algebraic k theory of group rings and the cyclotomic trace map we prove that the farrell jones assembly map for connective alge braic k theory is rationally injective under mild homological niteness condi tions on the group and assuming that a weak version of the leopoldt schneider conjecture holds for cyclotomic elds this generalizes a result of bokstedt hsiang and madsen and leads to a concrete description of a large direct summand of kn z g z q in terms of group homology in many cases the number theoretic conjectures are true so we obtain rational injectivity results about assembly maps in particular for whitehead groups under niteness assumptions on the group only the proof uses the cyclotomic trace map to topological cyclic homology bokstedt hsiang madsen s functor c and new general injectivity results about the assembly maps for topological hochschild homology and c

using dual approximation algorithms for scheduling problems theoretical and practical results the problem of scheduling a set of n jobs on m identical machines so as to minimize the makespan time is perhaps the most well studied problem in the theory of approximation algorithms for np hard optimization problems in this paper the strongest possible type of result for this problem a polynomial approximation scheme is presented more precisely for each e an algorithm that runs in time o n e 1 e 2 and has relative error at most e is given in addition more practical algorithms for e 1 5 2 k and e 1 6 2 k which have running times o n k log n and o n km 4 log n are presented the techniques of analysis used in proving these results are extremely simple especially in comparison with the baroque weighting techniques used previously the scheme is based on a new approach to constructing approximation algorithms which is called dual approximation algorithms where the aim is to find superoptimal but infeasible solutions and the performance is measured by the degree of infeasibility allowed this notion should find wide applicability in its own right and should be considered for any optimization problem where traditional approximation algorithms have been particularly elusive

reduction of the hall paige conjecture to sporadic simple groups abstract a complete mapping of a group g is a permutation g g such that g g g is also a permutation complete mappings of g are equivalent to transversals of the cayley table of g considered as a latin square in 1953 hall and paige proved that a finite group admits a complete mapping only if its sylow 2 subgroup is trivial or noncyclic they conjectured that this condition is also sufficient we prove that it is sufficient to check the conjecture for the 26 sporadic simple groups and the tits group

the admissibility of sporadic simple groups abstract in 1955 hall and paige conjectured that a finite group is admissible i e admits complete mappings if its sylow 2 subgroup is trivial or noncyclic in a recent paper wilcox proved that any minimal counterexample to this conjecture must be simple and further must be either the tits group or a sporadic simple group in this paper we improve on this result by proving that the fourth janko group is the only possible minimal counterexample to this conjecture john bray reports having proved that this group is also not a counterexample thus completing a proof of the hall paige conjecture

superpotential algebras and manifolds we study a special class of calabi yau algebras in the sense of ginzburg those arising as the fundamental group algebras of acyclic manifolds motivated partly by the usefulness of superpotential descriptions in motivic donaldson thomas theory we investigate the question of whether these algebras admit superpotential presentations we establish that the fundamental group algebras of a wide class of acyclic manifolds including all hyperbolic manifolds do not admit such descriptions disproving a conjecture of ginzburg regarding them we also describe a class of manifolds that do admit such descriptions and discuss a little their motivic donaldson thomas theory finally some links with topological field theory are described

equivariant operads string topology and tate cohomology from an operad fancyscript c with an action of a group g we construct new operads using the homotopy fixed point and orbit spectra these new operads are shown to be equivalent when the generalized g tate cohomology of fancyscript c is trivial applying this theory to the little disk operad fancyscript c 2 which is an s 1 operad we obtain variations on getzler s gravity operad which we show governs the chas sullivan string bracket

batalin vilkovisky algebras and two dimensional topological field theories by a batalin vilkovisky algebra we mean a graded commutative algebraa together with an operator a a 1 such that 2 0 and a a is a graded derivation ofa for alla a in this article we show that there is a natural structure of a batalin vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions we make use of a technique from algebraic topology the theory of operads

