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W. Casselman* (Vancouver) and D. Wigner (Ann Arbor) 0. Let G be the group of ~p-rational points on a connected semi- simple group defined over ~v, ff its Lie algebra, H* (G+ Qp) the continuous cohomology of G with coefficients in Qp. When G is compact, a result of Lazard's ([7], Chapter V, Theot'em 2.4.10) and an argument about Zariski-closure (see w 3) imply that H* (G, (I)p)~ H* (fr Qp). The original motivation for most of the results in this paper was the question asked by Serre ([II], p. 119): Does Lazard's result hold for more general G? We show this to be so (Theorem 1 in w 3). We include a largely self-contained exposition of continuous cohomol- ogy theory for locally compact groups. Our main result here is a form of Shapiro's Lemma (Propositions3 and 4 in w 1). We also include a discussion of the Hochschild-Serre spectral sequence (in w 2). We have drawn largely on a paper of Hochschild and Mostow [-4] which treats the case of G-modules which are real vector spaces, but our emphasis is quite different. In many cases our cohomology agrees with that constructed by Calvin Moore (described in [10]), and a number of our results are implied by results of his. In w 3 we apply Shapiro's Lemma and the Bruhat-Tits building to prove Serre's conjecture. In w 4 we deal with the cohomology of p-adic groups with coefficients in real vector spaces, including some remarks about the cohomology of smooth representations over more general fields. In w 5 we answer a question of Serre's about the analytic cohomol- ogy of a p-adic semi-simple group.

We deduce from Deligne's form of the Riemann hypothesis and the hard Lefschetz theoremin l-adic cohomology the corresponding facts for any “reasonable” cohomology theory, in particular for crystalline cohomology, and give some applications to algebraic cycles.

for a e T(v A V ) , X1? . . . , Xk + 1 e T(T). Since the curvature tensor of V restricted to T is identically zero we have that d o d = 0. Denote the homology of this complex by F*(T; V). This is the cohomology of the Lie algebra of vector fields tangent to the foliation with coefficients in sections of the normal bundle, the representation being given by the connection [GF]. In general the groups F\x ; v) are not finitely generated (the complex is not elliptic) but they satisfy the following. (i) F* is a functor from the category of foliated manifolds and transverse maps to the category of abelian groups and homomorphisms. (ii) If ƒ : N -» M is an embedded transverse submanifold, we can define relative cohomology groups F*(T; v, ƒ) and obtain the usual long exact sequence. (iii) F* is an invariant of the diffeomorphism type of the foliation. However, F* is not an invariant of the integrable homotopy type of the foliation when M is an open manifold.

We consider algebras A of continuous complex valued functions, which are given as the set of global sections of a sheaf S \mathcal {S} on a topological space X. Under the hypothesis that all the higher cohomology groups of the sheaf are zero, we investigate the relationship between ideals in A, kernels of algebra homomorphisms of A into the complex numbers C, and sets of functions vanishing at a point of X. As applications, we obtain some simple proofs of theorems about ideals in certain algebras of holomorphic functions.

It is well known that for X a CW-complex, K(X) and Hev(X) are isomorphic modulo finite groups, although the "isomorphism" is not natural. The purpose of this paper is to improve this result for X a finite CW-complex. 1. Preliminaries. For the basic definitions and theory (1) of X-rings, we refer to [21, [121. The ring Z of integers has a X-ring structure with Am: Z -Z the function Xmn) (fn)n(n1) .(n-m+ 1) m ml Definition. An augmented X-ring is a X-ring R together with X-ring homomorphisms i: Z -R and e: R -*Z such that ci = 1. Since the Definition implies that i: Z -4 R is a monomorphism, we think of Z C R as the multiples of the identity. Let $3 denote the category of augmented X-rings and X-ring homomorphisms which commute with the augmentation. If B e Ob 3, we write B7y for the nth term of the y-filtration on B, and P(B) for the associated graded ring. We note that ['(B)o = Z. Let d be the category of commutative graded rings A with Ao = Z. We define A: d $ as follows: If A E Ob , then as a set A(A)= l An. If a EA(A), we denote the component in An by an and for convenience of notation define ao = 1 E AO and write a as either of the formal expressions 1 +al +a 2+ *+an + * or ?a.. oro ^10~~~~~~~~~~ If a, b E A(A), we define their sum, a @ b, componentwise: (a $ b) = i a.b. n i+j=n Z so that '@' is analogous with multiplication of formal power series. This operation makes A(A) into an Abelian group, and we denote the inverse of a by e a. In particular if a is an element with an = 0 for n > 2, then Received by the editors December 13, 1972 and, in revised form, July 9, 1973. AMS (MOS) subject classications (1970). Primary 55B15; Secondary 55B40, 55F40. (I) We are indebted to Atiyah and Tall [21 for explicit proofs, in particular those based on 'formal algebra'. Copyright

