lie theory for nilpotent l algebras the deligne groupoid is a functor from nilpotent differential graded lie algebras concentrated in positive degrees to groupoids in the special case of lie algebras over a field of characteristic zero it gives the associated simply connected lie group we generalize the deligne groupoid to a functor from l algebras concentrated in degree n to n groupoids we actually construct the nerve of the n groupoid which is an enriched kan complex the construction of gamma is quite explicit it is based on dupont s proof of the de rham theorem and yields higher dimensional analogues of holonomy and of the campbell hausdorff formula in the case of abelian l algebras i e chain complexes the functor is the dold kan simplicial set

topological obstructions to nonnegative curvature we find new obstructions to the existence of complete riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups in particular we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metrics

some group actions on homotopy spheres of dimension seven and fifteen 0 introduction this paper is based on the simple observation that every 4 plane bundle over s4 admits a natural action of 50 3 by bundle maps and that every 8 plane bundle over 58 admits a natural action of the compact lie group g2 by bundle maps these actions are easy to see once one remembers that so 3 is the group of automorphisms of the quaternions and that g2 is the group of automorphisms of the cayley numbers the study of these actions is closely connected to several well known phenomena in differential topology and compact transformation groups the most obvious connection is to milnor s original construction of exotic 7 spheres as 3 sphere bundles over 54 26 milnor proved that if the euler class of such a bundle is a generator of h4 s4 z then its total space is homeomorphic to 57 he also defined a numerical invariant of the diffeomorphism type and used it to detect an exotic differential structure on some of these sphere bundles subsequently eells and kuiper 12 introduced a refinement of this invariant called the it invariant using the u invariant they proved that the sphere bundle m7 milnor s notation is a generator for the group of homotopy 7 spheres these constructions also work for 7 sphere bundles over 58 32 and the manifold m15 is a generator for bp16 the group of homotopy 15 spheres which bound wn manifolds it is a routine matter to check that milnor s arguments and their subsequent refinements work g equivariantly where g 5so 3 or g2 and we shall do this in sections 2 and 3 in particular each sphere bundle with the correct euler class is g homeomorphic to an orthogonal action on s2 1 where 2n 1 7 or 15 moreover distinct sphere bundles have distinct oriented g diffeomorphism types the proof of the second fact uses an equivariant version of the it invariant as originally defined this invariant takes values in q z however it is well known that in the presence of a g action with s c g its value in q is well defined using

transitive holonomy group and rigidity in nonnegative curvature in this note we examine the relationship between the twisting of a vector bundle xi over a manifold m and the action of the holonomy group of a riemannian connection on xi for example if there is a holonomy group which does not act transitively on each fiber of the corresponding unit sphere bundle then for any f s n to m the pullback f xi of xi admits a nowhere zero cross section these facts are then used to derive a rigidity result for complete metrics of nonnegative sectional curvature on noncompact manifolds

descent for n bundles abstract given a lie group g one constructs a principal g bundle on a manifold x by taking a cover u x specifying a transition cocycle on the cover and descending the trivialized bundle u g along the cover we demonstrate the existence of an analogous construction for local n bundles for general n we establish analogues for simplicial lie groups of moore s results on simplicial groups these imply that bundles for strict lie n groups arise from local n bundles our construction leads to simple finite dimensional models of lie 2 groups such as string n from cocycle data

non negative pinching moduli spaces and bundles with infinitely many souls we show that in each dimension n 10 there exist infinite sequences of homotopy equivalent but mutually non homeomorphic closed simply connected riemannian n manifolds with 0 sec 1 positive ricci curvature and uniformly bounded diameter we also construct open manifolds of fixed diffeomorphism type which admit infinitely many complete nonnegatively pinched metrics with souls of bounded diameter such that the souls are mutually non homeomorphic finally we construct examples of noncompact manifolds whose moduli spaces of complete metrics with sec 0 have infinitely many connected components