loop products and closed geodesics the critical points of the length function on the free loop space m of a compact riemannian manifold m are the closed geodesics on m the length function gives a filtration of the homology of m and we show that the chas sullivan product hi hj hi j n is compatible with this filtration we obtain a very simple expression for the associated graded homology ring grh m when all geodesics are closed or when all geodesics are nondegenerate we also interpret sullivan s coproduct see su1 su2 on c as a product in cohomology h 0 h 0 hi j n 1 0 where 0 m is the constant loop we show that is also compatible with the length filtration and we obtain a similar expression for the ring grh 0 the nonvanishing of products n and n is shown to be determined by the rate at which the morse index grows when a geodesic is iterated we determine the full ring structure h 0 for spheres m s n 3

the structure and complexity of nash equilibria for a selfish routing game in this work we study the combinatorial structure and the computational complexity of nash equilibria for a certain game that models selfish routing over a network consisting of m parallel links we assume a collection of n users each employing a mixed strategy which is a probability distribution over links to control the routing of her own traffic in a nash equilibrium each user selfishly routes her traffic on those links that minimize her expected latency cost given the network congestion caused by the other users the social cost of a nash equilibrium is the expectation over all random choices of the users of the maximum over all links latency through a link we embark on a systematic study of several algorithmic problems related to the computation of nash equilibria for the selfish routing game we consider in a nutshell these problems relate to deciding the existence of a pure nash equilibrium constructing a nash equilibrium constructing the pure nash equilibria of minimum and maximum social cost and computing the social cost of a given mixed nash equilibrium our work provides a comprehensive collection of efficient algorithms hardness results and structural results for these algorithmic problems our results span and contrast a wide range of assumptions on the syntax of the nash equilibria and on the parameters of the system

the price of anarchy for polynomial social cost in this work we consider an interesting variant of the well studied kp model for selfish routing on parallel links which reflects some influence from the much older wardrop model j g wardrop some theoretical aspects of road traffic research proc inst of civil eng part ii 1 1956 325 378 in the new model user traffics are still unsplittable and links are identical social cost is now the expectation of the sum over all links of latency costs each latency cost is modeled as a certain polynomial latency cost function evaluated at the latency incurred by all users choosing the link the resulting social cost is called polynomial social cost or monomial social cost when the latency cost function is a monomial all considered polynomials are of degree d where d 2 and have non negative coefficients we are interested in evaluating nash equilibria in this model and we use the monomial price of anarchy mpoa and the polynomial price of anarchy ppoa as our evaluation measures through establishing some remarkable relations of these costs and measures to some classical combinatorial numbers such as the stirling numbers of the second kind and the bell numbers we obtain a multitude of results for the special case of identical users the fully mixed nash equilibrium where all probabilities are strictly positive maximizes polynomial social cost the mpoa is no more than bd the bell number of order d this immediately implies that the ppoa is no more than 1 t dbt for the special case of two links the mpoa is no more than 2d 2 1 1 n d 1 and this bound is tight for n 2 the mpoa is exactly 2d 1 d d 1 2d 2 d 1 d 1 d d for pure nash equilibria this immediately implies that the ppoa is no more than 2 t d 2t 1 t t 1 2t 2 t 1 t 1 t t

universality of nash equilibria every real algebraic variety is isomorphic to the set of totally mixed nash equilibria of some three person game and also to the set of totally mixed nash equilibria of ann person game in which each player has two pure strategies from the nash tognoli theorem it follows that every compact differentiable manifold can be encoded as the set of totally mixed nash equilibria of some game moreover there exist isolated nash equilibria of arbitrary topological degree

a homotopy theoretic realization of string topology let m be a closed oriented manifold of dimension d let lm be the space of smooth loops in m in 2 chas and sullivan defined a product on the homology h lm of degree d they then investigated other structure that this product induces including a batalin vilkovisky structure and a lie algebra structure on the s1 equivariant homology h s 1 lm these algebraic structures as well as others came under the general heading of the string topology of m in this paper we will describe a realization of the chas sullivan loop product in terms of a ring spectrum structure on the thom spectrum of a certain virtual bundle over the loop space we also show that an operad action on the homology of the loop space discovered by voronov has a homotopy theoretic realization on the level of thom spectra this is the cactus operad defined in 6 which is equivalent to operad of framed disks in this operad action realizes the chas sullivan bv structure on h lm we then describe a cosimplicial model of this ring spectrum and by applying the singular cochain functor to this cosimplicial spectrum we show that this ring structure can be interpreted as the cup product in the hochschild cohomology hh c m c m