Let C denote the category of compact Hausdorff spaces and H : C → H C H:C \to HC be the homotopy functor. Let S : C → S C S:C \to SC be the functor of shape in the sense of Holsztyński for the projection functor H. Let X be a continuum and H n ( X ) {H^n}(X) denote n-dimensional Čech cohomology with integer coefficients. Let A x = char H 1 ( X ) {A_x} = {\text {char}}\;{H^1}(X) be the character group of H 1 ( X ) {H^1}(X) considering H 1 ( X ) {H^1}(X) as a discrete group. In this paper it is shown that there is a shape morphism F ∈ Mor S C ( X , A X ) F \in {\text {Mor}_{SC}}(X,{A_X}) such that F ∗ : H 1 ( A X ) → H 1 ( X ) {F^\ast }:{H^1}({A_X}) \to {H^1}(X) is an isomorphism. It follows from the results of a previous paper by the author that there is a continuous mapping f : X → A X f:X \to {A_X} such that S ( f ) = F S(f) = F and thus that f ∗ : H 1 ( A X ) → H 1 ( X ) {f^\ast }:{H^1}({A_X}) \to {H^1}(X) is an isomorphism. This result is applied to show that if X is locally connected, then H 1 ( X ) {H^1}(X) has property L. Examples are given to show that X may be locally connected and H n ( X ) {H^n}(X) not have property L for n > 1 n > 1 . The result is also applied to compact connected topological groups. In the last section of the paper it is shown that if X is compact and movable, then for every integer n, H n ( X ) / Tor H n ( X ) {H^n}(X)/{\operatorname {Tor}}\;{H^n}(X) has property L. This result allows us to construct peano continua which are nonmovable. An example is given to show that H n ( X ) {H^n}(X) itself may not have property L even if X is a finite-dimensional movable continuum.

Given a cotriple 𝔾 = (G, e, δ) on a category X and a functor E:X Opp→A into an abelian category A, there exists the cohomology theory of Barr and Beck: Hn(X, E) e |A| (n ≥ 0, X e |X|), ([1], p.249). Almost all the important cohomology theories in mathematics have been shown to be special instances of such a general theory (see [1], [2] and [3]). Usually E arises from an abelian group object Y in X in the following manner: it is the contravariant functor from X into the category Ab of abelian groups that associates to each object X in X the abelian group X(X, Y) of maps from X to Y. In such a situation we shall write Hn(X, Y)𝔾 instead of Hn(X, E)G. Barr and Beck [2] have shown that the Eilenberg-MacLane cohomology groups Hn(π, A), n ≥ 2, can be re-captured as follows. One considers the free group cotriple 𝔾′ on the category Gps of groups, which induces in a natural manner a cotriple 𝔾 on the category (Gps, π) of groups over a fixed group π.

Let n \mathfrak {n} be the maximal nilpotent ideal of a Borel subalgebra of a complex semisimple Lie algebra g \mathfrak {g} . Under the adjoint action n , g / n \mathfrak {n},\mathfrak {g}/\mathfrak {n} , and n ′ \mathfrak {n}’ (the dual space of n \mathfrak {n} ) are n \mathfrak {n} modules. Laplace operators for these three modules are computed by techniques which extend those introduced by B. Kostant in [6]. The kernels of these operators are then determined and, in view of the existence of a Hodge decomposition, the detailed structure of the first degree cohomology groups of n \mathfrak {n} with coefficients in n , g / n \mathfrak {n},\mathfrak {g}/\mathfrak {n} , and n ′ \mathfrak {n}’ is obtained. These cohomology groups (spaces) are described, in fact, as completely reducible modules of a Cartan subalgebra h \mathfrak {h} of g \mathfrak {g} .

This note is concerned with stable G-equivariant homology and cohomology theories (G a compact Lie group). In important cases, when H-equivariant theories are defined naturally for all closed subgroups H of G, we show that the G-(co)homology groups of G xH X are isomorphic with H-(co)homology groups of X. We introduce the concept of orientability of G-vector bundles and manifolds with respect to an equivariant cohomology theory and prove a duality theorem which implies an equivariant analogue of Poincare-Lefschetz duality.

