the determination of the lie algebra associated to the lower central series of a group in this paper we determine the lie algebra associated to the lower central series of a finitely presented group in the case where the defining relators satisfy certain independence conditions other central series such as the lower p central series are treated as well 1 statement of results the main purpose of this paper is to determine the lie algebra associated to the lower central series of a group thus extending the results of 6 7 to groups defined by more than one relator the methods apply to other central series such as the tower p central series the lower central series of a group g is the sequence of subgroups g 1 n 1 defined inductively by g g gi1 g g where g gn denotes the subgroup of g generated by the commutators x y xy 1xy with x e g y c gn the associated graded abelian group gr g dn 1grn g where grn g gn gn 1 has the structure of a graded lie algebra over the ring z of integers the bracket operation in gr g being induced by the commutator operation in g cf 2 9 11 12 the construction of the lie algebra gr g uses only the fact that gn is a sequence of subgroups of g with the following properties i g1 g ii gn 1 c gn iii gn9gkl c gn k such a family of subgroups of g is called a central filtration of g let f be the free group on the n letters x19 xn and let f be the lower central series of f if i is the image of xi in gr1 f f f f then the lie algebra received by the editors may 10 1983 invited paper presented at the special session in combinatorial group theory at the 803rd meeting of the ams in new york city april 14 15 1983 1980 mathematics subject classification primary 20f14 20f40 secondary 20e18

orientability of fredholm families and topological degree for orientable nonlinear fredholm mappings abstract we construct a degree theory for oriented fredholm mappings of index zero between open subsets of banach spaces and between banach manifolds our approach is based on the orientation of fredholm mappings it does not use fredholm structures on the domain and target spaces we provide a computable formula for the change in degree through an admissible homotopy that is necessary for applications to global bifurcation the notion of orientation enables us to establish rather precise relationships between our degree and many other degree theories for particular classes of fredholm maps including the elworthy tromba degree which have appeared in the literature in a seemingly unrelated manner

cohomologically generic 2 complexes and 3 dimensional poincare complexes this is part of a series of papers devoted to topological applications of deformation theory imp mp2 especially in low dimensional topology bp mp1 our lowdimensional program consists in fixing accessible homological invariants like the betti numbers or the cup product structure of the cohomology and then apply the machinery of deformation theory in order to both compute and distinguish more interesting invariants such as the graded lie algebra associated to the fundamental group gr l itself the entire homotopy type here we address the problem of computing gr ttl s a difficult task in general in terms of h s z the forthcoming bp and imp1 deal with the other above mentioned invariants leading in particular to invariants associated to links and 3 manifolds via deformation theory one may associate to any connected space s assuming h s z to be free of finite rank a graded lie algebra l defined as the quotient of the free lie algebra on has by the ideal generated by the image of the dual of the cup product h sah s h2s see sect 1 our main result reads

golod shafarevich algebras free subalgebras and noetherian images abstract it is shown that golod shafarevich algebras of a reduced number of defining relations contain noncommutative free subalgebras in two generators and that these algebras can be homomorphically mapped onto prime noetherian algebras with linear growth it is also shown that golod shafarevich algebras of a reduced number of relations cannot be nil

non commutative graded algebras and their hilbert series abstract this paper begins to develop a theory of non commutative graded algebras and their hilbert series which parallels the extensive already existing theory on commutative graded algebras we begin with a discussion of order ideals of monomials which give us vector space bases for these algebras we develop next the concept of weak summand which is a special relationship a subalgebra may have to an algebra and the concept of strongly free sets strongly free sets have many properties which are analogous to the properties of regular sequences in commutative algebras we discuss combinatorially free sets which are useful for constructing examples of many of the preceding ideas we conclude with an application of our results to algebraic topology by showing that there is a finite cw complex with only nine positive dimensional cells whose loop space has an irrational poincare series

pure virtual braids resonance and formality we investigate the resonance varieties lower central series ranks and chen ranks of the pure virtual braid groups and their upper triangular subgroups as an application we give a complete answer to the 1 formality question for this class of groups in the process we explore various connections between the alexander type invariants of a finitely generated group and several of the graded lie algebras associated to it and discuss possible extensions of the resonance chen ranks formula in this context

