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We study the Hurewicz map h from the homotopy groups of a spectrum X to the R-homology of its 0th space X(0), where R is a connective commutative S-algebra. We prove that the decreasing filtration of the domain of h associated to an R-based Adams resolution is compatible with the augmentation ideal filtration of the range associated to the suspension spectrum of X(0)_+, an augmented commutative S-algebra. The proof makes use of the interplay of this filtration with Topological Andre Quillen Homology. An application is a Connectivity Theorem: Localize at a prime p and suppose X is (c-1)-connected for some positive c. If f in pi_*(X) has Adams filtration s and |f| < cp^s, then f maps to zero in R_*(X(0)). An application of that is a Finiteness Theorem: If the mod p cohomology of X is finitely presented as a module over the Steenrod algebra, then the image of the mod p Hurewicz map for X(0) is finite. We illustrate these theorems with calculations of the mod 2 Hurewicz image of BO, its connected covers, and tmf(0), and the mod p Hurewicz image of all the spaces in the BP and BP spectra. Enroute, we get new proofs of theorems of Milnor and Wilson. In the special case when X is a suspension spectrum and R = HZ/2, we recover results announced by Lannes and Zarati in the 1980s (with a totally different proof), and generalizations to all primes p. We get a chromatic version of this for the Hurewicz map for Morava E theory and all X., Comment: 2018 version: final version with much technical background added, 35 pages. To appear in Invent. Math. 2017 version: 26 pages. This is a major rewrite of the earlier version. Technical bits are simplified by making use of [Kuhn-Pereira, AGT 17(2017), 1105--1130]. Many new applications have been discovered and added. 2014 version: 21 pages

Let A and B be two connected graded commutative k–algebras of finite type, where k is a perfect field of positive characteristic p. We prove that the quasi–shuffle algebras generated by A and B are isomorphic as Hopf algebras if and only if A and B are isomorphic as graded k–vector spaces equipped with a Frobenius (pth–power) map. For the hardest part of this analysis, we work with the dual construction, and are led to study connected graded cocommutative Hopf algebras H with two additional properties: H is free as an associative algebra, and the projection onto the indecomposables is split as a morphism of graded k–vector spaces equipped with a Verschiebung map. Building on work on non–commutative Witt vectors by Goerss, Lannes, and Morel, we classify such free, ‘split’ Hopf algebras. A topological consequence is that, if X is a based path connected space, then the Hopf algebra H ⁎ ( Ω Σ X ; k ) is determined by the stable homotopy type of X. We also discuss the much easier analogous characteristic 0 results, and give a characterization of when our quasi–shuffle algebras are polynomial, generalizing the so–called Ditters conjecture.

In the early 1990's, Lionel Schwartz gave a lovely characterization of the Krull filtration of U, the category of unstable modules over the mod p Steenrod algebra. Soon after, this filtration was used by the author as an organizational tool in posing and studying some topological nonrealization conjectures. In recent years the Krull filtration of U has been similarly used by Castellana, Crespo, and Scherer in their study of H--spaces with finiteness conditions, and Gaudens and Schwartz have given a proof of some of my conjectures. In light of these topological applications, it seems timely to better expose the algebraic properties of the Krull filtration., 21 pages

If O is a reduced operad in symmetric spectra, an O-algebra I can be viewed as analogous to the augmentation ideal of an augmented algebra. Implicit in the literature on Topological Andre-Quillen homology is that such an I admits a canonical (and homotopically meaningful) decreasing O-algebra filtration I > I^2 > I^3 > ... satisfying various nice properties analogous to powers of an ideal in a ring. In this paper, we are explicit about these constructions. With R a commutative S-algebra, we study derived versions of the circle product M o_O I, where M is an O-bimodule, and I is an O-algebra in R-modules. Letting M run through a decreasing O-bimodule filtration of O itself then yields the augmentation ideal filtration as above. The composition structure of the operad induces algebra maps from (I^i)^j to I^{ij}, fitting nicely with previously studied structure. As a formal consequence, an O-algebra map from I to J^d induces compatible maps from I^n to J^{dn}, for all n. This is an essential tool in the first author's study of Hurewicz maps for infinite loop spaces, and its utility is illustrated here with a lifting theorem., 21 pages

We study H^*(P), the mod p cohomology of a finite p--group P, viewed as an Out(P)--module. In particular, we study the conjecture, first considered by Martino and Priddy, that, if e_S in Z/p[Out(P)] is a primitive idempotent associated to an irreducible Z/p[Out(P)]--module S, then the Krull dimension of e_SH^*(P) equals the rank of P. The rank is an upper bound by Quillen's work, and the conjecture can be viewed as the statement that every irreducible Z/p[Out(P)]--module occurs as a composition factor in H^*(P) with similar frequency. In summary, our results are as follows. A strong form of the conjecture is true when p is odd. The situation is much more complex when p=2, but is reduced to a question about 2--central groups (groups in which all elements of order 2 are central), making it easy to verify the conjecture for many finite 2--groups, including all groups of order 128, and all groups that can be written as the product of groups of order 64 or less. Featured is the nilpotent filtration of the category of unstable A--modules. Also featured are unstable algebras of cohomology primitives associated to central group extensions., 29 pages

