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Fix a prime p p , and let A A be the polynomial part of the dual Steenrod algebra. The Frobenius map on A A induces the Steenrod operation P ~ 0 \widetilde {\mathscr {P}}^{0} on cohomology, and in this paper, we investigate this operation. We point out that if p = 2 p=2 , then for any element in the cohomology of A A , if one applies P ~ 0 \widetilde {\mathscr {P}}^{0} enough times, the resulting element is nilpotent. We conjecture that the same is true at odd primes, and that “enough times” should be “once.” The bulk of the paper is a study of some quotients of A A in which the Frobenius is an isomorphism of order n n . We show that these quotients are dual to group algebras, the resulting groups are torsion-free, and hence every element in Ext over these quotients is nilpotent. We also try to relate these results to the questions about P ~ 0 \widetilde {\mathscr {P}}^{0} . The dual complete Steenrod algebra makes an appearance.

Let G be a group and let R be a commutative ring. Let T rP icR(RG) be the group of isomorphism classes of standard self-equivalences of the derived category of bounded complexes of RG-modules. The subgroup HDR(G) of T rP icR(RG) consisting of self-equivalences xing the trivial RG-module acts on the cohomology ring H (G; R). The action is functorial with respect to R. The self-equivalences which are 'splendid' in a sense dened by J. Rickard act natural with respect to transfer and restriction to centralizers of p-subgroups in case R is a eld of characteristic p. In the present paper we prove that this action of self-equivalences on H (G; R) commutes with the action of the Steenrod algebra and study the behaviour of the action of splendid self-equivalences with respect to Lannes' T -functor. Introduction In an earlier joint paper (9) with RaphaRouquier I dened the group T rP icR(A) of standard self-equivalences of a derived module category D b (A) for an R-algebra A which is projective as an R-module. For any A-module M let HDM(A) be the subgroup of T rP icR(A) which is formed by the self-equivalences mapping M to an isomorphic copy. Then, in an earlier paper (11) I showed that, under some hypothesis on M, the group HDM(A) acts in a natural way on the Ext-algebra Ext A (M; M). In case of A being a group algebra RG, with R being a eld of characteristic p and G being a nite group, then J. Rickard denes in (8) what is a splendid equivalence by some technical conditions basically by asking that the homogeneous components of a tilting complex are p-permutation modules induced from diagonal p-subgroups, and by some invertibility condition in the homotopy category. These splendid equivalences induce self-equivalences of the derived categories of centralizers of p-subgroups by the Brauer construction. In (12) I showed that then, for M = R being the trivial module, the action of those splendid equivalences commute with transfer and restriction from and to the cohomology rings of centralizers of p-subgroups. In the present paper we enlarge these properties still further. The action of self-equivalences of the bounded derived category D b (RG) on H (G; R) commutes with the action of the Steenrod algebra on H (G; R) for any prime p. As consequence of the above statements, any cohomology ring H (G; Fp) denes a functor H (G; Fp) p HD p (B0( p G)) from the modules over the group of derived self-equivalences of the principal block of the group ring Fp HD p (FpG) xing the trivial module to the category of unstable modules Up and similarly in the opposite direction. Obviously, we may restrict to splendid self-equivalences. We shall describe the composition of Lannes' T -functor with the rst functor and the im- age of free unstable modules by the second. Moreover, by a result of Lannes TV (H (G; Fp)) decomposes into direct product of cohomology rings as unstable modules. We shall prove that this decomposition is also a decomposition as modules over the action of splendid self- equivalences. This will give evidence that splendid equivalences are the correct objects to study in this context. The paper is organized as follows. Section 1 recalls the necessary denitions and properties of Even's norm map as it is used here and the denition of the Steenrod operation. In Section 2 it is shown that the normalized part of the outer automorphism group of the

We study jet cohomology of isolated hypersurface singularities defined by partial differential forms and prove formulas to compute jet cohomology groups by linear algebra. We use jet sheaves of finite order (or equivalently infinitesimal neighbourhood of finite order) and related exterior differential forms to define cohomologies for singularities of mappings and varieties. (See [2], [3],[4], [5],[6],[7] and this paper.) They are new invariances. They were first defined and studied in [2] for smooth functions. In this paper we study jet cohomology HP(QVk. 0) defined by partial differential forms with respect to y for isolated hypersurface singularities.They can be computed by linear algebra (See Theorem 1 and Theorem 2.) and can distinguish singularities whose classical cohomological invariances and Milnor numbers coincide. For example, if X is a quasi-homogeneous hypersurface with isolated singularity 0. QX , is the complex defined by ordinary exterior differential forms. HP(Q> o) = 0, p > 1, and H0(Q0o) = C. (See [1].) But the cohomology groups defined by jets can tell. For example: (1) f = x 5 + x4. Milnor number ,u = 12. If the order of jets k = 3, dimcH0(Q;3-.0) = 6. (2) f = x3 + x7. ,a = 12. If k = 3, dimcHo (Q'V3 .,0) = 7. It is interesting that the cohomological groups defined by jets are determined by higher derivatives of f. (See Theorem 1 and Theorem 2.) All results and proofs in this paper are true not only for complex holomorphic cases but also for real analytic cases. The author would like to express his gratitude to Phillip A. Griffiths for enlightening discussions with him and various supports he and the Institute for Advanced Study offered during the time the author visited IAS. The main results of this paper are as follows. Let F= _~ 1 ak)If | ka! i9xa (

In this paper we introduce a new elliptic complex on an odd-dimensional manifold with a self-dual line field. The notion of a self-dual line field is a generalization of the notion of a conformal line field. Ellipticity, Fredholm properties and Hodge decompositions of these new complexes are proved both in the case of a closed manifold and in the case of a manifold with boundary. The cohomology groups of these elliptic complexes are computed in some cases. In addition, in this paper, we generalize the notion of an anti-self-dual connection on a smooth 4-manifold to a 3-manifold with a line field and a smooth 5-manifold with a line field. The above new elliptic complexes can be twisted by anti-self-dual connections in dimensions 3 and 5, but only by flat connections in dimensions above 5. This reveals a special feature of dimensions 3 and 5.

Let A A be a regular local ring of positive characteristic. This paper is concerned with the local cohomology modules of A A itself, but with respect to an arbitrary ideal of A A . The results include that all the Bass numbers of all such local cohomology modules are finite, that each such local cohomology module has finite set of associated prime ideals, and that, whenever such a local cohomology module is Artinian, then it must be injective. (This last result had been proved earlier by Hartshorne and Speiser under the additional assumptions that A A is complete and contains its residue field which is perfect.) The paper ends with some low-dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic 0 0 are true.

This is an old paper put here for archeological purposes. We compute the second cohomology of current Lie algebras of the form $L\otimes A$, where $L$ belongs to some class of Lie algebras which includes classical simple and Zassenhaus algebras, and of some modular semisimple Lie algebras. The results are largely superseded by subsequent papers, though, perhaps, some tricks and observations used here remain of minor interest., v3: fixed a typo

In this paper we study differential forms satisfying a Dirac type equation and taking values in a Clifford algebra. For them we establish a Cauchy representation formula and we compute winding numbers for pairs of nonintersecting cycles in R m {\mathbb {R}^m} as residues of special differential forms. Next we prove that the cohomology spaces for the complex of monogenic differential forms split as direct sums of de Rham cohomology spaces. We also study duals of spaces of monogenic differential forms, leading to a general residue theory in Euclidean space. Our theory includes the one established in our paper [11] and is strongly related to certain differential forms introduced by Habetha in [4].

