A subset M of a topological vector space X is said to be dense-lineable in X if there exists an infinite dimensional linear manifold in M∪{0} and dense in X. We give sufficient conditions for a lineable set to be dense-lineable, and we apply them to prove the dense-lineability of several subsets of C[a,b]. ...

Journal: Topology, vol. 48, no. 2, pp. 149-156, 2009]]>For an r=0,1,…,∞, by Dr(R), D+r(R), Dcr(R) we denote respectively the groups of Cr diffeomorphisms, orientation-preserving Cr diffeomorphisms, and compactly supported Cr diffeomorphisms of the real line. We think of these groups as bitopologies spaces endowed with the compact-open Cr topology and the Whitney Cr topology. We prove that all the ...

Journal: Topology, vol. 48, no. 2, pp. 119-129, 2009]]>In this paper some characterizations for generalized invexity and generalized monotonicity, under separable Hilbert spaces, are provided. The results established are useful for application in many problems in pure and applied analysis.

Journal: Topology, vol. 48, no. 2, pp. 66-79, 2009]]>We show that the capacity monad has a functional representation, i.e. the space of capacities on a compactum X can be naturally embedded (with preservation of the monad structure) in some space of functionals on C(X). We also describe this space of functionals in terms of properties of functionals. Using such a representation we obtain some results about ...

Journal: Topology, vol. 48, no. 2, pp. 100-104, 2009]]>In this paper, we study the symplectic geometry of singular conifolds of the finite group quotient Wr={(x,y,z,t)∣xy−z2r+t2=0}/μr(a,−a,1,0),r≥1, which we call orbi-conifolds. The related orbifold symplectic conifold transition and orbifold symplectic flops are constructed. Let X and Y be two symplectic orbifolds connected by such a flop. We ...

Journal: Topology, vol. 48, no. 1, pp. 1-22, 2009]]>In this paper we prove that the Hirzebruch surface F2,(2,2) embedded in CP17 supports the conjecture on the structure and properties of fundamental groups of complement of branch curves of generic projections, as laid out in [M. Teicher, New Invariants for surfaces, Contemp. Math. 231 (1999) 271–281]. We use the regeneration from [M. Friedman, M. Teicher, The ...

Journal: Topology, vol. 48, no. 1, pp. 23-40, 2009]]>We consider V.I. Arnold’s manifold of self-adjoint operators with fixed multiplicity of eigenvalues and K. Uhlenbeck’s manifold of eigenvectors. Our aim is to consider the local analysis and the connection between these manifolds. We present the topological description of the spectrum perturbation problem, specifically the finite-multiple eigenvalue splitting problem. For investigation of manifolds, we use the local ...

Journal: Topology, vol. 48, no. 2, pp. 213-223, 2009]]>Let X be a metric space. We study the free Banach space B(X) over X, that is a predual space of the Banach space of all Lipschitz functions on X which preserve a marked point θ∈X. Some applications to the extension theory of Lipschitz and two-Lipschitz functions are obtained.

Journal: Topology, vol. 48, no. 2, pp. 203-212, 2009]]>For a complex Banach algebra E, let HL(E) be the space of the mappings from E into E that are analytic in the sense of Lorch. We will show that HL(E) is a closed subalgebra of Hb(E,E) and we will give a description of its spectrum. As an application we will show that E is semi-...

Journal: Topology, vol. 48, no. 2, pp. 91-99, 2009]]>We propose an analogue of the classical polarization formula for any nonhomogeneous polynomial and analytic mapping on complex Banach spaces.

Journal: Topology, vol. 48, no. 2, pp. 197-202, 2009]]>Given a non-degenerate Peano continuum X, a dimension function D:2∗X→[0,∞] defined on the family 2∗X of compact subsets of X, and a subset Γ⊂[0,∞), we recognize the topological structure of the system 〈2X,D≤γ(X)〉α∈Γ, where 2X is the hyperspace of non-empty compact subsets of X and D≤γ(X) is the subspace ...

Journal: Topology, vol. 48, no. 2, pp. 43-53, 2009]]>Given an ordinal α and a pointed topological space X, we endow Xα=∪{Xβ:βα} with the strongest topology that coincides with the product topology on every subset Xβ of Xα, βα. It turns out that many important model spaces of infinite-dimensional topology (including the topology of non-metrizable manifolds) can be obtained as spaces of the form ...

