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111
Conjugation spaces
, 2004
"... There are classical examples of spaces X with an involution τ whose mod2comhomology ring resembles that of their fixed point set X τ: there is a ring isomorphism κ: H 2 ∗ (X) ≈ H ∗ (X τ). Such examples include complex Grassmannians, toric manifolds, polygon spaces. In this paper, we show that the ..."
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Cited by 185 (2 self)
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There are classical examples of spaces X with an involution τ whose mod2comhomology ring resembles that of their fixed point set X τ: there is a ring isomorphism κ: H 2 ∗ (X) ≈ H ∗ (X τ). Such examples include complex Grassmannians, toric manifolds, polygon spaces. In this paper, we show that the ring isomorphism κ is part of an interesting structure in equivariant cohomology called an H ∗frame. An H ∗ frame, if it exists, is natural and unique. A space with involution admitting an H ∗ frame is called a conjugation space. Many examples of conjugation spaces are constructed, for instance by successive adjunctions of cells homeomorphic to a disk in C k with the complex conjugation. A compact symplectic manifold, with an antisymplectic involution compatible with a Hamiltonian action of a torus T, is a conjugation space, provided X T is itself a conjugation space. This includes the coadjoint orbits of any semisimple compact Lie group, equipped with the Chevalley involution. We also study conjugateequivariant complex vector bundles (“real bundles ” in the sense of Atiyah) over a conjugation space and show that the isomorphism κ maps the Chern classes onto the StiefelWhitney classes of the fixed bundle.
The Alexander polynomial of a 3manifold and the Thurston norm on cohomology
 Ann. Sci. École Norm. Sup
, 2001
"... Let M be a connected, compact, orientable 3manifold with b1 (M) > 1, whose boundary (if any) is a union of tori. Our main result is the inequality kkA kkT between the Alexander norm on H 1 (M;Z), dened in terms of the Alexander polynomial, and the Thurston norm, dened in terms of the Euler ..."
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Cited by 70 (3 self)
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Let M be a connected, compact, orientable 3manifold with b1 (M) > 1, whose boundary (if any) is a union of tori. Our main result is the inequality kkA kkT between the Alexander norm on H 1 (M;Z), dened in terms of the Alexander polynomial, and the Thurston norm, dened in terms of the Euler characteristic of embedded surfaces. (A similar result holds when b1 (M) = 1.) Using this inequality we determine the Thurston norm for most links with 9 or fewer crossings. Contents 1
Optimal system of loops on an orientable surface
 DISCRETE COMPUT. GEOM
, 2005
"... Every compact orientable boundaryless surface M can be cut along simple loops with a common point v0, pairwise disjoint except at v0, so that the resulting surface is a topological disk; such a set of loops is called a system of loops for M. The resulting disk may be viewed as a polygon in which the ..."
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Cited by 43 (4 self)
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Every compact orientable boundaryless surface M can be cut along simple loops with a common point v0, pairwise disjoint except at v0, so that the resulting surface is a topological disk; such a set of loops is called a system of loops for M. The resulting disk may be viewed as a polygon in which the sides are pairwise identified on the surface; it is called a polygonal schema. Assuming that M is a combinatorial surface, and that each edge has a given length, we are interested in a shortest (or optimal) system of loops homotopic to a given one, drawn on the vertexedge graph of M. We prove that each loop of such an optimal system is a shortest loop among all simple loops in its homotopy class. We give an algorithm to build such a system, which has polynomial running time if the lengths of the edges are uniform. As a byproduct, we get an algorithm with the same running time to compute a shortest simple loop homotopic to a given simple loop.
Stable Systolic Inequalities and Cohomology Products
, 2001
"... On a compact Riemannian manifold (X; g) the real homology H (X; R) is naturally endowed with the stable norm. Briefly, if h 2 H k (X; R) then the stable norm of h is the infimum of the Riemannian k volumes of real cycles representing h. The stable ksystole is the minimum of the stable norm ove ..."
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Cited by 32 (11 self)
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On a compact Riemannian manifold (X; g) the real homology H (X; R) is naturally endowed with the stable norm. Briefly, if h 2 H k (X; R) then the stable norm of h is the infimum of the Riemannian k volumes of real cycles representing h. The stable ksystole is the minimum of the stable norm over nonzero elements in the lattice of integral classes in H k (X; R). Relying on results from the geometry of numbers due to W. Banaszczyk, and extending work by M. Gromov and J. Hebda, we prove metricindependent inequalities for products of stable systoles, where the product can be as long as the cup length of X. 1.
Algorithmic Properties of Relatively Hyperbolic Groups
, 2001
"... The following discourse is inspired by the works on hyperbolic groups of Epstein, and Neumann/Reeves. In [E], it is shown that geometrically finite hyperbolic groups are biautomatic. In [NR1], it is shown that virtually central extensions of word hyperbolic groups are biautomatic. We prove the follo ..."
