Geometric descriptions of polygon and chain spaces

We give a few simple methods to geometically describe some polygon and chain-spaces in R^d. They are strong enough to give tables of m-gons and m-chains when m <= 6

Polygon spaces and Grassmannians

We study the moduli spaces of polygons in R^2 and R^3, identifying them with subquotients of 2-Grassmannians using a symplectic version of the Gel'fand-MacPherson correspondence. We show that the bending flows defined by Kapovich-Millson arise as a reduction of the Gel'fand-Cetlin system on the Grassmannian, and with these determine the pentagon and hexagon spaces up to equivariant symplectomorphism. Other than invocation of Delzant's theorem, our proofs are purely polygon-theoretic in nature.Comment: plain TeX, 21 pages, submitted to Journal of Differential Geometr

A limit of toric symplectic forms that has no periodic Hamiltonians

We calculate the Riemann-Roch number of some of the pentagon spaces defined in [Klyachko,Kapovich-Millson,HK1]. Using this, we show that while the regular pentagon space is diffeomorphic to a toric variety, even symplectomorphic to one under arbitrarily small perturbations of its symplectic structure, it does not admit a symplectic circle action. In particular, within the cohomology classes of symplectic structures, the subset admitting a circle action is not closed.Comment: 7 pages, 2 external figure

Maximal Hamiltonian tori for polygon spaces

We study the poset of Hamiltonian tori for polygon spaces. We determine some maximal elements and give examples where maximal Hamiltonian tori are not all of the same dimension.Comment: 15 pages, Latex, 1 figur

Conjugation spaces and 4-manifolds

We show that 4-dimensional conjugation manifolds are all obtained from branched 2-fold coverings of knotted surfaces in Z/2-homology 4-spheres.Comment: 23 pages. (v3) Revisions to the Introduction and section order; references added. (v4) Final version, to appear in Math. Zeitschrif

Equivariant Bundles and Isotropy Representations

We introduce a new construction, the isotropy groupoid, to organize the orbit data for split $\Gamma$-spaces. We show that equivariant principal $G$-bundles over split $\Gamma$-CW complexes $X$ can be effectively classified by means of representations of their isotropy groupoids. For instance, if the quotient complex $A=\Gamma\backslash X$ is a graph, with all edge stabilizers toral subgroups of $\Gamma$, we obtain a purely combinatorial classification of bundles with structural group $G$ a compact connected Lie group. If $G$ is abelian, our approach gives combinatorial and geometric descriptions of some results of Lashof-May-Segal and Goresky-Kottwitz-MacPherson.Comment: Final version: to appear in "Groups, Geometry and Dynamics

Conjugation spaces and edges of compatible torus actions

Duistermaat introduced the concept of ``real locus'' of a Hamiltonian manifold. In that and in others' subsequent works, it has been shown that many of the techniques developed in the symplectic category can be used to study real loci, so long as the coefficient ring is restricted to the integers modulo 2. It turns out that these results seem not necessarily to depend on the ambient symplectic structure, but rather to be topological in nature. This observation prompts the definition of ``conjugation space'' in a paper of the two authors with V. Puppe. Our main theorem in this paper gives a simple criterion for recognizing when a topological space is a conjugation space.Comment: 19 page

The cell-dispensability obstruction for spaces and manifolds

We compare two properties for a CW-space $X$: (1) being homotopy equivalent to a CW-complex without $j$-cells for $k\leq j\leq \ell$ (($k,\ell$)-cellfree) and (2) $H^j(X;R)=0$ for any $\mathbb Z\pi_1(X)$-module $R$ when $k\leq j\leq \ell$ (cohomogy ($k,\ell$)-silent). Using the technique of Wall's finiteness obstruction, we show that a connected CW-space $X$ of finite type which is cohomogy ($k,\ell$)-silent determines a "cell-dispensability obstruction'' $w_k(X)\in\tilde K_0(\mathbb Z\pi_1(X))$ which vanishes if and only if $X$ is ($k,\ell$)-cellfree ($k\geq 4$). Any class in $\tilde K_0(\mathbb Z\pi)$ may occur as the cell-dispensability obstruction $w_k(X)$ for a CW-space $X$ with $\pi_1(X)$ identified with $\pi$. Using projective surgery, a similar theory is obtained for manifolds, replacing "cells" by "handles" (antisimple manifolds).Comment: 42 page

On the conjecture of Kevin Walker

In 1985 Kevin Walker in his study of topology of polygon spaces 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