In the mid 1980's, Pete Bousfield and I constructed certain p--local `telescopic' functors Phi_n from spaces to spectra, for each prime p and each positive integer n. These have striking properties that relate the chromatic approach to homotopy theory to infinite loopspace theory: roughly put, the spectrum Phi_n(Z) captures the v_n periodic homotopy of a space Z. Recently there have been a variety of new uses of these functors, suggesting that they have a central role to play in calculations of periodic phenomena. Here I offer a guide to their construction, characterization, application, and computation., Comment: 30 pages

In this paper, we show that Goodwillie calculus, as applied to functors from stable homotopy to itself, interacts in striking ways with chromatic aspects of the stable category. Localized at a fixed prime p, let T(n) be the telescope of a v_n self map of a finite S--module of type n. The Periodicity Theorem of Hopkins and Smith implies that the Bousfield localization functor associated to T(n) is independent of choices. Goodwillie's general theory says that to any homotopy functor F from S--modules to S--modules, there is an associated tower under F, {P_dF}, such that F --> P_dF is the universal arrow to a d--excisive functor. Our first theorem says that P_dF --> P_{d-1}F always admits a homotopy section after localization with respect to T(n) (and so also after localization with respect to Morava K--theory K(n)). Thus, after periodic localization, polynomial functors split as the product of their homogeneous factors. This theorem follows from our second theorem which is equivalent to the following: for any finite group G, the Tate spectrum t_G(T(n)) is weakly contractible. This strengthens and extends previous theorems of Greenlees--Sadofsky, Hovey--Sadofsky, and Mahowald--Shick. The Periodicity Theorem is used in an essential way in our proof. The connection between the two theorems is via a reformulation of a result of McCarthy on dual calculus., 25 pages. AmsLatex. Uses XYpics

If Fq is the nite eld of characteristic p and order q = p s , let F(Fq) be the category whose objects are functors from nite dimensional Fq{vector spaces to Fq{vector spaces, and with morphisms the natural trans- formations between such functors. We dene an innite lattice of thick subcategories of F(Fq). Our main re- sult then identies various subquotients as categories of modules over products of symmetric groups, via recollement diagrams. Our lattice of thick subcategories is a renemen t of the Eilenberg{MacLane polynomial degree ltration F 0 (Fq) F 1 (Fq) F 2 (Fq) : : : of F(Fq) which has been extensively studied and used in the algebraic K{theory lit- erature. Our main theorem implies a description of F d (Fq)=F d 1 (Fq) that renes and extends earlier results of Pirashvili and others. If q r, one of the subcategories is the Friedlander{Suslin category P r of 'strict polynomial functors of degree r', equivalent to the category of modules over the Schur algebra S(n; r) with n r. Our results can thus also be viewed as rening and extending the classic relationship between S(n; r){modules and r{modules via the Schur functor. In fact, (essentially) all our subcategories are equivalent to categories of modules over various nite dimensional algebras, and our lattice can be interpreted in terms of lattices of idempotent two sided ideals in these generalized Schur algebras. Applications include a simple proof, free of algebraic group theory, of a generalized Steinberg Tensor Product Theorem. This then implies the classic theorem for GLn(Fq), shedding some new light on this classic result. Our tensor product theorem is then used to study when various of our generalized Schur algebras are Morita equivalent. We use two technical tricks which may be of some independent interest. Firstly, we 'twist' by an action of the Galois group Gal(Fq; Fp) to be able to work entirely with vector spaces over the prime eld Fp, making discussions of 'base change' unnecessary. Secondly, we systematically use 'functors with product', a.k.a. lax symmetric monoidal functors, to dene our subcategories.

Let Rep(F;K) denote the category of functors from finite dimensional F-vector spaces to K-modules, where F is a field and K is a commutative ring. We prove that, if F is a finite field, and Char F is invertible in K, then the K-linear abelian category Rep(F;K) is equivalent to the product, over all k=0,1,2, ..., of the categories of K[GL(k,F)]-modules. As a consequence, if K is also a field, then small projectives are also injective in Rep(F;K), and will have finite length. Even more is true if Char K = 0: the category Rep(F;K) will be semisimple. In a last section, we briefly discuss "q=1" analogues and consider representations of various categories of finite sets. The main result follows from a 1992 result by L.G.Kovacs about the semigroup ring K[M_n(\F)]., 11 pages revised version: reference to Helmstutler's work is updated

