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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

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.

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

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).

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.

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

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 .

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).

We give a complete diffeomorphism classification of 1-connected manifolds (of dimension different from 4) whose integral homology is H(M)=Z+Z+Z., Comment: To appear in Journal of Differential Geometry

A conference with the title Fiber bundles and differential geometry was held at Cornell University from May 3 to May 7, 1953.* It was supported by a grant from the National Science Foundation. The purpose of the present paper is to record those problems presented at the conference which concern differentiable, almost-complex, complex, and algebraic manifolds. Several of the problems have been solved since the conference. These solutions gave rise to new problems which we will discuss. This report is divided into three sections: differentiable manifolds, almostcomplex manifolds, complex manifolds. The last part contains also the problems on algebraic varieties. Since some problems of the first two sections were motivated by theorems on algebraic varieties, there are occasional references in Sections 1 and 2 to Section 3.1. The author wishes to thank Professors K. Kodaira and D. C. Spencer heartily for many discussions and valuable suggestions during the period of preparation of this report.

In this paper, we prove a number of theorems which give some insight into the structure of differentiable manifolds. The methods, results and some notation of [13], hereafter referred to as GPC, and [12] will be used. These two papers and [14] can be considered as a starting point for this one. The main theorems in these papers are special cases of the theorems here. Among the most important theorems in this paper are 1. 1 and 6. 1. Some conversations with A. Haefliger were helpful in the preparation of parts of this paper. Everything will be considered from the differentable, equivalently C*, point of view; manifolds, imbeddings, and isotopes will be C*.