Happy New Year! This is the one-year anniversary of this modest service. We have three new papers at Clarence Wilkerson's archive. Instructions at the end. Mark Hovey Papers uploaded to Hopf between December 8,1995 and January 6, 1996: 1. /pub/Lesh/uass The Unstable Adams Spectral Sequence for Two-Stage Towers Kathryn Lesh University of Toledo Let KP denote a mod 2 Eilenberg-MacLane space. In this paper we study the mod 2 unstable Adams spectral sequence (UASS) of a space Y with polynomial cohomology which is obtained as the fiber of a map between simply connected spaces X \rightarrow KP to which the Massey-Peterson theorem applies. By using specially constructed Adams resolutions, we study the relationship between the UASS for X and for Y. Given conditions on X, we give information on the E_{2} term of the UASS for Y which actually gives a splitting of Ext under some circumstances. In the case that the UASS for X collapses at E_{2}, we show how to use a primary level calculation to compute almost complete information about the d_{2} differentials in the UASS for Y. 2./pub/Lesh/inf-loop Infinite Loop Spaces from Group Theory Kathryn Lesh University of Toledo Abstract: We generalize the Barratt-Priddy-Quillen theorem \Omega B (\coprod B\Sigma_{n}) is homotopy equivalent to QS^{0} by using tom Dieck's classifying spaces for a family of subgroups of a group. We show how to take a compatible choice of families F_{n} of subgroups of \Sigma_{n} and obtain an infinite loop space by group completing \coprod BF_{n}. The spaces QS^{0} and K(Z,0) are recovered as extreme cases and the infinite loop spaces obtained from other families can be thought of as interpolating between stable homotopy and integral homology. We study two special cases: the family of subgroups of the alternating groups, and the family generated by elementary abelian p-subgroups whose generators are disjoint p-cycles. We compute the infinite loop spaces which are formed in these cases and show that the latter is closely related to the cofiber of the transfer map. This paper will appear in Mathematische Zeitschrift. 3. /pub/Lesh/hybrids (The following paper is a revision of a paper already on the archive) Hybrid Spaces with Interesting Cohomology Kathryn Lesh University of Toledo Abstract: Let p be an odd prime, and let R be a polynomial algebra over the Steenrod algebra with generators in dimensions prime to p. To such an algebra is associated a p-adic pseudoreflection group W, and we assume that W is of order prime to p and irreducible. Adjoin to R a one-dimensional element z, and give R[z] an action of the Steenrod algebra by \beta z = 0 and \beta x = (|x|/2) zx for an even dimensional element x. We show that the subalgebra of elements of R[z] consisting of elements of degree greater than one is realized uniquely, up to homotopy, as the cohomology of a p-complete space. This space can be thought of as a cross between spaces studied by Aguade, Broto, and Notbohm, and the Clark-Ewing examples, further studied by Dwyer, Miller, and Wilkerson. This paper has appeared as TAMS 347, 3247-3262 (1995) ----------------- Five new papers this time. For some reason titles and author names are missing in most of these abstracts. I preferred them with titles and author names myself. Remember the way your abstract looks is entirely up to you. Instructions at the end. Mark Hovey Papers uploaded to Hopf between January 7,1996 and February 11, 1996: 1. /pub/Brown-Wensley/ind-nrm4 Computing crossed modules induced by an inclusion of a normal subgroup, with applications to homotopy 2-types Ronald Brown and Christopher D. Wensley School of Mathematics University of Wales, Bangor Gwynedd LL57 1UT, U.K. email: r.brown, c.d.wensley --- bangor.ac.uk We obtain some explicit determinations of crossed Q-modules induced from a crossed module over a normal subgroup P of Q. By virtue of theorems of Brown and Higgins, this enables the computation of the homotopy 2-types, and so of the second homotopy modules, of homotopy pushouts of maps of classifying spaces of discrete groups induced by certain morphisms of the groups. In some cases, the first k-invariant is calculated, using methods of free crossed resolutions. 