We study a spectral sequence converging to the cohomology of the configuration space of n ordered points in a manifold. A chain complex is constructed with homology equal to the E2 term. If the field is the rationals and the manifold is formal then the spectral sequence is shown to collapse. The results are applied to compute the Anderson spectral sequence converging to the cohomology of a function space. 0. INTRODUCTION From the point of view of homotopy theory, a very natural problem is to describe the topology of function spaces. In 1956, R. Thom [13] wrote a seminal paper. In 1972, D. Anderson [1], using the notion of a cosimplicial space introduced by Bousfield and Kan [4], obtained a spectral sequence for the homology of function spaces. In 1969-1970, Gelfand and Fuks [6] wrote a series of papers on the continuous cohomology of the Lie algebra of vector fields on a manifold M. They introduced a pair of spectral sequences for such a study. Based on the results of Gelfand-Fuks Bott and Fuks conjectured that the continuous cohomology of the Lie algebra of vector fields was the usual cohomology of a certain function space, namelyS that of the space of sections of a certain associated bundle to the tangent bundle of M. Haefliger [7] and Bott and Segal [3] in 1978> using entirely different methods, proved the Bott-Fuks conjecture. The method followed by Bott and Segal was to utilize the ideas of Anderson. The cohomology of "configuration spaces' appears in the computation of the El-term of the above spectral sequences. In 1978 F. Cohen and L. Taylor [5] were able to compute the cohomology of the configuration spaces for a certain class of manifolds and applied these results to describe the Gelfand-Fuks cohomology of these manifolds. Also, there has been interest in the configuration spaces as they appear as approximations to certain function spaces (see [2, 9]). In this paper we study the configuration spaces and the function spaces as outlined by Bott-Segal. In both instances the cohomology can be obtained from a Received by the editors October 3, 1987 and in revise(l form, July 1, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 54C35.

Let X be the flag variety of a compact Lie group and let h* be a complex-oriented generalized cohomology theory. We introduce operators on h*(X) which generalize operators introduced by Bernstein, Gel'fand, and Gel'fand for rational cohomology and by Demazure for K-theory. Using the Becker-Gottlieb transfer, we give a formula for these operators, which enables us to prove that they satisfy braid relations only for the two classical cases, thereby giving a topological interpretation of a theorem proved by the authors and extended by Gutkin. One of the central issues in Lie theory is the geometry of the flag variety associated to a compact Lie group G. An important problem concerning the flag variety is the Schubert calculus, which studies the ring structure of the cohomology of the flag variety. Work initiated by Borel, Bott and Kostant, which culminated in a paper by Bernstein, Gel'fand and Gel'fand [BGG], gave a complete solution to the problem. Demazure studied the same problem for K-theory. Moreover, these techniques have been generalized to the Kac-Moody situation by Kac-Peterson, Kostant-Kumar, and others. This work has focussed on algebro-geometric properties of the flag variety. Here, on the other hand we study the flag variety from the point of view of algebraic topology. As a consequence, not only do we recover the classical results described above, but we extend these results to a certain class of cohomology theories-those which are complex-oriented. Examples of complex-oriented theories include ordinary cohomology, K-theory, complex cobordism, and elliptic cohomology. Since the context we have chosen in very general, the proofs are universal and are often simpler than the classical arguments. In the work of BGG, a crucial role is played by operators Ai associated to each simple reflection si of the Weyl group of G (defined by Demazure in K-theory). These operators Ai satisfy the braid relations, which are the relations between pairs of simple reflections. In this paper, we generalize the A, to give operators D, acting on h*(G/T) for any complex-oriented theory h*. We prove that braid relations are satisfied only for cohomology theories with the formal group law of rational cohomology or of K-theory (Theorem Received by the editors June 21, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 55N20, 57T15. The second author was supported by an NSF graduate fellowship. (3 1990 American Mathematical Society 0002-9947/90 $1.00 + $.25 perpage

In this paper, a type of algebra is introduced and studied from a rational homotopy point of view, using differential graded Lie algebras. The main aim of the paper is to establish whether or not such an algebra is the rational cohomology algebra of a unique rational homotopy type of spaces. That is, in the language of rational homotopy, whether or not such an algebra is intrinsically formal. Examples are given which show that, in general, this is not so-7.8 and 7.9. However, whilst it is true that not all such algebras are intrinsically formal, some of them are. The main results of this paper show a certain class of these algebras to be intrinsically formal-Theorem 2 (6.1); and a second, different type of algebra also to be intrinsically formal-Theorem 1 (5.2), which type of algebra overlaps with the first type in many examples of interest. Examples are given in ?7. 1. PRELIMINARIES, NOTATION AND AN ACKNOWLEDGMENT In [D-Gr-Mo-Su] it is proven that compact Kahler manifolds are formal spaces. From the rational homotopy point of view, this means that compact Kahler manifolds are particularly interesting as examples. This paper is concerned with a more algebraic approach than that of [D-Gr-Mo-Su]. The cohomological properties of compact Kahler manifolds are abstracted out-see 3.1-and algebras satisfying these properties are studied, with a particular view to deciding whether or not such algebras are intrinsically formal-see 2.1. The main results of this paper are Theorems 1 and 2-5.2 and 6.1; many examples are also given in ?7. Also in ?7, examples are given of algebras that are not intrinsically formal, and yet satisfy the properties referred to above-see 7.8, 7.9. Rather than give a step by step breakdown of the paper here, I have put one or two sentences heading each section. These section headings can be read in order, at this stage, so as to give an overview of the contents of the paper. For the fundamentals of rational homotopy theory, the basic references are, in alphabetical order, [Bou-Gu, Gr-Mo, HI, Q, Su,, and Su2]. This paper will, throughout, only be concerned with the "simply connected with rational homology of finite type" case. For references which give all the results needed for this Received by the editors July 29, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 55P62, 55P15; Secondary 14M10, 53C55.

First cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts, including Wiles's famous paper (1995). Internal to group theory, 1-cohomology plays a role in the general theory of maximal subgroups of finite groups, as developed by Aschbacher and Scott (1985).One can pass to the case where the group acts faithfully and the underlying module is absolutely irreducible. In this case, R. Guralnick (1986) conjectured that there is a universal constant bounding all of the dimensions of these cohomology groups. This paper provides the first general positive results on this conjecture, proving that the generic 1-cohomology H 1 gen (G,L) := lim H 1 (G(q), L) (see Cline, Parshall, Scott, and van der Kallen) (1977) of a finite group G(q) of Lie type, with absolutely irreducible coefficients L (in the defining characteristic of G), is bounded by a constant depending only on the root system. In all cases, we are able to improve this result to a bound on H 1 (G(q),L) itself, still depending only on the root system. The generic H 1 result, and related results for Ext 1 , emerge here as a consequence of a general study, of interest in its own right, of the homological properties of certain rational modules Δ red (λ),∇ red (λ), indexed by dominant weights λ, for a reductive group G. The modules Δ red (λ) and ∇ red (λ) arise naturally from irreducible representations of the quantum enveloping algebra U ζ (of the same type as G) at a pth root of unity, where p > 0 is the characteristic of the defining field for G. Finally, we apply our Ext 1 -bounds, results of Bendel, Nakano, and Pillen (2006), as well as results of Sin (1993), (1992), (1994) on the Ree and Suzuki groups to obtain the (non-generic) bounds on H 1 (G(q),L).

A decade ago, J. F. Carlson proved that there are finitely many cohomology rings of finite 2 2 -groups of fixed coclass, and he conjectured that this result ought to be true for odd primes. In this paper, we prove the non-twisted case of Carlson’s conjecture for any prime and we show how to proceed in the twisted case.

In order to analyze the singularities of the solutions of certain partial differential equations, Hormander, in his paper on Fourier integral operators, extends the method of stationary phase by introducing the class of nondegenerate phase functions. Each phase function, in turn, defines a lagrangian submanifold of the cotangent bundle of the manifold which is the domain of the corresponding differential operator. Given a lagrangian submanifold of a cotangent bundle, when is it globally defined by a nondegenerate phase function? A necessary and sufficient condition is here found to be the vanishing of two topological obstructions; one in the cohomology and the other in the k-theory of the given lagrangian submanifold. The object of this paper is to establish necessary and sufficient conditions for the existence and uniqueness of a phase function which globally defines a given immersed lagrangian submanifold A of the cotangent space T*M of a smooth manifold M. For example, the condition for existence is the vanishing of two obstructions, one in H'(A; R) and one in KO'(A). The condition for uniqueness generalizes those given locally by Hormander and Weinstein. We also establish a covering homotopy property for lagrangian immersions defined by phase functions. Recall that T*M admits a natural sympletic structure: Q =-dwM, where WM is the canonical 1-form on T*M. A smooth immersion A: A-T*M is called lagrangian if A*Q = 0, that is, if X*wM is closed. Recall also Hormander's definition of a nondegenerate phase function: A smooth function o: M x RN R is called a nondegenerate phase function if the smooth map (x, a) > da4(x, a) of M X RN to the fiber RN is nonsingular on the stationary set Y. = {(x, a)Ida4(x, a) = 0). Somewhat more generally, we will say that 4 is nondegenerate on a collection of components Y.' of Y. if this condition holds on Y.'. Y.' is then a smooth submanifold of M X RN of dimension n = dim M, and AX: 2' > T*M defined by AX,(x, a) = (x, d 4(x, a)) is a lagrangian immersion. We say that X: A -> T*M is defined by 4 if XA = X Et It is shown in [Wl, 6.2] that A: A-T*M extends to a symplectic Received by the editors November 8, 1977. AMS (MOS) subject classifications (1970). Primary 58G15, 58F05; Secondary 55B15. ? 1979 American Mathematical Society 0002-9947/79/0000-025 8/$03.50 213 This content downloaded from 207.46.13.11 on Tue, 18 Oct 2016 05:43:10 UTC All use subject to http://about.jstor.org/terms

This paper proves two equivariant generalizations of the Conner- Floyd isomorphism relating unitary cobordism and K-theory. It extends a previous result of Okonek for abelian groups to all compact Lie groups. We also show that the result for finite groups is true using either the geometric or homotopical versions of cobordism. 1. Introduction. In (6) Conner and Floyd established a relation between cobordism and .rf-theory. They proved that MU*(X)®MU.K* = k*(X), where MU is unitary cobordism and K is complex A"-theory. A generalization of this result to the equivariant context was proved by Okonek (15). This paper improves upon that generalization in two ways. First, it expands the class of groups considered, from abelian groups to all compact Lie groups. Second, it shows that both of the usual generalizations of cobordism to the equivariant context can be used. The .rf-theory that we use is KG(—), the equivariant K-theory of Segal (17), which is the obvious generalization of K-theory to the equivariant world. Atiyah (2) proved that this theory has Bott periodicity; it is periodic with respect to Spinc-representations of the group. There are two common ways of generalizing cobordism; we will define these carefully in §2. One gives geometric cobordism, which we denote by UG( — ), and the other homotopical cobordism, MUG(—). Both of these are multiplicative cohomology theories. There are multiplicative maps p: U*G(-) -> K*G(-) and p: MUG(-) -* KG(-). Writing U*G for the ring UG(*), and similarly for the other theories, the two main results we will show are THEOREM A. p: UG(X) ®w KG —► KG(X) is an isomorphism when G is a finite group. THEOREM B. p: MUG(X) ®mu* Kg ~* KG(X) is an isomorphism for every compact Lie group G. It should be pointed out that these results use integer grading and not RO(G) grading. We will use integer grading throughout this paper, except on those occa- sions when it is necessary or particularly convenient to use RO(G); we will make it clear at these points that we are using the larger grading. Theorem B has already been shown for G abelian in (15), and we will use that result as a starting point. Theorem A follows from Theorem B, as shown in §3, by

Geometric interpretations of Zeeman's filtrations of the homology and cohomology of a triangulable space are given, using an analysis of his spectral sequence for Poincare duality. The failure of a space to satisfy Poincare duality is reflected by a topologically invariant filtration of its homology groups. In some sense, the filtration of a homology class measures the "degree of freedom" of its cycles. This filtration was first studied by Zeeman, who defined it using a spectral sequence for Poincare duality. (This spectral sequence was also discovered independently by Cartan and by Fary.) Zeeman started a geometric investigation of the spectral sequence for spaces such as singular algebraic varieties, for which it reveals a wealth of information by relating their local and global homology [30]. I was introduced to Zeeman's spectral sequence by Sullivan (cf. [25, p. 202] and [26]). In this paper I show that the Zeeman filtration of a homology class a in a triangulable space is equal to the largest integer q such that a is represented by a cycle in the complement of any closed subspace of dimension less than q (Theorem 8.3). If S is a stratification of the space X, then the homology class a of X has filtration > q if and only if a is represented by a cycle whose intersection with each stratum S of S has codimension > q in S (Theorem 8.4). This new geometric interpretation of the Zeeman filtration has been generalized recently by M. Goresky and R. MacPherson to define "intersection homology theories" for stratified spaces [9]. Zeeman also defined a filtration of the cohomology of a space. Here I prove his conjecture that the filtration of a cohomology class ,8 of a triangulable space is the smallest integer q such that /8 is represented by a cocycle whose support has dimension q (Theorem 8.8). (An incomplete proof of Zeeman's conjecture is given in [7].) In another paper [21] I will show that there is a simple relation between the Zeeman filtration and the Deligne weight filtration of the rational cohomology of a complex projective variety. Received by the editors October, 31, 1977. AMS (MOS) subject classifications (1970). Primary 55C05; Secondary 55H99, 57C99, 55B10.

The purpose of this paper is to compare the cohomological dimension of a group, relative to finite modules, with the cohomological dimension, in the usual sense, of its profinite completion. The basic tool used to perform this comparison is certain stable cohomology of the group. The reason is that there exists a spectral sequence which relates the continuous cohomology of the profinite completion, with coefficients in this stable cohomology, to the ordinary cohomology of the group. Moreover, the direct method of connecting the cohomology of the group with the profinite cohomology of its completion arises from the edge effects on the base of this spectral sequence. The problem of comparing dimensions considered here was proposed by Gruenberg [2, ?8.12, p. 177], to whom the author wishes to express his thanks. This paper is divided into two-parts. ? 1 deals with the stable cohomology of a group, and contains the results of qualitative nature. In particular, the spectral sequence mentioned above is included in this section, as well as its connection with the comparison method used by Serre in [4, Chapitre 1, Exercises du ?2.6, p. 1-15]. ?2 is essentially quantitative. Its heart is the comparison theorem, which is preceded by some preparatory results on dimensions of the three cohomologies involved in the question. 1. Stable cohomology. Throughout this paper, G is an arbitrary group, and d denotes the profinite group associated with G [4, Chapitre I, Exemple 3, p. I-l], that is, G = proj limu G/U, where the limit is taken over all normal subgroups U of finite index in G, with respect to the quotient morphisms of the identity map of G. Let h: G -+ d be the group morphism defined by the projections of G onto its finite quotients G/U; explicitly, h(s) = (sU)u, for all s E G. Recall that (6,h) can be regarded as the separate completion of G, Received by the editors September 19, 1974. AMS (MOS) subject classifications (1970). Primary 18H10, 20J05.

This paper is divided into two parts. The first part proves a number of general theorems on the cohomology of the classifying spaces of compact Lie groups. These theorems are proved by transfer methods, relying heavily on the double coset theorem [F1J. Several of these results are well known while others are quite new. For the most part the proofs of the theorems are independent of each other and are quite short. Nevertheless they are true in great generality. Several are proven for arbitrary compact Lie groups and arbitrary cohomology theories. Perhaps the most interesting of the new results relates the cohomology of the classifying space of an arbitrary compact Lie group with that of the normalizer of a maximal torus. The second part of the paper generalizes many theorems to certain equivariant cohomology theories. Some of these theorems appear in [F2]. I. Let h be any general cohomology theory in the sense of [Dy]. As is well known, a map p: X -* Y induces a homomorphism p*: h(Y) -* h(X). In certain situations a transfer homomorphism t*: h(X) -> h(Y) exists also [BG], [DO]. In this paper G is an arbitrary compact Lie group. H and K are closed subgroups of G. T is a maximal torus of G with normalizer N. We shall refer often to [FO] for notation. If p(H, G): BH -* BG is the natural projection of classifying spaces, a transfer homomorphism exists, T(H, G): h(BH) -* h(BG) [FO, II.5]. (Technically T(H, G) is only defined for finite-dimensional approximations of p(H, G) (Definition I.6) but this restriction should be overcome in the near future. Until this restriction is removed all the theorems quoted in this paper will technically only be proven for the finite-dimensional approximations of p(H, G).2) h5 is stable cohomotopy. This theory, represented by the sphere spectrum, acts on all others via the smash product. We use two properties of the transfer to prove various theorems relating h(BG) and h(BH) for arbitrary cohomology theories. The following is the most famous property of the transfer. We state it in a slightly different form than usual to facilitate its applications. PROPERTY I.1. Let B be a paracompact and locally compact connected space. Let ': Y -> B be a locally trivial fibration with fiber F a compact manifold. Then a transfer exists t*: h( Y) -* h(B) and t* o 7T* is multiplication by the universal fixed point index Received by the editors July 12, 1978 and, in revised form, March 6, 1980. 1980 Mathematics Subject Classification. Primary 55R40; Secondary 55N20, 57S15. 'Research was partially supported by a grant from NSF. 2Peter May has informed me that Gaunce Lewis has accomplished this. I also understand that M. Prieto's thesis extends the definition of the transfer correctly. O 1981 American Mathematical Society 0002-9947/8 1 /0000-0103/$03.50 49 This content downloaded from 157.55.39.215 on Wed, 31 Aug 2016 04:53:29 UTC All use subject to http://about.jstor.org/terms

Let g be a finite dimensional, complex, semisimple Lie algebra and let V be a finite dimensional, irreducible g-module. By computing a certain Lie algebra cohomology we show that the generalized versions of the weak and the strong Bemstein-Gelfand-Gelfand resolutions of V obtained by H. Garland and J. Lepowsky are identical. Let G be a real, connected, semisimple Lie group with finite center. As an application of the equivalence of the generalized Bernstein-Gelfand-Gelfand resolutions we obtain a complex in terms of the degenerate principal series of G, which has the same cohomology as the de Rham complex. Introduction. Let g be a finite dimensional, complex, semisimple Lie algebra and let V be a finite dimensional, irreducible g-module. In [2, Theorem 9.9], I. N. Bernstein, I. M. Gelfand and S. I. Gelfand constructed a resolution of V by certain a-modules with Verma composition series. In the same paper another resolution of V is constructed which resolves V by direct sums of Verma modules, improving Theorem 9.9. (Cf. [2, Theorem 10.1'].) In their work, Bernstein, Gelfand and Gelfand study systematically a certain category of g-modules known as the category 0. Both the weak and the strong resolutions were generalized by H. Garland and J. Lepowsky in the papers [9] and [14] where generalized Verma modules play the same role as Verma modules in [2]. We will refer to these resolutions as the generalized weak BGG resolution and the generalized strong BGG resolution. In this paper we prove that the two generalized BGG resolutions are identical. Our main theorem, Theorem 9.3 shows that each g-module in the generalized weak BGG resolution splits into a direct sum of generalized Verma modules. Consequently each such g-module is isomorphic to the g-module at the corresponding level in the generalized strong BGG resolution. One of our methods consists in studying a certain category of a-modules and later using such category as a framework in order to obtain a key lemma, Lemma 9.1. We would like to point out that, in the light of Yoneda's interpretation of cohomology, Lemma 9.1 implies vanishing theorems on Lie algebra cohomology. (See the remark following the proof of Lemma 9.1.) Received by the editors November 21, 1978 and, in revised form, May 21, 1979. 1980 Mathematics Subject Classification. Primary 22E47, 17B 10, 17B35, 22E46. ? 1980 American Mathematical Society 0002-9947/80/0000-0551 /$09.00

The main purpose of this paper is to describe how a torus group may act on a space, X X , whose rational cohomology ring is isomorphic to that of a product of three odd-dimensional spheres, in such a way that the fixed point set is nonempty, and X X is not totally nonhomologous to zero in the associated X X -bundle, X T → B T {X_T} \to {B_T} . In the first section of the paper some general results on the cohomology theory of torus actions are established. In the second section the cohomology theory of the above type of action is described; and in the third section the results of the first two sections are used to prove a Golber formula for such actions, which, under certain conditions, bears an interesting interpretation in terms of rational homotopy.

