constructions in sasakian geometry we first generalize the join construction described previously by the first two authors 4 for quasi regular sasakian einstein orbifolds to the general quasi regular sasakian case this allows for the further construction of specific types of sasakian structures that are preserved under the join operation such as positive negative or null sasakian structures as well as sasakian einstein structures in particular we show that there are families of sasakian einstein structures on certain 7 manifolds homeomorphic to s 2 s 5 we next show how the join construction emerges as a special case of lerman s contact fibre bundle construction 32 in particular when both the base and the fiber of the contact fiber bundle are toric we show that the construction yields a new toric sasakian manifold finally we study toric sasakian manifolds in dimension 5 and show that any simply connected compact oriented 5 manifold with vanishing torsion admits regular toric sasakian structures this is accomplished by explicitly constructing circle bundles over the equivariant blow ups of hirzebruch surfaces

k homology asymptotic representations and unsuspended e theory connes and higson defined a bivariant homology theory e a b for separable c algebras the elements of e a b are taken to be homotopy classes of asymptotic momorphisms from sa k to sb k in symbols e a b sa sb k if a is k nuclear then e theory agrees with kasparov s bivariant k theory we show that in many cases one need not take suspensions to calculate the e theory group e a b for many a we show e a b a b k for all b among the a for which this is true are c0 x pt for x with the homotopy type of a finite connected cw complex this gives a concrete realization of k homology related to the brown douglas fillmore description for example k0 x arises as asymptotic representations k0 x c0 x k other a for which our isomorphic holds include the nonunital dimension drop intervals in this case there is no distinction between homomorphisms and asymptotic morphisms so we have succeeded in classifying all homomorphisms from a dimension drop interval to a stable c algebra this was subsequently used by elliott j reine angew math 443 1993 179 219 in the classification of certain c algebras the dimension drop interval may also be used to describe k x z n in terms of paths of asymptotic representations of c0 x

abelian mathrm c ast subalgebras of mathrm c ast algebras of real rank zero and inductive limit mathrm c ast algebras

a diffeomorphism classification of 7 dimensional homogeneous einstein manifolds with mathrm su 3 times mathrm su 2 times mathrm u 1 symmetry on donne des exemples qui montrent que des espaces homogenes homeomorphes ne sont pas necessairement diffeomorphes

classification ofc algebras of real rank zero and unsuspendede equivalence types abstract in this article examples are given to prove that the graded scaled ordered k group is not the complete invariant for a c algebra in the class of unital separable nuclear c algebras of real rank zero and stable rank one even for a c algebra in the subclass which consists of those real rank zero stable rank one c algebras being expressed as inductive limits of k n i 1 m n i c x n i where x n i are two dimensional finite cw complexes and n i are positive integers in the case of simple such c algebras it has been proved that the above invariant is the complete invariant by george elliott and the author these examples prove that the classification conjecture of elliott for the case of non simple real rank zero c algebras should be revised one needs extra invariants the obstruction preventing two such c algebras with the same graded scaled ordered k group from being isomorphic is that they have different unsuspended e equivalence types a refinement of kk equivalence type of c algebras due to connes and higson in this article it is proved that for the above class of inductive limit c algebras the obstruction of unsuspended e equivalence type is the only obstruction i e if two c algebras in the class are unsuspended e equivalence then they are isomorphic it is a surprise that in the case of simple such c algebras or even the case of c algebras with finitely many ideals the obstruction will disappear see section 4

the topology of certain 3 sasakian 7 manifolds we calculate the integer cohomology ring and stable tangent bundle of a family of compact 3 sasakian 7 manifolds constructed by boyer galicki mann and rees the most important part of the cohomology ring is a previously undetermined torsion group that we describe explicitly and whose order we compute there is a surprising connection with the combinatorics of trees

surgery and duality surgery as developed by browder kervaire milnor novikov sullivan wall and others is a method for comparing homotopy types of topological spaces with difieomorphism or homeomorphism types of manifolds of dimension 5 in this paper a modiflcation of this theory is presented where instead of flxing a homotopy type one considers a weaker information roughly speaking one compares n dimensional compact manifolds with topological spaces whose k skeletons are flxed where k is at least n 2 a particularly attractive example which illustrates the concept is given by complete intersections by the lefschetz hyperplane theorem a complete intersection of complex dimension n has the same n skeleton as n and one can use the modifled theory to obtain information about their difieomorphism type although the homotopy classiflcation is not known the theory reduces this classiflcation result to the determination of complete intersections in a certain bordism group this was under certain restrictions carried out in tr the restrictions are if d d1 dr is the total degree of a complete intersection x n1 dr of complex dimension n then the assumption is that for all primes p with p pi1 n 1 the total degree d is divisible by p 2n 1 2pi1 1 theorem a two complete intersections x n