Journal: Topology, vol. 48, no. 2, pp. 186-196, 2009]]>Quantized operators acting on approximable Wiener type algebras of analytic functions with infinitely many variables are researched. For such operators a sufficient condition of a sectorial property is established and a holomorphic calculus is constructed.

Journal: Topology, vol. 48, no. 2, pp. 105-110, 2009]]>In this paper we give general conditions on a countable family V of weights on an unbounded open set U in a complex Banach space X such that the weighted space HV(U) of holomorphic functions on U has a Fréchet algebra structure. For such weights it is shown that the spectrum of HV(U) has a natural analytic manifold...

Journal: Topology, vol. 48, no. 2, pp. 54-65, 2009]]>We consider the problem of whether a given interpolating sequence for a uniform algebra yields linear interpolation. A positive answer is obtained when we deal with dual uniform algebras. Further we prove that if the Carleson generalized condition is sufficient for a sequence to be interpolating on the algebra of bounded analytic functions on the unit ball of c0, then ...

Journal: Topology, vol. 48, no. 2, pp. 111-118, 2009]]>We introduce a fuzzy ultrametric on the set of probability measures with compact support defined on a fuzzy metric space. The construction is a counterpart, in the realm of fuzzy ultrametric spaces, of the construction due to Vink and Rutten of an ultrametric on the set of probability measures with compact supports on an ultrametric space.It is proved that ...

Journal: Topology, vol. 48, no. 2, pp. 130-136, 2009]]>Let G+′=G′(R+d) stand for Roumieu ultradistributions with supports in the positive cone R+d. Throughout P(G+′) denotes the algebra of continuous scalar polynomials on the space G+′. We investigate the dual pair 〈P′(G+′)∣P(G+′)〉 generated by the algebra P(G+′) and by its strong dual P′(G+′). Properties of the polynomially extended operational calculus and ...

Journal: Topology, vol. 48, no. 2, pp. 80-90, 2009]]>Journal: Topology, vol. 48, 2009]]>

We introduce homotopical techniques in the frameworks of two-graded absolute valued algebras and absolute valued triple systems, which will simplify the study of these structures. To this end, we previously refine and concrete the known descriptions of two-graded absolute valued algebras and absolute valued triple systems, as well as characterize the fact that an absolute valued triple system ...

Journal: Topology, vol. 48, no. 2, pp. 157-168, 2009]]>Here we study the solution set of a nonlinear operator equation in a Banach subspace Ln⊂C(X) by reducing it to a Leray–Schauder type fixed point problem. The subspace Ln is of finite codimension n∈Z+ in C(X), with X an infinite compact Hausdorff space, and is defined by conditions αi∗(f)≔∫Xf(x)dμi(x)=0,f∈C(X), ...

Journal: Topology, vol. 48, no. 2, pp. 182-185, 2009]]>The Sugeno integral of an upper semicontinuous function from a compactum to a compact Hausdorff–Lawson lattice with respect to a lattice-valued capacity is introduced, and its characterization and properties are presented. It is proved that in a family of fuzzy integrals, the Sugeno integral is unique that provides a functional representation of the monad of lattice-valued capacities.

Journal: Topology, vol. 48, no. 2, pp. 137-148, 2009]]>Hardy type spaces of complex analytic functions with infinitely many variables, defined on the interior of irreducible unitary orbits of topological groups with an invariant measure are investigated. Spaces of Taylor coefficients for such classes of analytic functions are described by the associated symmetric Fock spaces.

Journal: Topology, vol. 48, no. 2, pp. 169-177, 2009]]>In the previous paper [Hiroshi Iritani, Quantum D-modules and equivariant Floer theory for free loop spaces, Math. Z. 252 (3) (2006) 577–622], the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D-module after a mirror transformation when the first Chern class c1(...

Journal: Topology, vol. 47, no. 4, pp. 225-276, 2008]]>We discuss separability properties of discrete groups, and introduce a new property of groups that is motivated by a geometric proof of separability of geometrically finite subgroups of Kleinian groups. This property appears natural in that it provides a general framework for old questions in the geometry and topology of hyperbolic manifolds and discrete groups.

Journal: Topology, vol. 47, no. 3, pp. 137-159, 2008]]>Let Σ be a closed, orientable surface of genus g. It is known that the SU(2,1) representation variety of π1(Σ) has 2g−3 components of (real) dimension 16g−16 and two components of dimension 8g−6. Of special interest are the totally loxodromic, faithful (that is quasi-Fuchsian) representations. In this paper we give global real analytic coordinates on a ...