A system of axioms for the Stiefel-Whitney classes of certain type of singular spaces is established. The main examples of these singular spaces are Euler manifolds mod 2 and homology manifolds mod 2. As a consequence, it is shown that on homology manifolds mod 2 the generalized Stiefel conjecture holds. The purpose of this paper is to set up a system of axioms which describe in a unique way the Stiefel-Whitney (S.W.) homology classes of some class of singular spaces, called in this paper allowable class. The main examples include Euler manifolds mod 2 and homology manifolds mod 2. This axiomatic characterization of S.W. homology classes then gives as corollaries affirmative answers to generalized Stiefel conjectures. The classical Stiefel conjecture says that on a smooth manifold Mn, the Poincare dual of the ith cohomology S.W. class is represented by the cycle mod 2 which is equal to the sum (mod 2) of all (n i)-simplices in the first barycentric subdivision of some triangulation of M [21]. In the generalized Stiefel conjecture we deal with Steenrod squares of Wu classes instead of ordinary cohomology S.W. classes. The question about a possible axiomatic description of S.W. classes for Euler manifolds or for more general spaces was posed by Blanton and Schweitzer [4]. Our corollary on homology manifolds mod 2 was also obtained by Taylor [19], but from a completely different viewpoint. The main tools in proving that our axioms determine a unique class are block bundle transversality and a description of cohomology classes as morphisms on bordism groups. These techniques then provide the "transversality classes" T(99) of an embedding 9p of our singular space X into the interior of a PL manifold M. T(99) lies in H*(M, aM; Z2) and determines characteristics (mod 2) of transversal intersections of X with singular manifolds in M. These ideas are based on the work of Latour [13]. In ?1 we give some basic facts about Euler manifolds mod 2 and their S.W. homology classes, which then motivates ??2 and 3 where we introduce allowable classes of spaces and introduce axioms for their S.W. classes. In ?4 we prove that the axioms determine unique classes and derive the above-mentioned consequences. Received by the editors July 10, 1980 and, in revised form, April 26, 1982. 1980 Mathematics Subject Classification. Primary 57P05, 55N40, 55R40; Secondary 55R60, 57R20.

In this paper we shall apply the cohomology groups constructed in [14] to a variety of problems in analysis. We show that cohomology classes admit direct integral decompositions, and we obtain as a special case a new proof of the existence of direct integral decompositions of unitary representations. This also leads to a Frobenius reciprocity theorem for induced modules, and we obtain splitting theorems for direct integrals of tori analogous to known results for direct sums. We also obtain implementation theorems for groups of automorphisms of von Neumann algebras. We show that the splitting group topology on the two-dimensional cohomology groups agrees with other naturally defined topologies and we find conditions under which this topology is T 2 {T_2} . Finally we resolve several questions left open concerning splitting groups in a previous paper [13].

In his paper, G. Shimura quotes a proposition [2, Proposition 2.5] which is derived directly from a transformation formula of theta functions due to KrazerPrym and suggests proving this proposition by means of a generalized Poisson summation formula obtained by A. Weil [4, Theoreme 4]. The purpose of this paper is to execute his idea. Let us explain briefly the result. Let V be a 2n dimensional real vector space and let D be a discrete subgroup of rank 2n in V. Suppose that we have a nondegenerate alternate bilinear form A on V x V which assumes integral values on D x D. The form A represents an integral cohomology class on the torus T= VID. A complex vector space structure J on V induces a complex structure on the torus, which may be denoted by the same notation. Such a complex structure J is said to be admissible if there is a positive divisor on the complex torus (T, J) whose cohomology class is A. Making use of the theory of theta functions on the complex vector space (V, J), we assign to each admissible complex structure J on T a holomorphic map 0(J) of the complex torus (T, J) into a complex projective space P of a certain dimension depending only on the form A, in such a way that the cohomology class A is a rational multiple of the image of the cohomology class of hyperplane sections on P under the cohomology homomorphism 0(J)*. Consider the group S(D) of real linear transformations of V which leave the bilinear form A and the subgroup D invariant. If a E S(D), then a induces an isomorphism a of T onto itself. If J is an admissible complex structure on T, there is a unique admissible complex structure Ja on T such that a: (T, Ja) -(T, J) is a holomorphic isomorphism of these two complex tori. Now, the result asserts that if we make a suitable choice of the assignment 0(J) to each admissible complex structure J and if a belongs to a congruence subgroup in S(D) of sufficiently high level with respect to an appropriate base of D, then the following diagram is commutative

Let π : X → S \pi :X \to S be a flat proper morphism of analytic spaces. π \pi may be thought of as providing a family of compact analytic spaces, X s {X_s} , parametrized by the space S S . Let F \mathcal {F} be a coherent sheaf on X X flat over S S . F \mathcal {F} may be thought of as a family of coherent sheaves, F s {\mathcal {F}_s} , on the family of spaces X s {X_s} . Let o ∈ S o \in S be a fixed point, ξ o ∈ H q ( X o , F o ) {\xi _o} \in Hq({X_o},{\mathcal {F}_o}) . In this paper, we consider the problem of extending ξ o {\xi _o} to a cohomology class ξ ∈ H q ( π − 1 ( U ) , F ) \xi \in Hq({\pi ^{ - 1}}(U),\mathcal {F}) where U U is some neighborhood of o o in S S . Extension problems of this type were first considered by P. A. Griffiths who obtained some results in the case in which the morphism π \pi is simple and the sheaf F \mathcal {F} is locally free. We obtain generalizations of these results without the restrictions. Among the applications of these results is a necessary and sufficient condition for the existence of a space of moduli for a compact manifold. This application was discussed in an earlier paper by the author. We use the Grauert “direct image” theorem, the theory of Stein compacta, and a generalization of a result of M. Artin on solutions of analytic equations to reduce the problem to an algebraic problem. In §2 we discuss obstructions to deforming ξ o {\xi _o} ; in §3 we show that if no obstructions exist, ξ o {\xi _o} may be extended; in §4 we give a useful criterion for no obstructions; and in §5 we discuss some examples.

Introduction. The main purpose of this paper is to draw attention to certain functors, exactly analogous to the functors "Tor" and "Ext" of Cartan-Eilenberg [2], but applicable to a module theory that is relativized with respect to a given subring of the basic ring of operators. In particular, we shall show how certain relative cohomology theories for groups, rings, and Lie algebras can be subsumed under the theory of the relative Ext functor, just as (in [2]) the ordinary cohomology theories have been subsumed under the theory of the ordinary Ext functor. Among the various relative cohomology groups that have been considered so far, some can be expressed in terms of the ordinary Ext functor; these have been studied systematically within the framework of general homological algebra by M. Auslander (to appear). A typical feature of these relative groups is that they appear naturally as terms of exact sequences whose other terms are the ordinary cohomology groups of the algebraic system in question, and of its given subsystem. There is, however, another type of relative cohomology theory whose groups are not so intimately linked to the ordinary cohomology groups and exhibit a more individualized behaviour. Specifically, the relative cohomology groups for Lie algebras, as defined (in [3]) by Chevalley and Eilenberg, and the relative cohomology groups of groups, defined and investigated by I. T. Adamson [1], are of this second type. It is these more genuinely "relative" cohomology theories that fall in our present framework of relative homological algebra. Our plan here is to sketch the general features of the relative Tor and Ext functors (?2) and to illustrate some of their possible uses or interpretations by a selection of unelaborated examples. Thus, ?3 illustrates the use of the relative Ext functor in extending the cohomology theory for algebras. ?4 deals with relative homology and relative cohomology of groups, and involves both the relative Tor functor and the relative Ext functor. ?5 discusses the role played by the relative Ext functor in the cohomology theory for Lie algebras. Since this paper is intended to serve as a preliminary survey, and since the topics dealt with are supplementary to the corresponding topics of the nonrelative theory (contained in [2]), we feel justified in presupposing that the