a homotopy classification of certain 7 manifolds this paper gives a homotopy classification of wallach spaces and a partial homotopy classification of closely related spaces obtained by free sl actions on su 3 and on s3 x s5

eta invariants of dirac operators on locally symmetric manifolds fibres de dirac determinant de la transformee de cayley operateurs de dirac sur des espaces localement symetriques trace de l operateur de la chaleur impair interpretation cohomologique la formule de la fonction zeta invariants wistes et applications

discrete time linear periodic systems gramian and modal criteria for reachability and controllability this paper deals with the reachability and controllability of periodic discrete time systems first we supply two necessary and sufficient complete reachability conditions which apply to reversible and non reversible systems respectively then a necessary and sufficient complete controllability condition is provided this condition as well as the complete reachability criteria is given in terms of the reachability gramian matrix equivalent modal criteria for reachability and controllability are established in the second part of the paper

adiabatic limits of the η invariants the odd dimensional atiyah patodi singer problem we study the invariant of boundary value problems of atiyah patodi singer type we prove the formula for the spectral flow of the families overs1 assuming a product structure in a collar neighbourhood of the boundary we show that the invariant behaves the same way as on a closed manifold we also study the adiabatic limit of the spectral invariants in nice cases we are able to split them into a contribution from the interior one from the cylinder and an error term then we show that the error term disappears with the increasing length of the cylinder

nonconnected moduli spaces of positive sectional curvature metrics for a closed manifold m let 9 m resp 9 ic m be the space of riemannian metrics on m with positive sectional resp ricci cur vature and let diff m be the diffeomorphism group of m which acts on these spaces we construct examples of 7 dimensional manifolds for which the moduli space 9 m diff m is not connected and others for which 9 c m diff m has infinitely many connected components the examples are obtained by analyzing a family of positive sectional curvature metrics on homogeneous spaces constructed by aloff and wallach on which su 3 acts transitively respectively a family of positive einstein metrics constructed by wang and ziller on homogeneous spaces on which su 3 x su 2 x u i acts transitively max planck institut fur mathematik gottfried claren strasse 26 5300 bonn 3 germany and fachbereich mathematik universitat mainz 6500 mainz germany current address johannes gutenberg universitat mainz fachbereich mathematik staudinger weg 9 6500 mainz germany e mail address kreck topologie mathematik uni mainz de department of mathematics university of notre dame notre dame indiana 46556 e mail address stolz l nd edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

k group and similarity classification of operators abstract let h be a complex separable hilbert space and l h denote the collection of bounded linear operators on h an operator a in l h is said to be a cowen douglas operator if there exist a connected open subset of complex plane c and n a positive integer such that a a z c a z is not invertible b ran a z y a z x y x h h for z in c z ker a z h and d dim ker a z n for z in in the paper we give a similarity classification of cowen douglas operators by using the ordered k group of the commutant algebra as an invariant and characterize the maximal ideals of the commutant algebras of cowen douglas operators the theorem greatly generalizes the main result in canada j math 156 4 2004 742 by simply removing the restriction of strong irreducibility of the operators the research is also partially inspired by the recent classification theory of simple ah algebras of elliott gong in documenta math 7 2002 255 on the classification of simple inductive limit c algebras ii the isomorphism theorem preprint also see j funct anal 1998 1 ann math 144 1996 497 amer j math 1996 187

the maslov cocycle smooth structures and real analytic complete integrability this paper proves two main results first it is shown that if is a smooth manifold homeomorphic to the standard n torus t n r n z n and h is a real analytically completely integrable convex hamiltonian on t then is diffeomorphic to tn second it is proven that for some topological 7 manifolds the cotangent bundle of each smooth structure admits a real analytically completely integrable riemannian metric hamiltonian

the classification of compact simply connected biquotients in dimension 6 and 7 we classify all compact simply connected biquotients of dimension 6 and 7 for each 6 dimensional biquotient all pairs of groups g h and homomorphisms h rightarrow g times g giving rise to it are classified

