the units of a ring spectrum and a logarithmic cohomology operation recall that if r is a commutative ring then the set r r of invertible elements of r is naturally an abelian group under multiplication this construction is a functor from commutative rings to abelian groups in general there is no obvious relation between the additive group of a ring r and the multiplicative group of units r however under certain circumstances one can define a homomorphism from a subgroup of r to a suitable completion of r e g the natural logarithm q 0 r or the p adic logarithm 1 pzp zp the logarithmic cohomology operation is a homotopy theoretic analogue of the above where r is a commutative s algebra and completion is bousfield localization with respect to a morava k theory the purpose of this paper is to give a formula for the logarithmic operation in certain contexts in terms of power operations before giving our results we briefly explain some of the concepts involved

witten s invariants of rational homology spheres at prime values of k and trivial connection contribution we establish a relation between the coefficients of asymptotic expansion of the trivial connection contribution to witten s invariant of rational homology spheres and the invariants that t ohtsuki extracted from witten s invariant at prime values ofk we also rederive the properties of primek invariants discovered by h murakami and t ohtsuki we do this by using the bounds on taylor series expansion of the jones polynomial of algebraically split links studied in our previous paper these bounds are enough to prove that ohtsuki s invariants are of finite type the relation between ohtsuki s invariants and trivial connection contribution is verified explicitly for lens spaces and seifert manifolds

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

asymptotic expansions of witten reshetikhin turaev invariants for some simple 3 manifolds the link l this paper investigates the invariant z m 0 when g sk for a simple family of rational homology 3 spheres obtained by integer surgery around 2 n type torus knots in particular we find a closed formula for a formal power series z m eq h in h q 1 from which zf m 0 may be derived for all sufficiently large primes r we show that this formal power series may be viewed as the asymptotic expansion around 4 1 of a multivalued holomorphic function of q with 1 contained on the boundary of its domain of definition for these particular manifolds most of which are not z homology spheres this extends work of ohtsuki and murakami in which the existence of power series with rational coefficients related to z m 0 was demonstrated for rational homology spheres the coefficients in the formal power series z m are expected to be identical to those obtained from a perturbative expansion of the witten chem simons path integral formula for z m 0 0 199 5 american institute of physics

on 4 manifolds with singular torus actions this paper studies the homeomorphism classification problem for closed oriented 4 manifolds which support effective codimension two torus actions with suitable restrictions on the actions a classification is obtained in terms of numerical invariants computable from orbit data these invariants are related to the fundamental groups and the cohomology rings of the underlying manifolds they do not distinguish among all 4 manifolds with actions as above and so the general classification problem remains open

sur les distributions covariantes dans les algebres de lie reductives abstract we study the possibility of factoring a covariant distribution on reductive lie algebras as finite sum of products of an invariant distribution by a covariant polynomial

presentation and central extensions of mapping class groups we give a presentation of the mapping class groupm of a possibly bounded surface considering either all twists or just non separating twists as generators we also study certain central extensions of m one of them plays a key role in studying tqft functors namely the mapping class group of a p1 structure surface we give a presentation of this extension

profinite completions of burnside type quotients of surface groups using quantum representations of mapping class groups we prove that profinite completions of burnside type surface group quotients are not virtually prosolvable in general further we construct infinitely many finite simple characteristic quotients of surface groups

the second homology group of the mapping class group of an orientable surface in i 7 mumford shows that the picard group p ic is isomorphic to h2 f 2 and conjectures the latter is rank one g 3 we prove this below for g 5 another interpretation of this theorem may be obtained by identifying h2 f as bordism classes of fiber bundles f w4 t where t is a closed oriented surface sect 0 when f is closed every such bundle is bordant to f w t a bundle admitting a section s t w the theorem then says that

consistent systems of correlators in non semisimple conformal field theory based on the modular functor associated with a not necessarily semisimple finite non degenerate ribbon category d we present a definition of a consistent system of bulk field correlators for

residue formulas for the large k asymptotics of witten s invariants of seifert manifolds the case of su 2 we derive the largek asymptotics of the surgery formula forsu 2 witten s invariants of general seifert manifolds the contributions of connected components of the moduli space of flat connections are identified the contributions of irreducible connections are presented in the residue form this allows us to express them in terms of intersection numbers on their moduli spaces