Journal: Topology, vol. 47, no. 2, pp. 101-135, 2008]]>We calculate certain homotopy groups of the moduli spaces for representations of a compact oriented surface in the Lie groups GL(n,C) and U(p,q). Our approach relies on the interpretation of these representations in terms of Higgs bundles and uses Bott–Morse theory on the corresponding moduli spaces.

Journal: Topology, vol. 47, no. 4, pp. 203-224, 2008]]>Gromov proposed an averaged version of the Dehn function and claimed that in many cases it should be subasymptotic to the Dehn function. Using results on random walks in nilpotent groups, we confirm this claim for most nilpotent groups. In particular, if a nilpotent group satisfies the isoperimetric inequality δ(l)Clα for α>2, then it satisfies the ...

Journal: Topology, vol. 47, no. 5, pp. 351-367, 2008]]>This paper takes up the systematic study of the Gottlieb groups Gn+k(Sn) of spheres for k≤13 by means of the classical homotopy theory methods. We fully determine the groups Gn+k(Sn) for k≤13 except for the 2-primary components in the cases: k=9,n=53;k=11,n=115. In particular, we show [ιn,ηn2σn+2]=0 if n=...

Journal: Topology, vol. 47, no. 6, pp. 399-430, 2008]]>The Berline–Vergne integral localization formula for equivariantly closed forms ([N. Berline, M. Vergne, Classes caractéristiques équivariantes. Formules de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris 295 (1982) 539–541], Theorem 7.11 in [N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac Operators, Springer-Verlag, 1992]) is well-known and requires the acting Lie group ...

Journal: Topology, vol. 47, no. 1, pp. 1-39, 2008]]>The forcing relation of braids has been introduced for a 2-dimensional analogue of the Sharkovskii order on periods for maps of the interval. In this paper, by making use of the Nielsen fixed point theory and a representation of braid groups, we deduce a trace formula for the computation of the forcing order.

Journal: Topology, vol. 47, no. 1, pp. 51-70, 2008]]>We study the behavior of the Nil-subgroups of K-groups under localization. As a consequence of our results, we obtain that the relative assembly map from the family of finite subgroups to the family of virtually cyclic subgroups is rationally an isomorphism. Combined with the equivariant Chern character, we obtain a complete computation of the rationalized source of the ...

Journal: Topology, vol. 47, no. 3, pp. 160-202, 2008]]>We prove that 2r+1⋅π∗(Pm(2r))=0 provided m≥4 and r≥6. This is the best possible result. As well, for 2≤r≤5 we obtain upper bounds on the homotopy exponent of Pm(2r).

Journal: Topology, vol. 47, no. 6, pp. 369-398, 2008]]>To a proper generic immersion of a finite number of copies of the unit interval in a 2-disc, called a divide, A’Campo associates a link in S3. From the more general notion of ordered Morse signed divides, one obtains a braid presentation of links of divides. In this paper, we prove that every strongly invertible link is isotopic to ...

Journal: Topology, vol. 47, no. 5, pp. 316-350, 2008]]>We study 2-dimensional Jacobian maps using so-called Newton–Puiseux charts. These are multi-valued coordinates near divisors of resolutions of indeterminacies at infinity of the Jacobian map in the source space as well as in the target space. The map expressed in these charts takes a very simple form, which allows us to detect a series of new ...

Journal: Topology, vol. 47, no. 6, pp. 431-469, 2008]]>Journal: Topology, vol. 47, no. 6, pp. 471-472, 2008]]>

A surface in a smooth 4-manifold is called flexible if, for any diffeomorphism ϕ on the surface, there is a diffeomorphism on the 4-manifold whose restriction on the surface is ϕ and which is isotopic to the identity. We investigate a sufficient condition for a smooth 4-manifold M to include flexible knotted surfaces, and introduce a local ...

Journal: Topology, vol. 47, no. 1, pp. 41-50, 2008]]>If a simple 3-manifold M admits a reducible and a toroidal Dehn filling, the distance between the filling slopes is known to be bounded by three. In this paper, we classify all manifolds which admit a reducible Dehn filling and a toroidal Dehn filling with distance 3.

Journal: Topology, vol. 47, no. 5, pp. 277-315, 2008]]>In a very general setting, we show that a 3-manifold obtained as the orbit space of the basin of a topological attractor is either S2×S1 or irreducible.We then study in more detail the topology of a class of 3-manifolds which are also orbit spaces and arise as invariants of gradient-like diffeomorphisms (in dimension 3). Up to ...

