Characterizations of spectra with -injective cohomology which satisfy the Brown-Gitler property

We work in the stable homotopy category of -complete connective spectra having mod homology of finite type. means cohomology with coefficients, and is a left module over the Steenrod algebra .

A spectrum is called spacelike if it is a wedge summand of a suspension spectrum, and a spectrum satisfies the Brown-Gitler property if the natural map is onto, for all spacelike .

It is known that there exist spectra satisfying the Brown-Gitler property, and with isomorphic to the injective envelope of in the category of unstable -modules.

Call a spectrum standard if it is a wedge of spectra of the form , where is a stable wedge summand of the classifying space of some elementary abelian -group. Such spectra have -injective cohomology, and all -injectives appear in this way.

Working directly with the two properties of stated above, we clarify and extend earlier work by many people on Brown-Gitler spectra. Our main theorem is that, if is a spectrum with -injective cohomology, the following conditions are equivalent:

(A) there exist a spectrum whose cohomology is a reduced -injective and a map that is epic in cohomology, (B) there exist a spacelike spectrum and a map that is epic in cohomology, (C) is monic in cohomology, (D) satisfies the Brown-Gitler property, (E) is spacelike, (F) is standard. ( is reduced if it has no nontrivial submodule which is a suspension.)

As an application, we prove that the Snaith summands of are Brown-Gitler spectra-a new result for the most interesting summands at odd primes. Another application combines the theorem with the second author's work on the Whitehead conjecture.

Of independent interest, enroute to proving that (B) implies (C), we prove that the homology suspension has the following property: if an -connected space admits a map to an -fold suspension that is monic in mod homology, then is onto in mod homology.

     

Positive definite spherical functions on Olshanskii domains

Let be a simply connected complex Lie group with Lie algebra , a real form of , and the analytic subgroup of corresponding to . The symmetric space together with a -invariant partial order is referred to as an Olshanskii space. In a previous paper we constructed a family of integral spherical functions on the positive domain of . In this paper we determine all of those spherical functions on which are positive definite in a certain sense.

     

Embeddings in generalized manifolds

We prove that a ()-connected map from a compact PL -manifold to a generalized -manifold with the disjoint disks property, , is homotopic to a tame embedding. There is also a controlled version of this result, as well as a version for noncompact and proper maps that are properly ()-connected. The techniques developed lead to a general position result for arbitrary maps , , and a Whitney trick for separating submanifolds of that have intersection number 0, analogous to the well-known results when is a manifold.

     

Sums of squares of regular functions on real algebraic varieties

Let be an affine algebraic variety over (or any other real closed field ). We ask when it is true that every positive semidefinite (psd) polynomial function on is a sum of squares (sos). We show that for the answer is always negative if has a real point. Also, if is a smooth non-rational curve all of whose points at infinity are real, the answer is again negative. The same holds if is a smooth surface with only real divisors at infinity. The ``compact' case is harder. We completely settle the case of smooth curves of genus : If such a curve has a complex point at infinity, then every psd function is sos, provided the field is archimedean. If is not archimedean, there are counter-examples of genus .

     

Double coset density in classical algebraic groups

We classify all pairs of reductive maximal connected subgroups of a classical algebraic group that have a dense double coset in . Using this, we show that for an arbitrary pair of reductive subgroups of a reductive group satisfying a certain mild technical condition, there is a dense -double coset in precisely when is a factorization.

     

On the non-vanishing of cubic twists of automorphic -series

Let be a normalised new form of weight for over and , its base change lift to . A sufficient condition is given for the nonvanishing at the center of the critical strip of infinitely many cubic twists of the -function of . There is an algorithm to check the condition for any given form. The new form of level is used to illustrate our method.

     

Cohomology of uniformly powerful -groups

In this paper we will study the cohomology of a family of -groups associated to -Lie algebras. More precisely, we study a category of -groups which will be equivalent to the category of -bracket algebras (Lie algebras minus the Jacobi identity). We then show that for a group in this category, its -cohomology is that of an elementary abelian -group if and only if it is associated to a Lie algebra.