quantum invariant for torus link and modular forms we consider an asymptotic expansion of kashaev s invariant or of the colored jones function for the torus link t 2 2m we shall give q series identity related to these invariants and show that the invariant is regarded as a limit of q being n th root of unity of the eichler integral of a modular form of weight 3 2 which is related to the character

computer calculation of witten s 3 manifold invariant witten s 2 1 dimensional chern simons theory is exactly solvable we compute the partition function a topological invariant of 3 manifolds on generalized seifert spaces thus we test the path integral using the theory of 3 manifolds in particular we compare the exact solution with the asymptotic formula predicted by perturbation theory we conclude that this path integral works as advertised and gives an effective topological invariant

quantum invariants modular forms and lattice points ii we study the su 2 witten reshetikhin turaev wrt invariant for the seifert fibered homology spheres with m exceptional fibers we show that the wrt invariant can be written in terms of differential of the eichler integrals of modular forms with weight 1 2 and 3 2 by use of nearly modular property of the eichler integrals we shall obtain asymptotic expansions of the wrt invariant in the large n limit we further reveal that the number of the gauge equivalent classes of flat connections which dominate the asymptotics of the wrt invariant in n is related to the number of integral lattice points inside the m dimensional tetrahedron

sommes de dedekind associees a un corps de nombres totalement reel we define dedekind sums attached to a totally real number field of class number one we prove that they satisfy some reciprocity law then we relate them to special values of hecke l functions we conclude that they are ruled by stark s conjecture

centrally extended mapping class groups from quantum teichmuller theory abstract the central extension of the mapping class groups of punctured surfaces of finite type that arises in quantum teichmuller theory is 12 times the meyer class plus the euler classes of the punctures this is analogous to the result obtained in 12 for the thompson groups

chern simons theory matrix integrals and perturbative three manifold invariants the universal perturbative invariants of rational homology spheres can be extracted from the chern simons partition function by combining perturbative and nonperturbative results we spell out the general procedure to compute these invariants and we work out in detail the case of seifert spaces by extending some previous results of lawrence and rozansky the chern simons partition function with arbitrary simply laced group for these spaces is written in terms of matrix integrals the analysis of the perturbative expansion amounts to the evaluation of averages in a gaussian ensemble of random matrices as a result explicit expressions for the universal perturbative invariants of seifert homology spheres up to order five are presented

chern simons witten invariants of lens spaces and torus bundles and the semiclassical approximation we derive explicit formulas for the chern simons witten invariants of lens spaces and torus bundles overs1 for arbitrary values of the levelk most of our results are for the groupg su 2 though some are for more general compact groups we explicitly exhibit agreement of the limiting values of these formulas ask with the semiclassical approximation predicted by the chern simons path integral

on the quantum invariants for the spherical seifert manifolds we study the witten reshetikhin turaev su 2 invariant for the seifert manifolds s3 gamma where is a finite subgroup of su 2 we show that the wrt invariants can be written in terms of the eichler integral of modular forms with half integral weight and we give an exact asymptotic expansion of the invariants by use of the nearly modular property of the eichler integral we further discuss that those modular forms have a direct connection with the polyhedral group by showing that the invariant polynomials of modular forms satisfy the polyhedral equations associated to

three dimensional quantum gravity chern simons theory and the a polynomial we study three dimensional chern simons theory with complex gauge group sl 2 which has many interesting connections with three dimensional quantum gravity and geometry of hyperbolic 3 manifolds we show that in the presence of a single knotted wilson loop in an infinite dimensional representation of the gauge group the classical and quantum properties of such theory are described by an algebraic curve called the a polynomial of a knot using this approach we find some new and rather surprising relations between the a polynomial the colored jones polynomial and other invariants of hyperbolic 3 manifolds these relations generalize the volume conjecture and the melvin morton rozansky conjecture and suggest an intriguing connection between the sl 2 partition function and the colored jones polynomial

a classification of multiplicity free actions abstract the action of a complex algebraic groupgon an affine varietyvis said to be multiplicity free if each irreducible representation ofgoccurs at most once in the ring c v of regular functions onv this paper concerns the classical setting wherevis a vector space andgacts linearly onv the irreducible linear multiplicity free actions have been classified by v kac the focus here is on nonirreducible linear multiplicity free actions we exhibit many such actions which do not decompose as direct products of irreducible actions our main results provide a classification for all linear multiplicity free actions on a vector space