on the negative cyclic homology of shc algebras let mathbb k be a field of characteristic p geq 0 and s1 the unit circle we prove that the shc structure on a cochain algebra a da induces an associative product on the negative cyclic homology hc a when the cochain algebra a da is the algebra of normalized cochains of the simply connected topological space x with coefficients in mathbb k then hc a is isomorphic as a graded algebra to h s 1 lx mathbb k the s1 equivariant cohomology algebra of lx the free loop space of x we use the notion of shc formality introduced in topology 41 85 106 2002 to compute the s1 equivariant cohomology algebras of the free loop space of the complex projective space mathbb c p n when n 1 0 p and of the even spheres s2n when p 2

hochschild homology criteria for trivial algebra structures we prove two similar results by quite different methods the first one deals with augmented artinian algebras over a field we characterize the trivial algebra structure on the augmentation ideal in terms of the maximality of the dimensions of the hochschild homology or cyclic homology groups for the second result let x be a 1 connected finite cw complex we characterize the trivial algebra structure on the cohomology algebra of x with coefficients in a fixed field in terms of the maximality of the betti numbers of the free loop space

the homotopy theory of cyclic sets the aim of this note is to show that the homotopy theory of the cyclic sets of connes 3 is equivalent to that of s0 2 spaces i e spaces with a circle action and hence to that of spaces over k z 2

a chern character in cyclic homology we show that inner derivations act trivially on the cyclic cohomol ogy of the normalized cyclic complex w cl 3s c where i2 is a differential graded algebra this is then used to establish the fact that the map introduced in gj defines a chern character in k theory j m bismut recently extended the classical theory of chern characters to the equivariant theory of the free loop space lx he explicitly constructed an s invariant equivariantly closed form a e v on lx whose restriction to x the space of constant loops is ch f tr expv2 closely related to the equivariant theory of the loop space of x is a connes cyclic theory of the differential graded algebra q of differential forms on x more precisely chen s theory of iterated integrals allows us to view cyclic cycles of forms on x as equivariantly closed forms on the loop space lx this is a generalization of the point of view that a one form co on x can be viewed as a function on lx by y fy this suggests the possibility of interpreting the classical theory of chern char acters in the context of cyclic homology i e of extending ch e off the fixed point set x of the circle action on lx by way of a cyclic cycle over q in gjp getzler jones and petrack gave an explicit formula for a cyclic cy cle y e e which via iterated integrals maps to the bismut chern character il e over lx a priori their construction of y e e appears to depend on the choice of idempotent e describing the vector bundle e but in fact it is shown that y e e depends only on the isomorphism class of e in kq x and hence defines the desired chern character this is essentially proved by establishing the triviality of certain group actions on the cyclic homology of a differential graded algebra q these actions are natural generalizations of the usual adjoint action ad g given by conjugation by an invertible element g and its infinitesimal analogue ad given by inner derivation to connes cyclic bar complex of q 1 cyclic homology in this section we review the definition of the cyclic homology of a dga according to c kassel by a mixed complex we mean a triplet fe d b where g7 is a graded algebra and where d w ff x and b w w x are derivations of degrees 1 and 1 respectively and subject to the rela

quantization of the lie bialgebra of string topology let m be a smooth simply connected closed oriented manifold and lm the free loop space of m using a poincare duality model for m we show that the reduced equivariant homology of lm has the structure of a lie bialgebra and we construct a hopf algebra which quantizes the lie bialgebra

