Vector fields on the real flag manifolds ℝ<Emphasis Type="Italic">F</Emphasis>(1, 1, <Emphasis Type="Italic">n</Emphasis> − 2)

We obtain the span of the real flag manifolds ℝF(1, 1, n−2), n ≥ 3, for the cases n ≡ 2 (mod 4), n ≡ 4 (mod 8) and n ≡ 8 (mod 16) and use the results to deduce that certain Stiefel-Whitney classes of the manifold are zero.      

On 4-manifolds and span-related numbers for cat manifolds

LetM ⁿ denote any closed connected CAT manifold of positive dimensionn. We define CATs(Mⁿ) to be the smallest positive dimension of all closed connected CAT manifoldsN for which the CAT span ofM×N is strictly greater than the CAT span ofN. We determine a formula for this characteristic number which involves only the Kirby-Siebenmann numberks(M) ofM and a Stiefel-Whitney number. Several results on splitting the tangent bundles of closed 4-manifolds are obtained. For example, both the Euler number ofM ⁴ andks(M⁴) represent the total obstruction to positive CAT span for a non-smoothable closed connected 4-manifold. Dedicated to the memory of Professor Otto Endler     

On the Betti number of the image of a generic map

Let be a differentiable map of a closed m-dimensional manifold into an (m + k)-dimensional manifold with k > 0. We show, assuming that f is generic in a certain sense, that f is an embedding if and only if the (m - k + 1)-th Betti numbers with respect to the Čech homology of M and f(M) coincide, under a certain condition on the stable normal bundle of f. This generalizes the authors' previous result for immersions with normal crossings [BS1]. As a corollary, we obtain the converse of the Jordan-Brouwer theorem for codimension-1 generic maps, which is a generalization of the results of [BR, BMS1, BMS2, Sae1] for immersions with normal crossings. Received: January 3, 1996     

Curvature, Diameter and Bounded Betti Numbers

In this paper, we introduce the notion of bounded Betti numbers, and show that the bounded Betti numbers of a closed Riemannian n-manifold (M, g) with Ric (M) ≥ -(n - 1) and Diam (M) ≤ D are bounded by a number depending on D and n. We also show that there are only finitely many isometric isomorphism types of bounded cohomology groups (H*(M), || · ||∞) among closed Riemannian manifold (M, g) with K(M) ≥-1 and Diam (M) ≤ D.     

Metrics with harmonic spinors

We show that every closed spin manifold of dimensionn 3 mod 4 with a fixed spin structure can be given a Riemannian metric with harmonic spinors, i.e. the corresponding Dirac operator has a non-trivial kernel (Theorem A). To prove this we first compute the Dirac spectrum of the Berger spheresS ⁿ ,n odd (Theorem 3.1). The second main ingredient is Theorem B which states that the Dirac spectrum of a connected sumM ₁#M ₂ with certain metrics is close to the union of the spectra ofM ₁ and ofM ₂.Partially supported by SFB 256 and by the GADGET program of the EU     

Generalized Thrackle Drawings of Non-bipartite Graphs

A graph drawing is called a generalized thrackle if every pair of edges meets an odd number of times. In a previous paper, we showed that a bipartite graph G can be drawn as a generalized thrackle on an oriented closed surface M if and only if G can be embedded in M. In this paper, we use Lins’ notion of a parity embedding and show that a non-bipartite graph can be drawn as a generalized thrackle on an oriented closed surface M if and only if there is a parity embedding of G in a closed non-orientable surface of Euler characteristic χ(M)−1. As a corollary, we prove a sharp upper bound for the number of edges of a simple generalized thrackle.     

The Minimal Length of a Closed Geodesic Net on a Riemannian Manifold with a Nontrivial Second Homology Group

Let Mⁿ be a closed Riemannian manifold with a nontrivial second homology group. In this paper we prove that there exists a geodesic net on Mⁿ of length at most 3 diameter(Mⁿ). Moreover, this geodesic net is either a closed geodesic, consists of two geodesic loops emanating from the same point, or consists of three geodesic segments between the same endpoints. Geodesic nets can be viewed as the critical points of the length functional on the space of graphs immersed into a Riemannian manifold. One can also consider other natural functionals on the same space, in particular, the maximal length of an edge. We prove that either there exists a closed geodesic of length ≤ 2 diameter(Mⁿ), or there exists a critical point of this functional on the space of immersed θ-graphs such that the value of the functional does not exceed the diameter of Mⁿ. If n=2, then this critical θ-graph is not only immersed but embedded.Mathematics Subject Classifications (2000). 53C23, 49Q10     