a unified witten reshetikhin turaev invariant for integral homology spheres we construct an invariant j m of integral homology spheres m with values in a completion widehat mathbb z q of the polynomial ring q such that the evaluation at each root of unity gives the the su 2 witten reshetikhin turaev invariant m of m at thus j m unifies all the su 2 witten reshetikhin turaev invariants of m it also follows that m as a function on behaves like an analytic function defined on the set of roots of unity

simultaneous kummer congruences and mathbb e _ infty e orientations of ko and tmf building on results of ando hopkins and rezk we show the existence of uncountably many mathbb e infty string orientations of real k theory ko and of topological modular forms tmf generalizing the hat a resp the witten genus furthermore the obstruction to lifting an mathbb e infty string orientations from ko to tmf is identified with a classical iwasawa theoretic condition the common key to all these results is a precise understanding of the classical kummer congruences imposed for all primes simultaneously this result is of independent arithmetic interest

witten reshetikhin turaev invariants of seifert manifolds for seifert homology spheres we derive a holomorphic function of k whose value at integer k is the sl2 witten reshetikhin turaev invariant zk at q exp 2 i k this function is expressed as a sum of terms which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the witten chern simons path integral the trivial connection contribution is found to have an asymptotic expansion in powers of k 1 which for k an odd prime power converges k adically to the exact total value of the invariant zk at that root of unity evaluations at rational k are also discussed using similar techniques an expression for the coloured jones polynomial of a torus knot is obtained providing a trivial connection contribution which is an analytic function of the colour this demonstrates that the stationary phase expansion of the chern simons witten theory is exact for seifert manifolds and for torus knots in s3 the possibility of generalising such results is also discussed

invariant generalized functions on sl 2 r with values in a sl 2 r module abstract let g be a finite dimensional real lie algebra let g end v be a representation of g in a finite dimensional real vector space let c v end v s g g be the algebra of end v valued invariant differential operators with constant coefficients on g let u be an open subset of g we consider the problem of determining the space of generalized functions on u with values in v which are locally invariant and such that c v is finite dimensional in this article we consider the case g sl 2 r let n be the nilpotent cone of sl 2 r we prove that when u is sl 2 r invariant then is determined by its restriction to u n where is analytic cf theorem 6 1 in general this is false when u is not sl 2 r invariant and v is not trivial moreover when v is not trivial is not always locally l1 thus this case is different and more complicated than the situation considered by harish chandra amer j math 86 1964 534 publ math 27 1965 5 where g is reductive and v is trivial to solve this problem we find all the locally invariant generalized functions supported in the nilpotent cone n we do this locally in a neighborhood of a nilpotent element z of g cf theorem 4 1 and on an sl 2 r invariant open subset u sl 2 r cf theorem 4 2 finally we also give an application of our main theorem to the superpfaffian superpfaffian prepublication e print math gr 0402067 2004

higher order terms in the melvin morton expansion of the colored jones polynomial we formulate a conjecture about the structure of upper lines in the expansion of the colored jones polynomial of a knot in powers of q 1 the melvin morton conjecture states that the bottom line in this expansion is equal to the inverse alexander polynomial of the knot we conjecture that the upper lines are rational functions whose denominators are powers of the alexander polynomial we prove this conjecture for torus knots and give experimental evidence that it is also true for other types of knots

finiteness results for modular curves of genus at least 2 a curve x over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is modular if it is dominated by x 1 n for some n if in addition the image of its jacobian in j 1 n is contained in the new subvariety of j 1 n then x is called a new modular curve we prove that for each g 2 the set of new modular curves over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i of genus g is finite and computable for the computability result we prove an algorithmic version of the de franchis severi theorem similar finiteness results are proved for new modular curves of bounded gonality for new modular curves whose jacobian is a quotient of j 0 n new with n divisible by a prescribed prime and for modular curves new or not with levels in a restricted set we study new modular hyperelliptic curves in detail in particular we find all new modular curves of genus 2 explicitly and construct what might be the complete list of all new modular hyperelliptic curves of all genera finally we prove that for each field k of characteristic zero and g 2 the set of genus g curves over k dominated by a fermat curve is finite and computable