local rigidity of symmetric spaces we show that on a symmetric space of noncompact or compact type the metric is locally rigid in the sense that if one changes the metric locally but preserves the curvature bounds then the new metric is isometric to the old one we also prove an analytic continuation property for symmetric spaces of rank 3 the aim of this paper is to show that a symmetric space is rigid under a local modification of the metric which preserves suitable curvature bounds our results were motivated by the following theorem of gromov bgs 5 theorem 1 gromov let x be a simply connected symmetric space of rank 2 and with nonpositive sectional curvature and q the complement of a compact set in x if f q y is a totally geodesic isometric embedding where y is a complete simply connected riemannian manifold with nonpositive sectional curvature then there exists a unique extension of f to a totally geodesic isometric embedding tf x y we will give a slightly different proof of this theorem which will then enable us to also prove the following dual result theorem 2 let x be a simply connected symmetric space of rank 2 and with sectional curvature 0 0 and dim y dim x and if q is the complement of a ball of radius 7r 30 in x then any isometric embedding f q y extends uniquely to an isometry 7 x y if in the above two theorems we choose y x and f id we obtain the desired local rigidity of symmetric spaces theorem 2 applies in particular to s2 x s and shows that the hopf conjecture is at least locally correct the optimal radius for the ball in theorem 2 would be the radius of a convex ball 7r 2 under those assumptions indeed in the case of nonpositive curvature gromov bgs 5 proved the following local version of theorem 1 received by the editors november 4 1987 and in revised form september 12 1988 1980 mathematics subject classification 1985 revision primary 53c35 secondary 53c20 the first author is supported by the schweizerischer nationalfond the second author is partially supported by a grant from the national science foundation and would like to thank the ihes and the max planck institut for its hospitality 1990 american mathematical society 0002 9947 90 1 00 25 per page

volumes of small balls on open manifolds lower bounds and examples question under what curvature assumptions on a complete open manifold is the volume of balls of a fixed radius bounded below independent of the center point xx two theorems establish such assumptions and two examples sharply limit their weakening in particular we give an example of a metric on r4 extending to higher dimensions of positive ricci curvature whose sectional curvatures decay to 0 and such that the volume of balls goes uniformly to 0 as the center goes to infinity 0 introduction the following question came up in discussions of the first author with peter li under what curvature assumptions on a complete open manifold m can one find a constant c m r such that the volume of balls b x r can be bounded from below independent of the midpoint x 0 1 vol b x r c m r note that in the presence of a lower bound on the ricci curvature one needs to prove the above inequality only for some r since the bishop gromov volume comparison theorem gr p 124 then gives a bound c m r for r c n rn in dimension 2 theorem a below is a satisfyingpositive answer since the assumptions do not imply a lower bound on the injectivity radius while any weakening of the assumptions leads to immediate counterexamples theorem a if m2 is complete with curvature k o then there exist c m such that for radii r c m r2 in higher dimensions however we have example 1 there are convex hypersurfaces md in rd 1 such that for d 3 no lower bounds as in 0 1 exist for md in dimensions 2 we therefore assume an upper curvature bound but then one gets uniform control even on the injectivity radius the following theorem has been rediscovered several times received by the editors august 10 1987 1980 mathematics subject classification 1985 revision primary 53c20 secondary 53a07 research supported by nsf grant mcs 7g01780 and the sloan foundation both authors were supported by invitations to the i h e s and the m s r i r 1988 american mathematical society 0002 9947 88 1 00 25 per page

presenting higher stacks as simplicial schemes abstract we show that an n geometric stack may be regarded as a special kind of simplicial scheme namely a duskin n hypergroupoid in affine schemes where surjectivity is defined in terms of covering maps yielding artin n stacks deligne mumford n stacks and n schemes as the notion of covering varies this formulation adapts to all hag contexts so in particular works for derived n stacks replacing rings with simplicial rings we exploit this to describe quasi coherent sheaves and complexes on these stacks and to draw comparisons with kontsevich s dg schemes as an application we show how the cotangent complex controls infinitesimal deformations of higher and derived stacks

compact differentiable transformation groups on exotic spheres 0 introduction 355 1 statement of the main results 357 2 higher dimensional subgroups of o n 1 358 3 higher dimensional transformation groups on homology spheres geometric weight systems and orbit structures 360 4 higher dimensional transformation groups on homotopy spheres and the proof of theorem b 375 5 degree of symmetry of exotic spheres revisited 381 6 large transformation groups on exotic spheres in bp 1 384

exotic spheres with positive ricci curvature we show that a certain class of manifolds admit metrics of positive ricci curvature this class includes many exotic spheres including all homotopy spheres which represent elements of bp2n