We then proceed to study the exponent of in the case that is associated to a Lie algebra . To do this, we use the Bockstein spectral sequence and derive a formula that gives in terms of the Lie algebra cohomologies of . We then expand some of these results to a wider category of -groups. In particular, we calculate the cohomology of the -groups which are defined to be the kernel of the mod reduction

     

A bracket power characterization of analytic spread one ideals

The main theorem characterizes, in terms of bracket powers, analytic spread one ideals in local rings. Specifically, let be regular nonunits in a local (Noetherian) ring and assume that , the integral closure of , where . Then the main result shows that for all but finitely many units in that are non-congruent modulo and for all large integers and it holds that for and not divisible by , where is the -th bracket power of . And, conversely, if there exist positive integers , , and such that has a basis such that , then has analytic spread one.

     

Low-dimensional linear representations of

We classify all complex representations of the automorphism group of the free group of dimension Among those representations is a new representation of dimension which does not vanish on the group of inner automorphisms.

     

The -Lagrangian of a convex function

At a given point , a convex function is differentiable in a certain subspace (the subspace along which has 0-breadth). This property opens the way to defining a suitably restricted second derivative of at . We do this via an intermediate function, convex on . We call this function the -Lagrangian; it coincides with the ordinary Lagrangian in composite cases: exact penalty, semidefinite programming. Also, we use this new theory to design a conceptual pattern for superlinearly convergent minimization algorithms. Finally, we establish a connection with the Moreau-Yosida regularization.

     

A construction of homologically area minimizing hypersurfaces with higher dimensional singular sets

We show that a large variety of singular sets can occur for homologically area minimizing codimension one surfaces in a Riemannian manifold. In particular, as a result of Theorem A, if is smooth, compact dimensional manifold, , and if is an embedded, orientable submanifold of dimension , then we construct metrics on such that the homologically area minimizing hypersurface , homologous to , has a singular set equal to a prescribed number of spheres and tori of codimension less than . Near each component of the singular set, is isometric to a product , where is any prescribed, strictly stable, strictly minimizing cone. In Theorem B, other singular examples are constructed.

     

Spin L-functions on

The ``spin' L-function of an automorphic representation of is an Euler product of degree associated with the spin representation of the L-group . If or , and the automorphic representation is generic in the sense of having a Whittaker model, the analytic properties of these L-functions are studied by the Rankin-Selberg method.

     

coefficients

Let be a system of real smooth vector fields, satisfying Hörmander's condition in some bounded domain (). We consider the differential operator

where the coefficients are real valued, bounded measurable functions, satisfying the uniform ellipticity condition:

for a.e. , every , some constant . Moreover, we assume that the coefficients belong to the space VMO (``Vanishing Mean Oscillation'), defined with respect to the subelliptic metric induced by the vector fields . We prove the following local -estimate:

for every , . We also prove the local Hölder continuity for solutions to for any with large enough. Finally, we prove -estimates for higher order derivatives of , whenever and the coefficients are more regular.

     

Random intersections of thick Cantor sets

Let , be Cantor sets embedded in the real line, and let , be their respective thicknesses. If , then it is well known that the difference set is a disjoint union of closed intervals. B. Williams showed that for some , it may be that is as small as a single point. However, the author previously showed that generically, the other extreme is true; contains a Cantor set for all in a generic subset of . This paper shows that small intersections of thick Cantor sets are also rare in the sense of Lebesgue measure; if , then contains a Cantor set for almost all in .

     

Tight closure, plus closure and Frobenius closure in cubical cones

We consider tight closure, plus closure and Frobenius closure in the rings , where is a field of characteristic and . We use a -grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular ring . We show that Frobenius closure is the same as tight closure in certain classes of ideals when . Since , we conclude that for these ideals. Using injective modules over the ring , the union of all th roots of elements of , we reduce the question of whether for -graded ideals to the case of -graded irreducible modules. We classify the irreducible -primary -graded ideals. We then show that for most irreducible -primary -graded ideals in , where is a field of characteristic and . Hence for these ideals.

     

Hankel Operators on Bounded Analytic Functions

For a domain in the complex plane and a bounded measurable function on , the generalized Hankel operator on is the operator of multiplication by followed by projection into . Under certain conditions on we show that either is compact or there is an embedded on which is bicontinuous. We characterize those 's for which is compact in the case that is a Behrens roadrunner domain.