finite type III 1 factors. Given a Riemaniann string manifold M n , i.e. an n-dimensional manifold with prescribed lifts of the (deriva-tives of the) coordinate changes to String(n), we define a classical 2-dimensional conformal field theory. The fields on space-time, a con-formal surface, are... Show More
Abstract: In memory of Raoul Bott, friend and mentor. Abstract. There are many models for the K-theory spectrum known to- day, each one having its own history and applications. The purpose of this note is to give an elementary description of eight such models (and their completions) and relate all of them by canonical maps, most of which are homeomorphisms (rather then just homotopy equivalences). The first model are Raoul Bott's iterated spaces of minimal geodesic in orthogonal groups. This model... Show More
Abstract: Slava Krushkal and Frank Quinn recently brought to our attention misstate- ments in the proof of our linear grope height raising procedure which we pub- lished in 1995 (FT). This appendix replaces pages 518-522 of that paper with a proof along the same lines but with correct details. The main difference is that we are more careful in which order we add surface stages. This resolves in particular the problem of how to deal with intersections that involve a dual pair of circles on a surface... Show More
Abstract: The main result of this paper is a four-dimensional stable version of Kneser's conjecture on the splitting of three-manifolds as connected sums. Namely, let M be a topological respectively smooth compact connected four-manifold (with orientation or Spin-structure). Suppose that π 1 (M) splits as * n i=1 Γ i such that the image of π 1 (C) in π 1 (M) is subconjugated to some Γ i for each component C of ∂M . Then M is stably homeomorphic respectively diffeomorphic (preserving the orientation or... Show More
Abstract: We construct a connected closed orientable smooth four-manifold whose fundamental group is the free product of two non-trivial groups such that it is not homotopy equivalent to M0)M1 unless M0 or M1 is homeomorphic to S 4 . Let N be the nucleus of the minimal elliptic Enrique surface V1(2; 2) and put M = N(@N N. The fundamental group of M splits asZ=2 Z=2. We prove that M)k(S 2 S 2 ) is dieomorphic to M0)M1 for non-simply connected closed smooth four-manifolds M0 and M1 if and only if k 8.... Show More
Abstract: We discuss existence and uniqueness of the *-construction which reverses the Kirby-Siebenmann invariant of a topological 4-manifold while fixing the homotopy type. In particular, we point out an error in Theorem 10.3 of [M. H. Freedman and F. S. Quinn, Topology of 4-manifolds Princeton Math. Ser. 39 (1990; Zbl 0705.57001)] where a certain *-partner was mistakenly proven not to exist. The existence of this 4-manifold will follow from our Theorem 1 which proves that for fundamental group ℤ/2... Show More
Abstract: An obstruction theory for representing homotopy classes of surfaces in 4– manifolds by immersions with pairwise disjoint images is developed using the theory of non-repeated Whitney towers. Generalizations and geometric proofs of some results of Milnor and of Casson are given...
Abstract: In this paper we classify nonorientable topological closed 4-mani-folds with fundamental group 212 up to homeomorphism. Our results give a complete list of such manifolds, and show how they can be distinguished by explicit invariants including characteristic numbers and the q-invariant associ-ated to a normal Pinc-structure by the spectral asymmetry of a certain Dirac operator. In contrast to the oriented case, there exist homotopy equivalent nonorientable topological 4-manifolds which are... Show More
Full-text available · Article · Feb 1994 · Transactions of the American Mathematical Society
Abstract: We construct a connected closed orientable smooth four-manifold whose fundamental group is the free product of two non-trivial
groups such that it is not homotopy equivalent toM
0#M
1 unlessM
0 orM
1 is homeomorphic toS
4. LetN be the nucleus of the minimal elliptic Enrique surfaceV
1(2, 2) and putM=N∪
∂NN. The fundamental group ofM splits as ℤ/2 * ℤ/2. We prove thatM#k(S
2×S2) is diffeomorphic toM
0#M
1 for non-simply connected closed smooth four-manifoldsM
0 andM
1 if and only ifk≥8. On... Show More
Full-text available · Article · Jan 1995 · Commentarii Mathematici Helvetici
Abstract: Even when the fundamental group is intractable (i.e. not good) many interesting 4-dimensional surgery problems have topological solutions. We unify and extend the known examples and show how they compare to the (presumed) counterexamples by reference to Dwyer's filtration on second homology. The development brings together many basic results on the nilpotent theory of links. As a special case, a class of links only slightly smaller than homotopically trivial links is shown to have (free)... Show More
Abstract: The technical lemma underlying the 5-dimensional topologicals-cobordism conjecture and the 4-dimensional topological surgery conjecture is a purely smooth category statement about locating p1-null immersions of disks. These conjectures are theorems precisely for those fundamental groups (“good groups”) where the p1-null disk lemma (NDL) holds. We expand the class of known good groups to all groups of subexponential growth and those that can be formed from these by a finite number of... Show More
Abstract: We show that if the lower central series of the fundamental group of a closed oriented $3$-manifold stabilizes then the maximal nilpotent quotient is a cyclic group, a quaternion $2$-group cross an odd order cyclic group, or a Heisenberg group. These groups are well known to be precisely the nilpotent fundamental groups of closed oriented $3$-manifolds.
