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We investigate the relation between holomorphic torus actions on complex manifolds of locally conformally Kahler (LCK) type and the existence of special LCK metrics. We show that if the group of biholomorphisms of such a manifold (M, J) contains a compact torus which is not totally real, then there exists a Vaisman metric on the manifold, generalising a result of Kamishima–Ornea. Also, we obtain a new obstruction to the existence of LCK structures on a given complex manifold in terms of its automorphism group. As an application, we obtain a classification of manifolds of LCK type among all the manifolds having the structure of a holomorphic principal torus bundle. Moreover, we show that if the group of biholomorphisms contains a compact torus whose dimension is half the real dimension of M, then (M, J) admits an LCK metric with positive potential. Finally, we obtain new non-existence results for LCK metrics on certain products of complex manifolds.

In this survey, symmetry provides a framework for classification of manifolds with differential-geometric structures. We highlight pseudo-Riemannian metrics, conformal structures, and projective structures. A range of techniques have been developed and successfully deployed in this subject, some of them based on algebra and dynamics and some based on analysis. We aim to illustrate this variety., Comment: 15 pages, 3 figures; submitted to Notices of the American Mathematical Society; references are limited by the journal; comments welcome

Let (M, g, ∇) be a pseudo-Riemannian manifold with a torsion free linear connection and let Jg be the generalized complex structure on M defined by g; see [13], [14]. We prove that in the case Jg is ∇- integrable the ±i-eigenbundles of Jg, E1,0 Jg and E0,1 Jg are complex Lie algebroids.Moreover E0,1 Jg and (E1,0 Jg)* are canonically isomorphic, thus we define the concept of generalized ∂̅-operator of (M, g, ∇) and we describe a class of generalized holomorphic sections of T(M) ⊕ T*(M). Also we relate the Lie bialgebroid property of (E1,0 Jg , (E1,0 Jg )*) to conditions on the metric g in the case of affine Hessian manifolds.

Paper brings classification of manifolds determined as Minkowski mixed triple combinations of three different curve segments represented parametrically by their vector maps. Some intrinsic differential properties of resulting manifolds are derived, namely those based on the properties of original curve segments chosen as operands in the respective Minkowski point set operations. Minkowski mixed triple combination can be determined in more different forms considering equal or not equal parameterization of the three basic curve segments, while resulting manifolds are found to be of dimension 1 to 3 consequently. Parameters of these combinations are powerful shaping tools influencing considerably geometric and aesthetic characteristics of the resulting objects. Various forms of manifolds determined by means of this generating principle are presented, including examples of manifolds determined as various Minkowski triple combinations of three circles positioned into different planes in the Euclidean space, and existence of their singular points is discussed.

We study natural mechanical systems describing the motion of a particle on a two-dimensional Riemannian manifold of revolution in the field of a central smooth potential. We obtain a classification of Riemannian manifolds of revolution and central potentials on them that have the strong Bertrand property: any nonsingular (that is, not contained in a meridian) orbit is closed. We also obtain a classification of manifolds of revolution and central potentials on them that have the 'stable' Bertrand property: every parallel is an 'almost stable' circular orbit, and any nonsingular bounded orbit is closed. Bibliography: 14 titles.

We study the Rarita-Schwinger operator on compact Riemannian spin manifolds. In particular, we find examples of compact Einstein manifolds with positive scalar curvature where the Rarita-Schwinger operator has a non-trivial kernel. For positive quaternion K\"ahler manifolds and symmetric spaces with spin structure we give a complete classification of manifolds admitting Rarita-Schwinger fields. In the case of Calabi-Yau, hyperk\"ahler, $G_2$ and Spin(7) manifolds we find an identification of the kernel of the Rarita-Schwinger operator with certain spaces of harmonic forms. We also give a classification of compact irreducible spin manifolds admitting parallel Rarita-Schwinger fields., Comment: 21 pages

We introduce the notion of even Clifford structures on Riemannian manifolds, a framework generalizing almost Hermitian and quaternion-Hermitian geometries. We give the complete classification of manifolds carrying parallel even Clifford structures: K\"ahler, quaternion-K\"ahler and Riemannian products of quaternion-K\"ahler manifolds, several classes of 8-dimensional manifolds, families of real, complex and quaternionic Grassmannians, as well as Rosenfeld's elliptic projective planes, which are symmetric spaces associated to the exceptional simple Lie groups. As an application, we classify all Riemannian manifolds whose metric is bundle-like along the curvature constancy distribution, generalizing well-known results in Sasakian and 3-Sasakian geometry., Comment: Final version, 28 pages

We study mappings with bounded ( p , q ) -distortion associated to Sobolev spaces on Carnot groups. Mappings of such type have applications to the Sobolev type embedding theory and classification of manifolds. For this class of mappings, we obtain estimates of linear distortion, and a geometrical description. We prove also Liouville type theorems and give some sufficient conditions for removability of sets.