2. /pub/Chacholski/SF12/ The outcome of this paper is two surprising facts: Theorem 1 Let F-->E-->B be any fibration over a connected space B, and h* be any homology theory. If F-->E is h* isomorphism, then B is h* acyclic Theorem 2 Let A and Y be pointed spaces and A-->X be any map. Let F be the homotopy fiber of the cofiber map X-->X/A. If A is weakly equivalent to a suspension of a connected space, then the following statements about Y are equivalent: -) map*(A,Y) is weakly contractible -) map*(F,Y) is weakly contractible 3. /pub/Johnson-Wilson/johnson-wilson-bu If $V$ is an elementary abelian $2$-group, Ossa proved that the connective $K$-theory of $BV$ splits into copies of ${\bf Z}/2$ and of the connective $K$-theory of the infinite real projective space. We give a brief proof of Ossa's theorem. 4. /pub/Moller/deterministic Deterministic p-compact group (J.M. Moller) A p-compact group is said to be deterministic if it is determined by the normalizer of a maximal torus. It is said to have N-determined automorphisms if any two of its automorphisms are determined by their restrictions to the normalizer of the maximal torus. We consider the class of p-compact groups with N-determined automorphisms and the, not unrelated, class of deterministic p-compact groups. The paper also contains some speculations that these two classes in fact comprise all p-compact groups. 5. /pub/Silverman/strip \noindent Let $\Tof{k}{f} = \sq{2^{k-1}f} \cdot \sq{2^{k-2}f} \cdot \ldots \cdot \sq{2f} \cdot \sq{f}$ in the mod-2 Steenrod algebra $\stnstar$, and let $\chi$ denote the canonical antiautomorphism of $\stnstar$. Given positive integers $k$, $\Lambda$ and $j$ with $1 \leq j \leq \Lambda$, we prove that \begin{eqnarray*} \chi \Tof{k}{2^{\Lambda}-j} & = & \Tof{\Lambda-(j-1)}{2^{j-1}(\spike{k})} \cdot \chi \Tof{k}{2^{j-1}-j}, \end{eqnarray*} generalizing formulae of Davis and the author. Given a positive integer $f$, denote by $\mu(f)$ the minimal number of summands in any representation of $f$ in the form $\sum (\spike{i_k})$. The antiautomorphism formula above implies that for $f = 2^{\Lambda}-j$,\ $1 \leq j \leq \Lambda +2$, the excess of $\chi \Tof{k}{f}$ satisfies $\ex{\chi \Tof{k}{f}} = (\spike{k})\mu(f)$ for all $k$, confirming the conjecture of the author ({\em Proc. Amer. Math. Soc.}, 119(2):657-661, 1993) for such $f$. We also prove that $\ex{\chi \Tof{k}{f}} \leq (\spike{k}) \mu(f)$ for all $f$ and $k$. ----------------- Two new papers, a revised version, and an improved abstract this time. Clarence asks me to tell everyone to please send him e-mail when you upload a file to the archive. I have given up on trying to make html versions of these messages. This has the advantage that my home page will now always have the latest messages, though in text form. Mark Hovey Papers uploaded to Hopf between February 11,1996 and April 4, 1996: 1. /pub/Brown-Golasinski-Porter-Tonks/equivcrs2.dvi Spaces of maps into classifying spaces for equivariant crossed complexes by R Brown, M Golasi\'{n}ski, T Porter, and A Tonks. ABSTRACT We give an equivariant version of the homotopy theory of crossed complexes. The applications generalize work on equivariant Eilenberg-Mac Lane spaces, including the non abelian case of dimension 1, and on local systems. It also generalizes the theory of equivariant 2-types, due to Moerdijk and Svensson. Further, we give results not just on the homotopy classification of maps but also on the homotopy types of certain equivariant function spaces. R. Brown, T.Porter School of Mathematics University of Wales, Bangor Gwynedd LL57 1UT United Kingdom r.brown,t.porter---bangor.ac.uk M. Golasinski Department of Mathematics Nicholas Copernicus University Torun Poland marek---mat.uni.torun.pl A. Tonks Institut Matematicas Univerdidad Autonoma Barcelona 08193 Bellaterra Barcelona Spain tonks---mat.uab.es 2. /pub/Greenlees/s1q (Note from Mark: as explained below, this paper can be downloaded whole or in 4 parts, with obvious titles) Title: Rational $S^1$-equivariant stable homotopy theory. Author: J.P.C.Greenlees Abstract: We make a systematic study of rational $S^1$-equivariant