Topological group extensions with Abelian kernels are analyzed using factor sets and following the pattern of the work of Eilenberg and MacLane on extensions of groups without topology. In this analysis, the Eilenberg‐MacLane cohomology is replaced by the Mackey‐Moore one, whose cochains are Borel mappings and which is especially suitable in the case of Polish groups (Hausdorff second countable complete groups). The connection between cohomology groups of degree 2 and equivalence classes of topological group extensions with Abelian kernels is established. A fundamental sequence of cohomology groups and group homomorphisms is proven to be exact, and it is shown that, in some interesting cases, the low degree cohomology groups of topological semidirect products are determined by the corresponding cohomology groups of the factors.

Using the localization technique for equivariant cohomology theory we prove a conjecture of G.Bredon (s. [4], p.381) which states that under certain conditions (s. the theorem below) the cohomology with Zp-coefficients of each component of the fixpoint set of a Zp-space can be generated as an algebra by (at most) the same number of elements as the cohomology of the space itself.

In this paper the concept of join of two permutation representations is defined and the cohomology of this join is computed and shown to have trivial cup-products. This computation is then used to compute the cohomology groups of the p p -Sylow subgroup of a symmetric group of order n n acting on the set of n n elements, and it is shown that the ring structure on these groups is not finitely generated, although it is transitive.

A. H. Clifford and S. MacLane [2] considered in 1941 the group of factor-sets H2(F, U) of a finite group F over its abstract unit group U. They proved the main theorem to the effect that H2(F, U) is isomorphic to the multiplicator M of F defined by I. Schur and also several other theorems under the assumption that F is a solvable group. They conjectured that these should hold for general finite groups r. In 1942 A. Weil proved the main theorem for general finite groups F, but this result was not published.' In this short note we shall prove that all the theorems in [2] are valid for general finite groups r, and also we shall extend their results for all (positive, zero, and negative) dimensional cohomology groups.2 1. We shall first prove a general lemma on cohomology groups. Let A be a finite group, and let E be a A-module. Suppose that Al, A2 are two A-submodules which are disjoint: AflrA2=0. Then we have the following commutative diagram such that each row and each column are exact

Cohomology theories of various algebraic structures have been investigated by several authors. The most noteworthy are due to Hochschild, MacLane and Eckmann, Chevalley and Eilenberg, who developed the theory of cohomology groups of associative algebras, abstract groups, and Lie algebras respectively. In this paper we are concerned primarily with a characterization of the third cohomology group of a Lie algebra by its extension properties.

A method is given of constructing a new class of hypo-Dirichlet algebras of given real codimension.

Let G be an algebraic linear group over a field F of characteristic 0, and let H be an algebraic subgroup of G. Let A, M be rational G-modules. In [4], we defined (A, M), and, in particular, relative cohomology groups Hn(G, H, M) were defined as (F, M).

On such a space one can define a “Hopf” invariant homomorphism h : π q n − 1 ( K ) → Z h:{\pi _{qn - 1}}(K) \to Z in two ways. We prove both definitions are equivalent and show that p π i ( K ) ≃ p π i − 1 ( S n − 1 ) ⊕ p π i ( S q n − 1 ) {}_p{\pi _i}(K) \simeq {}_p{\pi _{i - 1}}({S^{n - 1}}) \oplus {}_p{\pi _i}({S^{qn - 1}}) if and only if there is an α ∈ π q n − 1 ( K ) \alpha \in {\pi _{qn - 1}}(K) such that ( h ( α ) , p ) = 1 (h(\alpha ),p) = 1 . As immediate corollaries of this we get a result of Toda on the homotopy groups of the reduced product spaces of spheres and a well-known result of Serre on the odd primary parts of the homotopy groups of spheres.

A characterisation is given of the cohomology functor for Banach modules over a fixed Banach algebra. The result is used to simplify the proofs of two known isomorphism theorems.