Let BG be the classifying space of a finite group G. Given a multiplicative cohomology theory E ⁄ , the assignment G 7i! E ⁄ (BG) is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for complex oriented cohomology theories E ⁄ , using the theory of complex representations of finite groups as a model for what one would like to know. An analogue of Artin's Theorem is proved for all complex oriented E ⁄ : the abelian subgroups of G serve as a detecting family for E ⁄ (BG), modulo torsion dividing the order of G. When E ⁄ is a complete local ring, with residue field of characteristic p and associated formal group of height n, we construct a character ring of class functions that computes 1 E ⁄ (BG). The domain of the characters is Gn,p, the set of n-tuples of elements in G each of which has order a power of p. A formula for induction is also found. The ideas we use are related to the Lubin Tate theory of formal groups. The construction applies to many cohomology theories of current interest: completed versions of elliptic cohomology, E ⁄ n- theory, etc. The nth Morava K-theory Euler characteristic for BG is computed to be the number of G-orbits in Gn.p. For various groups G, including all symmetric groups, we prove that K(n)⁄(BG) concentrated in even degrees. Our results about E⁄(BG) extend to theorems about E⁄(EG◊GX), where X is a finite G-CW complex.

In the mid 1970s Mark Mahowald constructed a new infinite family of elements in the 2-component of the stable homotopy groups of spheres, ηj∈πSj2 (S0)(2) [M]. Using standard Adams spectral sequence terminology (which will be recalled in Section 3 below), ηj is detected by h1hj∈Ext2,*[Ascr ] (Z/2, Z/2). Thus he had found an infinite family of elements all having the same Adams filtration (in this case, 2), thus dooming the so-called Doomsday Conjecture. His constructions were ingenious: his elements were constructed as composites of pairs of maps, with the intermediate spaces having, on one hand, a geometric origin coming from double loopspace theory and, on the other hand, mod2 cohomology making them amenable to Adams Spectral Sequence analysis and suggesting that they were related to the new discovered Brown–Gitler spectra [BG].In the years that followed, various other related 2-primary infinite families were constructed, perhaps most notably (and correctly) Bruner's family detected by h2h2j∈ Ext3,*[Ascr ](Z/2, Z/2) [B]. An odd prime version was studied by Cohen [C], leading to a family in πS∗(S0)(p) detected by h0bj∈ Ext3,*[Ascr ] (Z/p, Z/p) and a filtration 2 family in the stable homotopy groups of the odd prime Moore space. Cohen also initiated the development of odd primary Brown–Gitler spectra, completed in the mid 1980s, using a different approach, by Goerss [G], and given the ultimate ‘modern’ treatment by Goerss, Lannes and Morel in the 1993 paper [GLM]. Various papers in the late 1970s and early 1980s, e.g. [BP, C, BC], related some of these to loopspace constructions.Our project originated with two goals. One was to see if any of the later work on Brown–Gitler spectra led to clarification of the original constructions. The other was to see if taking advantage of post Segal Conjecture knowledge of the stable cohomotopy of the classifying space BZ/p would help in constructing new families at odd primes, in particular a conjectural family detected by h0hj∈ Ext2,*[Ascr ] (Z/p, Z/p). (This followed a paper [K1] by one of us on 2 primary families from this point of view.)

We work in the stable homotopy category of p p –complete connective spectra having mod p p homology of finite type. H ∗ ( X ) H^*(X) means cohomology with Z / p \mathbf {Z}/p coefficients, and is a left module over the Steenrod algebra A \mathcal {A} . A spectrum Z Z is called spacelike if it is a wedge summand of a suspension spectrum, and a spectrum X X satisfies the Brown–Gitler property if the natural map [ X , Z ] → Hom A ⁡ ( H ∗ ( Z ) , H ∗ ( X ) ) [X,Z] \rightarrow \operatorname {Hom}_{\mathcal {A}}(H^*(Z),H^*(X)) is onto, for all spacelike Z Z . It is known that there exist spectra T ( n ) T(n) satisfying the Brown–Gitler property, and with H ∗ ( T ( n ) ) H^*(T(n)) isomorphic to the injective envelope of H ∗ ( S n ) H^*(S^n) in the category U \mathcal {U} of unstable A \mathcal {A} –modules. Call a spectrum X X standard if it is a wedge of spectra of the form L ∧ T ( n ) L \wedge T(n) , where L L is a stable wedge summand of the classifying space of some elementary abelian p p –group. Such spectra have U \mathcal {U} –injective cohomology, and all U \mathcal {U} –injectives appear in this way. Working directly with the two properties of T ( n ) T(n) stated above, we clarify and extend earlier work by many people on Brown–Gitler spectra. Our main theorem is that, if X X is a spectrum with U \mathcal {U} –injective cohomology, the following conditions are equivalent: (A) there exist a spectrum Y Y whose cohomology is a reduced U \mathcal {U} –injective and a map X → Y X \rightarrow Y that is epic in cohomology, (B) there exist a spacelike spectrum Z Z and a map X → Z X \rightarrow Z that is epic in cohomology, (C) ϵ : Σ ∞ Ω ∞ X → X \epsilon :\Sigma ^{\infty }\Omega ^{\infty }X \rightarrow X is monic in cohomology, (D) X X satisfies the Brown–Gitler property, (E) X X is spacelike, (F) X X is standard. ( M ∈ U M \in \mathcal {U} is reduced if it has no nontrivial submodule which is a suspension.) As an application, we prove that the Snaith summands of Ω 2 S 3 \Omega ^2S^3 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 n n –connected space X X admits a map to an n n –fold suspension that is monic in mod p p homology, then ϵ : Σ n Ω n X → X \epsilon : \Sigma ^n\Omega ^n X \rightarrow X is onto in mod p p homology.

The N-terminal 15 amino acids of the major protein associated with inorganic pyrophosphatase activity in Bacillus subtilis WB600 are identical to those of B. subtilis ORF yybQ. This ORF was amplified from B. subtilis WB600 DNA by PCR and cloned into an overexpression vector in Escherichia coli. Induction of overexpression produced a soluble protein of 34000 Da by SDS-PAGE and by matrix-assisted laser desorption and ionization mass spectrometry. The overexpressed protein had a high specific activity for the hydrolysis of magnesium pyrophosphate, and was specifically and reversibly activated by Mn2+ ions. These properties are identical to those of inorganic pyrophosphatase purified from B. subtilis WB600. No significant similarity was found between the derived sequence of the B. subtilis yybQ-encoded protein and published sequences of identified inorganic pyrophosphatases of Eukarya, Bacteria or Archaea domains. However, there is significant similarity to three putative proteins of unknown function from the archaea Methanococcus jannaschii and Archaeoglobus fulgidus, and from Streptococcus gordonii. The genomes of B. subtilis, M. jannaschii and A. fulgidus do not contain sequences similar to those of hitherto known soluble inorganic pyrophosphatases. The present findings, together with a survey of the properties of inorganic pyrophosphatases from 38 different sources, suggest that the B. subtilis yybQ-encoded protein is the first fully characterized member of a new class of inorganic pyrophosphatase.

With F q a finite field of characteristic p , let [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]( q ) be the category whose objects are functors from finite dimensional F q -vector spaces to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]-vector spaces. Friedlander and Suslin have introduced a category [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] of "strict polynomial functors" which has the same relationship to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]( q ) that the category of rational GL m -modules has to the category of GL m ( F q )-modules. Our main theorem says that, for all finite objects F, G ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /], and all s , the natural restriction map from [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="06i" /] ( F ( k ) , G ( k ) ) to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="07i" /] ( F, G ) is an isomorphism for all large enough k and q . Here F ( k ) denotes F twisted by the Frobenius k times. This combines with an analogous theorem of Cline, Parshall, Scott, and van der Kallen to show that, for all finite F, G ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="08i" /], and all s , evaluation on an m dimensional vector space V m induces an isomorphism from [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="09i" /] ( F, G ) to Ext s GL m ( F q ) ( F ( V m ), F ( V m )) for all large enough m and q . Thus group cohomology of the finite general linear groups has often been identified with MacLane (or Topological Hochschild) cohomology.

I fFq is the nite eld of order q and characteristic p ,l etF(q) be the category whose objects are functors from nite dimensional Fq{vector spaces to Fq{vector spaces, and with morphisms the natural transformations between such functors. Important families of objects in F(q) include the families Sn;S n ; n ; S n ,a ndcT n ,w ithc 2 Fq(n), dened by Sn(V )= (V n )n ,S n (V )= V n =n , n (V )= n th exterior power of V , S(V )= S(V )=(p th powers), and cT n (V )= c(V n ). Fixing F , we discuss the problem of computing HomF(q)(Sm;FG), for all m, given knowledge of HomF(q)(Sm;G) for all m .W henq = p ,w e get a complete answer for any functor F chosen from the families listed above. Our techniques involve Steenrod algebra technology, and, indeed, our most striking example, when F = S n , arose in recent work on the homology of iterated loopspaces.

In the early 1980's the author proved G.W. Whitehead's conjecture about stable homotopy groups and symmetric products. In the mid 1990's, Arone and Mahowald showed that the Goodwillie tower of the identity had remarkably good properties when specialized to odd dimensional spheres. In this paper we prove that these results are linked, as has been long suspected. We give a state-of-the-art proof of the Whitehead conjecture valid for all primes, and simultaneously show that the identity tower specialized to the circle collapses in the expected sense. Key to our work is that Steenrod algebra module maps between the primitives in the mod p homology of certain infinite loopspaces are determined by elements in the mod p Hecke algebras of type A. Certain maps between spaces are shown to be chain homotopy contractions by using identities in these Hecke algebras., 27 pages. As accepted for publication by the Journal of Topology. New: section 2 has been expanded, section 8 has been improved, and a dedication has been added

We define a category, H p n \mathcal {H}_p^n (for each n n and p p ), of spaces with strong homotopy commutativity properties. These spaces have just enough structure to define the mod p \bmod p Dyer-Lashof operations for n n -fold loop spaces. The category H p n \mathcal {H}_p^n is very convenient for applications since its objects and morphisms are defined in a homotopy invariant way. We then define a functor, P p n P_p^n , from H p n \mathcal {H}_p^n to the homotopy category of spaces and show P p n P_p^n to be left adjoint to the n n -fold loop space functor. We then show how one can exploit this adjointness in cohomological calculations to yield new results about iterated loop spaces.

