M. Kapovich, J. Millson, The symplectic geometry of polygons in Euclidean space, Journal of Di#. Geometry, 44 (1996), 479-513.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... cohomology ring of polygon spaces Jean Claude HAUSMANN Allen KNUTSON Abstract We compute the integer cohomology rings of the polygon spaces introduced in [Kl, KM]. This is done by embedding them in certain toric varieties; the restriction map on cohomology is surjective and we calculate its kernel using ideas from the theory of Grobner bases. Since we do not invert the prime 2, we can tensor with Z 2 ; halving all degrees we show this produces the Z 2 ....

....Recherche Scientifique for its support. 3. when the ff i are integral) the geometric invariant theory quotient of the Grassmannian of 2 planes in C n by T n , where the fff i g specify an action of T n on the canonical bundle. The connection of the first to the second is made in [Kl] and [KM]; the second to the third in [GM] GGMS] and the first to the third in [HK] This paper draws much from the polygonal intuition and will concentrate on the first. In this paper we compute the integer cohomology rings of these spaces, in the (generic) case that they are smooth. There are partial ....

[Article contains additional citation context not shown here]

Kapovich, M. & Millson, J. The symplectic geometry of polygons in Euclidean space. J. of Diff. Geometry 44 (1996), 479--513.

.... 19 Oct 1998 A limit of toric symplectic forms that has no periodic Hamiltonians Jean Claude HAUSMANN Allen KNUTSON yz October 19, 1998 Abstract We calculate the Riemann Roch number of some of the pentagon spaces defined in [Kl, KM, 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 ....

....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. 1 The result In [Kl, HK1, KM] are introduced polygon spaces, which are a family of symplectic manifolds often possessing many interesting circle actions. In the case of pentagons with distinct edge lengths, they are actually (symplectomorphic to) toric varieties. So it is rather a surprise that the regular pentagon space, ....

[Article contains additional citation context not shown here]

Kapovich, M. & Millson, J. The symplectic geometry of polygons in Euclidean space. Jour. Diff. Geom. 44 (1996), no. 3, 479--513

....be used. Most prominent is a symplectic version of the Gel 0 fand MacPherson correspondence identifying the spaces m P 3 (ff) as symplectic quotients of the Grassmannian of 2 planes in C m . Earlier occurrences of symplectic geometry in the study of polygon spaces can be found in [Kl] and [KM2]. While this paper illustrates many phenomena in symplectic geometry, the proofs are entirely polygon theoretic and involve only classical differential topology. Nonetheless, many of the examples are new, interesting in their own right and instructive for both fields. Among our results: 1. The ....

....then an illustration of a theorem of Duistermaat ( Du] As is always true, and yet al..ways mysterious, it is helpful for studying the real case here planar polygons to extend to the complex case here polygons in R 3 . 2. Identification of the densely defined bending flows ( Kl] and [KM2] on the polygon spaces with the reduction of the Gel 0 fand Cetlin system [GS1] on the Grassmannian. 3. In some cases, the bending flows are globally defined, and by Delzant s reconstruction theory the spaces are equivariantly symplectomorphic to toric varieties (for instance when m 6, as noted ....

[Article contains additional citation context not shown here]

Kapovich, M. & Millson, J. The symplectic geometry of polygons in Euclidean space. Preprint, 1995.

....to think of the E i as vectors in R 3 . The constraint submanifold C then consists of configurations of 4 vectors that close to form the sides of a (not necessarily planar) quadrilateral in R 3 . This description in terms of quadrilaterals allows us to apply the results of Kapovich, Millson [17], Hausmann and Knutson [16] However, for the sake of a self contained treatment we redo some of their work. Consider a symplectic leaf = fjE i j = r i g in (so(3) 4 . Its intersection with C consists of all quadrilaterals with sides having fixed lengths r 1 ; r 4 . The ....

M. Kapovich and J. Millson, The symplectic geometry of polygons in Euclidean space, Jour. Diff. Geom. 44 (1996) 479-51.

No context found.

M. Kapovich, J. J. Millson, The symplectic geometry of polygons in Euclidean space, Journal of Diff. Geometry, vol. 44, (1996), p. 479--513.

....Hodge theory and L 2 cohomology we study the singularities and topology of configuration and moduli spaces of polygonal linkages in the 2 sphere. As a consequence we describe the local deformation space of a folded paper cone in R 3 . 1 Introduction. This is a part of a series of our papers [KM2], KM3] KM4] KM5] where we study interrelations between members of the following diagram: Configuration spaces of geometric objects Gamma Gamma Gamma Gamma Gamma Algebraic varieties Representation varieties of groups Examples of geometric objects that we consider are: linkages in ....

....S 3 ) the resulting algebraic varieties have a complex analytic structure and in fact coincide with moduli spaces of complex algebraic objects. Our general goal to to see how properties of Geometric objects and Groups are reflected in local and global topology of Algebraic varieties . In [KM2] we relate This research was partially supported by NSF grant DMS 96 26633 at University of Utah (Kapovich) and NSF grant DMS 95 04193 the University of Maryland (Millson) 1 configuration spaces of n gon linkages in R 3 , relative representation varieties of the fundamental groups of the ....