periodic free resolutions from twisted matrix factorizations abstract the notion of a matrix factorization was introduced by eisenbud in the commutative case in his study of bounded periodic free resolutions over complete intersections since then matrix factorizations have appeared in a number of applications in this work we extend the notion of homogeneous matrix factorizations to regular normal elements of connected graded algebras over a field next we relate the category of twisted matrix factorizations to an element over a ring and certain zhang twists we also show that in the setting of a quotient of a ring of finite global dimension by a normal regular element every sufficiently high syzygy module is the cokernel of some twisted matrix factorization furthermore we show that in the noetherian as regular setting there is an equivalence of categories between the homotopy category of twisted matrix factorizations and the singularity category of the hypersurface following work of orlov

experimental studies of separation on a two dimensional airfoil at low reynolds numbers the laminar separation transition and turbulent reattachment near the leading edge of a two dimensional naca 663 018 airfoil were investigated using a low speed smoke visualization wind tunnel lift and drag force measurements were made using an external strain gage balance for a chord reynolds number range of 40 goo400 000 an extensive flow visualization study was performed and correlated with the force measurements experiments were also conducted with distributed surface roughness at the leading edge and external acoustic excitation to influence the development of the airfoil boundary layer this study delineates the effects of angle of attack and chord reynolds number on the separation characteristics and airfoil performance nomenclature c model chord cd section profile drag coefficient uncorrected cf section lift coefficient uncorrected cp pressure coefficient acoustic frequency hz r reattachment location rc reynolds number based on chord length u civ s separation location t location of approximate end of transition freestream velocity x c nondimensional distance along chord a angle of attack v kinematic viscosity

optimal 5 step nilpotent quadratic algebras by the golod shafarevich theorem an associative algebra r given by n generators and d n23 homogeneous quadratic relations is not 5 step nilpotent we prove that this estimate is optimal namely we show that for every positive integer n there is an algebra r given by n generators and n23 homogeneous quadratic relations such that r is 5 step nilpotent

separated shear layer development on an airfoil at low reynolds numbers flow transition in the separated shear layer on the upper surface of a naca 0025 airfoil at low reynolds numbers was investigated the study involved wind tunnel experiments and linear stability analysis detailed measurements were conducted for reynolds numbers of 100 000 and 150 000 at 0 5 and 10 degree angles of attack for all cases examined laminar boundary layer separation takes place on the upper surface of the airfoil the separated shear layer fails to reattach to the airfoil surface for the lower reynolds number but reattachment occurs for the higher reynolds number despite this difference in flow development experimental results show that a similar transition mechanism is attendant for both reynolds number flow regimes flow transition occurs due to the amplification of natural disturbances in the separated shear layer within a band of frequencies centered at some fundamental frequency the initial growth of disturbances centered at the fundamental frequency is followed by the growth of a subharmonic component eventually leading to flow transition the growing disturbances also cause shear layer roll up and the formation of roll up vortices the results show that inviscid stability theory can be employed to adequately estimate such salient characteristics as the frequency of the most amplified disturbances and their propagation speed this implies that the roll up vortices can be attributed to inviscid instability however the results suggest that viscous and nonparallel effects need to be accounted for to effectively model the convective growth of the disturbances in the separated shear layer

nonlinear fredholm operators with noncompact fibers and applications to elliptic problems on rn under suitable conditions an equation f x y between banach spaces involving a nonlinear fredholm mapping f of nonnegative index is shown to have a noncompact and hence infinite set of solutions for almost every y for which the equation is solvable the proof of this nonuniqueness but not existence result relies on an entirely new line of arguments in which the concept of generalized critical value plays a central role when f w2 p rn lp rn is associated with a quasilinear elliptic pde on rn with constant coefficient it often happens that a crucial denseness hypothesis in the abstract theorem is equivalent to the existence of a nontrivial solution to the equation f x f 0 and hence can be verified in practice generalizations exist for some classes of problems with nonconstant coefficients and for problems on exterior domains

superlinear convolution equations in rn abstract we give existence results for convolution equations u g u v on r n based on degree theory for fredholm mappings of index 0 existence is obtained in orlicz sobolev spaces w 1 l a adapted to the nonlinearity g and by a limiting process and under weaker conditions in l loc 1 the case of r n volterra equations is also treated and applications to pdes of the form p d u q d g u f are discussed

mild pro p groups and galois groups of p extensions of ℚ in this paper we introduce a new class of finitely presented pro pgroups g of cohomological dimension 2 called mild groups if d g r g are respectively the minimal number of generators and relations of g we give an infinite family of mild groups g with r g d g and d g 2 arbitrary these groups can be constructed with g g g finite answering a question of kuzmin if g gs p is the galois group of the maximal p extension of q unramified outside a finite set of primes s and p 6 2 we show that g is mild for a co final class of sets s even in the case p s