cohomology theories, or rather of their representing objects, rational $S^1$-spectra. In Part I we construct a complete algebraic model for the homotopy category of $S^1$-spectra, reminiscent of the localization theorem. The model is of homological dimension one, and simple enough to allow practical calculations; in particular we obtain a classification of rational $S^1$-equivariant cohomology theories. The model for semifree spectra is the derived category of the abelian category whose objects are $\Q [c]$-modules $N$ with a graded vector space $V$ giving an isomorphism $N[c^{-1}] = \Q [c , c^{-1}] \tensor V$; the model for arbitrary spectra is an appropriate generalization. In Part II we identify the algebraic counterparts of all the usual $S^1$-spectra and constructions on $S^1$-spectra. This enables us in Part III to give a rational analysis of a number of interesting phenomena, such as the Atiyah-Hirzebruch spectral sequence, the Segal conjecture, $K$-theory and topological cyclic cohomology. For reasons of size this is broken into four parts Part I (including Introduction and Table of Contents) Part II Part III Appendices (including Conventions and Index). Also available The whole thing (s1q123.dvi) Introduction and contents only (s1qIntro.dvi). 3./pub/Hovey-Palmieri-Strickland/axiomatic Axiomatic stable homotopy theory Mark Hovey, John Palmieri, and Neil Strickland This is the final version of a paper that was already on the archive. It is quite long. There are the usual bug fixes, plus an improved thick subcategory theorem, better understanding of morphisms of stable homotopy categories, a complete rewrite of the Noetherian section, with better results, and a couple of new sections on the Bousfield lattice. 4. /pub/Silverman/strip (This paper was announced last time, but the abstract was untexable. This version of the abstract is self-contained, and should be texable with a begin and end document thrown in. The paper itself has not changed.) Stripping and Conjugation in the Steenrod Algebra Judith H. Silverman Indiana University --- Purdue University Columbus judith---iu-math.math.indiana.edu To appear in Journal of Pure and Applied Algebra \noindent Let $S(k;f) = Sq^{2^{k-1}f} \cdot Sq^{2^{k-2}f} \cdot \ldots \cdot Sq^{2f} \cdot Sq^{f}$ in the mod-2 Steenrod algebra $A^*$, and let $\chi$ denote the canonical antiautomorphism of $A^*$. Given positive integers $k$, $\Lambda$ and $j$ with $1 \leq j \leq \Lambda$, we prove that \begin{eqnarray*} \chi S(k; 2^{\Lambda}-j) & = & S(\Lambda-(j-1); 2^{j-1}(2^k-1)) \cdot \chi S(k; 2^{j-1}-j), \end{eqnarray*} generalizing formulae of Davis and the author. Given a positive integer $f$, denote by $\mu(f)$ the minimal number of summands in any representation of $f$ in the form $\sum (2^{i_k}-1)$. The antiautomorphism formula above implies that for $f = 2^{\Lambda}-j$,\ $1 \leq j \leq \Lambda +2$, the excess of $\chi S(k;f)$ satisfies ${\mbox {ex}}(\chi S(k;f)) = (2^{k}-1)\mu(f)$ for all $k$, confirming the conjecture of the author ({\em Proc. Amer. Math. Soc.}, 119(2):657-661, 1993) for such $f$. We also prove that ${\mbox {ex}}(\chi S(k;f)) \leq (2^{k}-1) \mu(f)$ for all $f$ and $k$. ------------------- Six new papers this time. I remind you again that abstracts should be human-readable, and that you should send e-mail to Clarence when you upload a paper to hopf. Mark Hovey Papers uploaded to Hopf between April 4,1996 and Jun 3, 1996: 1. /pub/Broto-Crespo/hhhh H-spaces with noetherian mod two cohomology algebra by CARLOS BROTO and JUAN A.CRESPO Abstract: The object of this paper is to analyse the structure of connected H-spaces with noetherian mod two cohomology algebra. We will show that, up to 2-completion, they are, essentially, finite mod~2 $H$-spaces and their 3-connected covers, $\C P^\infty$, $B\Z/2^r$ and certain extensions of these. 