η invariant and a problem of berard bergery on the existence of closed geodesics we use the invariant of atiyah patodi singer to compute the eells kuiper invariant for the eells kuiper quaternionic projective plane by combining with a known result of berard bergery it shows that every eells kuiper quaternionic projective plane carries a riemannian metric such that all geodesics passing through a certain point are simply closed and of the same length

exotic structures on 4 manifolds detected by spectral invariants on etudie le spectre de l operateur de dirac twiste sur l espace projectif exotique q 4 construit par cappell et shaneson on demontre que les spectres de l operateur de dirac twiste sur rp 4 repect q 4 ne sont jamais les memes independemment des metriques choisies

non commutative spheres iii irrational rotations leta be the irrational rotation algebra i e thec algebra generated by two unitariesu v satisfyingvu e 2 i uv with irrational and consider the fixed point subalgebrab under the flip automorphismu u 1 v v 1 we prove thatb is an af algebra

the longitudinal cocycle and the index of toeplitz operators abstract let d be a self adjoint leafwise elliptic operator on a foliated manifold compressing multiplication operators to the range of the positive spectral projection for d yields the class of leafwise toeplitz operators the extension generated by these operators is constructed a topological formula for the index of a toeplitz operator with invertible symbol is given this index can also be obtained by pairing the k theory class of the symbol with a certain cyclic cocycle if one lifts an elliptic operator on a closed manifold to a leafwise elliptic operator on an associated flat foliated principal bundle then this cocycle can be used to obtain refined invariants of the original operator

on the classification of simple inductive limit c algebras ii the isomorphism theorem in this article it is proved that the invariant consisting of the scaled ordered k group and the space of tracial states together with the natural pairing between them is a complete invariant for the class of unital simple c algebras which can be expressed as the inductive limit of a sequence a 1 to a 2 to cdots to a n to cdots with a n bigoplus i 1 t n p n i m n i c x n i p n i where xn i is a compact metrizable space and pn i is a projection in m n i c xn i for each n and i and the spaces xn i are of uniformly bounded finite dimension note that the c algebras in the present class are not assumed to be of real rank zero as they were in eg2

7 dimensional compact riemannian manifolds with killing spinors using a link between einstein sasakian structures and killing spinors we prove a general construction principle of odd dimensional riemannian manifolds with real killing spinors in dimensionn 7 we classify all compact riemannian manifolds with two or three killing spinors finally we classify nonflat 7 dimensional riemannian manifolds with parallel spinor fields

block decoupling of linear omega periodic discrete time systems the paper considers block decoupling control of a linear omega periodic discrete time system with state feedback the notion of the maximal reachability subspace and the method to calculate it are extended to the periodic case under the assumption that the input transformation is nonsingular a necessary and sufficient solvability condition for block decoupling is represented in terms of the maximal reachability subspace

ancient solutions of the ricci flow on bundles abstract we generalize the circle bundle examples of ancient solutions of the ricci flow discovered by bakas kong and ni to a class of principal torus bundles over an arbitrary finite product of fano kahler einstein manifolds studied by wang and ziller in the context of einstein geometry as a result continuous families of collapsed and noncollapsed ancient solutions of type i are obtained on circle bundles for all odd dimensions 7 in dimension 7 such examples moreover exist on pairs of homeomorphic but not diffeomorphic manifolds continuous families of collapsed ancient solutions of type i are also obtained on torus bundles for all dimensions 8

similarity classification of holomorphic curves abstract let h be a complex separable hilbert space for an open connected subset of c we shall say that a map f gr n h is a holomorphic curve if there exist n holomorphic h valued functions 1 2 n on such that f 1 n where gr n h denotes the grassmann manifold the set of all n dimensional subspaces of h in the paper we give a similarity classification of some holomorphic curves by using the k group of its commutant algebra as an invariant

the classification of certain non simple c algebras of tracial rank zero abstract it is proved that the z 2 graded ordered k theory group with order unit k a k a 1 a is a complete invariant for the class of unital c algebras which are inductive limits of finite direct sums of unital amenable separable simple c algebras with tracial rank zero which satisfy the uct