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Cited by 30 (0 self)
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The following discourse is inspired by the works on hyperbolic groups of Epstein, and Neumann/Reeves. In [E], it is shown that geometrically finite hyperbolic groups are biautomatic. In [NR1], it is shown that virtually central extensions of word hyperbolic groups are biautomatic. We prove the following generalisation: Theorem 1. Let H be a geometrically finite hyperbolic group. Let σ ∈ H 2 (H) and suppose that σP = 0 for any parabolic subgroup P of H. Then the extension of H by σ is biautomatic We also prove another generalisation of the result in [E]. Theorem 2. Let G be hyperbolic relative to H, with the bounded coset penetration property. Let H be a biautomatic group with a prefixclosed normal form. Then G is biautomatic. Based on these two results, it seems reasonable to conjecture the following (which the author believes can be proven with a simple generalisation of the argument in 1): Let G be hyperbolic relative to H, where H has a prefixed closed biautomatic structure.
Inequalities between Entropy and Index of Coincidence derived from Information Diagrams
 IEEE Trans. Inform. Theory
, 2001
"... To any discrete probability distribution P we can associate its entropy H(P) = − � pi ln pi and its index of coincidence IC(P) = � p 2 i. The main result of the paper is the determination of the precise range of the map P � (IC(P), H(P)). The range looks much like that of the map P � (Pmax, H(P ..."
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Cited by 25 (11 self)
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To any discrete probability distribution P we can associate its entropy H(P) = − � pi ln pi and its index of coincidence IC(P) = � p 2 i. The main result of the paper is the determination of the precise range of the map P � (IC(P), H(P)). The range looks much like that of the map P � (Pmax, H(P)) where Pmax is the maximal point probability, cf. research from 1965 (Kovalevskij [18]) to 1994 (Feder and Merhav [7]). The earlier results, which actually focus on the probability of error 1 − Pmax rather than Pmax, can be conceived as limiting cases of results obtained by methods here presented. Ranges of maps as those indicated are called Information Diagrams. The main result gives rise to precise lower as well as upper bounds for the entropy function. Some of these bounds are essential for the exact solution of certain problems of universal coding and prediction for Bernoulli sources. Other applications concern Shannon theory (relations betweeen various measures of divergence), statistical decision theory and rate distortion theory. Two methods are developed. One is topological, another involves convex analysis and is based on a “lemma of replacement ” which is of independent interest in relation to problems of optimization of mixed type (concave/convex optimization).
Coarse Alexander duality and duality groups
 JOURNAL OF DIFFERENTIAL GEOMETRY
, 1999
"... We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n − 1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplic ..."
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Cited by 22 (4 self)
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We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n − 1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplicial action G � X determines a collection of “peripheral ” subgroups H1,..., Hk ⊂ G so that the group pair (G, {H1,..., Hk}) is an ndimensional Poincare duality pair. In particular, if G is a 2dimensional 1ended group of type F P2, and G � X is a free simplicial action on a coarse P D(3) space X, then G contains surface subgroups; if in addition X is simply connected, then we obtain a partial generalization of the Scott/Shalen compact core theorem to the setting of coarse P D(3) spaces. In the process we develop coarse topological language and a formulation of coarse Alexander duality which is suitable for applications involving quasiisometries and geometric group theory.
RATIONAL SURFACES WITH MANY NODES
, 2000
"... Let X be a smooth rational projective algebraic surface over an algebraically closed field k of characteristic ̸ = 2. It is known that for any nodal curve C on X there exists a birational morphism f: X → X ′ such that the image of C is an ordinary double point (a node). Let ..."
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Cited by 22 (7 self)
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Let X be a smooth rational projective algebraic surface over an algebraically closed field k of characteristic ̸ = 2. It is known that for any nodal curve C on X there exists a birational morphism f: X → X ′ such that the image of C is an ordinary double point (a node). Let
On the duality between filtering and NevanlinnaPick interpolation
 SIAM J. Control and Optimization
, 2000
"... Abstract. Positive real rational functions play a central role in both deterministic and stochastic linear systems theory, as well as in circuit synthesis, spectral analysis, and speech processing. For this reason, results about positive real transfer functions and their realizations typically have ..."
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Cited by 20 (14 self)
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Abstract. Positive real rational functions play a central role in both deterministic and stochastic linear systems theory, as well as in circuit synthesis, spectral analysis, and speech processing. For this reason, results about positive real transfer functions and their realizations typically have many applications and manifestations. In this paper, we study certain manifolds and submanifolds of positive real transfer functions, describing a fundamental geometric duality between filtering and Nevanlinna–Pick interpolation. Not surprisingly, then, this duality, while interesting in its own right, has several corollaries which provide solutions and insight into some very interesting and intensely researched problems. One of these is the problem of parameterizing all rational solutions of bounded degree of the Nevanlinna–Pick interpolation problem, which plays a central role in robust control, and for which the duality theorem yields a complete solution. In this paper, we shall describe the duality theorem, which we motivate in terms of both the interpolation problem and a fast algorithm for Kalman filtering, viewed as a nonlinear dynamical system on the space of positive real transfer functions. We also outline a new proof of the recent solution to the rational Nevanlinna–Pick interpolation problem, using an algebraic topological generalization of Hadamard’s global inverse function theorem. Key words. Nevanlinna–Pick interpolation, filtering, positive real functions, foliations, degree constraint AMS subject classifications. 30E05, 47N70, 58C99, 93B29, 93E11