signatures and higher signatures of s 1 quotients we define and study the signature widehat a genus and higher signatures of the quotient space of an s 1 action on a closed oriented manifold we give applications to questions of positive scalar curvature and to an equivariant novikov conjecture

the stable rank of topological algebras and a problem of r g swan abstract various notions of stable ranks are studied for topological algebras some partial answers to r g swan s problem have two banach or good frechet algebras as in the density theorem in k theory the same stable rank are obtained for example a frechet dense subalgebra a of a c algebra b closed under c functional calculus of self adjoint elements has the same bass stable rank as b

cheeger goreski macpherson s conjecture for the varieties with isolated singularities in 2 cheeger goreski macpherson asked the following question let vclpn ie be a projective variety and let v be the set of regular points of v is the l 2 de rham cohomology group of v with respect to the fubini study metric canonically isomorphic to the intersection cohomology group of v in virtue of the works of cheeger 1 hsian pati 3 and nagase 4 we know that the conjecture is true if dim v 2 our proof will be based on the previous work 5 which complements a result of l saper asserting that there exists a complete kfihler metric ds 2 on v such that the l 2 de rham cohomology group of v with respect to ds 2 is canonically isomorphic to the intersection cohomology group of v to relate saper s l 2 cohomology group to the original one the two metrics ds 2 and say ds 2 shall be connected by a path ds 2 e d s 2 eel0 1 then we shall see that a sharpened version of h6rmanderdonnelly fefferman s l 2 estimate is true for this family cf prop 10 which enables us to compare the space of harmonic forms with respect to ds2v and ds 2 in short it will be shown that the ordinary l 2 cohomology is canonically isomorphic to certain l 2 cohomology which is already known to be isomorphic to the intersection cohomology the manuscript of the present work was written during the author s stay at bergische universitfit gesamthochschule wuppertal on the occasion of aktivitfit in komplex analysis in mpi at bonn he would like to express deep gratitude to the hospitality of these institutions

rapidly decreasing functions in reduced algebras of groups let r be a group we associate to any length function l on r the space h r of rapidly decreasing functions on r with respect to l which coincides with the space of smooth functions on the k dimensional torus when r zk we say that r has property rd if there exists a length function l on r such that h r is contained in the reduced c algebra c r of r we study the stability of property rd with respect to some constructions of groups such as subgroups over groups of finite index semidirect and amalgamated products finally we show that the following groups have property rd 1 finitely generated groups of polynomial growth 2 discrete cocompact subgroups of the group of all isometries of any hyperbolic space

simplicial differential forms with poles shadow forms consider a simplicial complex k whitney associated to each simplicial cochain 4 a differential form v 4 on k whitney s map v from simplicial cochains to differential forms is a map of cochain complexes which induces the derham isomorphism we call the forms v 4 whitney forms they have been useful in several contexts most notably integral geometry and rational homotopy theory in this paper we introduce a generalization of whitney forms called shadow forms we associate to certain simplicial chains g with respect to the first derived subdivision k of k a shadow form w on k let us illustrate the construction of shadow forms when k is a single n simplex which we now denote by a in this case w g is defined for k simplices g of ax which do not lie in the boundary of a or for linear combinations of these for example if a is the 2 simplex then either of the 1 simplices or 2 pictured below has an associated shadow form

l cohomology is intersection cohomology abstract let x be a subanalytic compact pseudomanifold we show a de rham theorem for l forms on the nonsingular part of x we prove that their cohomology is isomorphic to the intersection cohomology of x in the maximal perversity

local isometries of function spaces abstract we show that for a large class of function spaces any isometry that coincides locally with a surjective isometry must be automatically surjective this class includes finite codimensional subspaces of c left x right and spaces c left x e right of e valued continuous functions for finite dimensional or uniformly convex and algebraically reflexive e

spectral invariant subalgebras of reduced crossed product c algebras abstractletg beadiscretegroupwithaproperlengthfunctionl inthispaper weprovethattheschwartz space s l2 g a is a spectral invariant dense subalgebra of the reduced crossedproduct c nr g a for all commutative c n algebra a with a g action if and only if g haspolynomialgrowthwithrespecttol r 2002elsevierscience usa allrightsreserved keywords reducedcrossedproductc n algebras spectralinvariantsubalgebras property mrd 0 introductionletg beadiscretegroupwithalengthfunctionl in 9 jolissaintprovedthatthe schwartzspace s l2 g is a spectralinvariant dense subalgebra of the reducedgroupc n algebrac n r g ifg haspolynomialgrowthorg isthefundamentalgroupofanegativelycurvedcompactmanifold in 4 delaharpegeneralizedjolissaint stheorem to all hyperbolic groups jolissaint and de la harpe s result plays animportantroleintheworkofconnes moscovicionthenovikovconjecture 2 andtheworkoflafforgueonthebaum connesconjecture 10 inthispaper wegiveanecessaryandsuf cientconditionfortheschwartzspaces

good wannier bases in hilbert modules associated to topological insulators for a large class of physically relevant operators on a manifold with discrete group action we prove general results on the non existence of a basis of well localized wannier functions for their spectral subspaces this turns out to be equivalent to the freeness of a certain hilbert module over the group c algebra canonically associated with the spectral subspace this brings into play k theoretic methods and justifies their importance as invariants of topological insulators in physics