In this paper we obtain several results on immersing manifolds into Euclidean spaces. For example, a spin manifold M n {M^n} immerses in R 2 n − 3 {R^{2n - 3}} for dimension n ≡ 0 mod 4 n \equiv 0\bmod 4 and n not a power of 2. A spin manifold M n {M^n} immerses in R 2 n − 4 {R^{2n - 4}} for n ≡ 7 mod 8 n \equiv 7\bmod 8 and n > 7 n > 7 . Let M n {M^n} be a 2-connected manifold for n ≡ 6 mod 8 n \equiv 6\bmod 8 and n > 6 n > 6 such that H 3 ( M ; Z ) {H_3}(M;Z) has no 2-torsion. Then M immerses in R 2 n − 5 {R^{2n - 5}} and embeds in R 2 n − 4 {R^{2n - 4}} . The method of proof consists of expressing k-invariants in Postnikov resolutions for the stable normal bundle of a manifold by means of higher order cohomology operations. Properties of the normal bundle are used to evaluate the operations.

Let L/k be a Galois extension of fields with group G and let A be the category of k-algebras isomorphic to finite products of finite field subextensions of L/k. It is known that, with appropriately defined covers, A is dual to the underlying category of a Grothendieck topology T [5, Ch. I, Theorem 4.2] and that (strict) cohomological dimension of G may be characterized via TCech cohomology with coefficients in either additive (product-preserving) functors or sheaves [5, Ch. I, Theorems 4.3 and 5.9].

This paper has three purposes: (1) to develop the machinery of a commutative cohomology theory for commutative algebras, (2) to apply this cohomology theory to give a working theory of ring extensions for commutative algebras, and (3) to employ and test the theory by relating certain natural cohomological conditions to corresponding conditions in algebraic geometry. The first purpose is approached in the first and third sections. Our cohomology has several properties not enjoyed by the Hochschild theory (an explicit relation for the tensor product, a workable sequence for changes from an algebra to a factor algebra, and an explicit formula for going local); however, unlike the Hochschild theory, we have no dimension shifting techniques. For this reason, we restrict consideration to the first, second, and third cohomology modules. The second purpose is considered in the second section. For B and A commutative algebras, we let M(A) denote HomA(A, A), taken modulo the ideal of multiplications by elements in A. Following the prototype theory which Eilenberg and Mac Lane developed for groups, we associate to each algebra homomorphism from B into M(A), an element in a third cohomology module. The homomorphism is called unobstructed if this element is zero. The unobstructed homomorphisms, together with the elements of a certain second cohomology module, are associated in a one-one fashion with the commutative algebra extensions of A by B. The purpose of the fourth and last section is best expressed by considering a point p on an affine variety V over a perfect field k. Let A be the coordinate ring of V, , be the prime ideal associated with p, and Q be the quotient field of A/p. We prove that p is a simple point if and only if the second cohomology module of A with coefficients in Q is zero. In this manner we give a characterization of regular local rings. We also prove that p will be a complete intersection if and only if the difference between the dimensions of the first and second cohomology modules (both of A with coefficients in Q) is the dimension of V. Our third main result is that V will be nonsingular if and only if A has trivial second cohomology with respect to all finitely generated Amodules.

Introduction. In his papers [5; 6] Eichler demonstrated the significance for the study of automorphic forms of Bol 's discovery [3 ] of some remarkably simple differential operators taking automorphic forms into automorphic forms. In [6] in particular Eichler discussed a relation between the automorphic forms associated to a transformation group 9 on a Riemann surface D and some purely algebraic constructions involving the group 9, the first cohomology groups of 9 with certain modules of polynomials as coefficients; the cocycles appeared as the periods of the automorphic forms under iterated indefinite integration, generalizing the classical interpretation of the periods of the abelian integrals on D/g (which can of course be considered as automorphic forms on D) as cocycles of the group 9 or alternatively of the space ID/9. The object of studying such a relation is the development of tools for calculating the dimensions of spaces of automorphic forms and the traces of the Hecke operators on automorphic forms. The aim of the present paper is the study of a more general form of this relation in somewhat greater detail for one complex variable, but in such a manner that the results can be extended to several complex variables; the actual extension to several complex variables, as well as the application to the study of the Hecke operators, will be discussed elsewhere. As for the contents of this paper, ? 1 is devoted to an exposition of Bol's differential operators in a form more useful in the present context than that of [3]. In ?2 these differential operators are applied to give an exact cohomology sequence containing, in a rather more transparent form, the relation of Eichler discussed above. The interpretations of the terms appearing in this exact sequence are discussed in ??3 through 5; the only point of difficulty arises in ?4, Theorem 3 of that section really being a form of the Serre duality theorem [12] appropriate to the occasion. These results are combined in ?6 to give a formal statement of the fundamental result of the paper. 1. Differential operators preserving automorphic forms. Let SC be the group of 2 X 2 real matrices