The structure set of a given manifold fits into a surgery exact sequence, which is the main tool for classification of manifolds. In the present paper, we describe relations between various structure sets and groups of obstructions which naturally arise for triples of manifolds. The main results are given by commutative braids and diagrams of exact sequences.

Complete PL and topological classification and partial smooth classification of manifolds homotopy equivalent to a Wall's manifold (defined as a mapping torus of a Dold manifold), introduced by Wall in his 1960 Annals paper on cobordism, have been done by determining: (1) the normal invariants of Wall's manifolds, (2) the surgery obstruction of a normal invariant and (3) the action of the Wall surgery obstruction groups on the smooth, PL and homeomorphism classes of homotopy Wall's manifolds (to be made precise in the body of the paper). Consequently classification results of automorphisms (self homeomorphisms, and self PL-homeomorphisms) of Dold manifolds follow.

Complete PL and topological classification and partial smooth classification of manifolds homotopy equivalent to a real Milnor manifold (hypersurface of type (1,1) of a product of two real projective spaces) have been done by determining: (1) the normal invariants of the real Milnor manifolds, (2) the surgery obstruction of a normal invariant, and (3) the action of the Wall surgery obstruction groups on the smooth, PL and homeomorphism classes of homotopy real Milnor manifolds (to be made precise in the body of the paper).

We study the twistor equation on pseudo-Riemannian $Spin^c-$manifolds whose solutions we call charged conformal Killing spinors (CCKS). We derive several integrability conditions for the existence of CCKS and study their relations to spinor bilinears. A construction principle for Lorentzian manifolds admitting CCKS with nontrivial charge starting from CR-geometry is presented. We obtain a partial classification result in the Lorentzian case under the additional assumption that the associated Dirac current is normal conformal and complete the Classification of manifolds admitting CCKS in all dimensions and signatures $\leq 5$ which has recently been initiated in the study of supersymmetric field theories on curved space., 26 pages v2: typos corrected, minor changes

Hornology boundary links have become an increasingly important class of links, largely due to their significance in the ongoing concordance classification of links. Tim Cochran and Jerome Levine defined an algebraic object called a pattern associated to a homology boundary link which can be used to study the deviance of a homology boundary link from being a boundary link. Since a pattern is a set of nx elements which normally generates the free group of rank nxS any invariants which detect non-trivial patterns can be applied to the purely algebraic question of when such a set is a set of conjugates of a generating set for the free group. We will give a constructive geometric proof that all patterns are realized by some homology boundary link Lrl in srl+2. We shall also prove an analogous existence theorem for calibrations of E-links, a more general and less understood class of links tha homology boundary links. 1. 1[NTRODUCTION A lin1z of m components is a smooth, oriented, submanifold L = {X1, , Km} of Sn+2 that is an ordered disjoint union of m manifolds, each of which is piecewiselinearly homeomorphic to sn. If m = 1, L is usually called a 1fnot. L is a boundary lin1z if there exist m disjoint Seifert surfaces, i.e. oriented submanifolds V1, . . ., Vm of Sn+2 such that AVi = Ki for i = 1, . . ., m. Knots and links are of interest since they repeatedly arise in the classification of manifolds. An especially significant equivalence relation on links is c oncordance. Two links Lo and L1 are concordant if there exists a smooth, oriented submanifold of m components C = {C1, . . ., Cm} of Sn+2 X I such that: (i) C is piecewise-linearly homeomorphic to Lo x I, and (ii) aC n (Sn+2 X {i})-lLi for i C {0,1}. The classification of knot concordance groups was obtained in the rnid 1960's by M. Kervaire and J. Levine [12], [14]. Among the things they prove is that the knot concordance group is trivial when n is even and is an infinitely generated group when n is odd. The techniques used in the concordance classification of knots have been found to extend in a compatible manner to the class of bou:ndary links [2], [13], [19]. However, the extension of these ideas to links in general has been a much more difficult and less successful task. In the process, another class of links, called homology boundary links, has arisen. A hornology boundary link of m components is a link which admits m disjoint generalized Seifelzt surfaces {Y1, . . ., Ym} such that Received by the editors May 16, 1995 and, in revised form, October 30, 1995. 1991 Mathematics S?lbject Classification. Primary 57Q45, 57M07, 57M15. (a)1998 American Mathematic.1l So iety 87 This content downloaded from 207.46.13.28 on Tue, 30 Aug 2016 05:27:13 UTC All use subject to http://about.jstor.org/terms

A new method of constructing exotic 11-dimensional structures, as solutions for the cancellation of global gravitational anomalies, is revealed. The construction makes use of link manifolds. Of relevant interest is the fact that the geometry and topology of the exotic 11-dimensional manifolds are encoded in the classical invariants of knots and links in dimension three.