codimension two souls and cancellation phenomena abstract for each m 0 we find an open 4 m 9 dimensional simply connected manifold with complete nonnegatively curved metrics whose souls are nondiffeomorphic homeomorphic and have codimension 2 we give a diffeomorphism classification of the pairs n soul when n is the total space of a nontrivial complex line bundle over s 7 cp 2 up to diffeomorphism there are precisely three such pairs distinguished by their nondiffeomorphic souls

diffeomorphism type of the berger space so 5 so 3 we compute the eells kuiper invariant of the berger manifold so 5 so 3 and determine that it is diffeomorphic to the total space of an s 3 bundle over s 4 this answers a question raised by k grove and w ziller

the l 2 index theorem for homogeneous spaces of lie groups the l2 index theorem for covering spaces of atiyah 3 and singer 20 asserts that given a discrete group f acting smoothly and freely on a manifold m with compact quotient m f m and an elliptic differential operator d on m which is f invariant and thus descends to an elliptic differential operator d on the compact manifold m then the p index indrd dimrkerd dimrkerd of d coincides with the ordinary index ind d dim ker d dim ker d of d here ker d is the space of l2 solutions of du 0 and dim r denotes the dimension function corresponding to the trace on the commutant of f acting on the hilbert space of l2 sections over m naturally associated to p the importance of this theorem lies in the fact that indd 0 implies the existence of nontrivial l2 solutions for the equation du 0 and as such it was used in a crucial way by atiyah and schmid 5 to construct explicit realizations of the discrete series representations for semisimple lie groups indeed if g is a lie group which possesses a discrete torsion free cocompact subgroup f and h is a compact subgroup of g then the l2 index theorem applied to the covering space m g h of m f g h combined with the index formula of atiyah singer 6 yields existence results for l2 solutions of g invariant elliptic equations on the homogeneous space g h it is relevant to note that with d denoting this time a g invariant elliptic differential operator on g h the ratio between the f index of d and the covolume of f gives a number independent of f this real number indg d can be in fact intrinsically defined as the difference of the two formal degrees corresponding to the representations of g on ker d and ker d respectively and hence makes sense for any unimodular lie group g even when it has no discrete cocompact subgroups which outside the semisimple case is the generic

classification of simple c algebras inductive limits of matrix algebras over one dimensional spaces abstract in this article we will give a complete classification of simple c algebras which can be written as inductive limits of algebras of the form a n i 1 k n m n i c x n i where x n i are arbitrary variable one dimensional compact metrizable spaces the results unify and generalize the previous results for the case x n i s 1 and for the case of x n i being trees we obtain our classification results by reducing the case of general one dimensional spaces to the case of circles the techniques in this paper play important roles in the study of the case of higher dimensional spaces

the kk lifting problem for dimension drop interval algebras abstract in this paper we investigate the kk theory of generalized dimension drop interval algebras with possibly different dimension drops at the endpoints with special emphasis on the problem which kk class is representable by a homomorphism between two such c algebras allowing the tensor product with a matrix algebra for the codomain algebra this lifting problem makes sense in its own right in kk theory and also has application to the classification of c algebras which are inductive limits of these building blocks it turns out that when the dimension drops at the two endpoints are different there exist kk elements which preserve the order structure defined by m dadarlat and t a loring in 4 on the mod p k theory but cannot be lifted to a homomorphism this is different from the equal dimension drop case as shown by s eilers in 6

kreck stolz invariants for quaternionic line bundles we generalise the kreck stolz invariants s2 and s3 by defining a new invariant the t invariant for quaternionic line bundles over closed spin manifolds m of dimension 4k 1 with h 3 m q h 4 m q 0 the t invariant classifies closed smooth oriented 2 connected rational homology 7 spheres up to almost diffeomorphism and detects exotic homeomorphisms between such manifolds the t invariant also provides information about quaternionic line bundles over a fixed manifold and we use it to give a new proof of a theorem of feder and gitler about the values of the second chern classes of quaternionic line bundles over hp k the t invariant for s 4k 1 is closely related to the adams e invariant on the 4k 5 stem

on inductive limits of matrix algebras over the two torus it will be shown in this paper that certain real rank zero c algebras which are inductive limits of c algebras of the form i m k i c inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i can be expressed as inductive limits of c algebras of the form i m k i c s 1 in particular if both a and b are of real rank zero and are inductive limits of c algebras of the form i m k i c s 1 then also a b is an inductive limit of c algebras of the form i m k i c s 1 hence a b can be classified by its k theory this is a key step in the general classification theory of inductive limit c algebras