the isocohomological property higher dehn functions and relatively hyperbolic groups abstract the property that the polynomial cohomology with coefficients of a finitely generated discrete group is canonically isomorphic to the group cohomology is called the weak isocohomological property for the group in the case when a group is of type hf i e that has a classifying space with the homotopy type of a polyhedral complex with finitely many cells in each dimension we show that the isocohomological property is geometric and is equivalent to the property that the universal cover of the classifying space has polynomially bounded higher dehn functions if a group is hyperbolic relative to a collection of subgroups each of which is polynomially combable respectively hf and isocohomological then we show that the group itself has these respective properties combining with the results of connes moscovici and dru u sapir we conclude that a group satisfies the strong novikov conjecture if it is hyperbolic relative to subgroups which are of property rd of type hf and isocohomological

morita equivalence of smooth noncommutative tori we show that in the generic case the smooth noncommutative tori associated with two n n real skew symmetric matrices are morita equivalent if and only if the matrices are in the same orbit of the natural so n n z action

local isometries on spaces of continuous functions let x and y be banach spaces and s a subset of the space of linear continuous operators from x to y we say that an operator t belongs locally to s if for every x x there is s s possibly depending on x such that tx sx pointwise should be better than locally but we have followed tradition if each operator that belongs locally to s belongs in fact to s we say that s is algebraically reflexive when y x and s iso x is the group of isometries of x we say that t is a local isometry of x in this paper isometry means linear surjective isometry similarly a local automorphism of a banach algebra is an operator which agrees at every point with some automorphism also we will consider approximate local isometries and automorphisms these are operators having with the following property given x x and e 0 there is an isometry respectively an automorphism s of x such that tx sx e the study of local isometries and automorphisms of banach algebras spurred a considerable interest in recent years see the bibliography of the dissertation 16 in this paper we deal with local isometries and automorphisms of the algebras c0 l as usual we write c0 l or c k 0 l for the banach algebra of all continuous k valued functions on the locally compact space l vanishing at infinity where k is either c or r if l is compact the subscript will be omitted by the banach stone theorem if t is local isometry of c0 l then for each f there are a homeomorphism of l and a continuous unitary u l k such that tf u f

cohomology of flat currents on definable pseudomanifolds abstract we show that the cohomology of flat currents on definable pseudomanifolds in polynomially bounded o minimal structures is isomorphic to intersection cohomology in the top perversity

smooth dense subalgebras of reduced group c algebras schwartz cohomology of groups and cyclic cohomology abstract we introduce the notion of schwartz co homology of a discrete group with a length function which is the natural generalization of the bounded cohomology of a discrete group we prove that if the length function is chosen to be a word length on a group of polynomial growth then its schwartz cohomology is canonically isomorphic to the usual group cohomology with complex coefficients using this result we will show that the group cohomology of a discrete group of polynomial growth is of polynomial growth in the sense of connes and moscovici and that there are no nontrivial idempotents in the group c algebra of a torsion free group of polynomial growth by the method of cyclic cohomology of connes

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

hodge decompositions and dolbeault complexes on normal surfaces give the smooth subset of a normal singular complex projective surface the metric induced from the ambient projective space the l2 cohomology of this incomplete manifold is isomorphic to the surface s intersection cohomology which has a natural hodge decomposition this paper identifies dolbeault complexes whose 0 closed and 0 coclosed forms represent the classes of pure type in the corresponding hodge decomposition of l2 cohomology

realizing the analytic surgery group of higson and roe geometrically part iii higher invariants we construct an isomorphism between the geometric model and higson roe s analytic surgery group reconciling the constructions in the previous papers in the series on realizing the analytic surgery group of higson and roe geometrically with their analytic counterparts following work of lott and wahl we construct a chern character on the geometric model for the surgery group it is a delocalized chern character from which lott s higher delocalized rho invariants can be retrieved following work of piazza and schick we construct a geometric map from stolz positive scalar curvature sequence to the geometric model of higson roe s analytic surgery exact sequence

quantum hall effect on the hyperbolic plane in this paper we study both the continuous model and the discrete model of the quantum hall effect qhe on the hyperbolic plane the hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables we define a twisted analogue of the kasparov map which enables us to use the pairing between k theory and cyclic cohomology theory to identify this geometric invariant with a topological index thereby proving the integrality of the hall conductivity in this case

l2 cohomology of arithmetic varieties abstract the l2 cohomology of arithmetic quotients of bounded symmetric domains is studied we establish the conjecture of zucker equating the l2 cohomology of these spaces to the intersection cohomology of their baily borel compactifications

group cohomology and the cyclic cohomology of crossed products definite results are obtained in the conditions the group g is torsion free and the class h of the extension 0 zh g h n h 0 in h2 nh k is niipotent here gh for h g stands for the centralizer of h the cyclic homology groups hc a g decompose naturally as a direct sum of two subgroups called the homogeneous and the inhomogeneous parts ofhc a g the homogeneous part can be obtained from a spectral sequence converging to it with e2 q hp g hcq a the inhomogeneous part vanishes after inverting s for g z and in the topological situation we thus reobtain the result of nest 17 for general g but a k and trivial action thus for group rings k g we reobtain results of burghelea 4 the free groups satisfy our conditions see 10 for other classes of groups satisfying these conditions in this case our results are compatible with the results of pimsner and voiculescu 18 19 and suggest that for a large class of groups the part of ktor a r d g is obtained from a convergent spectral sequence with e q hp g k here r176 denotes the topological k theory functors see 2 a is a c algebra and a d g the reduced crossed product c algebra is a suitable completion of a g kasparov 13 succeeded to prove this for groups having a special manifold as classifying space we expect that our spectral sequence will give more insight in kasparov s spectral sequence we also mention

on the homotopy invariance of higher signatures for manifolds with boundary if m is a compact oriented manifold with boundary whose fundamental group is virtually nilpotent or gromov hyperbolic we show that the higher signatures of m are oriented homotopy invariants we give applications to the question of when higher signatures of closed manifolds are cut and paste invariant