These lectures might be more accurately titled “The application of characteristic classes to geometric problems on manifolds.” In particular we will be interested in studying vector fields on manifolds. This first lecture gives the basic definitions we will need throughout the course.

CONTENTS Introduction Chapter I. Classification of simply-connected topological four-dimensional manifolds § 1. Intersection forms of four-dimensional manifolds § 2. The Pontryagin-Whitehead and Novikov-Wall theorems § 3. Handle body theory. The h-cobordism theorem § 4. Casson handles § 5. Geometric control theorem. Casson handle design § 6. Freedman's theorem and its corollaries § 7. Classification of simply-connected topological four-dimensional manifolds Chapter II. Geometric methods § 1. Gauge fields and Donaldson's theorem § 2. Construction of cobordism § 3. Further results § 4. Floer homology References

In the early 70's Dennis Sullivan applied his theory of minimal models and surgery to the classification of 1-connected closed smooth manifolds of dimension ≥ 5 up to finite ambiguity [Su]. To a diffeomorphism class of such a manifold M he assigns the isomorphism class given by the real minimal model ℳ (M), the integral structure in form of various lattices and the real Pontryagin classes. If one controls the torsion of the manifolds by some bound, his result is that the map given by the triple above is finite-to-one ([Su], Theorem 13.1). He also proves a realization result for the rational minimal model and the Pontryagin classes but not for the lattices ([Su], Theorem 13.2).

So far in this book we have concerned ourselves with the problem of classifying knots (and, of course, links). Intrinsically, this is a knot theoretical problem. This book, however, is twofold in nature and we wish to balance the purely theoretical with some practical applications of knot theory. The various applications of knot theory are discussed in detail in the latter chapters of this book; we would, however, in this chapter like to consider what might be called the classic application of knot theory. One of the most important, even fundamental, problems in algebraic topology is the general classification of manifolds (see Definition 8.0.1 below). In this chapter we will show that it is possible to create from an arbitrary knot (or link) a 3-dimensional manifold (usually shortened to 3-manifold). Hence by studying the properties of knots we can gain insight into the properties of 3-manifolds.

The fixed point index of the Poincare translation operator associated to an ordinary differential equation is a very useful tool for establishing the existence of periodic solutions. In this chapter we focus on ODEs on differentiable manifolds embedded in Euclidean spaces. Our purpose is twofold: on the one hand we aim to provide a short and accessible introduction to some topological tools (such as the Topological Degree, the Degree of a tangent vector field and the Fixed Point Index) that are useful in Nonlinear Analysis; on the other hand we offer a unifying approach to several results about the fixed point index of the Poincare translation operator that were previously scattered among a number of publications. Our main concern will be a formula for the computation of the fixed point index of the flow operator induced on a manifold by a first order autonomous ordinary differential equation. Other formulas for the fixed point index of the translation operator associated with non-autonomous equations will be deduced as consequences. We emphasize that other results, unrelated to our approach, but still involving the fixed point index of the Poincare translation operator have been successfully exploited, for instance, by Srzednicki (see e.g. [Srz2, Srz3]).

3.1 Affine connections 3.2 The Levi-Civita connection 3.3 Tubular neighborhood 3.4 Curvature 3.5 E. Cartan structural equations of a connection

Wall's finiteness obstruction is an algebraic K-theory invariant which decides if a finitely dominated space is homotopy equivalent to a finite CW complex. The object of this survey is to describe the invariant (which was first formulated in 1965) and some of its many applications to the surgery classification of manifolds.

The main result of this paper is a four-dimensional stable version of Kneser’s conjecture on the splitting of three-manifolds as connected sums. Namely, let M be a topological respectively smooth compact connected four-manifold (with orientation or Spin-structure). Suppose that π1(M) splits as ∗i=1Γi such that the image of π1(C) in π1(M) is subconjugated to some Γi for each component C of ∂M . Then M is stably homeomorphic respectively diffeomorphic (preserving the orientation or Spin-structure) to a connected sum ]i=1Mi with Γi = π1(Mi). Stably means that one allows additional connected sums with some copies of S2 × S2 on both sides. We also prove a uniqueness statement. As a consequence we obtain the existence and uniqueness of the stable prime decomposition of compact connected four-manifolds (with orientation or Spin-structure). The main technical ingredients are the bordism approach to the stable classification of manifolds due to the first author and the Kurosh Subgroup Theorem.