Article · Jan 1997 · Mathematical Research Letters
Abstract: We describe a nonsingular hermitian form of rank 4 over the group ring Z(Z) which is not extended from the integers. Moreover, we show that under certain indefiniteness asumptions, every nonsingular hermitian form on a free Z(Z)- module is extended from the integers. As a corollary, there exists a closed oriented 4-dimensional manifold with fundamental group Z which is not the connected sum of S1 × S3 with a simply-connected 4-manifold.
Abstract: . We prove a geometric refinement of Alexander duality for certain 2complexes, the so-called gropes, embedded into 4-space. In addition, we give new proofs and extended versions of two lemmas from [2] which are of central importance in the A-B-slice problem, the main open problem in the classification theory of topological 4-manifolds. Our methods are group theoretical, rather than using Massey products and Milnor ¯-invariants as in the original proofs. 1. Introduction Consider a finite... Show More
Full-text available · Article · Feb 1998 · Geometry & Topology
Questions about extending various kinds of representations arise naturally in low dimensional topology and in group theory, e.g., [29] [17] [18] [19] [20] [6]. It is tempting to wonder if the present results can be profitably generalized to the combinatorial group theory setting.
[Show abstract] [Hide abstract] ABSTRACT: Results are obtained on extending flat vector bundles or equivalently general
representations from the fundamental group of S, a connected subsurface of the
connected boundary of a compact, connected, oriented 3-dimensional manifold, to
the whole manifold M. These are applied to representations of fundamental
groups of 3-dimensional rational homology cobordisms. The proofs use the
introduction and complete computation up to sign of new numerical invariants
which "count with multiplicities and signs" the number of representations up to
conjugacy of the fundamental group of M to the unitary group U(n) (resp., the
special unitary group SU(n)) which when restricted to S are conjugate to a
specified irreducible representation, rho, of the fundamental group of S. These
invariants are inspired by Casson's work on SU(2) representations of closed
manifolds. All the invariants treated here are independent of the choice of
rho.
If T equals the difference of the Euler characteristics of S and M and is
non-negative, then a T times (dim U(n)) (resp., dim SU(n)) cycle is produced
that carries information about the space of such U(n) (resp., SU(n))
representations. For T = 0, the above integer invariant results and it is
entirely computed up to sign.
For T > 0, under the assumption that rho sends each boundary component of S
to the identity, a list of invariants for U(n) (resp. SU(n)) results which are
expressed as a homogeneous polynomial in many variables, reminiscent of the
work of Donaldson on 4-manifolds.
Article · May 2014 · Algebraic & Geometric Topology
[Show abstract] [Hide abstract] ABSTRACT: This paper computes Whitney tower filtrations of classical links. Whitney
towers consist of iterated stages of Whitney disks and allow a tree-valued
intersection theory, showing that the associated graded quotients of the
filtration are finitely generated abelian groups. Twisted Whitney towers are
studied and a new quadratic refinement of the intersection theory is
introduced, measuring Whitney disk framing obstructions. It is shown that the
filtrations are completely classified by Milnor invariants together with new
higher-order Sato-Levine and higher-order Arf invariants, which are
obstructions to framing a twisted Whitney tower in the 4-ball bounded by a link
in the 3-sphere. Applications include computation of the grope filtration, and
new geometric characterizations of Milnor's link invariants.
