Dirk Schütz

The isomorphism problem for planar polygon spaces

We give a proof of a conjecture of Walker that states that one can recover the lengths of the bars of a circular linkage from the cohomology ring of the configuration space. For a large class of length vectors, this has been shown by Farber, Hausmann and Schutz. In the remaining cases, we use Morse theory and the fundamental group to describe a subring of the cohomology invariant under graded ring isomorphism. From this subring the conjecture can be derived by applying a result of Gubeladze on the isomorphism problem of monoidal rings.

Moving homology classes to infinity

Abstract Let be a regular covering over a finite polyhedron with free abelian group of covering translations. Each nonzero cohomology class ξ ∈ H 1(X;R) with q*ξ = 0 determines a notion of “infinity” of the noncompact space . In this paper we characterize homology classes z in which can be realized in arbitrary small neighborhoods of infinity in . This problem was motivated by applications in the theory of critical points of closed 1-forms initiated in [Farber M.: Zeros of closed 1-forms, homoclinic orbits and Lusternik-Schnirelman theory. Topol. Methods Nonlinear Anal. 19 (2002), 123–152], [Farber M.: Lusternik-Schnirelman theory and dynamics. Lusternik-Schnirelmann Category and Related Topics. Contemporary Mathematics 316 (2002), 95–111].

The Walker conjecture for chains in ℝd.

A chain is a configuration in ℝd of segments of length l1, . . ., ln−1 consecutively joined to each other such that the resulting broken line connects two given points at a distance ln. For a fixed generic set of length parameters the space of all chains in ℝd is a closed smooth manifold of dimension (n − 2)(d − 1) − 1. In this paper we study cohomology algebras of spaces of chains. We give a complete classification of these spaces (up to equivariant diffeomorphism) in terms of linear inequalities of a special kind which are satisfied by the length parameters l1, . . ., ln. This result is analogous to the conjecture of K. Walker which concerns the special case d=2.

Intersection homology of linkage spaces

We consider the moduli spaces ℳd(l) of a closed linkage with n links and prescribed lengths l ∈ ℝn in d-dimensional Euclidean space. For d > 3 these spaces are no longer manifolds generically, but they have the structure of a pseudomanifold. We use intersection homology to assign a ring to these spaces that can be used to distinguish the homeomorphism types of ℳd(l) for a large class of length vectors in the case of d even. This result is a high-dimensional analogue of the Walker conjecture which was proven by Farber, Hausmann and the author.

Intersection homology of linkage spaces in odd-dimensional Euclidean space

We consider the moduli spaces Md(l)ℳd(l) of a closed linkage with nn links and prescribed lengths l∈Rnl∈ℝn in dd–dimensional Euclidean space. For d>3d>3 these spaces are no longer manifolds generically, but they have the structure of a pseudomanifold. We use intersection homology to assign a ring to these spaces that can be used to distinguish the homeomorphism types of Md(l)ℳd(l) for a large class of length vectors. These rings behave rather differently depending on whether dd is even or odd, with the even case having been treated in an earlier paper. The main difference in the odd case comes from an extra generator in the ring, which can be thought of as an Euler class of a stratified bundle.

Homology of moduli spaces of linkages in high-dimensional Euclidean space

We study the topology of moduli spaces of closed linkages in ℝd depending on a length vector l ∈ ℝn. In particular, we use equivariant Morse theory to obtain information on the homology groups of these spaces, which works best for odd d. In the case d = 5 we calculate the Poincare polynomial in terms of combinatorial information encoded in the length vector.

CLOSED 1-FORMS IN TOPOLOGY AND DYNAMICS

This article surveys recent progress of results in topology and dynamics based on techniques of closed 1-forms. Our approach lets us draw conclusions about properties of flows by studying homotopical and cohomological features of manifolds. More specifically, a Lusternik-Schnirelmann type theory for closed 1-forms is described, along with the focusing effect for flows and the theory of Lyapunov 1-forms. Also discussed are recent results about cohomological treatment of the invariants and and their explicit computation in certain examples.

Framed cobordism and flow category moves

Framed flow categories were introduced by Cohen, Jones and Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW complex with one cell for each object of the category. The idea is that the Floer invariant should take the form of the stable homotopy type of the resulting complex, recovering the Floer cohomology as its singular cohomology. Such a framed flow category was produced, for example, by Lipshitz and Sarkar from the input of a knot diagram, resulting in a stable homotopy type generalising Khovanov cohomology. We give moves that change a framed flow category without changing the associated stable homotopy type. These are inspired by moves that can be performed in the Morse–Smale case without altering the underlying smooth manifold. We posit that if two framed flow categories represent the same stable homotopy type then a finite sequence of these moves is sufficient to connect the two categories. This is directed towards the goal of reducing the study of framed flow categories to a combinatorial calculus. We provide examples of calculations performed with these moves (related to the Khovanov framed flow category), and prove some general results about the simplification of framed flow categories via these moves.

ON THE RELATIVE LUSTERNIK–SCHNIRELMANN CATEGORY WITH RESPECT TO A REAL COHOMOLOGY CLASS

In this paper, we study a homotopy invariant cat( X , B , [ω]) on a pair ( X , B ) of finite CW complexes with respect to the cohomology class of a continuous closed 1-form ω. This is a generalisation of a Lusternik–Schnirelmann-category-type cat( X , [ω]), developed by Farber in [ 3, 4 ], studying the topology of a closed 1-form. This paper establishes the connection with the original notion cat( X , [ω]) and obtains analogous results on critical points and homoclinic cycles. We also provide a similar ‘cuplength’ lower bound for cat( X , B , [ω]).