Journal: Topology, vol. 47, no. 2, pp. 71-100, 2008]]>Journal: Topology, vol. 47, no. 6, pp. 369-398, 2008]]>

Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the “homotopy theory” of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a “homotopy theory of homotopy theories”. In this paper we show ...

Journal: Topology, vol. 46, no. 4, pp. 397-436, 2007]]>This paper gives a partial description of the homotopy type of K, the space of long knots in R3. The primary result is the construction of a homotopy equivalence K≃C2(P⊔{∗}) where C2(P⊔{∗}) is the free little 2-cubes object on the pointed space P⊔{∗}, where P⊂K is the subspace of prime knots, and ∗ is ...

Journal: Topology, vol. 46, no. 1, pp. 1-27, 2007]]>Using a cell model for the little discs operad in terms of spineless cacti we give a minimal common topological operadic formalism for three a priori disparate algebraic structures: (1) a solution to Deligne’s conjecture on the Hochschild complex, (2) the Hopf algebra of Connes and Kreimer, and (3) the string topology of Chas and Sullivan.

Journal: Topology, vol. 46, no. 1, pp. 39-88, 2007]]>The homotopy type of the complement of a complex coordinate subspace arrangement is studied by utilising some connections between its topological and combinatorial structures. A family of arrangements for which the complement is homotopy equivalent to a wedge of spheres is described. One consequence is an application in commutative algebra: certain local rings are proved to be Golod, that is, ...

Journal: Topology, vol. 46, no. 4, pp. 357-396, 2007]]>For Γ a relatively hyperbolic group, we construct a model for the universal space among Γ-spaces with isotropy on the family VC of virtually cyclic subgroups of Γ. We provide a recipe for identifying the maximal infinite virtually cyclic subgroups of Coxeter groups which are lattices in O+(n,1)=Isom(Hn). We use the information we obtain to ...

Journal: Topology, vol. 46, no. 6, pp. 527-553, 2007]]>We study the category of discrete modules over the ring of degree-zero stable operations in p-local complex K-theory, where p is an odd prime. We show that the K(p)-homology of any space or spectrum is such a module, and that this category is isomorphic to a category defined by Bousfield and used in his work ...

Journal: Topology, vol. 46, no. 2, pp. 139-154, 2007]]>We estimate the number of exceptional slopes for hyperbolic 3-manifolds with a torus boundary component and at least one other boundary component.

Journal: Topology, vol. 46, no. 5, pp. 437-468, 2007]]>Let (X,x0) be any one-pointed compact connected Riemann surface of genus g, with g≥3. Fix two mutually coprime integers r>1 and d. Let MX denote the moduli space parametrizing all logarithmic SL(r,C)-connections, singular over x0, on vector bundles over X of degree d. We prove that the isomorphism class of the variety MX ...

Journal: Topology, vol. 46, no. 3, pp. 295-317, 2007]]>A proper CAT(0) metric space X is cocompact if it has a compact generating domain with respect to its full isometry group. Any proper CAT(0) space, cocompact or not, has a compact metrizable boundary at infinity ∂∞X; indeed, up to homeomorphism, this boundary is arbitrary. However, cocompactness imposes restrictions on what the boundary can be. Swenson showed that ...

Journal: Topology, vol. 46, no. 2, pp. 129-137, 2007]]>Let G be a torsion-free discrete group with a finite-dimensional classifying space BG. We show that G has a dual-Dirac morphism if and only if a certain coarse (co-)assembly map is an isomorphism. Hence the existence of a dual-Dirac morphism for such groups is a metric, that is, coarse, invariant. We get results for groups ...

Journal: Topology, vol. 46, no. 2, pp. 185-209, 2007]]>A study of symplectic actions of a finite group G on smooth 4-manifolds is initiated. The central new idea is the use of G-equivariant Seiberg–Witten–Taubes theory in studying the structure of the fixed-point set of these symmetries. The main result in this paper is a complete description of the fixed-point set structure (and the ...

Journal: Topology, vol. 46, no. 2, pp. 103-128, 2007]]>Rasmussen introduced a knot invariant based on Khovanov homology theory, and showed that this invariant estimates the four-genus of knots. We compare his result with the sharper slice-Bennequin inequality for knots. Then we obtain a similar estimate of the Rasmussen invariant for this inequality.