higher massey products in the cohomology of mild pro p groups abstract translating results due to j labute into group cohomological language a schmidt proved that a finitely presented pro p group g is mild and hence of cohomological dimension cd g 2 if h 1 g f p u v as f p vector space and the cup product h 1 g f p h 1 g f p h 2 g f p maps u v surjectively onto h 2 g f p and is identically zero on v v this has led to important results in the study of p extensions of number fields with restricted ramification in particular in the case of tame ramification in this paper we extend labute s theory of mild pro p groups with respect to weighted zassenhaus filtrations and prove a generalization of the above result for higher massey products which allows to construct mild pro p groups with defining relations of arbitrary degree we apply these results to one relator pro p groups and obtain some new evidence of an open question due to serre

the entropy of graded algebras we define the entropy of a graded algebra a n an by limsupn diman this is related to the notion of entropy in symbolic dynamics and could serve as a natural dimension concept for algebras with exponential growth we study the entropy of quotients and subalgebras of free associative algebras and free lie algebras we also study the behavior of the entropy function under free products and obtain several characterizations of free algebras in terms of entropy

zeroth hochschild homology of preprojective algebras over the integers we determine the z module structure of the preprojective algebra and its zeroth hochschild homology for any non dynkin quiver and hence the structure working over any base commutative ring of any characteristic this answers and generalizes a conjecture of hesselholt and rains producing new p torsion classes in degrees 2pl l 1 we relate these classes by p th power maps and interpret them in terms of the kernel of verschiebung maps from noncommutative witt theory an important tool is a generalization of the diamond lemma to modules over commutative rings which we give in the appendix

ascoli s theorem for functions vanishing at infinity and selected applications abstract we give a new form of the ascoli theorem for functions on r n tending to some given closed subset z of a complete metric space e at infinity for instance when e is a normed space and z 0 the usual uniform decay requirement is replaced by the assumption that the 0 function is the only continuous function produced by some limiting process this formulation which has significant practical value in concrete applications is described in its general form but with emphasis on the case when z is totally disconnected variants in sobolev spaces and the properness of nonlinear ordinary differential operators are discussed

on monomial algebras of finite global dimension let g be an associative monomial k algebra if g is assumed to be finitely presented then either g contains a free subalgebra on two monomials or else g has polynomial growth if instead g is assumed to have finite global dimension then either g contains a free subalgebra or else g has a finite presentation and polynomial growth also a graded hopf algebra with generators in degree one and relations in degree two contains a free hopf subalgebra if the number of relations is small enough

loop space homology of spaces of small category only little is known concerning h omegax k the loop space homology of a finite cw complex x with coefficients in a field k a space x is called an r cone if there exists a filtration x0 subset of x1 subset of subset of x r x such that x i has the homotopy type of the cofibre of a map from a wedge of sphere into x i 1 denote by a x the sub hopf algebra image of h omegax1 we prove then that for a graded r cone r less than or equal to 3 there exists an isomorphism a x x t u congruent to h omegax

graph products of spheres associative graded algebras and hilbert series given a finite simple vertex weighted graph we construct a graded associative noncommutative algebra whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges we show that the hilbert series of this algebra is the inverse of the clique polynomial of the graph using this result it easy to recognize if the ideal is inert from which strong results on the algebra follow noncommutative grobner bases play an important role in our proof there is an interesting application to toric topology this algebra arises naturally from a partial product of spheres which is a special case of a generalized moment angle complex we apply our result to the loop space homology of this space

inert sets and the lie algebra associated to a group abstract independently j labute trans amer math soc 288 1985 51 57 has given a sufficient condition for the lie algebra associated with a discrete group s lower central series to have a certain presentation and s halperin and j m lemaire math scand in press have discussed the conditions under which some homogeneous elements in a graded algebra constitute an inert set as it turns out these conditions essentially coincide as a consequence various theorems about graded algebras may now be translated into results into group theory a nontrivial application to link groups is offered

coherent algebras and noncommutative projective lines abstract a well known conjecture says that every one relator group is coherent we state and partly prove a similar statement for graded associative algebras in particular we show that every gorenstein algebra a of global dimension 2 is graded coherent this allows us to define a noncommutative analogue of the projective line p 1 as a noncommutative scheme based on the coherent noncommutative spectrum qgra of such an algebra a that is the category of coherent a modules modulo the torsion ones this category is always abelian ext finite hereditary with serre duality like the category of coherent sheaves on p 1 in this way we obtain a sequence p n 1 n 2 of pairwise non isomorphic noncommutative schemes which generalize the scheme p 1 p 2 1

