intersection forms of non spin four manifolds 1 introductionit remains a central question in four manifold topology which quadratic formoccurs as the intersection form of an orientable four manifold in the topologicalcategory there is a beautiful answer the celebrated result of freedman assertsthat all the quadratic forms can be realized however in the smooth category wehave a rather different picture by the surprising result of donaldson d fs a among the de nite quadratic forms only the diagonalizable ones can berealized thereisalsoalongstanding11

the structure of generalized complexity cores structural properties of proper hard cores generalized complexity cores are studied the setting involves a countable class c of sets of strings that is closed under finite union and under finite variation for any set a let ca denote the collection of all c c such that c a any such a has a proper hard core h h a and for every c ca h c is finite the collection of all proper hard cores of a with respect to c modulo finite differences forms a lattice that is isomorphic to one of just three possible lattices a set b is c levelable if for any c ca there exists a c ca such that c c is infinite the structure of the lattice of hard cores of a with respect to c depends only on the relationship between a and c if a is finite or in c then the lattice is the trivial one point lattice if a is infinite not in c and not c levelable then the lattice is isomorphic to the lattice of all subsets of modulo finite subsets where lcub0 1rcub if a is c levelable then the lattice is isomorphic to a special lattice f f

a classification of complexity core lattices abstract lynch 1975 has shown that every recursive set a not in p contains an infinite polynomial complexity core a set of elements c a such that any algorithm deciding a needs more than polynomial time almost everywhere on c actually any a not in p contains infinitely many different cores the collection of which forms a lattice under inclusion we study the structure of this lattice proving that surprisingly there are only three possibilities assuming the lattice is not trivial which happens if a is in p its shape depends only on whether a is almost p immune or not it is known that the natural intractable sets usually do not have this property

a uniform approach to obtain diagonal sets in complexity classes abstract a uniform method for constructing sets which diagonalize over certain complexity classes while preserving other complexity properties is given we obtain some known results as well as some new ones as corollaries of our main theorem the new results concern the complexity classes p np co np pspace apt almost polynomial time r random polynomial time and the polynomial hierarchy

kolmogorov complexity and degrees of tally sets we show that either epm tally epbtt tally or epm tally ep1 tt tally ep2 tt tally ep3 tt tally where erp tally denotes the class of sets which are equivalent to a tally set under rp reductions furthermore the question of whether or not emp tally ebttp tally is equivalent to the question of whether or not ne predicates can be solved in deterministic exponential time the proofs use the techniques of generalized kolmogorov complexity as corollaries to some of the main results we obtain new results about the kolmogorov complexity of sets in p

on the size of classes with weak membership properties abstract it is shown that the following classes have resource bounded measure 0 in e the class of p selective sets the class of p multiselective sets the class of cheatable sets the class of easily countable sets the class of easily approximable sets the class of near testable sets the class of nearly near testable sets the class of locally self reducible sets these are corollaries of a more general result stating that the class of sets that are p isomorphic to p quasi approximable sets has measure 0 in e by considering the recent approach of allender and strauss for measuring in subexponential classes we obtain similar results with respect to p for classes having weak logarithmic time membership properties

padding commitment and self reducibility abstract a new property of np sets called commitability is introduced in this paper roughly a language l in np is commitable if given any instance x and any string y a string z can be found in polynomial time such that z is in l iff y is a prefix of a witness of x in the context of a ptime relation that defines l it is shown that all np sets complete under polynomial time many one reductions are commitable commitability can be seen as a genralization of d self reducibility every d self reducible set is commited and yet commitability retains an important aspect of d self reducibility the tree search method of berman is shown to be extendible to commitable sets therefore certain known results about reductions to co sparse sets and p selective sets generalize from d self reductible sets to commitable sets the connection with paddability is as follows every paddable and commitable set is shown to be d self reducible it is also shown that paddability in np implies kernel constructibility which was introduced elsewhere to capture a notion of self reducibility more general than d self reducibility

on p creative sets and p completely creative sets abstract we study p creative sets and p completely creative sets we first prove that for recursively enumerable sets p creativeness is equivalent to p complete creativeness and myhill s theorem still holds in the polynomial setting we then consider p creativity and p complete creativity for time complexity classes we prove that for p p creativeness is equivalent to p complete creativeness moreover we prove that a set a is p m complete for dext iff a is p creative for p in dext since every p m complete set for dext is p 1 complete berman 1977 we know that myhill s theorem still holds for p in dext these results can also be proved for np in next k creative sets and k completely creative sets in np are next studied k l creative sets and k l completely creative sets in a more general setting are defined and shown to exist it is known that k completely creative sets are np complete joseph and young 1985 but it is not known whether the converse is true we approach this problem based on our double diagonalization technique of showing that every p m complete set for dext is p creative for p a new class of k completely creative sets is constructed as well

a comparison of polynomial time completeness notions ladner lynch and selman 1975 showed the differences among the power of several types of polynomial time reductions for dext c 0dtime 2cn but questions concerning complete degrees w r t these reductions have remained open here we show in dext almost all the possible differences between the completeness notions w r t any pair of the following reductions pt ptt pp n tt pc pd pbtt pk tt pm differences between several types of truth table completenesses we introduce the notion f easy subset for each pr reduction we first prove the property that all pr complete sets for dext have f easy subsets where the class f depends on each reduction type r then the construction of a ps complete set which is not pr complete becomes a simple diagonalization argument these constructions also work for all deterministic superpolynomial complexity classes which have a pm complete set in order to construct a set which is pt complete for dext but not ptt complete for dext we use strings which are not polynomially producible from small sized inputs these strings are said to have large generalized kolmogorov complexity cf hartmanis 1983

resource bounded immunity and simplicity revisiting the 30 years old notions of resource bounded immunity and simplicity we investigate the structural characteristics of various immunity notions strong immunity almost immunity and hyperimmunity as well as their corresponding simplicity notions we also study limited immunity and simplicity called k immunity and feasible k immunity and their simplicity notions finally we propose the k immune hypothesis as a working hypothesis that guarantees the existence of simple sets in np

on one one polynomial time equivalence relations abstract two sets a and b are one one polynomial time equivalent one one p equivalent if there are polynomial time one one reductions from a to b and from b to a in this paper we show that all exptime complete sets are one one p equivalent by length increasing reductions moreover we show this result for many complexity classes which can be proved by a straightforward diagonalization to contain p properly we also obtain some nontrivial examples in exptime p which show the difference between one one and many one polynomial time equivalence relations

on fields of moduli of curves the field of moduli k of a curve x a priori defined over the separable closure k5 of k need not be a field of definition this paper shows that the obstruction is essentially the same as the obstruction to k being a field of definition of the cover x x aut x using previous results of debes douai we then obtain a cohomological measure of the obstruction this yields concrete criteria for the field of moduli to be a field of definition an interesting application is the following local global principle if a curve x together with all of its automorphisms is defined over p for all primes p then it is defined over

wilson s map operations on regular dessins and cyclotomic fields of definition dessins d enfants can be seen as bipartite graphs embedded in compact orientable surfaces 5 according to grothendieck and others a dessin uniquely determines a complex structure on the surface even an algebraic structure as a projective algebraic curve defined over a number field combinatorial properties of the dessin should therefore determine the equations and also structural properties of the curve such as the field of moduli or the field of definition however apart from a few series of examples very few general results concerning such correspondences 10 are known as a step in this direction we present a graph theoretic characterisation of certain quasiplatonic curves defined over cyclotomic fields based on wilson s operations on maps these leave invariant the graph but change the cyclic ordering of edges around the vertices therefore they change the embeddings and hence the dessins and hence the conformal and algebraic structure of the underlying curves under suitable assumptions satisfied by many series of regular 15 dessins these changes coincide with the effect of galois conjugation this coincidence allows one to draw conclusions about galois orbits and fields of definition of dessins the possibilities afforded by these techniques and their limitations are illustrated by a new look at some known examples and a study of dessins based on the regular embeddings of complete graphs 1 the mathematical framework 20 for the last twenty years embeddings of graphs into surfaces have become important for the theory of algebraic curves as an essential tool in the theory of dessins d enfants in this first section we briefly recall the necessary definitions and give an outline of the problems and results presented in this paper one may introduce dessins d enfants as hypermaps on compact oriented 2 manifolds these 25 objects first studied by cori 5 have several equivalent algebraic or topological definitions topologically they are a generalisation of maps whose underlying graphs are replaced with hypergraphs so that the hyperedges are now allowed to be incident with any finite number of hypervertices and hyperfaces one way to visualise this concept is to represent a hypermap d by its walsh map w d a connected bipartite graph b embedded in a compact oriented 30 surface dividing it into simply connected cells 26 in the language of hypermaps the white and black vertices represent the hypervertices and hyperedges of d the edges represent incidences between them and the cells represent the hyperfaces algebraic hypermaps are an alternative and equivalent way to describe dessins these are triples g x y consisting of a permutation group g sn the monodromy or hypercarto 35 graphic group of the dessin acting transitively on the n edges of the walsh map w d a nd generated by two elements x and y the generators x and y describe the anticlockwise cyclic permutations of the edges around their incident white and black vertices we call d a dessin