Journal: Topology, vol. 46, no. 1, pp. 29-38, 2007]]>We determine the possible eigenvalues of elliptic matrices A,B,C in PU(2,1) satisfying ABC=1. This is done by describing geometrically the image of a group-valued momentum map for the (non-compact) group action of PU(2,1) by conjugation on C1×C2 where C1 and C2 are fixed elliptic conjugacy classes in PU(2,1). Contrary ...

Journal: Topology, vol. 46, no. 2, pp. 155-183, 2007]]>In this paper we use fractal geometry to investigate boundary aspects of the first homology group for finite coverings of the modular surface. We obtain a complete description of algebraically invisible parts of this homology group. More precisely, we first show that for any modular subgroup the geodesic forward dynamic on the associated surface admits a canonical symbolic representation by ...

Journal: Topology, vol. 46, no. 5, pp. 469-491, 2007]]>In this paper we will estimate the smallest length of a minimal geodesic net on an arbitrary closed Riemannian manifold Mn in terms of the diameter of this manifold and its dimension. Minimal geodesic nets are critical points of the length functional on the space of immersed graphs into a Riemannian manifold. We prove that there exists a minimal geodesic ...

Journal: Topology, vol. 46, no. 4, pp. 343-356, 2007]]>In the rational category of nilpotent complexes, let E be an H-space acting on a space X. With mild hypotheses we show that the action on the base point w:E→X factors through a map ΓE:SE→X, where SE is a finite product of odd-dimensional spheres and ΓE is a homotopy monomorphism. Among others, the following consequences are ...

Journal: Topology, vol. 46, no. 5, pp. 493-506, 2007]]>We construct an obstruction theory for relative Hilbert schemes in the sense of [K. Behrend, B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1) (1997) 45–88] and compute it explicitly for relative Hilbert schemes of divisors on smooth projective varieties. In the special case of curves on a surface V, our obstruction theory determines a virtual fundamental class [[...

Journal: Topology, vol. 46, no. 3, pp. 225-294, 2007]]>A compactum X is an ‘absolute cone’ if, for each of its points x, the space X is homeomorphic to a cone with x corresponding to the cone point. In 1971, J. de Groot conjectured that each n-dimensional absolute cone is an n-cell. In this paper, we give a complete solution to that conjecture. In particular, we show ...

Journal: Topology, vol. 46, no. 2, pp. 89-102, 2007]]>Define the length of a finite presentation of a group G as the sum of lengths of all relators plus the number of generators. How large can the kth Betti number bk(G)= rank Hk(G) be providing that G has length ≤N and bk(G) is finite? We prove that for every k≥3 the maximum bk(N) of the ...

Journal: Topology, vol. 46, no. 2, pp. 211-223, 2007]]>The convexity theorem of Atiyah and Guillemin–Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar–Lerman proved that the Marsden–Weinstein reduction of a connected Hamitonian G-manifold is a stratified symplectic space. Suppose 1→A→G→T→1 is an exact ...

Journal: Topology, vol. 46, no. 6, pp. 554-576, 2007]]>Locally stable maps S3→R4 are classified up to homotopy through locally stable maps. The equivalence class of a map f is determined by three invariants: the isotopy class σ(f) of its framed singularity link, the generalized normal degree ν(f), and the algebraic number of cusps κ(f) of any extension of f to a locally stable map of ...

Journal: Topology, vol. 46, no. 6, pp. 577-597, 2007]]>We study nilpotent subgroups of automorphism groups in the category of groups and the homotopy category of spaces. We establish localization and completion theorems for nilpotent groups of automorphisms of nilpotent groups. We then apply these algebraic theorems to prove analogous results for certain groups of self-homotopy equivalences of spaces.

Journal: Topology, vol. 46, no. 3, pp. 319-341, 2007]]>We construct a sphere fibration over a finite aspherical Poincaré complex X, which we call the tangential end fibration, under the condition that the universal cover of X is forward tame and simply connected at infinity. We show that it is tangent to X if the formal dimension of X is even or, when the formal dimension is odd, if...

Journal: Topology, vol. 46, no. 5, pp. 507-525, 2007]]>We study relative Gromov–Witten theory via universal relations provided by the degeneration and localization formulas. We find relative Gromov–Witten theory is completely determined by absolute Gromov–Witten theory. The relationship between the relative and absolute theories is guided by a strong analogy to classical topology.As an outcome, we present a mathematical determination of the Gromov–Witten invariants (...