However, this last statement is currently only known to hold for links L with µ I (L) = 0 for any multi-index I = {i 1 i 2 . . . i n+2 } in which at most one index appears more than once (and at most twice) in I [12]. In 1985, Tim Cochran discovered a beautiful method of lifting certain Milnor invariants to well-defined integers [2]: Given a 2-component link L = (L 1 , L 2 ) with µ 12 (L) = 0, he first defined its derived link D(L) by forming a knot as the intersection of Seifert surfaces for the components (each in the complement of the other component), and then taking this knot in place of L 2 to yield the new 2- component link D(L).
[Show abstract] [Hide abstract] ABSTRACT: We show that Tim Cochran's invariants $\beta^i(L)$ of a $2$-component link $L$ in the $3$--sphere can be computed as intersection invariants of certain 2-complexes in the $4$--ball with boundary $L$. These 2-complexes are special types of twisted Whitney towers, which we call {\em Cochran towers}, and which exhibit a new phenomenon: A Cochran tower of order $2k$ allows the computation of the $\beta^i$ invariants for all $i\leq k$, i.e. simultaneous extraction of invariants from a Whitney tower at multiple orders. This is in contrast with the order $n$ Milnor invariants (requiring order $n$ Whitney towers) and consistent with Cochran's result that the $\beta^i(L)$ are integer lifts of certain Milnor invariants.
This leads to the notion of a non-repeated Whitney tower W which has also a non-repeated intersection tree λ(W) that generalizes the λ–invariant of Wall's intersection form. We shall explain these notions in a different paper [31] where we also prove the following beautiful application of the theory.
[Show abstract] [Hide abstract] ABSTRACT: We continue to develop an obstruction theory for embedding 2-spheres into 4-manifolds in terms of Whitney towers. The proposed intersection invariants take values in certain graded abelian groups generated by labelled trivalent trees, and with relations well known from the 3-dimensional theory of finite type invariants. Surprisingly, the same exact relations arise in 4 dimensions, for example the Jacobi (or IHX) relation comes in our context from the freedom of choosing Whitney arcs. We use the finite type theory to show that our invariants agree with the (leading term of the tree part of the) Kontsevich integral in the case where the 4-manifold is obtained from the 4-ball by attaching handles along a link in the 3-sphere.
Article · Mar 2004 · Algebraic & Geometric Topology
We remark that the manifold used in the second part of the proof above is known to not be smoothable [12] . A smooth example would come from a smoothable 4- manifold with fundamental group Z whose Z[Z] intersection form is not extended from the integers.
[Show abstract] [Hide abstract] ABSTRACT: We investigate certain $4$-dimensional analogues of the classical $3$-dimensional Dehn's lemma, giving examples where such analogues do or do not hold, in the smooth and topological categories. In particular, we show that an essential $2$-sphere $S$ in the boundary of a simply connected $4$-manifold $W$ such that $S$ is null-homotopic in $W$ need not extend to an embedding of a ball in $W$. However, if $W$ is simply connected (or more generally a $4$-manifold with abelian fundamental group) with boundary a homology sphere, then $S$ bounds a topologically embedded ball in $W$. Moreover, we give examples where such an $S$ does not bound any smoothly embedded ball in $W$. In a similar vein, we construct incompressible tori $T\subseteq \partial W$ where $W$ is a contractible $4$-manifold such that $T$ extends to a map of a solid torus in $W$, but not to any embedding of a solid torus in $W$. Moreover, we construct an incompressible torus $T$ in the boundary of a contractible $4$-manifold $W$ such that $T$ extends to a topological embedding of a solid torus in $W$ but no smooth embedding. As an application of our results about tori, we address a question posed by Gompf about extending certain families of diffeomorphisms of $3$-manifolds which he has recently used to construct infinite corks.
Namely, there are 4-manifolds simple homotopy equivalent to a connected sum which do not admit (topologically) corresponding connected sum splitting. For other results on the 4-dimensional Kneser-conjecture, see [16] and [17]. (3) The proof of theorem 1 is purely topological, but (b) and (f) also mean that all the manifolds can be smoothed after product with R 3 or connected sum with enough S 2 ×S 2 's.
[Show abstract] [Hide abstract] ABSTRACT: Recent computations of UNil-groups by Connolly, Ranicki and Davis are used to study splittability of homotopy equivalences between 4-dimensional manifolds with infinite dihedral fundamental groups.