endomorphism algebras of jacobians very little is known about the existence of curves and families of curves whose jacobians are acted on by large rings of endomorphisms in this paper we show the existence of curves x with an injection k hom jac x jac x z q where k is a subfield of even index at most 10 in a primitive cyclotomic field q p or a subfield of index 2 in q pq or q p this result generalizes previous work of brumer mestre and tautz top and verberkmoes our curves are constructed as branched covers of the projective line and the endomorphisms arise as quotients of double coset algebras of the galois groups of these coverings in certain cases we show the existence of continuous families of jacobians admitting the desired endomorphism algebras at the end we raise some questions about upper bounds for endomorphism algebras of the type described here and ask about a non galois version of hurwitz s theorem

sparse sets in np p exptime versus nexptime this paper investigates the structural properties of sets in np p and shows that the computational difficulty of lower density sets in np depends explicitly on the relations between higher deterministic and nondeterministic time bounded complexity classes the paper exploits the recently discovered upward separation method which shows for example that there exist sparse sets in np p if and only if exptime nexptime in addition the paper uses relativization techniques to determine logical possibilities limitations of these proof techniques and exhibits one of the first natural structural differences between relativized np and conp

genericity randomness and polynomial time approximations polynomial time safe and unsafe approximations for intractable sets were introduced by meyer and paterson technical report tm 126 laboratory for computer science mit cambridge ma 1979 and yesha siam j comput 12 1983 pp 411 425 respectively the question of which sets have optimal safe and unsafe approximations has been investigated extensively duris and rolim lecture notes in comput sci 841 springer verlag berlin new york 1994 pp 38 51 and ambos spies proc 22nd icalp springer verlag berlin new york 1995 pp 384 392 showed that the existence of optimal polynomial time approximations for the safe and unsafe cases is independent using the law of the iterated logarithm for p random sequences which has been recently proven in proc 11th conf computational complexity ieee computer society press piscataway nj 1996 pp 180 189 we extend this observation by showing that both the class of polynomial time delta levelable sets and the class of sets which have optimal polynomial time unsafe approximations have p measure 0 hence typical sets in e in the sense of p measure do not have optimal polynomial time unsafe approximations we will also establish the relationship between resource bounded genericity concepts and the polynomial time safe and unsafe approximation concepts

algebraic curves uniformized by congruence subgroups of triangle groups we construct certain subgroups of hyperbolic triangle groups which we call congruence subgroups these groups include the classical congruence subgroups of sl2 z hecke triangle groups and 19 families of arithmetic triangle groups associated to shimura curves we determine the eld of moduli of the curves associated to these groups and thereby realize the groups psl2 fq and pgl2 fq regularly as galois groups

exponential time and subexponential time sets abstract in this paper we prove that the symmetric difference of a p k parity hard set for e and a subexponential time computable set is still p k parity hard for e this remains true for p m hard set for e since a 1 parity reduction is a many one reduction in addition we show that this property fails to hold for some other types of reductions we introduce and study the notions of e complete kernel and e hard kernel

completeness approximation and density polynomial time approximations to exponential time computable problems exp are considered from the point of view of structure infinitely often speedable and almost everywhere complex problems are studied using the notions of polynomial time productivity and immunity in particular the existence of a polynomial time immune set which is not polynomial time approximable at all but which is polynomial time tt complete in exp is proven the relationship between completeness and approximability is also studied it is shown that being polynomial time m complete in exp does not provide any control of the probability of erroneous approximations

reductions among polynomial isomorphism types abstract a set a is polynomial many one reducible to a set b a is karp reducible to b if there is a polynomially computable function f such that for all x x a iff f x b arbitrary sets a and b are of the same polynomial many one degree if each is polynomial many one reducible to the other a and b are polynomially isomorphic if the function f can be taken one to one onto and polynomially invertible in classical recursive function theory all many one complete sets are recursively isomorphic berman and hartmanis have observed that all known np complete sets are polynomially isomorphic berman and hartmanis have observed that all known np complete sets are polynomially isomorphic and have conjectured that all np complete sets complete under karp reducibility are isomorphic in this paper we show that not just the complete degree but every polynomial many one degree consists either of a single isomorphism type or else contains infinitely many isomorphism types densely ordered under one one size increasing polynomially invertible reductions and also contains infinitely many isomorphisms types which are incomparable under one one invertible reductions in fact we show that every countable partial ordering can be embedded in any such many one degree we also exhibit polynomial degrees which have infinitely many isomorphism types no examples are known of degrees consisting of a single isomorphism type

the density and complexity of polynomial cores for intractable sets let a be a recursive problem not in p lynch has shown that a then has an infinite recursive polynomial complexity core this is a collection c of instances of a such that every algorithm deciding a needs more than polynomial time almost everywhere on c we investigate the complexity of recognizing the instances in such a core and show that every recursive problem a not in p has an infinite core recognizable in subexponential time we further study how dense the core sets for a can be under various assumptions about the structure of a our main results in this direction are that if p np then np complete problems have polynomially nonsparse cores recognizable in subexponential time and that exptime complete problems have cores of exponential density recognizable in exponential time