We study the spectral sequence that one obtains by applying mod 2 homology to the Goodwillie tower which sends a spectrum X to the suspension spectrum of its 0th space X_0. This converges strongly to H_*(X_0) when X is 0-connected. The E^1 term is the homology of the extended powers of X, and thus is a well known functor of H_*(X), including structure as a bigraded Hopf algebra, a right module over the mod 2 Steenrod algebra A, and a left module over the Dyer-Lashof operations. Hopf algebra considerations show that all pages of the spectral sequence are primitively generated, with primitives equal to a subquotient of the primitives in E^1. We use an operad structure on the tower and the Z/2 Tate construction to show how Dyer-Lashof operations and differentials interact. These then determine differentials that hold for any spectrum X. These universal differentials then lead us to construct, for every A-module M, an algebraic spectral sequence depending functorially on M. The algebraic spectral sequence for H_*(X) agrees with the topological spectral sequence for X for many spectra, including suspension spectra and almost all generalized Eilenberg-MacLane spectra, and appears to give an upper bound in general. The E^infty term of the algebraic spectral sequence has form and structure similar to E^1, but now the right A-module structure is unstable. Our explicit formula involves the derived functors of destabilization as studied in the 1980's by W. Singer, J. Lannes and S. Zarati, and P. Goerss., Comment: 45 pages, latest version has reorganized examples, and makes better use of Hopf algebra methods. Now appeared in AGT 13 (2013)

Suppose that A2 is the mod-2 Steenrod algebra, and that R = F2[x1, . . . , xn] is the polynomial ring in n variables over the prime field F2. Campbell and Selick [3] observed that the equations Sqxi = x 2 i−1 for 2≤ i≤ n, and Sqx1 = x 2 n, define an action of A2 on R that makes R isomorphic as an A2-module to the A2-module defined by the standard action on R. In that paper, they also pose the following question, due to Tom Hunter: does the equation Sqxi = ∑ jmijx 2 j (where M = [mij ] is any n-by-n matrix over F2) define an action of A2 on R? In this paper we give an affirmative answer to the above question and classify the actions defined in this way. More precisely, let S(V ) be the symmetric

We study bounds on nilpotence in H*(BG), the mod p cohomology of the classifying space of a compact Lie group G. Part of this is a report of our previous work on this problem, updated to reflect the consequences of Peter Symonds recent verification of Dave Benson's Regularity Conjecture. New results are given for finite p--groups, leading to good bounds on nilpotence in H*(BP) determined by the subgroup structure of the p--group P., Comment: 22 pages, better organized invariant theory results in section 4. Final version is in Proc. E.M.S. 56 (2013)

Manfred Stelzer has pointed out that part of Corollary 4.5 of our paper "Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology" [Alg. Geo. Top. 8 (2008), 2109--2129] was not sufficiently proved, and, indeed, is likely incorrect as stated. This necessitates a little more argument to finish the proof of the main theorem of the original paper. The statement of this theorem, and all the examples, remain unchanged., Comment: Notation in this correction corresponds to that in the published paper, when this differs from the original ArXiv submission

1. Manganese(II) buffers were set up with inorganic triphosphate, trimethylenediaminetetraacetate and tetramethylenediaminetetraacetate to study the Mn dependence of beta 1,4-galactosyltransferase (lactose synthetase) in preparations of rat mammary gland. 2. In intact particulate preparations, treated with the calcium ionophore A23187, lactose synthesis was abolished by chelators and restored by bivalent transition metal ions in a manner characteristic of activation site I of the pure enzyme. Ni(II) also activated, as did Mg at high concentration. 3. Only Mn(II) could restore endogenous rates, giving an apparent Km of 0.1-0.2 microM, and eliciting about 70% full activity without addition of a site II activator. 4. In purified Golgi membrane vesicles, Mn gave an apparent Km of 0.4 microM. This increased sharply to about 10 microM on permeabilization with filipin, lysis with detergents, solubilization with Triton X-100, or in the pure enzyme. Preparations of chemically undamaged Golgi vesicles, known to include a proportion of the enzyme on exposed membranes, exhibited both low-Km and high-Km components. 5. The response of particulate galactosyltransferase to apparently physiological concentrations of free Mn(II) ion is interpreted as either due to a sensitizing factor within the Golgi lumen, or to the accumulation of Mn at elevated concentrations. Alternatively, the high Km of the soluble enzyme may reflect proteolytic damage.