entropy and collapsing of compact complex surfaces we study the problem of existence of mathcal f structures in the sense of cheeger and gromov but not necessarily polarized on compact complex surfaces we give a complete classification of compact complex surfaces of kahler type admitting mathcal f structures in the non kahler case we give a complete classification modulo the gap in the classification of surfaces of class vii we then use these results to study the minimal entropy problem for compact complex surfaces we prove that modulo the gap in the classification of surfaces of class vii all compact complex surfaces of kodaira dimension less than or equal to 1 have minimal entropy zero and such a surface admits a smooth metric g with zero topological entropy if and only if it is mathbb c p 2 a ruled surface of genus 0 or 1 a hopf surface a complex torus a kodaira surface a hyperelliptic surface or a kodaira surface modulo a finite group the key result we use to prove this is a new topological obstruction to the existence of metrics with vanishing topological entropy finally we show that these results fit perfectly into wall s study of geometric structures on compact complex surfaces for instance we show that the minimal entropy problem can be solved for a minimal compact kahler surface s of kodaira dimension infty 0 or 1 if and only if s admits a geometric structure modelled on mathbb c p 2 s 2 times s 2 s 2 times mathbb e 2 or mathbb e 4

the hall paige conjecture and synchronization for affine and diagonal groups abstract the hall paige conjecture asserts that a finite group has a complete mapping if and only if its sylow subgroups are not cyclic the conjecture is now proved and one aim of this paper is to document the final step in the proof for the sporadic simple group j 4 we apply this result to prove that primitive permutation groups of simple diagonal type with three or more simple factors in the socle are non synchronizing we also give the simpler proof that for groups of affine type or simple diagonal type with two socle factors synchronization and separation are equivalent synchronization and separation are conditions on permutation groups which are stronger than primitivity but weaker than 2 homogeneity the second of these being stronger than the first empirically it has been found that groups which are synchronizing but not separating are rather rare it follows from our results that such groups must be primitive of almost simple type

dyer lashof operations in the string topology of spheres and projective spaces we compute the 2 primary dyer lashof operations in the string topology of several families of manifolds specifically spheres and a variety of projective spaces these operations while well known in the context of iterated loop spaces give a collection of homotopy invariants of manifolds new to string topology the computations presented here begin an exploration of these invariants

on string topology of classifying spaces let g be a compact lie group by work of chataur and menichi the homology of the space of free loops in the classifying space of g is known to be the value on the circle in a homological conformal field theory this means in particular that it admits operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces here we present a radical extension of this result giving a new construction in which diffeomorphisms are replaced with homotopy equivalences and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs the result is a novel kind of field theory which is related to both the diffeomorphism groups of surfaces and the automorphism groups of free groups with boundaries our work shows that the algebraic structures in string topology of classifying spaces can be brought into line with and in fact far exceed those available in string topology of manifolds for simplicity we restrict to the characteristic 2 case the generalization to arbitrary characteristic will be addressed in a subsequent paper

equivariant cohomology syzygies and orbit structure let x be a nice space with an action of a torus t we consider the atiyah bredon sequence of equivariant cohomology modules arising from the filtration of x by orbit dimension we show that a front piece of this sequence is exact if and only if the h bt module h t x is a certain syzygy moreover we express the cohomology of that sequence as an ext module involving a suitably defined equivariant homology of x one consequence is that the gkm method for computing equivariant cohomology applies to a poincare duality space if and only if the equivariant poincare pairing is perfect

structure and complexity of extreme nash equilibria we study extreme nash equilibria in the context of a selfish routing game specifically we assume a collection of n users each employing a mixed strategy which is a probability distribution over m parallel identical links to control the routing of its own assigned traffic in a nash equilibrium each user selfishly routes its traffic on those links that minimize its expected latency cost the social cost of a nash equilibrium is the expectation over all random choices of the users of the maximum over all links latency through a link we provide substantial evidence for the fully mixed nash equilibrium conjecture which states that the worst nash equilibrium is the fully mixed nash equilibrium where each user chooses each link with positive probability specifically we prove that the fully mixed nash equilibrium conjecture is valid for pure nash equilibria furthermore we show that under a certain condition the social cost of any nash equilibrium is within a factor of 2h 1 e of that of the fully mixed nash equilibrium where h is the factor by which the largest user traffic deviates from the average user traffic considering pure nash equilibria we provide a ptas to approximate the best social cost we give an upper bound on the worst social cost and we show that it is n p hard to approximate the worst social cost within a multiplicative factor better than 2 2 m 1