on the virasoro constraints for torus knots we construct a virasoro algebra of differential operators for the matrix model for torus knots these operators generate various relations between wilson loops then we discuss the operators constructed and corresponding relations in the stability limit we also give a series of examples

topological modular forms with level structure the cohomology theory known as mathrm tmf for topological modular forms is a universal object mapping out to elliptic cohomology theories and its coefficient ring is closely connected to the classical ring of modular forms we extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them along the way we produce a natural way to restrict to the cusps providing multiplicative maps from mathrm tmf with level structure to forms of k theory in particular this allows us to construct a connective spectrum mathrm tmf 0 3 consistent with properties suggested by mahowald and rezk this is accomplished using the machinery of logarithmic structures we construct a presheaf of locally even periodic elliptic cohomology theories equipped with highly structured multiplication on the log etale site of the moduli of elliptic curves evaluating this presheaf on modular curves produces mathrm tmf with level structure

bridged links and tangle presentations of cobordism categories abstract we develop a calculus of surgery data called bridged links which involves besides links also pairs of balls that describe one handle attachments as opposed to the usual link calculi of kirby and others this description uses only elementary local moves namely modifications and isolated cancellations and it is valid also on non simply connected and disconnected manifolds in particular it allows us to give a presentation of a 3 manifold by doing surgery on any other 3 manifold with the same boundary bridged link presentations on unions of handlebodies are used to give a cerf theoretical derivation of presentations of 2 1 dimensional cobordisms categories in terms of planar ribbon tangles and their composition rules as an application we give a different more natural proof of the matveev polyak presentations of the mapping class groups and furthermore find systematically surgery presentations of general mapping tori we discuss a natural extension of the reshetikhin turaev invariant to the calculus of bridged links invariance follows now similar as for knot invariants from simple identifications of the elementary moves with elementary categorical relations for invariances or cointegrals respectively hence we avoid the lengthy computations and the unnatural fenn rourke reduction of the original proofs moreover we are able to start from a much weaker modularity condition which implies the one of turaev generalizations of the presentation to cobordisms of surfaces with boundaries are outlined

on denominators of the kontsevich integral and the universal perturbative invariant of 3 manifolds the integrality of the kontsevich integral and perturbative invariants is discussed it is shown that the denominator of the degree n part of the kontsevich integral of any knot or link is a divisor of 2 3 n 4 n 1 we prove this by establishing the existence of a drinfeld s associator in the space of chord diagrams with special denominators we also show that the denominator of the degree n part of the universal perturbative invariant of homology 3 spheres is not divisible by any prime greater than 2n 1

a simple proof of the fundamental theorem of kirby calculus on links in this paper we relate surgeries on links and heegaard decompositions relate framed links and surface mapping classes and give a simple proof of the fundamental theorem of kirby calculus on links by the presentation of the surface mapping class groups

skein theory and witten reshetikhin turaev invariants of links in lens spaces we study the behavior of the witten reshetikhin turaev su 2 invariants of an arbitrary link in l p q as a function of the level r 2 they are given by times one of p laurent polynomials evaluated at the congruence class of r modulo p determines which polynomial is applicable if p 0 mod 4 the meridian of l p q is non trivial in the skein module but has trivial witten reshetikhin turaev su 2 invariants on the other hand we show that one may recover the element in the kauffman bracket skein module of l p q represented by a link from the collection of the wrt invariants at all levels if p is a prime or twice an odd prime by a more delicate argument this is also shown to be true for p 9

a contribution of the trivial connection to the jones polynomial and witten s invariant of 3d manifolds ii we extend the results of our previous paper 1 from knots to links by using a formula for the jones polynomial of a link derived recently by n reshetikhin we establish a relation between the parameters of this formula and the multivariable alexander polynomial this relation is illustrated by an example of a torus link we check that our expression for the alexander polynomial satisfies some of its basic properties finally we derive a link surgery formula for the loop corrections to the trivial connection contribution to witten s invariant of rational homology spheres

chern simons perturbation theory ii we study the perturbation theory for three dimensional chern simons quantum field theory on a general compact three manifold without boundary we show that after a simple change of variables the action obtained by brs gauge fixing in the lorentz gauge has a superspace formulation the basic properties of the propagator and the feynman rules are written in a precise manner in the language of differential forms using the explicit description of the propagator singularities we prove that the theory is finite finally the anomalous metric dependence of the 2 loop partition function on the riemannian metric which was introduced to define the gauge fixing can be cancelled by a local counterterm as in the 1 loop case in fact the counterterm is equal to the chern simons action of the metric connection normalized precisely as one would expect based on the framing dependence of witten s exact solution invited talk at the xxth conference on differential geometric methods in physics new york june 1991

distributions covariantes sur les espaces hyperboliques resume soient g k p une algebre de lie symetrique reelle de rang 1 k un groupe de lie d algebre de lie k et e une representation complexe irreductible de dimension finie de k l objet de cet article est de montrer que toute distribution k covariante sur p a valeurs dans e est produit d une distribution k invariante et d un polynome k covariant

modular cocycles and linking numbers it is known that the 3 manifold sl 2 z sl 2 r is diffeomorphic to the complement of the trefoil knot in s 3 e ghys showed that the linking number of this trefoil knot with a modular knot is given by the rademacher symbol which is a homogenization of the classical dedekind symbol the dedekind symbol arose historically in the transformation formula of the logarithm of dedekind s eta function under sl 2 z in this paper we give a generalization of the dedekind symbol associated to a fixed modular knot this symbol also arises in the transformation formula of a certain modular function it can be computed in terms of a special value of a certain dirichlet series and satisfies a reciprocity law the homogenization of this symbol which generalizes the rademacher symbol gives the linking number between two distinct symmetric links formed from modular knots

orbifold genera product formulas and power operations we generalize the definition of orbifold elliptic genus and introduce orbifold genera of chromatic level h using h tuples rather than pairs of commuting elements we show that our genera are in fact orbifold invariants and we prove integrality results for them if the genus arises from an h8 map into the morava lubin tate theory eh then we give a formula expressing the orbifold genus of the symmetric powers of a stably almost complex manifold m in terms of the genus of m itself our formula is the p typical analogue of the dijkgraaf moore verlinde verlinde formula for the orbifold elliptic genus r dijkgraaf et al elliptic genera of symmetric products and second quantized strings comm math phys 185 1 1997 197 209 it depends only on h and not on the genus

quantum invariants at the sixth root of unity a general topological formula is given for thesu 2 quantum invariant of a 3 manifoldm at the sixth root of unity it is expressed in terms of the homology witt invariants and signature defects of the various 2 fold covers ofm and thus ties in with basic 4 dimensional invariants a discussion of the range of values of these quantum invariants is included and explicit evaluations are made for lens spaces

on covariant functions and distributions under the action of a compact group abstract let g be a compact subgroup of gl n r acting linearly on a finite dimensional complex vector space e b malgrange has shown that the space c r n e g of c and g covariant functions is a finite module over the ring c r n g of c and g invariant functions first we generalize this result for the schwartz space s r n e g of g covariant functions secondly we prove that any g covariant distribution can be decomposed into a sum of g invariant distributions multiplied with a fixed family of g covariant polynomials this gives a generalization of an oksak result proved in 4

the sigma orientation is an h map in an earlier paper the authors constructed a natural map called the sigma orientation from the thom spectrum mu 6 to any elliptic spectrum mu 6 is an h ring spectrum and in this paper we show that if e c t is the elliptic spectrum associated to the universal deformation of a supersingular elliptic curve over a perfect field of characteristic p 0 then the sigma orientation is a map of h ring spectra

power operations in elliptic cohomology and representations of loop groups the first part describes power operations in elliptic cohomology in terms of isogenies of the underlying elliptic curve the second part discusses a relationship between equivariant elliptic cohomology and representations of loop groups the third part investigates the representation theoretic considerations which give rise to the power operations discussed in the first part

a unified quantum so 3 invariant for rational homology 3 spheres given a rational homology 3 sphere m with h1 m z b and a link l inside m colored by odd numbers we construct a unified invariant im l belonging to a modification of the habiro ring where b is inverted our unified invariant dominates the whole set of the so 3 witten reshetikhin turaev invariants of the pair m l i fb 1a ndl im coincides with habiro s invariant of integral homology 3 spheres for b 1 the unified in variant defined by the third author is determined by im important applica tions are the new ohtsuki series perturbative expansions of im dominating quantum so 3 invariants at roots of unity whose order is not a power of a prime these series are not known to be determined by the lmo invariant