instanton homology of seifert fibred homology three spheres for an oriented integral homology 3 sphere 2 a casson has introduced an integer invariant a 2 that is defined by using the space 2 of conjugacy classes of irreducible representations of 2 into su 2 see 1 this invariant a 2 can be computed from a surgery or heegard description of 2 and satisfies a 2 ju 2 mod 2 where i 2 is the kervaire milnor rochlin invariant of 2 this powerful new invariant was used to settle an outstanding problem in 3 manifold topology namely showing that if 2 is a homotopy 3 sphere then ju 2 o c taubes 17 utilizing gauge theoretic considerations has reinterpreted casson s invariant in terms of flat connections refining this approach a floer 13 has recently defined another invariant of 2 its instanton homology which takes the form of an abelian group 2 with a natural z8 grading that is an enhancement of a 2 in that

the existence and density of generalized complexity cores if c is a class of sets and a is not in c then an infinite set h is a proper hard core for a with respect to c if h a and for every c e c such that c a c c h is finite it is shown that if c is a countable class of sets of strings that is closed under finite union and finite variation then every infinite set not in c has a proper hard core with respect to c in addition the density of such generalized complexity cores is studied

on one way functions and polynomial time isomorphisms it is shown that if one way functions exist then there are sets a and b such that a and b are equivalent under one one and length increasing polynomial time reductions and such that a is not polynomial time isomorphic to b furthermore sets a and b can be constructed such that they are polynomial time truth table complete for the class of exponential time computable sets

some remarks on witness functions for nonpolynomial and noncomplete sets in np abstract we present two results about witness functions for sets in np and conp first any set that has a polynomially computable function which witnesses that it is not in conp must be at least np hard it follows from this result that any set in np conp that has a polynomially computable function which witnesses this fact must already be complete for np second if b is any set for which there is a polynomially computable function which witnesses that it is not complete for np by witnessing that some fixed set in np is not in p b then b must already be in np conp thus for two sets in np conp there are no polynomially computable functions which witness that one is not polynomially reducible to the other in proving the first result we introduce the notion of a k creative set and prove that all k creative sets are np complete since these sets seem not to be all polynomially isomorphic we counter the conjecture of berman and hartmanis that all np complete sets are isomorphic to sat with our own conjecture that not all k creative sets are isomorphic to sat the proofs we give are recursion theoretic in style but straightforward

on inefficient special cases of np complete problems abstract every intractable set a has a polynomial complexity core a set h such that for any p subset s of a or of s h is finite a complexity core h of a is proper if h a it is shown here that if p np then every currently known i e either invertibly paddable or k creative np complete set a and its complement have proper polynomial complexity cores that are nonsparse and are accepted by deterministic machines in time 2 cn for some constant c turning to the intractable class d ext c 0 d time 2 cn it is shown that every set that is p m complete for d ext has an infinite proper polynomial complexity core that is nonsparse and recursive

a first order isomorphism theorem we show that for most complexity classes of interest all sets complete under first order projections fops are isomorphic under first order isomorphisms that is a very restricted version of the berman hartmanis conjecture holds since natural complete problems seem to stay complete via fops this indicates that up to first order isomorphism there is only one natural complete problem for each nice complexity class

optimal approximations and polynomially levelable sets a set a not in p is polynomially levelable if any algorithm for a has speedup to a polynomial infinitely often in a precisely if given any algorithm m for a and polynomial p it is possible to find another algorithm m for a and polynomial p such that m runs in time p x on infinitely many inputs x in a on which the running time of m exceeds p x intuitively this condition states that among polynomial time computable approximations to a there is no optimal one or one giving correct answers on a maximally large subset of a it appears that most naturally occurring intractable sets are polynomially levelable we prove this for sets not in p that are either paddable self reducible or complete for a deterministic time class we also discuss levelability preserving reductions and give a simple reducibility characterization of nonlevelable sets

the inverse galois problem and rational points on moduli spaces we reduce the regular version of the inverse galois problem for any finite group g to finding one rational point on an infinite sequence of algebraic varieties as a consequence any finite group g is the galois group of an extension l p x with l regular over any pac field p of characteristic zero a special case of this implies that g is a galois group over fp x for almost all primes p

one way functions and the isomorphism conjecture abstract the isomorphism conjecture is investigated for exponential time and other complexity classes if one way functions exist then we show that there are one way functions such that a p f a where a is a standard complete set for np or e or ne if one way functions exist we also show that there are k completely creative sets in np with one way productive functions but which are p isomorphic to standard complete sets we then present a type of one way functions whose existence is equivalent to the failure of the isomorphism conjecture for e finally we show that the isomorphism conjecture holds for e ne if and only if it holds for exp nexp