2. /pub/Brown-Szczarba/szczarba \magnification=1200 \nologo \documentstyle{amsppt} \NoBlackBoxes \topmatter \title On the Rational Homotopy Type of Function Spaces\endtitle \author Edgar H. Brown, Jr. and Robert H. Szczarba\endauthor \abstract The main result of this paper is the construction of a minimal model for the function space $\Cal F(X,Y)$ of continuous functions from a finite type, finite dimensional space $X$ to a finite type, nilpotent space $Y$ in terms of minimal models for $X$ and $Y$. For the component containing the constant map, $\pi_*(\Cal F(X,Y))\otimes Q =\pi_*(Y)\otimes H^{-*}(X;Q)$ in positive dimensions. When $X$ is formal, there is a simple formula for the differential of the minimal model in terms of the differential of the minimal model for $Y$ and the coproduct of $H_*(X;Q)$. We also give a version of the main result for the space of cross sections of a fibration. \endabstract \endtopmatter 3. /pub/Intermont/vkampen An Equivariant van Kampen Spectral Sequence Michele Intermont Mesa State College Grand Junction, CO 81502 mintermo---mesa5.mesa.colorado.edu This paper constructs an equivariant homotopy spectral sequence for any finite group G, finite dimensional representation V, and two suitably connected G-CW complexes X and Y. The spectral sequence converges to the collection of equivariant homotopy groups of the wedge of X and Y, while the E^2 term depends only on the equivariant homotopy groups of X and of Y, along with primary homotopy operations. The edge homomorphism of the spectral sequence is actually an isomorphism in a range, which is the equivariant van Kampen theorem of L.G. Lewis, Jr. When G is the trivial group, the spectral sequence reduces to that of C.R. Stover. 4. /pub/Lindenstrauss/ram26 \nopagenumbers \magnification=\magstep1 \centerline{\bf Abstract of the paper:} \centerline{\bf The Topological Hochschild Homology of the Gaussian Integers} \smallskip \centerline{by Ayelet Lindenstrauss} \bigskip The calculation in this paper gives the $2$-torsion in the topological Hochschild homology of rings of integers in quadratic extensions of the rationals. In general, it is not hard to deduce the $p$-torsion in the topological Hochschild homology of rings of integers which do not ramify at $p$ from the corresponding torsion for the integers. Thus, for the ring of Gaussian integers, and also for the integers with the square root of $2$ or $-2$ adjoined, this paper completes the calculation of the topological Hochschild homology spectrum (which is a priori known to be a product of Eilenberg-MacLane spectra). Precisely because of this a priori knowledge, the homotopy groups can be found by doing a homology calculation. A spectral sequence arising from the filtration of topological Hochschild homology by simplicial `skeleta' converges to the desired homology. If one reduces the rings in question modulo $2$, the spectral sequence collapses at its second term. This comparison bounds the non-triviality of higher differentials in the spectral sequence for the original ring. It is explicitly demonstrated, using simplicial calculations, that the higher differentials are as non-trivial as they could be, given this bound. \end 5. /pub/Morava/Luminy2.abstract Quantum generalized cohomology Jack Morava \begin{center}{Abstract for {\bf Quantum generalized cohomology}}\bigskip \end{center} \noindent There is a variant of Segal's category of Riemann surfaces, in which morphisms are stable complex algebraic curves [i.e. double points are allowed], with some smooth points marked; composition is defined by glueing at marked points. The spaces of morphisms in this category are built from the compactified moduli spaces $\overline M_{g,n}$ of Deligne, Mumford, and Knudesen; here $g$ is the genus and $n$ is the number of marked points. A generalized topological field theory taking values in the category of module-spectra over a ring-spectrum $\bf R$ is a family $$\tau_{g,n} : \overline M_{g,n} \rightarrow {\bf M} \wedge_{\bf R} \dots \wedge_{\bf R} {\bf M} = {\bf M}^{\wedge n}$$ of maps, which respect composition of morphisms. More precisely, $\bf M$ is an $\bf R$-module spectrum, $\wedge_{\bf R}$ is the Robinson smash product, and $\bf M$ is endowed with a suitably nondegenerate bilinear form $${\bf M} \wedge_{\bf R} {\bf M} \rightarrow {\bf R}.$$ This data entails the existence of an $\bf R$-algebra structure on $\bf M$, such that $\tau_{g,1}$ is a morphism of monoids if the moduli space of curves is given the pair-of-pants product; it seems to define a natural context for quantum generalized cohomology.