[Article contains additional citation context not shown here]

M. Kapovich, J. J. Millson, The symplectic geometry of polygons in Euclidean space, Journal of Diff. Geometry (to appear).

....of bending deformations of polygons in E 3 , hypergeometric integrals and the Gassner representation Michael Kapovich and John J. Millson August 27, 1999 Abstract The Hamiltonian potentials of the bending deformations of n gons in E 3 studied in [KM] and [Kly] give rise to a Hamiltonian action of the Malcev Lie algebra Pn of the pure braid group Pn on the moduli space M r of n gon linkages with the side lengths r = r 1 ; r n ) in E 3 . If e 2 M r is a singular point we may linearize the vector fields in Pn at e. This linearization ....

....If e 2 M r is a singular point we may linearize the vector fields in Pn at e. This linearization yields a flat connection r on the space C n of n distinct points on C . We show that the monodromy of r is the dual of a quotient of a specialized reduced Gassner representation. 1 Introduction In [KM] and [Kly] certain Hamiltonian flows on the moduli space M r of n gon linkages in E 3 were studied. In [KM] these flows were interpreted geometrically and called bending deformations of polygons. In [Kly] Klyachko pointed out that the Hamiltonian potentials of the bending deformations gave ....

[Article contains additional citation context not shown here]

M. Kapovich, J. J. Millson, The symplectic geometry of polygons in Euclidean space, Journal of Diff. Geometry, Vol. 44 (1996) p. 479--513.

....of bending deformations of polygons in E 3 , hypergeometric integrals and the Gassner representation Michael Kapovich and John J. Millson January 13, 1999 Abstract The Hamiltonian potentials of the bending deformations of n gons in E 3 studied in [KM] and [Kly] give rise to a Hamiltonian action of the Malcev Lie algebra Pn of the pure braid group Pn on the moduli space M r of n gon linkages with the side lengths r = r 1 ; r n ) in E 3 . If e 2 M r is a singular point we may linearize the vector fields in Pn at e. This linearization ....

....If e 2 M r is a singular point we may linearize the vector fields in Pn at e. This linearization yields a flat connection r on the space C n of n distinct points on C . We show that the monodromy of r is the dual of a quotient of a specialized reduced Gassner representation. 1 Introduction In [KM] and [Kly] certain Hamiltonian flows on the moduli space M r of n gon linkages in E 3 were studied. In [KM] these flows were interpreted geometrically and called bending deformations of polygons. In [Kly] Klyachko pointed out that the Hamiltonian potentials of the bending deformations gave ....

[Article contains additional citation context not shown here]

M. Kapovich, J. J. Millson, The symplectic geometry of polygons in Euclidean space, Journal of Diff. Geometry, Vol. 44 (1996) p. 479--513.

.... problem is based on our discovery of the connection between configuration spaces of elementary geometric objects and representation varieties of Coxeter, Shephard and Artin groups, developed in [KM2] KM3] KM5] KM6] The reader may also find our works on polygonal linkages [KM1] in R 2 ) [KM4] (in R 3 ) and [KM7] in S 2 ) to be of interest. We devote most of this paper to our most recent work [KM8] dealing with moduli spaces of planar linkages. A linkage (L; is a graph L with a positive real number (e) assigned to each edge e. We assume that we have chosen a distinguished ....

....degenerate realizations of a square. A square is the polygonal linkage where all four sides have equal length (see Figure 3) We have Lemma 3.4 The moduli space of the square is isomorphic to a union of three projective lines in general position in the projective plane. Proof: See [KM1, x12] and [KM4, x6]. Two of the components of the moduli space of the square consist of degenerate squares. We can eliminate the components consisting of degenerate squares by rigidifying the square as on Figure 6. We have Lemma 3.5 The moduli space of the rigidified square Q is isomorphic to RP 1 (i.e. a ....

M. Kapovich, J. J. Millson, The symplectic geometry of polygons in Euclidean space, Journal of Diff. Geometry, Vol. 44 (1996) 479--513.

.... D;C are parallel and A Gamma B = D Gamma C Therefore real points OE of M 0 ( Sigma) correspond to parallelograms: OE(v 1 ) Gamma OE(v 3 ) OE(v 6 ) Gamma OE(v 4 ) The assertion that the only singular real points of M 0 ( Sigma) are degenerate realizations can be proven analogously to [KM1]. An alternative proof follows from the discussion below. We first consider the case of a parallelogram S which is not a square. The moduli space M(S) for such S is described in [GN, Case II, page 120] The authors of [GN] describe the (projectivized) moduli space as a real projective subvariety ....

M. Kapovich, J. J. Millson, The symplectic geometry of polygons in Euclidean space, Journal of Diff. Geometry, Vol. 44 (1996) p. 479--513.

No context found.

M. Kapovich, J. Millson, The symplectic geometry of polygons in Euclidean space, Journal of Di#. Geometry, 44 (1996), 479-513.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC