Lefschetz Number Research Articles

Resolvent of vector fields and Lefschetz numbers

Resolvent of vector fields and Lefschetz numbers

Trace relations and open string vacua

We study to what extent, and in what form, the notion of gauge-string duality may persist at finite N. It is shown, in the half-BPS sector, that the states of D3 giant graviton branes in AdS5 × S5 are holographically dual to certain auxiliary ghosts that compensate for finite N trace relations in U(N) N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills. The complex formed from spaces of states of bulk D3 giants is observed to furnish a BRST-like resolution of the half-BPS Hilbert space of U(N) N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM at finite N. We argue that the identification between the states of certain bulk D-branes and the auxiliary ghosts in the boundary holds rather generally at vanishing ’t Hooft coupling λ = 0. We propose that a complex, which furnishes a BRST-like resolution of the finite N Hilbert space of a boundary U(N) gauge theory at λ = 0, should be identified as the space of states of the dual string theory in the α′ → ∞ limit. The Lefschetz trace formula provides the holographic map in this regime, where bulk observables are computed by taking the alternating sum of the expectation values in an ensemble of states built on each open string vacuum. The giant graviton expansion is recovered and generalized in a limit of the resolution.

Colored Jones polynomials and abelianized Lefschetz numbers

We show that colored Jones polynomials of the closure of a braid compute weighted sums of abelianized Lefschetz numbers associated with homeomorphisms on configuration spaces on punctured disks defined from the braid. The sum is over number of configuration points. Then we interpret this sum in terms of Poincaré–Lefschetz duality intersection pairing between homology classes.

Lefschetz numbers of Verdier monodromy and the motivic monodromy conjecture for toric varieties

Lefschetz numbers of Verdier monodromy and the motivic monodromy conjecture for toric varieties

The Reidemeister trace of an n-valued map

In topological fixed point theory, the Reidemeister trace is an invariant associated to a selfmap of a polyhedron which combines information from the Lefschetz and Nielsen numbers. In this paper we define the Reidemeister trace in the context of n-valued selfmaps of compact polyhedra. We prove several properties of the Reidemeister trace which generalize properties from the single-valued theory, and prove an averaging formula.

Parametric Topological Entropy for Multivalued Maps and Differential Inclusions with Nonautonomous Impulses

The main purpose of this paper is to investigate a parametric topological entropy for impulsive differential inclusions on tori. In this way, besides other matters, we would like to extend our recent results concerning impulsive differential equations as well as those on “nonparametric” topological entropy to impulsive differential inclusions. Parametric topological entropy, which is usually called a topological entropy for nonautonomous dynamical systems, is considered here via the compositions of associated multivalued Poincaré translation operators with the single-valued time-dependent impulsive maps. On compact polyhedra and, in particular on tori, parametric topological entropy for families of admissible multivalued maps can be estimated from below by means Ivanov-type inequality in terms of the asymptotic Nielsen and Lefschetz numbers which are, unlike the topological entropy, homotopy invariants. In the scalar case, an effective criterion for a positive parametric topological entropy can be given by topological degree arguments for equi-continuous impulsive maps. In a single-valued nonparametric case, a positive topological entropy usually signifies topological chaos. Some simple illustrative examples are provided.

Local terms for transversal intersections

The goal of this note is to show that in the case of ‘transversal intersections’ the ‘true local terms’ appearing in the Lefschetz trace formula are equal to the ‘naive local terms’. To prove the result, we extend the strategy used in our previous work, where the case of contracting correspondences is treated. Our new ingredients are the observation of Verdier that specialization of an étale sheaf to the normal cone is monodromic and the assertion that local terms are ‘constant in families’. As an application, we get a generalization of the Deligne–Lusztig trace formula.

Minimal homeomorphisms and topological $K$-theory

The Lefschetz fixed point theorem provides a powerful obstruction to the existence of minimal homeomorphisms on well-behaved spaces such as finite CW -complexes. We show that these obstructions do not hold for more general spaces. Minimal homeomorphisms are constructed on compact connected metric spaces with any prescribed finitely generated K -theory or cohomology. In particular, although a non-zero Euler characteristic obstructs the existence of a minimal homeomorphism on a finite CW -complex, this is not the case on a compact metric space. We also allow for some control of the map on K -theory and cohomology induced from these minimal homeomorphisms. This allows for the construction of many minimal homeomorphisms that are not homotopic to the identity. Applications to C^* -algebras will be discussed in another paper.