\medskip \noindent There is an interesting example of all this, associated to a smooth algebraic variety $V$. It is closely related to the Tate $\bf MU$-cohomology of the universal cover of the free loopspace of $V$, but it can be described more concretely in terms of the rational Novikov ring $\Lambda = {\Bbb Q} [H_{2}(V,{\Bbb Z})]$ of $V$ by setting ${\bf R} = {\bf MU} \otimes \Lambda$; then {\bf E} is the function spectrum $F(V,{\bf R})$ representing the cobordism of $V$ tensored with $\Lambda$, and the bilinear pairing is defined by Poincar\'e duality. In this case $\tau_{g,n}$ represents the cobordism class of the space of stable maps [in the sense of Kontsevich] from a curve of genus $g$, marked with $n$ ordered smooth points together with an indeterminate number of unordered smooth points, to $V$. A variant construction requires the unordered points to lie on a cycle $z$ in $V$; this defines a parameterized family of multiplications satisfying the analogue of the WDVV equation. When $V$ is a point, the resulting theory boils down to the version of topological gravity I advertised at the Adams Symposium; the coupling constant of the associated topological field theory is the cobordism analogue of Manin's exponential $$\sum_{n \geq 0} \overline M_{0,n+3} \frac {z^{n}}{n!} .$$ Although much of the machinery used here comes from fields adjacent to topology, this paper is concerned with the old problem of constructing complex cobordism out of Riemann surfaces by some analogue of the plus-construction. Having hacked through the physics background, I hope to produce a more topological account in the near future. \medskip \noindent This is to appear in Contemporary Math., in the Proceedings of the Hartford/Luminy Conference on the Renaissance of Operads, ed. J.-L. Loday, J. Stasheff, and A. A. Voronov. \end{document} 6. /pub/Rudyak/LScategory (The following abstract was unreadable--slight modifications were made by Mark to make it slightly more readable, but only slightly) SOME REMARKS ON CATEGORY WEIGHT Yu.B. Rudyak April 1996 Abstract. We develop and apply the conceptof category weight which was introduced by Fadell and Husseini. We remark that elements of maximal category weight enable us to control the Lusternik-Schnirelmann category of a space. For example, we prove that if f : N --> M is a map of degree 1 of closed stable parallelizable manifolds and dim M 2cat M4 (sic) then cat N cat M (sic). Furthermore, we prove that category weight of every non-trivial Massey product is at least 2. In particular, if the Massey product hu1; :;:u:ni,ui 2 eH(X) (sic), is defined and hu1; : :;:uni 6= 0 (sic) then catX > 1, even if 0 2 hu1; : :;:uni (sic). Using a result of Singhof, we also prove that cat(M Sm ) = catM + 1 provided (sic) ------------------ Two new papers this time. Mark Hovey Papers uploaded to Hopf between Jun 3,1996 and Jun 5, 1996: 1. /pub/Davis-Lueck/assembly Authors: James F. Davis and Wolfgang Lueck Title: Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory Email: jfdavis---indiana.edu, lueck---topologie.mathematik.uni-mainz.de We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic K- and L-theory of integral group rings and to the Baum-Connes Conjecture on the topological K-theory of reduced group C*-algebras. The approach is through spectra over the orbit category of a discrete group G. We give several points of view on the assembly map for a family of subgroups and describe such assembly maps by a universal property generalizing the results of Weiss and Williams to the equivariant setting. The main tools are spaces and spectra over a category and the study of the associated generalized homology and cohomology theories and homotopy limits. 