hochschild homology of structured algebras abstract we give a general method for constructing explicit and natural operations on the hochschild complex of algebras over any prop with a multiplication we think of such algebras as a algebras with extra structure as applications we obtain an integral version of the costello kontsevich soibelman moduli space action on the hochschild complex of open tcfts the tradler zeinalian and kaufmann actions of sullivan diagrams on the hochschild complex of strict frobenius algebras and give applications to string topology in characteristic zero our main tool is a generalization of the hochschild complex

expected length of the longest probe sequence in hash code searching an investigation ts made of the expected value of the maximum number of accesses needed to locate any element m a hashing file under various colhston resoluuon schemes this differs from usual worst case considerations winch for hashmg would be the largest sequence of accesses for the worst possible file asymptotic expressxons of these expected values are found for full and partly full tables for the open addressing scheme with a clustering free model these values are found to be 0 6315 x n for a full table and logan for a partly full table where n ts the number of records m is the size of the table and a n m for the open addressing scheme which reorders the insertions to minimize the worst case the lower bounds in n 1 077 and a in l a are found for full and partly full tables respectively fmauy for the separate chaining or direct chaining method both expected values are found to be f t n these results show that for these schemes the actual behawor of the worst case in hash tables is quite good on the average

a new model for selfish routing in this work we introduce and study a new potentially rich model for selfish routing over non cooperative networks as an interesting hybridization of the two prevailing such models namely the kpmodel e koutsoupias c h papadimitriou worst case equilibria in g meinel s tison eds proceedings of the 16th annual symposium on theoretical aspects of computer science in lecture notes in computer science vol 1563 springer verlag 1999 pp 404 413 and the wmodel j g wardrop some theoretical aspects of road traffic research proceedings of the of the institute of civil engineers 1 pt ii 1952 325 378 in the hybrid model each of nusers is using a mixed strategy to ship its unsplittable traffic over a network consisting of m parallel links in a nash equilibrium no user can unilaterally improve its expected individual cost to evaluate nash equilibria we introduce quadratic social cost as the sum of the expectations of the latencies incurred by the squares of the accumulated traffic this modeling is unlike the kp model where social cost e koutsoupias c h papadimitriou worst case equilibria in g meinel s tison eds proceedings of the 16th annual symposium on theoretical aspects of computer science in lecture notes in computer science vol 1563 springer verlag 1999 pp 404 413 is the expectation of the maximum latency incurred by the accumulated traffic but it is like the w model since the quadratic social cost can be expressed as a weighted sum of expected individual costs we use the quadratic social cost to define quadratic coordination ratio here are our main findings quadratic social cost can be computed in polynomial time this is unlike the p completeness d fotakis s kontogiannis e koutsoupias m mavronicolas p spirakis the structure and complexity of nash equilibria for a selfish routing game in p widmayer f triguero r morales m hennessy s eidenbenz r conejo eds proceedings of the 29th international colloquium on automata languages and programming in lecture notes in computer science vol 2380 springer verlag 2002 pp 123 134 of computing social cost for the kp model for the case of identical users and identical links the fully mixed nash equilibrium m mavronicolas p spirakis the price of selfish routing algorithmica 48 1 2007 91 126 where each user assigns positive probability to every link maximizes quadratic social cost as our main result we present a comprehensive collection of tight constant that is independent of m and n strictly less than 2 lower and upper bounds on the quadratic coordination ratio for several interesting special cases some of the bounds stand in contrast to corresponding super constant bounds on the coordination ratio previously shown in a czumaj b vocking tight bounds for worst case equilibria acm transactions on algorithms 3 1 2007 e koutsoupias m mavronicolas p spirakis approximate equilibria and ball fusion theory of computing systems 36 6 2003 683 693 e koutsoupias c h papadimitriou worst case equilibria in g meinel s tison eds proceedings of the 16th annual symposium on theoretical aspects of computer science in lecture notes in computer science vol 1563 springer verlag 1999 pp 404 413 m mavronicolas p spirakis the price of selfish routing algorithmica 48 1 2007 91 126 for the kp model