On the Frøyshov invariant and monopole Lefschetz number

Given an involution on a rational homology $3$-sphere $Y$ with quotient the $3$-sphere, we prove a formula for the Lefschetz number of the map induced by this involution in the reduced monopole Floer homology. This formula is motivated by a variant of Witten’s conjecture relating the Donaldson and Seiberg–Witten invariants of $4$-manifolds. A key ingredient is a skein-theoretic argument, making use of an exact triangle in monopole Floer homology, that computes the Lefschetz number in terms of the Murasugi signature of the branch set and the sum of Frøyshov invariants associated to spin structures on $Y$. We discuss various applications of our formula in gauge theory, knot theory, contact geometry, and $4$-dimensional topology.

Rational Reidemeister trace of an outer automorphism of finite order

Let H be a group of type FP over the rationals and let ϕ:H→H be an automorphism whose class in Out(H) has finite order and such that the Lefschetz number of some iterate ϕi is nonzero. We show that under a suitable Bass conjecture assumption, this iterate ϕi is conjugate to a finite order automorphism, and the intersection of the fixed point set of ϕ with the center of H is of bounded exponent. In the torsion-free case it follows that the center of the HNN extension 〈H,t|∀h∈H:tht−1=ϕ(h)〉 is infinite cyclic. These results are related to Geoghegan and Nicas' first order Euler characteristic of a group G with a finite Eilenberg-MacLane complex; this is a morphism χ1(G) from the center of G to H1(G;k) where k is a ring of coefficients. We give a complete determination of χ1(G) with rational coefficients.

Morse–Novikov cohomology for blow-ups of complex manifolds

The weight $\theta$-sheaf $\underline{\mathbb{R}}_{X,\theta}$ helps us to reinterpret Morse-Novikov cohomologies via sheaf theory. We give several theorems of K\"{u}nneth and Leray-Hirsch types. As applications, we prove that the $\theta$-Lefschetz number is independent of $\theta$ and calculate the Morse-Novikov cohomologies of projective bundles. Based on these results, we give two blow-up formulae on (\emph{not necessarily compact}) complex manifolds, where the self-intersection formulae play a key role in establishing the explicit expressions for them.

Thom condition and monodromy

We give the definition of the Thom condition and we show that given any germ of complex analytic function f:(X,x)rightarrow ({mathbb {C}},0) on a complex analytic space X, there exists a geometric local monodromy without fixed points, provided that fin {mathfrak {m}}_{X,x}^2, where {mathfrak {m}}_{X,x} is the maximal ideal of {mathcal {O}}_{X,x}. This result generalizes a well-known theorem of the second named author when X is smooth and proves a statement by Tibar in his PhD thesis. It also implies the A’Campo theorem that the Lefschetz number of the monodromy is equal to zero. Moreover, we give an application to the case that X has maximal rectified homotopical depth at x and show that a family of such functions with isolated critical points and constant total Milnor number has no coalescing of singularities.

The moduli space of nonnegatively curved metrics on quotients of S2 × S3 by involutions

We show that for an orientable non-spin manifold with fundamental group Z2 and universal cover S2×S3, the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. The representatives of the components are quotients of the standard metric on S3×S3 or metrics on Brieskorn varieties previously constructed using cohomogeneity one actions. The components are distinguished using the relative η invariant of the spinc Dirac operator computed by means of a Lefschetz fixed point theorem.

Lefschetz trace formula and $$\ell $$-adic cohomology of Rapoport–Zink tower for GSp(4)

Lefschetz trace formula and $$\ell $$-adic cohomology of Rapoport–Zink tower for GSp(4)

Concordance surgery and the Ozsváth–Szabó 4-manifold invariant

We compute the effect of concordance surgery, a generalization of knot surgery defined using a self-concordance of a knot, on the Ozsváth–Szabó 4-manifold invariant. The formula involves the graded Lefschetz number of the concordance map on knot Floer homology. The proof uses the sutured Floer TQFT, and a version of sutured Floer homology perturbed by a 2-form.