Journal: Topology, vol. 45, no. 5, pp. 887-918, 2006]]>It is shown that the fundamental groups of certain non-positively curved 2-complexes have the property that their quasiconvex subgroups are the intersections of finite index subgroups.As a consequence, every geometrically finite subgroup of the figure 8 knot group is the intersection of finite index subgroups. The same result holds for many other prime alternating link groups.

Journal: Topology, vol. 45, no. 3, pp. 421-463, 2006]]>We redefine the Baum–Connes assembly map using simplicial approximation in the equivariant Kasparov category. This new interpretation is ideal for studying functorial properties and gives analogues of the Baum–Connes assembly map for other equivariant homology theories. We extend many of the known techniques for proving the Baum–Connes conjecture to this more general setting.

Journal: Topology, vol. 45, no. 2, pp. 209-259, 2006]]>We prove that the twisted K-homology of a simply connected simple Lie group G of rank n is an exterior algebra on n−1 generators tensor a cyclic group. We give a detailed description of the order of this cyclic group in terms of the dimensions of irreducible representations of G and show that the congruences determining this cyclic order ...

Journal: Topology, vol. 45, no. 6, pp. 955-988, 2006]]>We say that an oriented contact manifold (M,ξ) is Milnor fillable if it is contactomorphic to the contact boundary of an isolated complex-analytic singularity (X,x). In this article we prove that any three-dimensional oriented manifold admits at most one Milnor fillable contact structure up to contactomorphism. The proof is based on Milnor open books: we associate ...

Journal: Topology, vol. 45, no. 3, pp. 673-689, 2006]]>Every element in the first cohomology group of a 3-manifold is dual to embedded surfaces. The Thurston norm measures the minimal ‘complexity’ of such surfaces. For instance the Thurston norm of a knot complement determines the genus of the knot in the 3-sphere. We show that the degrees of twisted Alexander polynomials give lower bounds on the Thurston ...

Journal: Topology, vol. 45, no. 6, pp. 929-953, 2006]]>We extend the W-construction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for well-pointed Σ-cofibrant operads. The standard simplicial resolution of Godement as well as the cobar–bar chain resolution are shown to be particular instances of this generalised W-construction.

Journal: Topology, vol. 45, no. 5, pp. 807-849, 2006]]>Classical homological algebra takes place in additive categories. In homotopy theory such additive categories arise as homotopy categories of “additive groupoid enriched categories”, in which a secondary analog of homological algebra can be performed. We introduce secondary chain complexes and secondary resolutions leading to the concept of secondary derived functors. As a main result we show that the E3-term ...

Journal: Topology, vol. 45, no. 2, pp. 295-324, 2006]]>A fake projective plane is a compact complex surface (a compact complex manifold of dimension 2) with the same Betti numbers as the complex projective plane, but not isomorphic to the complex projective plane. As was shown by Mumford, there exists at least one such surface.In this paper we prove the existence of a fake projective plane which is ...

Journal: Topology, vol. 45, no. 5, pp. 919-927, 2006]]>For a smooth map from a closed surface to the two-dimensional plane, its critical value set is called the apparent contour. In this paper we study first order local Vassiliev-type invariants of generic apparent contours.

Journal: Topology, vol. 45, no. 1, pp. 27-45, 2006]]>Using degree N isogenies of elliptic curves, we produce a spectrum Q(N). This spectrum is built out of spectra related to tmf. At p=3 we show that the K(2)-local sphere is built out of Q(2) and its K(2)-local Spanier–Whitehead dual. This gives a conceptual reinterpretation a resolution of Goerss, Henn, Mahowald, and ...

Journal: Topology, vol. 45, no. 2, pp. 343-402, 2006]]>This paper introduces a submanifold of the moduli space of unitary representations of the fundamental group of a punctured sphere with fixed local monodromy. The submanifold is defined via products of involutions through Lagrangian subspaces. We show that the moduli space of Lagrangian representations is a Lagrangian submanifold of the moduli of unitary representations.

Journal: Topology, vol. 45, no. 1, pp. 65-99, 2006]]>If G1 and G2 are finite groups with periodic Tate cohomology, then G1×G2 acts freely and smoothly on some product Sn×Sn.

Journal: Topology, vol. 45, no. 4, pp. 735-749, 2006]]>We prove that the last discrete deformation of the R-Fuchsian (4, 4, 4)-triangle group in PU(2,1) is a cocompact arithmetic lattice. We also describe an experimental method for finding the combinatorics of a Dirichlet fundamental domain, and apply it to the lattice in question.