We prove a strengthened version of a theorem of Lionel Schwartz that says that certain modules over the Steenrod algebra cannot be the mod 2 cohomology of a space. What is most interesting is our method, which replaces his iterated use of the Eilenberg--Moore spectral sequence by a single use of the spectral sequence converging to the mod 2 cohomology of Omega^nX obtained from the Goodwillie tower for the suspension spectrum of Omega^nX. Much of the paper develops basic properties of this spectral sequence., Comment: 18 pages

This paper assembles published and new material to trace out the compartmental aspects of milk sugar synthesis in the mammary gland. The generation of lactose, from glucose and UDP-galactose, in thetrans cisternae of the Golgi stack is well established from biochemical and electron microscopy work. New experiments show that in the rat essentially all this lactose is available for sialyllation to α-2,3-sialyllactose, which accumulates in the same compartment. UDP arising from utilization of UDP-galactose inhibits lactose synthetase, but is removed by nucleoside diphosphatase. Therefore lactose synthetase, sialyltransferase and nucleoside diphosphatase form a coupled enzyme system located on the luminal face of the Golgi membrane. Transport of UMP to the cytosol, followed by its re-conversion via UDP and UDP-glucose to UDP-galactose, defines the operation of a uridine nucleotide cycle that links Golgi and cytosol compartments in the support of milk sugar synthesis. Evidence for carrier-mediated uniport or antiport of UMP and UDP-galactose across the Golgi membrane is discussed, together with the evidence for transport of glucose and inositol by small water-filled pores that forbid back diffusion of lactose, galactinol or sialyllactose. Possible intra-Golgi signals, achieving short-term regulation of milk sugar synthesis, include glucose, submicromolar free Mn(II) ions and cationic activator proteins. Experimental evidence for intracellular and intra-Golgi glucose concentrations in the range 0.1–0.4 mM imply lack of saturation of lactose synthetase. It is shown that lactose synthetase within undamaged Golgi membrane vesicles is about 100 times more sensitive to Mn ions than in damaged or solubilized membranes. Both lactose synthetase and sialyltransferase are greatly stimulated by basic proteins, and similar lactose synthease-activating material is extractable from Golgi preparations. Development of these features may designate the sorts of control exerted upon these Golgi enzymes, and perhaps others. This may uncover the mechanisms by which hormonal and nutritional changes, and manipulation of mammary tissue in vitro, cause profound alterations in milk sugar production.

This article contains a collection of problems contributed during the course of the conference.

Henn, Lannes, and Schwartz have introduced two invariants, d_0(G) and d_1(G), of the mod p cohomology of a finite group G such that H^*(G) is detected and determined by H^d(C_G(V)) for d no bigger than d_0(G) and d_1(G), with V < G p-elementary abelian. We study how to calculate these invariants. We define a number e(G) that measures the image of the restriction of H^*(G) to its maximal central p-elementary abelian subgroup. Using Benson--Carlson duality, we show that when $G$ has a p-central Sylow subgroup P, d_0(G) = d_0(P) = e(P), and a similar exact formula holds for d_1(G). In general, we show that d_0(G) is bounded above by the maximum of the e(C_G(V))'s, if Benson's Regularity Conjecture holds. In particular, the inequality holds for all groups such that the p--rank of G minus the depth of H^*(G) is at most 2. When we look at examples with p=2, we learn that d_0(G) is at most 7 for all groups with 2--Sylow subgroup of order up to 64, unless the Sylow subgroup is isomorphic to that of either Sz(8) (and d_0(G) = 9) or SU(3,4) (and d_0(G)=14). Enroute we recover and strengthen theorems of Adem and Karagueuzian on essential cohomology, and Green on depth essential cohomology, and prove theorems about the structure of cohomology primitives associated to central extensions., 51 pages Prop. 8.1 now given a correct proof

This paper is based on talks I gave in Nagoya and Kinosaki in August of 2003. I survey, from my own perspective, Goodwillie's work on towers associated to continuous functors between topological model categories, and then include a discussion of applications to periodic homotopy as in my work and the work of Arone-Mahowald., Comment: This is the version published by Geometry & Topology Monographs on 29 January 2007

Let Map(K,X) denote the space of pointed continuous maps from a finite cell complex K to a space X. Let E_* be a generalized homology theory. We use Goodwillie calculus methods to prove that under suitable conditions on K and X, Map(K, X) will send a E_*--isomorphism in either variable to a map that is monic in E_* homology. Interesting examples arise by letting E_* be K--theory, K be a sphere, and the map in the X variable be an exotic unstable Adams map between Moore spaces., Comment: 13 pages

We begin this paper by noting that, in a 1969 paper in the Transactions, M.C. McCord introduced a construction that can be interpreted as a model for the categorical tensor product of a based space and a topological abelian group. This can be adapted to Segal’s very special Γ-spaces - indeed this is roughly what Segal did — and then to a more modern situation:K ⊗ RwhereKis a based space andRis a unital, augmented, commutative, associative S-algebra.