[Show abstract] [Hide abstract] ABSTRACT: This is a report on aspects of the theory and use of monoidal categories. The
first section introduces the main concepts through the example of the category
of vector spaces. String notation is explained and shown to lead naturally to a
link between knot theory and monoidal categories. The second section reviews
the light thrown on aspects of representation theory by the machinery of
monoidal category theory, such as braidings and convolution. The category
theory of Mackey functors is reviewed in the third section. Some recent
material and a conjecture concerning monoidal centres is included. The fourth
and final section looks at ways in which monoidal categories are, and might, be
used for new invariants of low-dimensional manifolds and for the field theory
of theoretical physics.
In the study of higher dimensional knotted objects in codimension 2, the notion of link-homotopy seem to have first been studied by W. S. Massey and D. Rolfsen for 2–component 2–links, i.e. two 2–spheres embedded in 4–space[13]. In the late nineties, the study of 2–links up to link-homotopy was definitively settled by A. Bartels and P. Teichner, who showed in[3]that all 2–links are link-homotopically trivial. Actually, their result is much stronger, as it holds in any dimension.
[Show abstract] [Hide abstract] ABSTRACT: We consider knotted annuli in 4-space, called 2-string-links, which are knotted surfaces in codimension two that are naturally related, via closure operations, to both 2-links and 2-torus links. We classify 2-string-links up to link-homotopy by means of a 4-dimensional version of Milnor invariants. The key to our proof is that any 2-string link is link-homotopic to a ribbon one; this allows to use the homotopy classification obtained in the ribbon case by P. Bellingeri and the authors. Along the way, we give a Roseman-type result for immersed surfaces in 4-space. We also discuss the case of ribbon k-string links, for $k\geq 3$.
[Show abstract] [Hide abstract] ABSTRACT: The macroscopic dimensions of space should not be input but rather output of a general model for physics. Here, dimensionality arises from a recently discovered mathematical bifurcation: positive versus indefinite manifold pairings. It is used to build an action on a formal chain of combinatorial space-times of arbitrary dimension. The context for such actions is 2-field theory where Feynman integrals are not over classical, but previously quantized configurations. A topologically enforced singularity of the action terminates the dimension at four and, in fact, the final fourth dimension is Lorentzian due to light-like vectors in the four dimensional manifold pairing. Our starting point is the action of causal dynamical triangulations but in a dimension-agnostic setting. It is encouraging that some hint of extra small dimensions emerges from our action.
Article · Aug 2010 · Algebraic & Geometric Topology
There is a similar result for the lower central series. If G [ω] = G [n] for some finite n then n ≤ 3, and G [ω] is finite, Z or is a nilpotent P D 3 -group [76]. If G is not virtually representable onto Z then G/G (ω) is either finite (and G (ω) is a perfect P D 3 -group) or is finitely generated, residually finite-solvable group and has one or infinitely many ends.
[Show abstract] [Hide abstract] ABSTRACT: We state a number of open questions on 3-dimensional Poincare duality groups and their subgroups, motivated by considerations from 3-manifold topology. AMS Subject classification (1985): Primary 57N10, Secondary 57M05, 20J05
Article · Jan 2011 · Algebraic & Geometric Topology
In fact, in [6], Corollary 3.2 is obtained under the condition that either the fundamental group is finite or the second rational homology of the 2–type is non–zero. Corresponding conditions were used in [5] for oriented PD 2n –complexes with (n − 1)–connected universal covers, and Teichner extended the approach of [6] to the non–oriented case in his thesis [19]. Our result shows that the conditions on finiteness and rational homology used in these papers are not necessary.
[Show abstract] [Hide abstract] ABSTRACT: We describe an algebraic structure on chain complexes yielding algebraic models which classify homotopy types of Poincare duality complexes of dimension 4. Generalizing Turaev's fundamental triples of Poincare duality complexes of dimension 3, we introduce fundamental triples for Poincare duality complexes of dimension n > 2 and show that two Poincare duality complexes are orientedly homotopy equivalent if and only if their fundamental triples are isomorphic. As applications we establish a conjecture of Turaev and obtain a criterion for the existence of degree 1 maps between n-dimensional manifolds.