a multiplication in cyclic homology we define a multiplication on the cyclic homology of a commutative cocommutative bialgebra h with superproduct in the case when h is a field of characteristic zero the cyclic homology becomes a polynomial algebra in one generator the loday quillen multiplication is trivial in that case in this paper we define an associative multiplication on the cyclic homology of certain commutative algebras h the definition is based on the cartesian product of sets instead of the shuffle product as in the case of lq and ga when h contains q our product is compatible with the hodge decomposition of hc h so it has the form hc n h hc j m h hc n m h the existence of this multiplication in the case h k is classical since it is just given by the tensor product of complex line bundle however its significance came to our attention in joint work with john klein as a consequence of our study of fiberwise morse theory on circle bundles ik and in other joint works with r penner and r hain by doing morse theory on circles john klein and i realized that there was a rationally nontrivial associative multiplication on the cyclic homology of the integral group ring on any cyclic group and that it had important consequences for pseudoisotopy theory this will be explained in a later paper in this paper we show that this multiplication in cyclic homotopy is defined first in the case when the algebra is h km k m where m is the underlying additive group of a ring then in the case when h is a hopf algebra with superproduct in the sense of gaucher ga and finally in the more general case of any algebra example 12 4 in fact we do not need additive inverses in m nor do we need an antipode for h so we can take m to be a monoid and h to be a bialgebra when h k i e m 1 the cyclic homology becomes a divided power algebra which is of course isomorphic to a polynomial algebra in one generator if k is a field of characteristic zero we assume throughout the paper that k is a commutative ring with 1 and that all tensor products are taken over k this includes those tensor products which appear in the definition of cyclic homology we also use connes original definition that a cyclic object in a category c is a covariant functor c where is connes cyclic category since this is the same as a contravariant functor c an outline of the paper is as follows in the first section we give a topological definition of the multiplication in hc km the construction is so elementary that received by the editors april 14 1994 1991 mathematics subject classification primary 18g60 secondary 16w30 this research was supported by nsf grant no dms 90 02512 c 1999 american mathematical society

topological structure of non contractible loop space and closed geodesics on real projective spaces with odd dimensions abstract in this paper we use chas sullivan theory on loop homology and leray serre spectral sequence to investigate the topological structure of the non contractible component of the free loop space on the real projective spaces with odd dimensions then we apply the result to get the resonance identity of non contractible homologically visible prime closed geodesics on such spaces provided the total number of distinct prime closed geodesics is finite

the fixed point theorem in equivariant cohomology in this paper we study the s equivariant de rham cohomology of infinite dimensional s manifolds our main example is the free loop space lx where x is a finite dimensional manifold with the circle acting by rotating loops we construct a new form of equivariant cohomology h which agrees with the usual periodic equivariant cohomology in finite dimensions and we prove a suitable analogue of the classical fixed point theorem which is valid for loop spaces lx this gives a cohomological framework for studying differential forms on loop spaces and we apply these methods to various questions which arise from the work of witten 16 atiyah 2 and bismut 5 in particular we show following atiyah in 2 that the a polynomial of x arises as an equivariant characteristic class in the theory h of the normal bundle to x considered as the space of constant loops in lx

the cyclic homology of crossed product algebras in their article 9 on cyclic homology feigin and tsygan have given a spectral sequence for the cyclic homology of a crossed product algebra generalizing burghelea s calculation 4 of the cyclic homology of a group algebra for an analogous spectral sequence for the hochschild homology of a crossed product algebra see brylinski 2 3 in this article we give a new derivation of this spectral sequence and generalize it to negative and periodic cyclic homology hc a and hp a the method of proof is itself of interest since it involves a natural generalization of the notion of a cyclic module in which the condition that the morphism n n is cyclic of order n 1 is relaxed to the condition that it be invertible we call this category the paracyclic category given a paracyclic module p we can define a chain complex c p with differentials b and b which respectively lower and raise degree the condition that the module p is paracyclic translates to the condition on c p that 1 bb bb is invertible our main result is to show that there is an analogue of the eilenberg zilber theorem for bi paracyclic modules it is then easy to obtain a new expression for the cyclic homology of a crossed product algebra which leads immediately to the spectral sequence of feigin and tsygan if m is a module over a commutative ring k we will denote by m k the iterated tensor product defined by m 0 k and m k 1 m k m if m and n are graded modules we will denote by m n their graded tensor product we would like to thank j block j l brylinski and c kassel for a number of interesting discussions on the results presented here