2. /pub/Henn/sl3 \title{The cohomology of $SL(3,{\Z}[1/2])$} \author{Hans-Werner Henn} \date{ } \maketitle \begin{abstract} We compute the cohomology of $SL(3,{\Z}[1/2])$ with coefficients in the prime fields and in the integers. On the way we obtain the cohomology of certain mod - $2$ congruence subgroups of $SL(3,{\Z})$ with coefficients in ${\FF}_p$ for $p>2$. Finally we compute the cohomology of $GL(3,{\Z}[1/2])$. \end{abstract} ----------------- Four new papers this time. Clarence will be out of town for a few weeks. Mark Hovey Papers uploaded to Hopf between Jun 5,1996 and Jun 26, 1996: 1. /pub/Dwyer/sharp Sharp homology decompositions for classifying spaces of finite groups W. Dwyer Suppose that G is a finite group. A homology decomposition for BG is a way of constructing BG up to mod p homology as a homotopy colimit of classifying spaces of subgroups K of G. The decomposition is said to be sharp if the mod p homology spectral sequence associated to the homotopy colimit collapses to give an isomorphism between the H^*BG and colim H^*BK. We develop techniques for showing that a decomposition is sharp, and apply the techniques to a number of examples. 2. /pub/Dwyer-Wilkerson/kahler-differentials/stein Kahler differentials, the T-functor, and a theorem of Steinberg William G. Dwyer and Clarence W. Wilkerson (Notre Dame University and Purdue University) It is shown that any component of the Lannes T-functor applied to a finitely generated connected graded polynomial algebra with an unstable action of the mod p Steenrod algebra yields a finitely generated polynomial algebra. This is a purely algebraic analogue of the observation that for a connected Lie group with no homology p-torsion, each centralizer of a sub-elementary abelian p-group also has no homology p-torsion. It has the algebraic consequence, previously proved by Nakajima, that if W is a subgroup of GL(V) such that Symm[V#]^W is polynomial algebra, then for any subset U of V, the stabilizer group W_U has the same property. Examples are given of reflection groups in positive characteristic for which stabilizer subgroups are not reflection groups. These include $W(F_4)$ at $p=3$, $W(E_8)$ for $p=5$, and $W( SU(pN)/Z/pZ)$ at $p=p$, for most N. 3. /pub/Kashiwabara/bpqs Abstract for "Brown-Peterson cohomology of QS^{2n}" by Takuji Kashiwabara In this paper we introduce the notion of $BP$-destabilization functor, and we show that BP cohomology of QS^{2n}'s (n>0) are naturally isomorphic to the destabilization of BP-cohomology of S^{2n}. This describes BP cohomology of QS^{2n} completely algebraically. In the passage we also determine their Morava K-theories. 4. /pub/Kashiwabara/bpqx Abstract for "On Brown-Peterson cohomology of QX" by Takuji Kashiwabara In this paper we compute BP cohomology of QX when BP cohomology of X satisfies the Landweber's exact functor theorem (under some other technical hypotheses). This includes the cases such as sphere's, Eilenberg-Maclane spaces, and many classifying spaces. ----------------- Clarence has been on vacation for a while, so we have 12 new papers this time. Please include the name and title of the paper in the abstract--otherwise Clarence has to sort through them all himself. Mark Hovey Papers uploaded to Hopf between Jun 26,1996 and Aug 9, 1996: 1. /pub/Arone-Mahowald/identity The Goodwillie tower of the identity functor and the unstable $v_n$-periodic homotopy of spheres. Greg Arone and Mark Mahowald Goodwillie's tower of the identity functor is a tower of fibrations converging to unstable homotopy, whose fibers are infinite loop spaces. The fibers in this tower were first described by B. Johnson. We reformulate this description and investigate the tower in the case of a sphere. The main result is that in the case of a sphere the tower is finite in $v_n$-periodic homotopy. The proof involves calculating the stable cohomology of the fibers in the tower, which may be of independent interest. It is possible that some changes will still be made. Comments on the manuscript are most welcome. 