On combinatorics of the Arthur trace formula, convex polytopes, and toric varieties

AbstractWe explicate the combinatorial/geometric ingredients of Arthur’s proof of the convergence and polynomiality, in a truncation parameter, of his noninvariant trace formula. Starting with a fan in a real, finite dimensional, vector space and a collection of functions, one for each cone in the fan, we introduce a combinatorial truncated function with respect to a polytope normal to the fan and prove the analogues of Arthur’s results on the convergence and polynomiality of the integral of this truncated function over the vector space. The convergence statements clarify the important role of certain combinatorial subsets that appear in Arthur’s work and provide a crucial partition that amounts to a so-called nearest face partition. The polynomiality statements can be thought of as far reaching extensions of the Ehrhart polynomial. Our proof of polynomiality relies on the Lawrence–Varchenko conical decomposition and readily implies an extension of the well-known combinatorial lemma of Langlands. The Khovanskii–Pukhlikov virtual polytopes are an important ingredient here. Finally, we give some geometric interpretations of our combinatorial truncation on toric varieties as a measure and a Lefschetz number.

Equivariant fixed point formulae and Toeplitz operators under Hamiltonian torus actions and remarks on equivariant asymptotic expansions

Suppose given a holomorphic and Hamiltonian action of a compact torus on a polarized Hodge manifold. Assume that the action lifts to the quantizing line bundle, so that there is an induced unitary representation of the torus on the associated Hardy space which decomposes into isotypes. The main result of this paper is the description of asymptotics along rays in weight space of traces of equivariant Toeplitz operators composed with quantomorphisms for the torus action. The main ingredient in the proof is the micro-local analysis of the equivariant Szegő kernels. As a particular case we obtain a simple approach for asymptotics of the Lefschetz fixed point formula and traces of Toeplitz operators in the setting of ladder representations. We also consider equivariant asymptotic when the decomposition given by the standard circle action is taken into account, in this case one can recall previous results of X. Ma and W. Zhang or of R. Paoletti. We address some explicit computations for the action of the special unitary group of dimension two.

Gromov–Witten theory and Noether–Lefschetz theory for holomorphic-symplectic varieties

Abstract We use Noether–Lefschetz theory to study the reduced Gromov–Witten invariants of a holomorphic-symplectic variety of $K3^{[n]}$ -type. This yields strong evidence for a new conjectural formula that expresses Gromov–Witten invariants of this geometry for arbitrary classes in terms of primitive classes. The formula generalizes an earlier conjecture by Pandharipande and the author for K3 surfaces. Using Gromov–Witten techniques, we also determine the generating series of Noether–Lefschetz numbers of a general pencil of Debarre–Voisin varieties. This reproves and extends a result of Debarre, Han, O’Grady and Voisin on Hassett–Looijenga–Shah (HLS) divisors on the moduli space of Debarre–Voisin fourfolds.

Factorization statistics and bug-eyed configuration spaces

A recent theorem of Hyde proves that the factorizations statistics of a random polynomial over a finite field are governed by the action of the symmetric group on the configuration space of $n$ distinct ordered points in $\mathbb R^3$. Hyde asked whether this result could be explained geometrically. We give a geometric proof of Hyde's theorem as an instance of the Grothendieck--Lefschetz trace formula applied to an interesting, highly nonseparated algebraic space. An advantage of our method is that it generalizes uniformly to an arbitrary Weyl group. In the process we study certain non-Hausdorff models for complements of hyperplane arrangements, first introduced by Proudfoot.

On the monopole Lefschetz number of finite-order diffeomorphisms

Let $K$ be a knot in an integral homology 3-sphere $Y$, and $\Sigma$ the corresponding $n$-fold cyclic branched cover. Assuming that $\Sigma$ is a rational homology sphere (which is always the case when $n$ is a prime power), we give a formula for the Lefschetz number of the action that the covering translation induces on the reduced monopole homology of $\Sigma$. The proof relies on a careful analysis of the Seiberg--Witten equations on 3-orbifolds and of various $\eta$-invariants. We give several applications of our formula: (1) we calculate the Seiberg--Witten and Furuta--Ohta invariants for the mapping tori of all semi-free actions of $Z/n$ on integral homology 3-spheres; (2) we give a novel obstruction (in terms of the Jones polynomial) for the branched cover of a knot in $S^3$ being an $L$-space; (3) we give a new set of knot concordance invariants in terms of the monopole Lefschetz numbers of covering translations on the branched covers.