Curvature-free upper bounds for the smallest area of a minimal surface

In this paper we will present two upper estimates for the smallest area of a possibly singular minimal surface in a closed Riemannian manifold Mⁿ with a trivial first homology group. The first upper bound will be in terms of the diameter of Mⁿ, the second estimate will be in terms of the filling radius of a manifold, leading also to the estimate in terms of the volume of Mⁿ. If n = 3 our upper bounds are for the smallest area of a smooth embedded minimal surface. After that we will establish similar upper bounds for the smallest volume of a stationary k-dimensional integral varifold in a closed Riemannian manifold Mⁿ with . The above results are the first results of such nature. Received: October 2004 Revision: May 2005 Accepted: June 2005     

Topological restrictions for circle actions and harmonic morphisms

 Let M ^m be a compact oriented smooth manifold admitting a smooth circle action with isolated fixed points which are isolated as singularities as well. Then all the Pontryagin numbers of M ^m are zero and its Euler number is nonnegative and even. In particular, M ^m has signature zero. We apply this to obtain non-existence of harmonic morphisms with one-dimensional fibres from various domains, and a classification of harmonic morphisms from certain 4-manifolds. Received: 16 May 2002 Published online: 14 February 2003 Mathematics Subject Classification (2000): 58E20, 53C43, 57R20.     

A remark on the space of metrics having nontrivial harmonic spinors

Let M be a closed spin manifold of dimension n ≡ 3 mod 4. We give a simple proof of the fact that the space of metrics on M with invertible Dirac operator is either empty or it has infinitely many path components.     

Mod 2 cohomology of combinatorial Grassmannians

Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles. It defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles. It then gives a transformation from matroid bundles to spherical quasifibrations, by showing that the geometric realization of a matroid bundle is a spherical quasifibration. The poset of oriented matroids of a fixed rank classifies matroid bundles, and the above transformations give a splitting from topology to combinatorics back to topology. A consequence is that the mod 2 cohomology of the poset of rank k oriented matroids (this poset classifies matroid bundles) contains the free polynomial ring on the first k Stiefel-Whitney classes.     

Insecurity for compact surfaces of positive genus

A pair of points in a riemannian manifold M is secure if the geodesics between the points can be blocked by a finite number of point obstacles; otherwise the pair of points is insecure. A manifold is secure if all pairs of points in M are secure. A manifold is insecure if there exists an insecure point pair, and totally insecure if all point pairs are insecure. Compact, flat manifolds are secure. A standing conjecture says that these are the only secure, compact riemannian manifolds. We prove this for surfaces of genus greater than zero. We also prove that a closed surface of genus greater than one with any riemannian metric and a closed surface of genus one with generic metric are totally insecure.     

Limits of functions and elliptic operators

We show that a subspaceS of the space of real analytical functions on a manifold that satisfies certain regularity properties is contained in the set of solutions of a linear elliptic differential equation. The regularity properties are thatS is closed inL ² (M) and that if a sequence of functions fn in ƒ_n converges inL ²(M), then so do the partial derivatives of the functions ƒ_n.     

Additivity of Higher Rho Invariants and Nonrigidity of Topological Manifolds

Let X be a closed oriented connected topological manifold of dimension n ≥ 5 . The structure group is the abelian group of equivalence classes of all pairs (f, M) such that M is a closed oriented manifold and f : M → X is an orientation‐preserving homotopy equivalence. The main purpose of this article is to prove that a higher rho invariant map defines a group homomorphism from the topological structure group of X to the analytic structure group of X . Here X is the universal cover of X , Γ = π₁X is the fundamental group of X , and is a certain C^* ‐algebra. In fact, we introduce a higher rho invariant map on the homology manifold structure group of a closed oriented connected topological manifold, and prove its additivity. This higher rho invariant map restricts to the higher rho invariant map on the topological structure group. More generally, the same techniques developed in this paper can be applied to define a higher rho invariant map on the homology manifold structure group of a closed oriented connected homology manifold. As an application, we use the additivity of the higher rho invariant map to study nonrigidity of topological manifolds. More precisely, we give a lower bound for the free rank of the algebraically reduced structure group of X by the number of torsion elements in π₁X . Here the algebraically reduced structure group of X is the quotient of modulo a certain action of self‐homotopy equivalences of X . We also introduce a notion of homological higher rho invariant, which can be used to detect many elements in the structure group of a closed oriented topological manifold, even when the fundamental group of the manifold is torsion free. In particular, we apply this homological higher rho invariant to show that the structure group is not finitely generated for a class of manifolds. © 2020 Wiley Periodicals LLC     