This method is based on the Cartan-Leray spectral sequence, [3; 1 ], and can be generalized to other algebraic situations, as will be shown in a forthcoming paper of Cartan-Eilenberg [2]. Since the details of the Cartan-Leray technique have not been published (other than in seminar notes of limited circulation), we develop them in Chapter I. The auxiliary theorems we need for this purpose are useful also in other connections. In Chapter II, which is independent of Chapter I, we obtain a spectral sequence quite directly by filtering the group of cochains for G. This filtration leads to the same group E2=H(G/K, H(K)) (although we do not know whether or not the succeeding terms are isomorphic to those of the first spectral sequence) and lends itself more readily to applications, because one can identify the maps which arise from it. This is not always the case with the first filtration, and it is for this reason that we have developed the direct method in spite of the somewhat lengthy computations which are needed for its proofs. Chapter III gives some applications of the spectral sequence of Chapter II. Most of the results could be obtained in the same manner with the spectral sequence of Chapter I. A notable exception is the connection with the theory of simple algebras which we discuss in ?5. Finally, let us remark that the methods and results of this paper can be transferred to Lie Algebras. We intend to take up this subject in a later paper.

Introduction. In a recent paper [2], E. H. Brown introduced the notion of a twisted tensor product. Briefly, the definition is as follows. Let K be a D.G.A. (differential, graded, augmented) coalgebra, A a D.G.A. algebra, and M a D.G.A. A-module. The twisted tensor product K 0 M of K with M is, except for the differential, the usual tensor product. The differential on K 0, 0 M is modified using a twisting cochain 4 in Hom(K, A). Now suppose p: E-*X is a fiber space (essentially of the Hurewicz type) with fiber F. Then C(X) is a D.G.A. coalgebra, C(QX) a D.G.A. algebra, and C(F) a D.G.A. C(QX)-module. Brown's main theorem states that there is a twisting cochain 4 in Hom(C(X), C(QX)) and a chain equivalence 41: C(X) X C(F) -> C(E). In another recent paper [1], Barratt, Gugenheim, and Moore define the twisted cartesian product of two simplicial sets (semi-simplicial complexes). If X and F are simplicial sets and G is a simplicial group acting on F, the twisted cartesian product of X and F, X XT F, coincides with the usual cartesian product except that the initial face is modified in terms of a twisting function r: X -G. It is proved in [1] that any simplicial fiber space p: EX with fiber F can, for the purposes of algebraic topology, be replaced by a twisted cartesian product X XT F. (The group G and the twisting function r: X >G are shown to exist.) Considering these two results, one might expect that an analogue of Brown's theorem could be proved for twisted cartesian products, explicitly defining the twisting cochain in terms of the twisting function(2). This is in fact done in Part I of this paper. In Part II, the explicit form of the twisting cochain is used to investigate fiber bundles over spheres. The homology and cohomology Wang sequences are derived and some partial results obtained describing the behavior of the maps in the cohomology Wang sequence with respect to cup products. I would like to express my gratitude to Professor Saunders MacLane for his patient assistance and encouragement during the preparation of this

Introduction. Most of the recent rapid development of the homology or cohomology theory of fibre spaces is based on the use of spectral sequences, as introduced by J. Leray [1]. In order to use spectral sequences successfully, it is almost always necessary to have a good deal of information about the term usually called for cohomology E2 or for homology E2 of the spectral sequence. In his original work, Leray proved that the term E2 in the spectral sequence for the Cech cohomology of a fibre space is naturally isomorphic to the cohomology of the base space with local coefficients in the cohomology of the fibre. The appropriate analogues of this result for cubical singular homology, and cohomology were proved by J.-P. Serre [3 ], and are fundamental in applications of homology theory to homotopy theory. The main object of this paper is to calculate the term E2 in the singular homology spectral sequence of a fibre space by the use of the theory of acyclic models of Eilenberg and MacLane [4]. Notice that we said the singular homology spectral sequence of a fibre space, and did not specify whether we meant simplicial or cubical singular theory. One of our results asserts essentially that there is a singular homology spectral sequence of a fibre space, and that it may be obtained by using either simplicial or cubical singular theory. Further, we show that in cubical singular homology theory a filtration somewhat different from the one used by Serre may be used to obtain the spectral sequence. This filtration is symmetric in all coordinates, and consequently much more convenient to use when dealing with fibre spaces wherein a multiplication is defined in the base space and total space, and such that projection map is a homomorphism. This fact will be exploited elsewhere. In the course of our work, it has been necessary to make a fairly extensive axiomatic investigation of the notion of a "singular theory," and of the notion of "local coefficients." One outcome of this work is a proof of the fact that cubical singular homology and simplicial singular homology coincide even with local coefficients. For ordinary coefficients this was proved by Eilenberg and MacLane [4]. Since much of our work is quite abstract, the general theory will be interspersed with examples from cubical singular theory. Technically this paper is divided into chapters, and eachi chapter into sections. The notation 11.3.4 refers to the 4th numbered statement in the

of the Steenrod squares for the Q2-cohomology rings of finite complexes whose integral homology is free of 2-torsion, where Q2 is the ring of rationals with odd denominators. These are not cohomology operations as they only satisfy generalized naturality properties, but under favourable conditions they provide information which it does not seem possible to obtain using either primary or secondary cohomology operations. The treatment of these generalized cohomology operations given in this note is not complete; only those properties are considered which are needed for the immediate applications. In later papers the author hopes to extend both the theory and the applications of these generalized operations as indicated in [13]. All the additive and several of the multiplicative results given in this paper have quite precise analogues for odd primes. These are not considered here, as the most elegant treatment of the corresponding generalized reduced power operations for odd primes involves the use of theorems of J. F. Adams which have not yet been published. I would like to thank Professor J. F. Adams and Dr. I. M. James for the considerable assistance they have given in the preparation of this paper, which essentially formed a part of the author's doctoral thesis.

Introduction. In [4] G. W. Whitehead gives a method by which all extraordinary homology and cohomology theories may be generated by fundamental homotopy constructions. Unfortunately the definition of the extraordinary theories used is not sufficiently general so as to include all theories that would normally be considered as (co) homology theories. An example of a theory not included is (co) homology with local coefficients. In this paper, we will show how to extend the methods of Whitehead so as to include such theories. Let G be a fixed continuous group. If we consider the category of principal G-bundles over C.W. complexes, we may define a connected sequence of functors from this category to the category of abelian groups, satisfying the usual axioms of extraordinary (co) homology theory, with principal G-bundles replacing spaces, and bundle maps and homotopies replacing maps and homotopies. It is the purpose of this paper to show that such theories may be generated by homotopy constructions. In particular, we will indicate how local coefficient theory may be generated in this manner. In order to develop the homology theories spoken of above, it will first be necessary to define a class of spaces closely related to C.W. complexes but with just enough additional generality so as to include spaces for which a useful C.W. decomposition may not be available, but which are structured enough to give us a good deal of the type of information usually associated with a C.W.-decomposition. The definition of cohomology groups used in this paper was proposed by Professor Israel Berstein of Cornell University and will be exploited by him in a forthcoming paper on the Thom Isomorphism. This paper represents part of a doctoral dissertation written under Professor Berstein at Cornell University. 1. Piecewise-C.W. complexes. We will need three lemmas from homotopy theory. The first two will not be proved as they are quite easy. In the following