on the sections of universal hyperelliptic curves abstract in this paper we will give an algebraic proof for determining the sections for the universal pointed hyperelliptic curves c h g n k h g n k when g 3 and the image of the l adic cyclotomic character g k z is infinite furthermore we will study the nonabelian phenomena associated to the universal hyperelliptic curves for example we will show that the section conjecture holds for c h g k h g k and the unipotent analogue of the conjecture holds for the pointed cases this work is an extension of hain s original work on the rational points of universal curves to the hyperelliptic case

the almost cyclicity of the fundamental groups of positively curved manifolds recall that a pure f structure is a kind of generalized torus action the main result asserts that if a compact positively curved manifold mn admits an invariant pure f structure such that each orbit has positive dimension then the fundamental group has a finite cyclic subgroup with index less than wn a constant depending only on n as an application we conclude that for all 0 1 the fundamental group of a pinched n manifold either has a cyclic subgroup with index less than wn or has order less than w n a constant depending only on n and in particular this substantially improves the main result in ro1

topology of riemannian submanifolds with prescribed boundary we prove that a smooth compact submanifold of codimension 2 immersed in r n 3 bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces this settles a question of guan and spruck related to a problem of yau analogous results for complete fillings of arbitrary riemannian submanifolds are obtained as well on the other hand we show that these finiteness theorems may not hold if the codimension is too high or the prescribed boundary is not sufficiently regular our proofs employ among other methods a relative version of nash s isometric embedding theorem and the theory of alexandrov spaces with curvature bounded below including the compactness and stability theorems of gromov and perelman

twisting and nonnegative curvature metrics on vector bundles over the round sphere a complete noncompact manifold m with nonnegative sectional curvature is diffeomorphic to the normal bundle of a compact submanifold s called the soul of m when s is a round sphere we show that the clutching map of this bundle is restricted this is used to deduce that there are at most finitely many isomorphism types of such bundles with sectional curvature lying in a fixed interval 0 we also examine the opposite question of how the twisting of the bundle limits the type of possible nonnegative curvature metrics on the bundle it turns out that if the bundle does not admit a nowhere zero section then the normal exponential map is necessarily a diffeomorphism onto m and the ideal boundary of m consists of a single point in their paper cg cheeger and gromoll raised the question of which vector bundles over the round sphere admit complete metrics with nonnegative sectional curvature the significance of this problem is that it attempts to determine to what extent a converse to the soul theorem holds recall that this theorem states that every open i e complete noncompact manifold m with nonnegative curvature km is diffeomorphic to a vector bundle over a compact totally geodesic submanifold s called a soul a natural question then is whether all such vector bundles admit complete metrics with km 0 in ow it was shown that when the soul is a bieberbach manifold nonnegative curvature metrics force the vanishing of the euler class of the vector bundle it follows that among oriented plane bundles over the torus only the trivial one admits such a metric the above case is fairly rigid a priori however since for example any bundle with curvature 0 over such a manifold must also admit a flat metric and the corresponding question for a simply connected base remains open an answer could provide insights on the topological structure of compact manifolds with nonnegative curvature since any open manifold with nonnegative curvature can after modifying its metric be isometrically 1991 mathematics subject classification primary 53c20

the metric projection onto the soul we study geometric properties of the metric projection r m s of an open manifold m with nonnegative sectional curvature onto a soul s ir is shown to be c up to codimension 3 in arbitrary codimensions small metric balls around a soul turn out to be convex so that the unit normal bundle of s also admits a metric of nonnegative curvature next we examine how the horizontal curvatures at infinity determine the geometry of m and stu dy the structure of sharafutdinov lines we conclude with regularity properties of the cut and conjugate loci of m the resolution of the soul conjecture of cheeger and gromoll by perelinan showed that the structure of open manifolds with nonnegative sectional curvature is more rigid than expected orne of the key results in 14 is that the metric projection wf m s which maps a point p in m to the point 7r p in s that is closest to p is a riemannian submersioni perelman also observed that this map is at least of class c1 and later it was shown in 9 that 7w is at least c2 and c at almnost every point the existence of such a riemannian submersion combined with the restriction on the sign of the curvature suggests that the geometry of m inust be special the purpose of this paper is to illustrate this in several different directions in fact we show that most natural geometric objects classically used in the study of these spaces are intrinsically related to the structure of the map w the paper is essentially structured as follows after introducing notation and recalling some basic results in sectioni 1 we establish in section 2 the existence of maps similar to w for any submanifold of m homologous and isometric to the soul and use this to prove the existence of convex tubular neighborhoods for them in particular this also proves theorem 2 5 the unit normal bundle vti s admits a metric with nonnegative sectional curvature as a consequence open manifolds with nonnegative curvature provide examples of compact manifolds with the same lower curvatuire bound see also 81 theorem 2 5 also implies that nontrivial plane bundles over a bieberbach maanifold do not admit nonnegatively curved metrics thereby providing a shorter proof of the main result in 13 we are grateful to the referee for pointing out this fact to us properties such as these are established via stanidard submersion techniques but the fact that 7r is not known to be smooth everywhere prevents us fromi using these techniques to study the geometry of m far away from the soul one way around received by the editors august 18 1997 1991 mathematics subject classification primary 53c20