This chapter deals with some of the basic homological properties of topological manifolds. Since the main result is the Poincare duality theorem, we begin with a simple example to establish an intuitive feeling for this classical result. This is followed by material on topological manifolds and a detailed proof of the theorem. The approach used follows the excellent treatment of Samelson [1965, pp. 323–336] and proceeds by way of the Thorn isomorphism theorem. Several applications of the theorem follow, including the determination of the cohomology rings of projective spaces and results on the index of topological manifolds and cobordism.

My aim in this chapter is to present some of the many recently proven results about 4-dimensional manifolds.

Quadratic forms on lattices (free modules of finite rank) over Z have received wide attention in recent years because of their interactions with group theory, coding theory, and topology; lattices with unimodular quadratic forms have been a subject of special interest in all these areas. For example, unimodular lattices are important to topologists in connection with the classification of manifolds: If M is a closed, oriented, simply connected 4-manifold, then the homology group H2(M; Z) is a unimodular 2-lattice with respect to the quadratic form induced by intersection numbers, and there is an orientationpreserving homotopy equivalence of two such manifolds if and only if the corresponding lattices are isometric (see [M-H, p. 103], or [Ki, p. 22]). Freedman ([F-Q, Sec. 10.1]) has shown that every unimodular 2-lattice occurs as a homology group of a simply connected, closed, topological 4-manifold in this way. If the form is even ("type II") then the isometry class of the lattice determines the manifold uniquely; if the form is odd ("type I") there are exactly two associated manifolds (and they are distinguished by their Kirby-Siebenmann invariants). Two indefinite unimodular 2-lattices are isometric if and only if they have the same rank, type, and signature (see [M-H, Chap. 2], or [S, Chap. 5]); but classification in the definite case remains a major open problem. In low dimensions the classes of definite unimodular lattices are known: the work of Kneser [Kn], Niemeier [Ni], and Borcherds [B] has produced an explicit listing of the classes of rank < 25. (Those of rank < 24 appear in Chapters 16 and 17 of [C-S].) For example, there are only 24 classes of even unimodular lattices of rank 24. But mass formula considerations show that there are more than 107 such classes of rank 32 and more than 1051 such classes in rank 40, suggesting that a complete set of computable invariants may be extraordinarily difficult to find. Recently Newman [Ne2] has developed a matrix algorithm that yields a surprisingly simple reduced form for unimodular lattices. We will look at his result in detail, but the upshot is that every unimodular lattice has a primitive sublattice of codimension 1 belonging to a rather special family, which we call

Almost-contact manifolds are considered as generalized almost-Hermitian manifolds of defect 1, which permits the use of the apparatus of the geometry of generalized almost-Hermitian manifolds to get a number of strong results, in particular to get a complete classification of important types of almost-contact manifolds. The theory of Q-algebras, which plays an important role in the apparatus mentioned, is presented in expanded form.

The work is an analytically systematic exposition of modern problems in the investigation of differentiable manifolds and the geometry of fields of geometric objects on such manifolds.

The Lusternik-Schnirelmann category of a manifold M M is the smallest integer k k such that M M can be covered by k k open sets each of which is contractible in M M . The classification up to cobordism of manifolds with category 3 3 was completed by the author in 1985. The object of this paper is to attempt a similar classification of manifolds with category 4 4 .

CONTENTS Introduction ??1. Hamiltonian systems and the class (H) ??2. The complexity of 3-manifolds ??3. The classification of manifolds of small complexity ??4. Computation of volumes of hyperbolic manifolds ??5. The list of 3-manifolds of complexity not greater than 6 References

Suppose the four dimensional torus T 4 acts effectively on a 6-manifold M so that the orbit space M ∗ is a closed 2-disk, and there exist no exceptional orbits, and the isotropy groups span T 4 . Then the fundamental group of M is a finite abelian group with at most two generators. In this paper, we obtain a homology classification of manifolds of this type under an additional hypothesis that one of the two generators is trivial. We then use this result to obtain a complete classification of simply connected 6-manifolds supporting effective T 4 -actions.

In [1] and [2] a classification of a manifoldMof the type(n,p,1)was given whereHp(M)=Hn−p(M)=ℤis the only non-trivial homology groups. In this paper we give a complete classification of manifolds of the type(n,p,2)and we extend the result to manifolds of type(n,p,r)whereris any positive integer andp=3,5,6,7mod(8).

The classification of (fξηρ)-structures on differentiable manifolds of even and odd dimensions is given.