laminar separation bubble characteristics on an airfoil at low reynolds numbers an experimental investigation was conducted in order to document the structure and behavior of laminar separation bubbles at low reynolds numbers data of this type are necessary if the currently insufficient analytical and numerical models are to be improved the laminar separation bubble that forms on a naca 663 018 airfoil model was surveyed at chord reynolds numbers of 50 000 200 000 at angles of attack of 8 12 deg the effects of the various testing conditions on the separation bubble were isolated and the data were analyzed in relation to existing separation bubble correlations in order to test their low reynolds number applicability this analysis indicated that the chord reynolds number and the disturbance environment strongly influence the experimental pressure distributions these effects must be included in any analytic prediction technique applied to the low reynolds number flight regime

a generalization of castelnuovo mumford regularity for representations of noncommutative algebras abstract we introduce and generalize the notion of castelnuovo mumford regularity for representations of noncommutative algebras effectively establishing a measure of complexity for such objects the grobner shirshov basis theory for modules over noncommutative algebras is developed by which a noncommutative analogue of schreyer s theorem is proved for computing syzygies by a repeated application of this theorem we construct free resolutions for representations of noncommutative algebras some interesting examples are included in which graded free resolutions and regularities are computed for representations of various algebras in particular using the bernstein gelfand gelfand resolutions for integrable highest weight modules over kac moody algebras we compute the projective dimensions and regularities explicitly for the cases of finite type and affine type a n 1

on the homology of associative algebras we present a new free resolution for k as an g module where g is an associative augmented algebra over a field k the resolution reflects the combinatorial properties of g introduction let k be a field and let g be an associative augmented k algebra for many purposes one wishes to have a projective resolution of k as a g module the bar resolution is always easy to define but it is often too large to use in practice at the other extreme minimal resolutions may exist but they are often hard to write down in a way that is amenable to calculations the main theorem of this paper presents a compromise resolution though rarely minimal it is small enough to offer some bounds but explicit enough to facilitate calculations as it relies heavily upon combinatorial constructions it is best suited for analyzing otherwise tricky algebras given via generators and relations since several results we get as consequences of the main theorem have been obtained before through other means this paper may be viewed as generalizing and unifying several seemingly unrelated ideas in particular we are generalizing priddy s results on koszul algebras 12 extending homology computations by govorov 9 and backelin 3 and complementing bergman s methods regarding the diamond lemma 6 three results may be of interest the homology of the mod p steenrod algebra is given in terms of the homology of a new chain complex smaller than the a algebra in theorem 3 5 formula 16 offers an efficient algorithm for the determination of hilbert series and theorem 4 2 asserts the existence of new bounds on the torsion groups of commutative graded rings 1 definitions and the main theorem throughout this paper k denotes any field and g is an associative k algebra with unity the field k embeds in g via 77 k g and we suppose that g has an augmentation i e a k algebra map s g k for which 77 is a right inverse s denotes a set of generators for g as a k algebra and k s is the free associative k algebra with unity on s there is a canonical surjection f k s g once s is chosen and the augmentation e is determined once we know s x for each x e s in particular this means that k s may be augmented such that f becomes a map of augmented algebras to s we associate a function e s z called a grading in the absence of a more compelling choice we often take e to be grading by length i e e x 1 for received by the editors may 23 1983 and in revised form february 22 1984 this paper was the subject of an invited one hour address in boulder colorado during the week long ams summer program on combinatorics and algebra june 1983 1980 mathematics subject classification 1985 revision primary 16a62 secondary 13d03 55s10 1986 american mathematical society 0002 9947 86 1 00 25 per page

ancestors graph and an upper bound for the subword complexity function a general upper bound for the subword complexity function of a fixed point of a primitive substitution is presented the approach relies on the introduction and discussion of the ancestors graph of a substitution more precisely we characterize the language of a fixed point of a primitive substitution as the strongly connected component of the letters in its ancestors graph we propose algorithms for the local construction of this graph we give an upper bound for the subword complexity function in terms of the hilbert series of some quotient of a noncommutative polynomial ring by a sum of automatic bilateral ideals

alexander invariants of complex hyperplane arrangements leta be an arrangement of n complex hyperplanes the funda mental group of the complement of a is determined by a braid monodromy homomorphism fi fs pn using the gassner representation of the pure braid group we flnd an explicit presentation for the alexander invariant ofa from this presentation we obtain combinatorial lower bounds for the ranks of the chen groups ofa we also provide a combinatorial criterion for when these lower bounds are attained