Let Map_T(K,X) denote the mapping space of continuous based functions between two based spaces K and X. If K is a fixed finite complex, Greg Arone has recently given an explicit model for the Goodwillie tower of the functor sending a space X to the suspension spectrum \Sigma^\infty Map_T(K,X). Applying a generalized homology theory h_* to this tower yields a spectral sequence, and this will converge strongly to h_*(Map_T(K,X)) under suitable conditions, e.g. if h_* is connective and X is at least dim K connected. Even when the convergence is more problematic, it appears the spectral sequence can still shed considerable light on h_*(Map_T(K,X)). Similar comments hold when a cohomology theory is applied. In this paper we study how various important natural constructions on mapping spaces induce extra structure on the towers. This leads to useful interesting additional structure in the associated spectral sequences. For example, the diagonal on Map_T(K,X) induces a `diagonal' on the associated tower. After applying any cohomology theory with products h^*, the resulting spectral sequence is then a spectral sequence of differential graded algebras. The product on the E_\infty -term corresponds to the cup product in h^*(Map_T(K,X)) in the usual way, and the product on the E_1-term is described in terms of group theoretic transfers. We use explicit equivariant S-duality maps to show that, when K is the sphere S^n, our constructions at the fiber level have descriptions in terms of the Boardman-Vogt little n-cubes spaces. We are then able to identify, in a computationally useful way, the Goodwillie tower of the functor from spectra to spectra sending a spectrum X to \Sigma ^\infty \Omega ^\infty X., Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-28.abs.html

Let T(j) be the dual of the j th stable summand of Ω2 S 3 (at the prime 2) with top class in dimension j. Then it is known that T(j) is a retract of a suspension spectrum, and that the homotopy colimit of a certain sequence T(j)→ T(2j)→… is an infinite wedge of stable summands of K(V,1)’s, where V denotes an elementary abelian 2 group. In particular, when one starts with T(1), one gets K(Z/2, 1) = RP ∞ as one of the summands.

Openreading frame mj0608 of the Methanococcus jannaschii genome, recognized by its sequence similarity to that of the gene coding for class C inorganic pyrophosphatase in Bacillus subtilis, was cloned and over-expressed in Escherichia coli. The protein was purified and characterized by SDS-PAGE, M(r), and N-terminal sequence. Under suitable conditions it catalyzed the specific hydrolysis of PPi at about 600 micromol x min(-1) x mg(-1) at 25 degrees C, and at 8000 micromol x min(-1) x mg(-1) at 85 degrees C. Therefore this protein is a specific inorganic pyrophosphatase. The activities of Mg(2+), Mn(2+), Co(2+), and Zn(2+) ions as cofactors for hydrolysis of PPi were compared at pH 7.5 and 9.0. Unlike the class C pyrophosphatase of B. subtilis, this enzyme required no prior activation by low concentrations of Mn(2+) or Co(2+) ions. However, prior exposure to these ions afforded striking protection against inhibition by sodium fluoride, to which the enzyme was otherwise very sensitive.

If F q is the finite field of characteristic p and order q= p s , let F(q)be the category whose objects are functors from finite dimensional F q -vector spaces to F q -vector spaces, and with morphisms the natural transformations between such functors. We survey the basic structure of this category and its close connections to the finite general linear groups, the symmetric groups, classical Schur algebras, algebraic K-theory, and the Steenrod algebra.

The hydrolysis of magnesium pyrophosphate by inorganic pyrophosphatase from Bacillus subtilis required its specific, time-dependent, and prior activation by Mn2+ ions. This was reversed when Mn2+ ions were removed with EDTA. Free Mn2+ ions were not required for catalysis. Pyrophosphatase purified to near homogeneity gave a single main band of apparent M(r) 36,000 by SDS-PAGE, but of M(r) 34,000 by matrix-assisted laser desorption ionization-mass spectrometry. The native enzyme equilibrated at pH 7 between three distinct molecular forms. Exposure to Mn2+ generated a catalytically active trimer of specific activity about 5000 mumol pyrophosphate hydrolyzed/min/mg protein. Exposure to EDTA generated two catalytically inactive forms, a dimer at low ionic strength and a separate form, of uncharacterized multimeric nature, at molar concentrations of Na2SO4 or Li2SO4. The latter form was an intermediate in the dimer-trimer transition caused by addition or removal of manganese ions. Mn2+ reacted with this "intermediate" form, apparently by reversible association with two noninteracting binding sites of Kd approximately 0.005 and 0.35 microM, respectively. The properties of this enzyme may account in part for the unusual manganese requirements of B. subtilis and related species.