1. Introduction. The secondary characteristic classes of a sphere bundle with vanishing Euler class were introduced by Massey [10]. These classes provide the additional information over and above the cohomology structure of the base needed to determine the mod 2 cohomology of the total space as ring, as module over the cohomology of the base, and as module over the mod 2 Steenrod algebra. It was for this purpose that these classes were originally defined. Subsequent work by Peterson and Stein [16], Meyer [13], and the author [14] led to many new properties of secondary characteristic classes, including analogues of the Wu and Whitney formulas and reinterpretation of these classes in terms of functional cohomology operations (yielding in turn simpler proofs of formulas of [16]). In this paper we define secondary characteristic classes for an n-plane bundle e such that the top k Whitney classes Wn-kk 1,. . ., wn of e vanish. For k = 1, these classes are the classes of Massey. By a theorem of Massey and Peterson [12], these classes are the additional information needed to determine the cohomology of the total space of the associated bundle with fibre Vn,k We then prove analogues of the Wu and Whitney formulas. Our secondary classes will be defined in terms of a functional cohomology operation, involving relative cup-product, defined by the basic method of Steenrod [17] on "Thom" classes of the bundle. While our formal treatment of the algebraic properties of such operations, as well as the presentation of the Wu and Whitney formulas, has been greatly influenced by the paper of Meyer [13], the operations we consider have smaller indeterminacy and the results obtained from these operations

paper, involves the application of a spectral sequence defined for any DGAalgebra U to the special case U = C(A), the cobar construction of A. In the second half of the paper we define Massey products in U and relate them to the differentials in the cited spectral sequence.

We define the categorical cohomology of a k k -graph Λ \Lambda and show that the first three terms in this cohomology are isomorphic to the corresponding terms in the cohomology defined in our previous paper. This leads to an alternative characterisation of the twisted k k -graph C ∗ C^* -algebras introduced there. We prove a gauge-invariant uniqueness theorem and use it to show that every twisted k k -graph C ∗ C^* -algebra is isomorphic to a twisted groupoid C ∗ C^* -algebra. We deduce criteria for simplicity, prove a Cuntz-Krieger uniqueness theorem and establish that all twisted k k -graph C ∗ C^* -algebras are nuclear and belong to the bootstrap class.

We give a formula relating the transfer maps for the cohomology theories E n E_{n} and C t C_t to the transchromatic generalized character maps of a previous paper by the author. We then apply this to understand the effect of the transchromatic generalized character maps on Strickland’s isomorphism between the Morava E E -theory of the symmetric group Σ p k \Sigma _{p^k} (modulo a transfer ideal) and the global sections of the scheme that classifies subgroups of order p k p^k in the formal group associated to E n E_{n} . This provides an algebro-geometric interpretation to the C t C_t -cohomology of the class of groups arising as centralizers of finite sets of commuting elements in symmetric groups.

Zagier introduced special bases for weakly holomorphic modular forms to give the new proof of Borcherds' theorem on the infinite product expansions of integer weight modular forms on $\SL_2(\ZZ)$ with a Heegner divisor. These good bases appear in pairs, and they satisfy a striking duality, which is now called the Zagier duality. After the result of Zagier, this type duality was studied broadly in various view points including the theory of a mock modular form. In this paper, we consider this problem with the Eichler cohomology theory, especially the supplementary function theory developed by Knopp. Using holomorphic Poincar�� series and their supplementary functions, we construct a pair of families of vector-valued harmonic weak Maass forms satisfying the Zagier duality with integer weights $-k$ and $k+2$ respectively, $k>0$, for a $H$-group. We also investigate the structures of them such as the images under the differential operators $D^{k+1}$ and $��_{-k}$ and quadric relations of the critical values of their $L$-functions.

In 2007 Phillips and Weaver showed that, assuming the Continuum Hypothesis, there exists an outer automorphism of the Calkin algebra. (The Calkin algebra is the algebra of bounded operators on a separable complex Hilbert space, modulo the compact operators.) In this paper we establish that the analogous conclusion holds for a broad family of quotient algebras. Specifically, we will show that assuming the Continuum Hypothesis, if $A$ is a separable algebra which is either simple or stable, then the corona of $A$ has nontrivial automorphisms. We also discuss a connection with cohomology theory, namely, that our proof can be viewed as a computation of the cardinality of a particular derived inverse limit.

In this paper we consider non-abelian 1-cohomology for groups with coefficients in other groups. We prove versions of the `five lemma' arising from this situation. We go on to show that a connected unipotent algebraic group Q acted on morphically by a connected algebraic group G admits a filtration with successive quotients having the structure of G-modules. From these results we deduce extensions to results due to Cline, Parshall, Scott and van der Kallen. Firstly, if G is a connected, reductive algebraic group with Borel subgroup B and Q a unipotent algebraic G-group, we show the restriction map H^1(G,Q)\to H^1(B,Q) is an isomorphism. We also show that this situation admits a notion of rational stability and generic cohomology. We use these results to obtain corollaries about complete reducibility and subgroup structure., Comment: 25 pages; v5; final version to appear in Transactions of the AMS

In this paper we determine the integral quandle homology groups of Alexander quandles of prime order. As a special case, this settles the delayed F ibonacci conjecture by M. Niebrzydowski and J. H. Przytycki in [7]. Moreover, we determine the cohomology group of the Alexander quandle and obtain relatively simple presentations of all higher degree cocycles which generate the cohomology group. Furthermore, we prove that the integral quandle homology of a finite connected Alexander quandle is annihilated by the order of the quandle.

In this paper we present a model to calculate the stringy product on twisted orbifold K-theory of Adem-Ruan-Zhang for abelian complex orbifolds. In the first part we consider the non-twisted case on an orbifold presented as the quotient of a manifold acted by a compact abelian Lie group. We give an explicit description of the obstruction bundle, we explain the relation with the product defined by Jarvis-Kaufmann-Kimura and, via a Chern character map, with the Chen-Ruan cohomology, and we explicitely calculate the stringy product for a weighted projective orbifold. In the second part we consider orbifolds presented as the quotient of a manifold acted by a finite abelian group and twistings coming from the group cohomology. We show a decomposition formula for twisted orbifold K-theory that is suited to calculate the stringy product and we use this formula to calculate two examples when the group is $(\integer/2)^3$., Comment: 23 pages. A chapter on the stringy product on twisted orbifold K-theory has been added, including a decomposition formula and examples with non trivial twisting. The title has been changed accordingly. To appear in TAMS

In this paper we answer a question of Zagier and find the dimensions of spaces of holomorphic second-order forms of even weight. We also establish a cohomological interpretation and prove an Eichler-Shimura-type isomorphism.

In this paper we present a method to compute the real cohomology of any finitely generated virtually nilpotent group. The main ingredient in our setup consists of a polynomial crystallographic action of this group. As any finitely generated virtually nilpotent group admits such an action (which can be constructed quite easily), the approach we present applies to all these groups. Our main result is an algorithmic way of computing these cohomology spaces. As a first application, we prove a kind of Poincare duality (also in the nontorsion free case) and we derive explicit formulas in the virtually abelian case.