extending the notion of rationality of selfish agents second order nash equilibria motivated by the increasing interest of the computer science community in the study and understanding of non cooperative systems we present a novel model for formalizing the rational behavior of agents with a more farsighted view of the consequences of their actions this approach yields a framework creating new equilibria which we call second order equilibria starting from a ground set of traditional ones by applying our approach to pure nash equilibria we define the set of second order pure nash equilibria and present their applications to the prisoner s dilemma game to an instance of braess s paradox in the wardrop model and to the kp model with identical machines

the unstable integral homology of the mapping class groups of a surface with boundary we construct a graph complex calculating the integral homology of the bordered mapping class groups we compute the homology of the bordered mapping class groups of the surfaces s 1 1 s 1 2 and s 2 1 using the circle action on this graph complex we build a double complex and a spectral sequence converging to the homology of the unbordered mapping class groups we compute the homology of the punctured mapping class groups associated to the surfaces s 1 1 and s 2 1 finally we use miller s operad to get the first kudo araki and browder operations on our graph complex we also consider an unstable version of the higher kudo araki dyer lashoff operations

cyclic homology and equivariant homology the purpose of this paper is to explore the relationship between the cyclic homology and cohomology theories of connes 9 11 see also loday and quillen 20 and if equivariant homology and cohomology theories here ii is the circle group the most general results involve the definitions of the cyclic homology of cyclic chain complexes and the notions of cyclic and cocyclic spaces so precise statements will be postponed until w 3 in this introduction we explain some of the formal similarities between the cyclic theory and the equivariant theory and give two examples where the general results apply let a be an associative algebra over a commutative ring k then one can form the cyclic homology hc a and cohomology hc a of a these groups have periodicity operators

on the complexity of constrained nash equilibria in graphical games a widely accepted rational behavior for non cooperative players is based on the notion of nash equilibrium although the existence of a nash equilibrium is guaranteed in the mixed framework i e when players select their actions in a randomized manner in many real world applications the existence of any equilibrium is not enough rather it is often desirable to single out equilibria satisfying some additional requirements in order for instance to guarantee a minimum payoff to certain players which we call constrained nash equilibria in this paper a formal framework for specifying these kinds of requirement is introduced and investigated in the context of graphical games where a player p may directly be interested in some of the other players only called the neighbors of p this setting is very useful for modeling large population games where typically each player does not directly depend on all the players and representing her utility function extensively is either inconvenient or infeasible based on this framework the complexity of deciding the existence and of computing constrained equilibria is then investigated in the light of evidencing how the intrinsic difficulty of these tasks is affected by the requirements prescribed at the equilibrium and by the structure of players interactions the analysis is carried out for the setting of mixed strategies as well as for the setting of pure strategies i e when players are forced to deterministically choose the action to perform in particular for this latter case restrictions on players interactions and on constraints are identified that make the computation of nash equilibria an easy problem for which polynomial and highly parallelizable algorithms are presented

