In this paper, we study topology of the variety of closed planar n-gons with given side lengths l 1 , . . . , l n . The moduli space M where = (l 1 , . . . , l n ), encodes the shapes of all such n-gons. We describe the Betti numbers of the moduli spaces M as functions of the length vector = (l 1 , . . . , l n ). We also find sharp upper bounds on the sum of Betti numbers of M depending only on the number of links n. Our method is based on an observation of a remarkable interaction between Morse functions and involutions under the condition that the fixed points of the involution coincide with the critical points of the Morse function.

In 1985 Kevin Walker in his study of topology of polygon spaces [15] raised an interesting conjecture in the spirit of the well-known question "Can you hear the shape of a drum?" of Marc Kac. Roughly, Walker's conjecture asks if one can recover relative lengths of the bars of a linkage from intrinsic algebraic properties of the cohomology algebra of its configuration space. In this paper we prove that the conjecture is true for polygon spaces in R 3 . We also prove that for planar polygon spaces the conjecture holds is several modified forms: (a) if one takes into account the action of a natural involution on cohomology, (b) if the cohomology algebra of the involution's orbit space is known, or (c) if the length vector is normal. Some of our results allow the length vector to be non-generic, the corresponding polygon spaces have singularities. Our main tool is the study of the natural involution and its action on cohomology. A crucial role in our proof plays the solution of the isomorphism problem for monoidal rings due to J. Gubeladze.where both inclined arrows are induced by the inclusions M ℓ → V and M ℓ ′ → V .

Abstract. Let q :X → X 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 spaceX. In this paper we characterize homology classes z inX which can be realized in arbitrary small neighborhoods of infinity inX. This problem was motivated by applications in the theory of critical points of closed 1-forms initiated in [2], [3].

This paper studies the homotopy invariant cat.X; / introduced by the first author in [6]. Given a finite cell-complex X , we study the function 7 ! cat.X; / where varies in the cohomology space H 1 .X I ‫./ޒ‬ Note that cat.X; / turns into the classical Lusternik-Schnirelmann category cat.X / in the case D 0. Interest in cat.X; / is based on its applications in dynamics where it enters estimates of complexity of the chain recurrent set of a flow admitting Lyapunov closed 1-forms, see [6; 7].In this paper we significantly improve earlier cohomological lower bounds for cat.X; / suggested in [6; 7]. The advantages of the current results (see Theorems 5, 6 and 7 below) are twofold: firstly, we allow cohomology classes of arbitrary rank (while in [6] the case of rank one classes was studied), and secondly, the theorems of the present paper are based on a different principle and give slightly better estimates even in the case of rank one classes. We introduce in this paper a new controlled version of cat.X; / and find upper bounds for it (Theorems 11 and 16). We apply these upper and lower bounds in a number of specific examples where we explicitly compute cat.X; / as a function of the cohomology class 2 H 1 .X I ‫./ޒ‬ 58E05; 55N25, 55U99

In this article we describe relations of the topology of closed 1-forms to the group theoretic invariants of Bieri-Neumann-Strebel-Renz. Starting with a survey, we extend these Sigma invariants to finite CWcomplexes and show that many properties of the group theoretic version have analogous statements. In particular we show the relation between Sigma invariants and finiteness properties of certain infinite covering spaces. We also discuss applications of these invariants to the Lusternik-Schnirelmann category of a closed 1-form and to the existence of a nonsingular closed 1-form in a given cohomology class on a high-dimensional closed manifold.To S.P. Novikov on the occasion of his 70-th birthday