The classification of curves of genus 2 over an algebraically closed field was studied by Clebsch and Bolza using invariants of binary sextic forms, and completed by Igusa with the computation of the corresponding three-dimensional moduli variety M 2 \mathcal M_2 . The locus of curves with group of automorphisms isomorphic to one of the dihedral groups D 8 D_8 or D 12 D_{12} is a one-dimensional subvariety. In this paper we classify these curves over an arbitrary perfect field k k of characteristic char ⁡ k ≠ 2 \operatorname {char} k\neq 2 in the D 8 D_8 case and char ⁡ k ≠ 2 , 3 \operatorname {char} k\neq 2,3 in the D 12 D_{12} case. We first parameterize the k ¯ \overline k -isomorphism classes of curves defined over k k by the k k -rational points of a quasi-affine one-dimensional subvariety of M 2 \mathcal M_2 ; then, for every curve C / k C/k representing a point in that variety we compute all of its k k -twists, which is equivalent to the computation of the cohomology set H 1 ( G k , Aut ⁡ ( C ) ) H^1(G_k,\operatorname {Aut}(C)) . The classification is always performed by explicitly describing the objects involved: the curves are given by hyperelliptic models and their groups of automorphisms represented as subgroups of GL 2 ⁡ ( k ¯ ) \operatorname {GL}_2(\overline k) . In particular, we give two generic hyperelliptic equations, depending on several parameters of k k , that by specialization produce all curves in every k k -isomorphism class.

In this paper we compute the integral cohomology of certain semi-direct products of the form Z" x G, arising from a linear G action on the n-torus, where G is a finite group. The main application is the complete calculation of torsion gerbes for six-dimensional examples arising in string theory.

Let Tr k be the algebraic transfer that maps from the coinvariants of certain GL k -representations to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer tr k : π S * ((BV k ) + ) → π S * (S 0 ). It has been shown that the algebraic transfer is highly nontrivial, more precisely, that Tr k is an isomorphism for k = 1, 2, 3 and that Tr = ○+ k Tr k is a homomorphism of algebras. In this paper, we first recognize the phenomenon that if we start from any degree d and apply Sq° repeatedly at most (k - 2) times, then we get into the region in which all the iterated squaring operations are isomorphisms on the coinvariants of the GL k -representations. As a consequence, every finite Sq 0 -family in the coinvariants has at most (k - 2) nonzero elements. Two applications are exploited. The first main theorem is that. Tr k is not a.n isomorphism for k > 5. Furthermore, for every k > 5, there are infinitely many degrees in which Tr k is not an isomorphism. We also show that if Tr detects a nonzero element in certain degrees of Ker(Sq 0 ), then it is not a monomorphism and further, for each k > , Tr k is not a monomorphism in infinitely many degrees. The second main theorem is that, the elements of any Sq 0 -family in the cohomology of the Steenrod algebra, except at most its first (k - 2) elements, are either all detected or all not detected by Tr k , for every k. Applications of this study to the cases k = 4 and 5 show that; Tr 4 does not detect the three families g, D 3 and p', and that Tr 5 does not. detect the family {h n +1g n |n > 1}.

Let p be an odd prime number and let G be an extraspecial pgroup. The purpose of the paper is to show that G has no non-zero essential mod-p cohomology (and in fact that H* (G, IFp) is Cohen-Macaulay) if and only if IGI = 27 and exp(G) = 3.

In this paper we study finite group extensions represented by special cohomology classes. As an application we obtain some restrictions on finite groups which can act freely on a product of spheres or on a product of real projective spaces. In particular, we prove that if (Z/p) acts freely on (S), then r ≤ k.

This paper studies the homology of finite modules over the exterior algebra E E of a vector space V V . To such a module M M we associate an algebraic set V E ( M ) ⊆ V V_E(M)\subseteq V , consisting of those v ∈ V v\in V that have a non-minimal annihilator in M M . A cohomological description of its defining ideal leads, among other things, to complementary expressions for its dimension, linked by a ‘depth formula’. Explicit results are obtained for M = E / J M=E/J , when J J is generated by products of elements of a basis e 1 , … , e n e_1,\dots ,e_n of V V . A (infinite) minimal free resolution of E / J E/J is constructed from a (finite) minimal resolution of S / I S/I , where I I is the squarefree monomial ideal generated by ‘the same’ products of the variables in the polynomial ring S = K [ x 1 , … , x n ] S=K[x_1,\dots ,x_n] . It is proved that V E ( E / J ) V_E(E/J) is the union of the coordinate subspaces of V V , spanned by subsets of { e 1 , … , e n } \{\,e_1,\dots ,e_n\,\} determined by the Betti numbers of S / I S/I over S S .

Generalized Witt algebras, over a field F F of characteristic 0 0 , were defined by Kawamoto about 12 years ago. Using different notations from Kawamoto’s, we give an essentially equivalent definition of generalized Witt algebras W = W ( A , T , φ ) W=W(A,T,\varphi ) over F F , where the ingredients are an abelian group A A , a vector space T T over F F , and a map φ : T × A → K \varphi :T\times A\to K which is linear in the first variable and additive in the second one. In this paper, the derivations of any generalized Witt algebra W = W= W ( A , T , φ ) W(A,T,\varphi ) , with the right kernel of φ \varphi being 0 0 , are explicitly described; the isomorphisms between any two simple generalized Witt algebras are completely determined; and the second cohomology group H 2 ( W , F ) H^2(W,F) for any simple generalized Witt algebra is computed. The derivations, the automorphisms and the second cohomology groups of some special generalized Witt algebras have been studied by several other authors as indicated in the references.

We study the asymptotic behavior as a function of e e of the lengths of the cohomology of certain complexes. These complexes are obtained by applying the e e -th iterated Frobenius functor to a fixed finite free complex with only finite length cohomology and then tensoring with a fixed finitely generated module. The rings involved here all have positive prime characteristic. For the zeroth homology, these functions also contain the class of Hilbert-Kunz functions that a number of other authors have studied. This asymptotic behavior is connected with certain intrinsic dimensions introduced in this paper: these are defined in terms of the Krull dimensions of the Matlis duals of the local cohomology of the module. There is a more detailed study of this behavior when the given complex is a Koszul complex.

S. Cappell and J. Shaneson (Stratifiable maps and topological invariants, J. Amer. Math. Soc. 4 (1991), 521-551) have recently developed a theory of homology L-classes, extending Goresky-MacPherson's homology Lclasses. In this paper we show that Cappell-Shaneson's homology L-classes for compact complex, possibly singular, algebraic varieties can be interpreted as a unique natural transformation from a covariant cobordism function Q to the Q-homology functor H ( ; Q) satisfying a certain normalization condition, just like MacPherson's Chern classes and Baum-Fulton-MacPherson's Todd classes.

Let Γ g {\Gamma _g} denote the mapping class group of genus g g . In this paper, we calculate p p -torsion of Farrell cohomology H ^ ∗ ( Γ p ) {\widehat {H}^*}({\Gamma _p}) for any odd prime p p .