Purification of human liver arginase by chromatography on DEAE-Sepharose, CM-Sepharose, hydroxylapatite, and MonoS yielded protein of greater than 95% purity by sodium dodecyl sulfate-gel electrophoresis. Detailed kinetic studies of the interconversion of active and inactive forms of arginase showed the effects of metal ion addition and withdrawal, metal ion type, time, temperature, and pH. At pH 7 and 37°C, removal of Mn 2+ caused a first-order deactivation with half-life of 1 h. Reactivation was completed within 0.5 min (1 mM Mn 2+ ) or 90 min (ca. 6 nM Mn 2+ ). Activation by Mn 2+ showed a hyperbolic response, with K d for Mn 2+ of about 36 nM. Mn 2+ apparently displaced about 2 H + , resulting in sigmoid dependence upon concentration of OH − . Both the maximal velocity of catalysis and the K m toward arginine were markedly pH-dependent in the physiological range. The findings lead to a model where Mn 2+ allosterically activates arginase by a sequential, and pH-sensitive, mechanism. The combined pH sensitivities of activation, V max , and K m are likely to give arginase a role in mediating the demonstrated pH control of the ornithine cycle and hence in the regulation of body pH.

The purpose of this announcement is to describe, on one hand, a new approach to the modular representation theory of the semigroup of n × n matrices M n (F q ) and its group of units GL n (F q ), and, on the other, a new approach to studying the category of unstable modules over the Steenrod “q th-power” operations. Perhaps most remarkable is that these two approaches are the same via a generalization of classical Morita equivalence.

This paper is intended to be an informal introduction to [HKR], where we study the Morava K–theories (which will be partially defined below) of the classifying space of a finite group G and related matters. It will be long on exposition and short on proofs, in the spirit of the third author’s lecture at the conference. In Section 1 we will recall Atiyah’s theorem relating the complex K–theory of BG to the complex representation ring R(G). We will also define classical characters on G. In Section 2 we will introduce the Morava K–theories K(n)∗ and the related theories E(n)∗. In Section 3 we will state our main results and conjectures. Most of the former are generalizations of the classical results stated in Section 1. In the remaining five sections we will outline the proofs of our results. Section 4 is purely group–theoretic, i.e., it makes no use of any topology. In it we will prove our formula for the number χn,p(G) of conjugacy classes of commuting n– tuples of elements of prime power order in a finite group G. We have discussed this material with several prominent group theorists, but we have yet to find one who admits to ever having considered this question. In Section 5 we equate this number with the Euler characteristic of K(n)∗(BG). Our generalized characters are functions on the set of such conjugacy classes with values in certain p–adic fields. In Section 6 we recall the Lubin–Tate construction from local algebraic number theory. It uses formal group laws to construct abelian extensions of finite extensions of the field of p–adic numbers. We need it in Section 7 where we describe the connection between E(n)∗(BG) and our generalized characters. In Section 8, we prove a theorem about wreath products. A corollary of this is that our main conjecture (3.5) about K(n)∗(BG) holds for all the symmetric ∗Partially supported by the National Science Foundation †Partially supported by the NSF, the Sloan Foundation and the SERC ‡Partially supported by the NSF

The manganese dependence of arginase was reinvestigated with extracts of mouse liver to see whether more physiological properties were displayed than have been reported for the purified enzyme. In a preincubation with Mn(II) ions at 37 °C the enzyme underwent a slow and reversible activation. At least 90–95% of the activation achieved was dependent on Mn 2+ . However, no Mn 2+ was required for catalytic activity in the assay. The activation showed little dependence upon pH over the range 6.5–9.5, whereas the catalytic activity increased 12-fold in apparent accord with the titration curve of an ionizable group of p K α 7.9. The Mn 2+ dependence of arginase activation obeyed Michaelis-Menten kinetics, with K d varying from 0.3 μ m at pH 6.8 to 0.08 μ m at pH 7.7. Free Mn 2+ concentrations were established in these assays with a trimethylenediaminetetraacetate-Mn buffer. V max increased about three-fold over this range. The calculated arginase activity at 0.05 μ m Mn 2+ increases about nine-fold over this physiological pH range. An enzyme model is proposed to explain these findings. The activity of arginase at “physiological” [Mn 2+ ] and the pronounced pH dependence conferred upon it are consistent with a recently revised role for the urea cycle in the control of bicarbonate and pH in the body. It appears possible that arginase loses Mn 2+ sensitivity during the usual purification.

The mod p cohomology of a space or spectrum X is a module over the mod p Steenrod algebra A. If X is a space, it has additional structure making it an unstable A-algebra, where we recall that an A-module M is unstable if, when p = 2, Sqi x = 0 for all i > IxI and x E M, and, when p > 2, 3ePix = 0 for all 2i + e > JxJ, e = 0 or 1, and x E M. Questions about realizing particular A-modules, algebras, and A-algebras as the cohomology of spaces or spectra are among the most fundamental in algebraic topology. The Hopf invariant one problem [A] and the problem of realizing polynomial algebras [DMW] are two famous examples of ongoing interest. If one regards the first of these as a problem about realizing finite A-modules, and the second of these as a problem about realizing rather large A-modules, it is the purpose of this paper to study the "middle ground". We begin with the following conjecture.