finiteness theorems for nonnegatively curved vector bundles we prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds more generally we obtain finiteness results for open riemannian manifolds whose topology is concentrated on compact domains of bounded geometry

obstructions to nonnegative curvature and rational homotopy theory we establish a link between rational homotopy theory and the problem which vector bundles admit a complete riemannian metric of nonnegative sectional curvature as an application we show for a large class of simply connected nonnegatively curved manifolds that if c lies in the class and t is a torus of positive dimension then most vector bundles over c x t admit no complete nonnegatively curved metrics

ricci curvature bounds and einstein metrics on compact manifolds where f mo m1 is a homeomorphism and dil f is the dilatation of f given by dil f supx x2 dist f x f x2 dist x1 x2 if mo and m1 are not homeomorphic define dl mo mi oo gromov 20 proves the remarkable result that the space of compact riemannian manifolds f a 3 d of sectional curvature iki 3 0 and diameter dm v and diameter dm c iki dm vm1 in particular gromov s compactness theorem may be strengthened to the statement that f a v d is c1 l compact in the lipschitz topology in this paper we study the question of lipschitz convergence of compact riemannian manifolds with bounds imposed on the ricci curvature ric in

dubins problem on surfaces ii nonpositive curvature let m be a complete connected two dimensional riemannian manifold with nonpositive gaussian curvature k we say that m satisfies the unrestricted complete controllability ucc property for the dubins problem if the following holds given any p 1 v 1 and p 2 v 2 in tm there exists for every eps 0 a curve gamma in m with geodesic curvature smaller than eps such that gamma connects p 1 to p 2 and for i 1 2 dot gamma is equal to v i at p i property ucc is equivalent to the complete controllability of a family of control systems of dubins type parameterized by eps it is well known that the poincare half plane does not verify property ucc in this paper we provide a complete characterization of the two dimensional nonpositively curved manifolds m with either uniformly negative or bounded curvature that satisfy property ucc more precisely if sup mk infty we show that ucc holds if and only if i m is of the first kind or ii the curvature satisfies a suitable integral decay condition at infinity

riemannian submersions from compact four manifolds we show that under certain conditions a nontrivial riemannian submersion from positively curved four manifolds does not exist this gives a partial answer to a conjecture due to fred wilhelm we also prove a rigidity theorem for riemannian submersions with totally geodesic fibers from compact four dimensional einstein manifolds

the virtual cohomological dimension of the mapping class group of an orientable surface let f f r be the mapping class group of a surface f of genus g with s punctures and r boundary components the purpose of this paper is to establish cohomology properties of f parallel to those of the arithmetic groups if g is a linear algebraic group defined over q x is the symmetric space associated to g and a is an arithmetic subgroup of g then a is virtually torsion free and acts properly discontinuously on x the rational homology of a is the same as that of x a furthermore there is a bordification of x bs to a manifold with corners 3 and an extension of the action of a to a properly discontinuous action on jf so that the quotient x a is compact the boundary of jf is homotopy equivalent to a wedge of spheres say of dimension d and the virtual cohomological dimension of a is n d 1 where n is the dimension of x in the case of the mapping class group there is no analog for g ivanov unpublished has proven that f is not arithmetic nevertheless f acts properly discontinuously on teichmiiller space r which is homeomorphic to euclidean space of dimension 6 g 6 2 s 3 will play the role of the symmetric space the quotient of 9by f is the moduli space of curves whose rational homology is then identified with that of e harvey har has constructed a borel serre bordification j of 3 by analytic methods in the case where f has punctures we will build g by a different combinatorial method in addition we will explicitly describe inside y a cell complex y of dimension 4 g 4 s onto which j may be f equivariantly retracted thus establishing an analog of the constructions for sl n by serre soul6 sol and ash a this complex will be of the lowest possible dimension because we will use j to prove our main result