Journal: Topology, vol. 45, no. 6, pp. 989-1020, 2006]]>The construction of 3-manifolds via Dehn surgery on links in S3 is an important technique in the classification of 3-manifolds. This paper describes a method of constructing infinite collections of distinct hyperbolic knots in S3 which admit a longitudinal surgery yielding the same manifold. In one case, the knots constructed each admit a longitudinal surgery yielding the same ...

Journal: Topology, vol. 45, no. 4, pp. 725-733, 2006]]>In this paper, we try to generalize to the case of compact Riemannian orbifolds Q some classical results about the existence of closed geodesics of positive length on compact Riemannian manifolds M. We shall also consider the problem of the existence of infinitely many geometrically distinct closed geodesics.In the classical case the solution of those problems involve the consideration ...

Journal: Topology, vol. 45, no. 3, pp. 611-641, 2006]]>We give an explicit formula for the Whitehead group of a three-dimensional crystallographic group Γ in terms of the Whitehead groups of the virtually infinite cyclic subgroups of Γ.

Journal: Topology, vol. 45, no. 1, pp. 1-25, 2006]]>The aim of this paper is to study compact 5-manifolds which admit fixed point free circle actions. The first result implies that the torsion in the second homology and the second Stiefel–Whitney class have to satisfy strong restrictions. We then show that for simply connected 5-manifolds these restrictions are necessary and sufficient.

Journal: Topology, vol. 45, no. 3, pp. 643-671, 2006]]>We show that if a homeomorphism between the ideal boundaries of two Fuchsian buildings preserves the combinatorial cross ratio almost everywhere, then it extends to an isomorphism between the Fuchsian buildings. Together with the results of Bourdon–Pajot and Kleiner, it implies the quasi-isometric rigidity for Fuchsian buildings: any quasi-isometry between two Fuchsian buildings that admit cocompact lattices ...

Journal: Topology, vol. 45, no. 1, pp. 101-169, 2006]]>In this paper we use fractal geometry to investigate boundary aspects of the first homology group for finite coverings of the modular surface. We obtain a complete description of algebraically invisible parts of this homology group. More precisely, we first show that for any modular subgroup the geodesic forward dynamic on the associated surface admits a canonical symbolic representation by ...

Journal: Topology, 2006]]>We define new bordism and spin bordism invariants of certain subgroups of the mapping class group of a surface. In particular, they are invariants of the Johnson filtration of the mapping class group. The second and third terms of this filtration are the well-known Torelli group and Johnson subgroup, respectively. We introduce a new representation in terms of spin ...

Journal: Topology, vol. 45, no. 5, pp. 851-886, 2006]]>It is well known that the signature operator on a manifold defines a K-homologyclass which is an orientation after inverting 2. Here we address the following puzzle: What is this class localized at 2, and what special properties does it have? Our answers include the following:•the K-homologyclass ΔM of the signature operator is a bordism invariant;•the ...

Journal: Topology, vol. 45, no. 1, pp. 47-63, 2006]]>We give an algebro-geometric derivation of the known intersection theory on the moduli space of stable rank 2 bundles of odd degree over a smooth curve of genus g. We lift the computation from the moduli space to a Quot scheme, where we obtain the intersections via equivariant localization with respect to a natural torus action.

Journal: Topology, vol. 45, no. 3, pp. 531-541, 2006]]>We present a practical algorithm to determine the minimal genus of non-orientable spanning surfaces for 2-bridge knots, called the crosscap numbers. We will exhibit a table of crosscap numbers of 2-bridge knots up to 12 crossings (all 362 of them).

Journal: Topology, vol. 45, no. 3, pp. 513-530, 2006]]>It is shown that Segal's theorem on the spaces of rational maps from CP1 to CPn can be extended to the spaces of continuous rational maps from CPm to CPn for any m⩽n. The tools are the Stone–Weierstrass theorem and Vassiliev's machinery of simplicial resolutions.

Journal: Topology, vol. 45, no. 2, pp. 281-293, 2006]]>A knot in the 3-sphere is called doubly slice if it is a slice of an unknotted 2-sphere in the 4-sphere. We give a bi-sequence of new obstructions for a knot being doubly slice. We construct it following the idea of Cochran-Orr-Teichner's filtration of the classical knot concordance group. This yields a bi-...