a canonical decomposition of linear periodic discrete time systems in this paper a linear periodic discrete time system is studied on the basis of simple properties of the subspaces of controllable states and of unreconstructible ones a canonical structure theorem is derived this generalizes 1o such a system the classical kalman decomposition while preserving the constant dimensionality of the four subsystems which arise when a periodic continuous time system is decomposed the dynamic matrices of the non controllable and or non reconstructible subsystems are shown to be non singular at each time instant as those for a time invariant discrete time system are

g2 manifolds and associative submanifolds via semi fano 3 folds we construct many new topological types of compact g 2 manifolds i e riemannian 7 manifolds with holonomy group g 2 to achieve this we extend the twisted connected sum construction first developed by kovalev and apply it to the large class of asymptotically cylindrical calabi yau 3 folds built from semi fano 3 folds constructed previously by the authors in many cases we determine the diffeomorphism type of the underlying smooth 7 manifolds completely we find that many 2 connected 7 manifolds can be realised as twisted connected sums in a variety of ways raising questions about the global structure of the moduli space of g 2 metrics many of the g 2 manifolds we construct contain compact rigid associative 3 folds which play an important role in the the higher dimensional enumerative geometry gauge theory calibrated submanifolds approach to defining deformation invariants of g 2 metrics by varying the semi fanos used to build different g 2 metrics on the same 7 manifold we can change the number of rigid associative 3 folds we produce

discrete time linear periodic systems a note on the reachability and controllability interval length abstract discrete time linear systems with periodic coefficients of period t are considered in this paper the reachability and controllability indices at time t for periodic systems rt and ct are defined it is shown that the reachability controllability subspace at time t does coincide with the reachability controllability subspace over the interval t rt t t t t ct t and properly includes the reachability controllability subspace over the interval t rt 1 t t t t cr 1 t

at structure of ah algebras with the ideal property and torsion free k theory abstract let a be an ah algebra that is a is the inductive limit c algebra of a 1 1 2 a 2 2 3 a 3 a n with a n i 1 t n p n i m n i c x n i p n i where x n i are compact metric spaces t n and n i are positive integers and p n i m n i c x n i are projections suppose that a has the ideal property each closed two sided ideal of a is generated by the projections inside the ideal as a closed two sided ideal suppose that sup n i dim x n i this condition can be relaxed to a certain condition called very slow dimension growth in this article we prove that if we further assume that k a is torsion free then a is an approximate circle algebra or an a t algebra that is a can be written as the inductive limit of b 1 b 2 b n where b n i 1 s n m n i c s 1 one of the main technical results of this article called the decomposition theorem is proved for the general case i e without the assumption that k a is torsion free this decomposition theorem will play an essential role in the proof of a general reduction theorem where the condition that k a is torsion free is dropped in the subsequent paper gong et al preprint 31 of course in that case in addition to space s 1 we will also need the spaces t ii k t iii k and s 2 as in gong 2002 29

on the asymptotic homotopy type of inductive limit c algebras let x y be compact connected metrisable spaces with base points xo yo and let denote the compact operators it is shown that co x xo is asymptotically homotopic or shape equivalent to co y yo yi r if and only if x and y have isomorphic k groups similar results are obtained for certain inductive limits of nuclear c algebras let r denote the category whose objects are all the separable c algebras and whose set of morphisms from a to b denoted a b consists of homotopy classes of asymptotic morphisms the construction of this category is due to connes and higson ch who defined a bivariant homology theory e a b lisa sb ff and have shown how to define the intersection product for arbitrary extensions of separable c algebras if a is k nuclear then e theory agrees with kasparov s bivariant k theory k on the other hand the asymptotic homotopy category d appears to be the right framework for the homotopy theory of separable c algebras this point of view is supported by results in h ch d chi ck d1 for instance we have shown in d1 that asymptotic homotopy is equivalent to a strong shape theory and hence is intimately related to the shape theories of ek1 and b which were intended as homotopy theories for noncommutative singular spaces in particular it turned out that two separable c algebras are shape equivalent if and only if they are asymptotically homotopic i e isomorphic in r the isomorphism class in sr of a separable c algebra a is called the asymptotic homotopy type of a in this note we exhibit large classes of projeetionless stable nuclear c algebras whose asymptotic homotopy type is determined by k theoretical data theorem 6 this is done via a suspension isomorphism