when strictly locally convex hypersurfaces are embedded in this paper we will prove hadamard stoker type theorems in the following ambient spaces mathcal m n times mathbb r where mathcal m n is a 1 4 pinched manifold and certain killing submersions e g berger spheres and heisenberg spaces that is under the condition that the principal curvatures of an immersed hypersurface are greater than some non negative constant depending on the ambient space we prove that such a hypersurface is embedded and we also study its topology

cohomogeneity one disk bundles with normal homogeneous collars we consider cohomogeneity one homogeneous disk bundles and adress the question when these admit a nonnegatively curved1 invariant metric with normal collar i e such that near the boundary the metric is the product of an interval and a normal homogeneous space if such a bundle is not the quotient of a trivial bundle then we show that its rank has to be in 2 3 4 6 8 moreover we give a complete classification of such bundles of rank 6 and 8 and a partial classification for rank 3

a classification of hyperpolar and cohomogeneity one actions an isometric action of a compact lie group on a riemannian manifold is called hyperpolar if there exists a closed connected submanifold that is flat in the induced metric and meets all orbits orthogonally in this article a classification of hyperpolar actions on the irreducible riemannian symmetric spaces of compact type is given since on these symmetric spaces actions of cohomogeneity one are hyperpolar i e normal geodesics are closed we obtain a classification of the homogeneous hypersurfaces in these spaces by computing the cohomogeneity for all hyperpolar actions this result implies a classification of the cohomogeneity one actions on compact strongly isotropy irreducible homogeneous spaces

curvature and symmetry of milnor spheres since milnor s discovery of exotic spheres mi one of the most intriguing problems in riemannian geometry has been whether there are exotic spheres with positive curvature it is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature hi on the other hand there are many examples of exotic spheres with positive ricci curvature cf ch1 he po and na and this work recently culminated in wr where it is shown that every exotic sphere that bounds a parallelizable manifold has a metric of positive ricci curvature this includes all exotic spheres in dimension 7 so far however no example of an exotic sphere with positive sectional curvature has been found in fact until now only one example of an exotic sphere with nonnegative sectional curvature was known the so called gromoll meyer sphere gm in dimension 7 as one of our main results we prove

rational points of universal curves in positive characteristics rational points of universal curves in positive characteristics by tatsunari watanabe department of mathematics duke university

the cut off covering spectrum we introduce the r cut off covering spectrum and the cut off covering spectrum of a metric space or a riemannian manifold the spectra measure the sizes of localized holes in the space and are defined using covering spaces called covers and r cut off covers they are investigated using 6 homotopies which are homotopies via grids whose squares are mapped into balls of radius on locally compact spaces we prove that these new spectra are subsets of the closure of the length spectrum we prove the r cut off covering spectrum is almost continuous with respect to the pointed gromov hausdorff convergence of spaces and that the cut off covering spectrum is also relatively well behaved this is not true of the covering spectrum defined in our earlier work which was shown to be well behaved on compact spaces we close by analyzing these spectra on riemannian manifolds with lower bounds on their sectional and ricci curvature and their limit spaces

strongly nonnegative curvature we prove that all currently known examples of manifolds with nonnegative sectional curvature satisfy a stronger condition their curvature operator can be modified with a 4 form to become positive semidefinite

conditions for nonnegative curvature on vector bundles and sphere bundles this paper addresses cheeger and gromoll s question of which vector bundles admit a complete metric of nonnegative curvature and relates their question to the issue of which sphere bundles admit a metric of positive curvature we show that any vector bundle which admits a metric of nonnegative curvature must admit a connection a tensor and a metric on the base space which together satisfy a certain differential inequality on the other hand a slight sharpening of this condition is sufficient for the associated sphere bundle to admit a metric of positive curvature our results sharpen and generalize walschap and strake s conditions under which a vector bundle admits a connection metric of nonnegative curvature