Journal: Topology, vol. 45, no. 3, pp. 543-566, 2006]]>We prove that in any nonzero cohomology class ξ∈H1(M;R) there always exists a closed 1-form having at most one zero.

Journal: Topology, vol. 45, no. 3, pp. 465-473, 2006]]>We give an alternative to the stable classification of p-completed homotopy types of classifying spaces of finite groups offered by Martino–Priddy. For a finite group G with Sylow subgroup S, we regard the stable p-completed classifying space Σ∞BGp∧ as an object under Σ∞BS via the canonical inclusion map. Thus we get a classification in terms of induced ...

Journal: Topology, vol. 45, no. 3, pp. 601-609, 2006]]>We construct a pair of transverse genuine laminations on an atoroidal 3-manifold admitting a transversely orientable uniform 1-cochain. The laminations are induced by the uniform 1-cochain and they are the straightening of the coarse laminations defined by Calegari, by using minimal surface techniques. Moreover, when we collapse these laminations, we get a topological pseudo-Anosov flow, as ...

Journal: Topology, vol. 45, no. 4, pp. 751-784, 2006]]>Let A be a noetherian Fp-algebra that is finitely generated as a module over the subring Ap of pth powers. We give an explicit formula for the de Rham–Witt complex of the power series ring A〚t〛 in terms of that of the ring A. We use this formula to show that, for every complete regular local Fp-algebra ...

Journal: Topology, vol. 45, no. 3, pp. 475-493, 2006]]>The aim of this paper is to show that the fundamental group of an oriented 3-manifold which satisfies Thurston's geometrization conjecture has a solvable conjugacy problem. In other words, for any such 3-manifold M, there exists an algorithm which can decide for any couple of elements u,v of π1(M) whether u and v are in ...

Journal: Topology, vol. 45, no. 1, pp. 171-208, 2006]]>We give some explicit bounds for the number of cobordism classes of real algebraic manifolds of real degree less than d, and for the size of the sum of Betti numbers with Z/2 coefficients for the real form of complex manifolds of complex degree less than d.

Journal: Topology, vol. 45, no. 2, pp. 261-280, 2006]]>We say a knot k in the 3-sphere S3 has PropertyIE if the infinite cyclic cover of the knot exterior embeds into S3. Clearly all fibred knots have Property IE.There are infinitely many non-fibred knots with Property IE and infinitely many non-fibred knots without property IE. Both kinds of examples are established here for the first ...

Journal: Topology, vol. 45, no. 4, pp. 691-705, 2006]]>Let a smooth vector field V on a smooth closed manifold M be given and let Z⊂M be an isolated invariant set for the flow of V. In this situation, we give a necessary and sufficient condition for the existence of a Lyapunov 1-form for (V,Z) in terms of the relative asymptotic cycles associated with certain invariant measures ...

Journal: Topology, vol. 45, no. 4, pp. 707-723, 2006]]>Building on our earlier work on toric residues and reduction, we give a proof of the mixed toric residue conjecture of Batyrev and Materov. We simplify and streamline our technique of tropical degenerations, which allows one to interpolate between two localization principles: one appearing in the intersection theory of toric quotients and the other in the calculus of toric residues. ...

Journal: Topology, vol. 45, no. 3, pp. 567-599, 2006]]>Let X be a nonsingular algebraic curve of genus g⩾3, and let Mξ denote the moduli space of stable vector bundles of rank n⩾2 and degree d with fixed determinant ξ over X such that n and d are coprime. We assume that if g=3 then n⩾4 and if g=4 then n⩾3, and suppose further that n0, d0 ...

Journal: Topology, vol. 45, no. 2, pp. 403-419, 2006]]>We propose to study a new kind of monodromy homomorphism for families of regular elliptic fibrations of a given differentiable fibration type to get a hold on topological properties of moduli stacks of elliptic surfaces.In specific cases, including the most significant one, when all singular fibres are nodal irreducible rational curves, we compute the corresponding monodromy group, a subgroup ...

Journal: Topology, vol. 45, no. 4, pp. 785-806, 2006]]>We show that there exists a Lipschitz almost-complex structure J on CP2, arbitrarily close to the standard one, and a compact lamination by J-holomorphic curves satisfying the following properties: it is minimal, it has hyperbolic holonomy and it is transversally Lipschitz. Its transverse Hausdorff dimension can be any number δ in an interval (0,δmax) where δmax=1....

Journal: Topology, vol. 45, no. 3, pp. 495-512, 2006]]>