on the topological hochschild homology of bu i the purpose of this paper and its sequel is to determine the homotopy groups of the spectrum thh here p is an odd prime is the adams summand of p local connective k theory see for example 25 and thh is the topological hochschild homology construction introduced by bokstedt in 3 in the present paper we will determine the mod p homotopy groups of thh and also the integral homotopy groups of thh l where l denotes the periodic adams summand in the sequel we will investigate the integral homotopy groups of thh using our present results as a starting point the thh construction appears to be of basic importance in algebraic k theory because it combines two useful properties it can be used to construct good approximations to the algebraic k theory functor and it is very accessible to calculation we shall review what is known about the first property in a moment the second property was demonstrated by bokstedt s calculation in his paper 4 of the homotopy groups of thh hz p and thh hz here hz p and hz denote the evident eilenberg mac lane spectra it is natural to ask about thh r for other popular ring spectra r and our work is a first step in this direction we pay special attention to the connective case because this is the case which is likely to be relevant in applications see subsection 1 4 below the calculation which we present in this paper is a homotopy theoretic one which uses the adams spectral sequence hereafter abbreviated ass this calculation has several interesting features in particular it is a pleasing example of an ass calculation in which although there are infinitely many differentials it is still possible to get the complete answer here is a summary of the contents of the paper in section 2 we review the facts we need to know about ordinary hochschild homology in section 3 we do the same

generalized string topology operations we show that the chas sullivan loop product a combination of the pontrjagin product on the fiber and the intersection product on the base makes sense on the total space homology of any fiberwise monoid e over a closed oriented manifold m more generally the thom spectrum e tm is a ring spectrum similarly a fiberwise module over e defines a module over e tm fiberwise monoids include adjoint bundles of principal bundles and the construction is natural with respect to maps of principal bundles this naturality implies homotopy invariance of the algebra structure on h lm arising from the loop product if m bg is the infinite dimensional classifying space of a compact lie group then we get a well defined pro ring spectrum which we define to be the string topology of bg if e has a fiberwise action of the little n cubes operad then e tm is an e n ring spectrum this gives homology operations combining dyer lashof operations on the fiber and the poincare duals of steenrod operations on the base we give several examples where the new operations give homological insight borrowed from knot theory complex geometry gauge theory and homotopy theory

string topology on gorenstein spaces the purpose of this paper is to describe a general and simple setting for defining g p q string operations on a poincare duality space and more generally on a gorenstein space gorenstein spaces include poincare duality spaces as well as classifying spaces or homotopy quotients of connected lie groups our presentation implies directly the homotopy invariance of each g p q string operation as well as it leads to explicit computations

equivariant localization and completion in cyclic homology and derived loop spaces abstract we prove an equivariant localization theorem over an algebraically closed field of characteristic zero for smooth quotient stacks by reductive groups x g in the setting of derived loop spaces as well as hochschild homology and its cyclic variants we show that the derived loop spaces of the stack x g and its classical z fixed point stack 0 x z g z become equivalent after completion along a semisimple parameter z g g implying the analogous statement for hochschild and cyclic homology of the dg category of perfect complexes perf x g we then prove an analogue of the atiyah segal completion theorem in the setting of periodic cyclic homology where the completion of the periodic cyclic homology of perf x g at the identity e g g is identified with a 2 periodic version of the derived de rham cohomology of x g together these results identify the completed periodic cyclic homology of a stack x g over a parameter z g g with the 2 periodic derived de rham cohomology of its z fixed points

decompositions into spanning rainbow structures a subgraph of an edge coloured graph is called rainbow if all its edges have distinct colours the study of rainbow subgraphs goes back more than two hundred years to the work of euler on latin squares and has been the focus of extensive research ever since euler posed a problem equivalent to finding properly n edge coloured complete bipartite graphs kn n which can be decomposed into rainbow perfect matchings while there are proper edge colourings of kn n without even a single rainbow perfect matching the theme of this paper is to show that with some very weak additional constraints one can find many disjoint rainbow perfect matchings in particular we prove that if some fraction of the colour classes have at most 1 o 1 n edges then one can nearly decompose the edges of kn n into edge disjoint perfect rainbow matchings as an application of this we establish in a very strong form a conjecture of akbari and alipour and asymptotically prove a conjecture of barat and nagy both these conjectures concern rainbow perfect matchings in edge colourings of kn n with quadratically many colours using our techniques we also prove a number of results on near decompositions of graphs into other rainbow structures like hamiltonian cycles and spanning trees most notably we prove that any properly coloured complete graph can be nearly decomposed into spanning rainbow trees this asymptotically proves the brualdi hollingsworth and kaneko kano suzuki conjectures which predict that a perfect decomposition should exist under the same assumptions