The Gauss-Bonnet theorem for vector bundles

We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle E of even rank over a closed compact orientable manifold M. This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special case when M is a Riemannian manifold and E is the tangent bundle of M endowed with the Levi-Civita connection. The proof is based on an explicit geometric construction of the Thom class for 2-plane bundles. Dedicated to the memory of Philip Bell Research partially supported by NSF grant DMS-9703852.     

Flat Riemannian Manifolds as Torus Bundles

Vasquez showed that for any finite group G there exists a number n(G) such that for every flat Riemannian manifold M with holonomy group G there exists a fiber bundle T → M → N, where T is a flat torus and N is a flat manifold of dimension less than or equal to n(G). We show that n(H) ≤ n(G) if H Δ leftG or G = N ? H and use this result to describe groups with the Vasquez number equal to 2 or 3.     

Fixed Points of Automorphisms of Graphs with 1 – Factorizations

We consider sets of fixed points of finite simple undirected connected graphs with 1 – factorizations. The maximum number of fixed points of complete graphs K_2n (n > 2) is n if n ≡ 0 mod 4, n — 1 if n ≡ 3 mod 4 or n ≡ 5 mod 12, n — 2 if n ≡ 2 mod 4, n — 3 if n ≡ 1 mod 4 and n ≢ 5 (mod 12). The maximum number of fixed points of 1 – factorizations of (non – complete) graphs with 2n vertices is less than or equal to n. If n is a prime number, then there are graphs with 2n vertices whose 1 – factorizations have automorphisms with n fixed points. Moreover, a result on the structure of a group of fixed – point – free automorphisms is presented.     

Volume,diameter and the minimal mass of a stationary 1-cycle

In this paper we present upper bounds on the minimal mass of a non-trivial stationary 1-cycle. The results that we obtain are valid for all closed Riemannian manifolds. The first result is that the minimal mass of a stationary 1-cycle on a closed n-dimensional Riemannian manifold Mⁿ is bounded from above by (n + 2)!d/4, where d is the diameter of a manifold Mⁿ. The second result is that the minimal mass of a stationary 1-cycle on a closed Riemannian manifold Mⁿ is bounded from above by where where is the filling radius of a manifold, and where is its volume.     

Poincaré-Hopf inequalities for periodic orbits

The interplay between the dynamics of a nonsingular Morse-Smale flow on a smooth, closed, n-dimensional manifold, M, and the topology of M, was exhibited in Franks (Comment Math Helv 53(2):279?C294, 1978), Smale (Bull Am Math Soc 66:43?C49, 1960), by means of a collection of inequalities, which we refer to as Morse-Smale inequalities. These inequalities relate the number of closed orbits of each index to the Betti numbers of M. These well-known inequalities provide the necessary conditions for a given dynamical data in the form of a specified number of closed orbits of a given index to be realized as a nonsingular Morse-Smale flow on M. In this article we provide two inequalities, hereby referred to as Poincaré-Hopf inequalities for periodic orbits, which imposes constraints on the dynamics of periodic orbits without reference to the Betti numbers of the manifold M. The main theorem establishes the necessity and sufficiency of the Poincaré-Hopf inequalities in order for the Morse-Smale inequalities to hold.     

Curvature and tachibana numbers

The purpose of this paper is to define the rth Tachibana number t _r of an n-dimensional closed and oriented Riemannian manifold (M,g) as the dimension of the space of all conformal Killing r-forms for r = 1, 2, . . . , n ? 1 and to formulate some properties of these numbers as an analog of properties of the rth Betti number b _r of a closed and oriented Riemannian manifold.