2. /pub/Blanc/mcw % % David Blanc % Mapping spaces and M-CW complexes % July 9, 1996 % abstract: The concept of ``homotopy groups with coefficients'', in which spheres are replaced by a Moore spaces as the representing objects, were first studied by Peterson, and in greater detail by Neisendorfer. Much of homotopy theory can be redone in this spirit, with an arbitrary but fixed space $\M$ and its suspensions replacing the spheres not only in the definition of homotopy groups, but also in that of a $CW$-complex, loop space, and so on. In particular, an M-CW complex is a space constructed inductively by successively attaching M-cells. Some of the properties of ordinary $CW$ complexes carry over to M-CW complexes - e.g., the Whitehead theorem - but others do not. In this note we address the question of recovering the space X from the mapping space X^M, for a special class of ``self-map resolvable'' spaces M, a question analogous to the classical one of recovering X from its n-fold loop space. Just as for loop spaces, one needs some additional structure on X^M in order to do so. Our procedure for recovering X is given recursively by a sequence of homotopy colimits. We may also think of this procedure as another construction of an M-CW approximation functor. Our approach can be made more explicit in the case of the mod k Moore space. 3. /pub/Blanc/rat % David Blanc % Homotopy operations and rational homotopy type % June 19, 1996 % abstract: % We describe a collection of higher homotopy operations which determine the rational homotopy type of a simply-connected CW complex X. The (integral) homotopy type of X is determined by its homotopy groups pi_*X, together with the action of all primary homotopy operations on it, and of certain higher homotopy operations. However, Whitehead products are the only non-trivial primary homotopy operations on the rational homotopy groups, and the relevant higher order operations are also simpler than in the integral case. Here we exhibit a collection of higher homotopy operations, which, together with the rational homotopy Lie algebra itself, determine the rational homotopy type of X. These higher operations are certain subsets of pi_* X which are indexed by elements in the homology of a certain inductively defined collection of differential graded Lie algebras (DGLs) defined below. Thus they take values in the corresponding cohomology groups, with coefficients in pi_* X. It is clear intuitively that cycles in the homology of a DGL L which are not generators, or products of other cycles, represent ``higher homotopy operations'' in L, in some sense. One of our objectives is to formalize this intuition within a more general framework. Moreover, if L represents the rational homotopy type of a topological space X, it is not always evident how to represent these rational operations as integral higher order operations. In order to address this problem, we must consider a somewhat ``flabbier'' model of rational homotopy than that provided by differential graded Lie algebras, namely a certain class of differential graded non-associative algebras. Thus we also provide a (somewhat incomplete) answer to the following question: what additional structure on the ordinary homotopy groups pi_* X of a simply-connected space X, beyond the Whitehead products, is needed to determine its homotopy type up to rational equivalence? 4. /pub/Christenson-Strickland/phantoms (There is a typo in Christensen's name, so this path may be corrected). Phantom Maps and Homology Theories J. Daniel Christensen and Neil P. Strickland (jdchrist---mit.edu and neil---pmms.cam.ac.uk) Keywords: phantom map, stable homotopy theory, spectrum, triangulated category Abstract: We study phantom maps and homology theories in a stable homotopy category $\cS$ via a certain Abelian category $\cA$. We express the group $\cP(X,Y)$ of phantom maps $X\ra Y$ as an $\Ext$ group in $\cA$, and give conditions on $X$ or $Y$ which guarantee that it vanishes. We also determine $\cP(X,HB)$. We show that any composite of two phantom maps is zero, and use this to reduce Margolis' axiomatisation conjecture to an extension problem. We show that a certain functor $\cS\ra\cA$ is the universal example of a homology theory with values in an AB 5 category, and compare this with some results of Freyd. 5. /pub/DDavis/pexp5 Elements of large order in pi_*(SU(n)) Donald M. Davis Last updated July 5, 1996. 50 pages long. Abstract It is proved that if p is an odd prime, then some homotopy group of SU(n) contains an element of order p^e, where e = n-1+[(n+2p-3)/p^2]+[(n+p^2-p-1)/p^3]. The method is to compute v1-periodic homotopy groups, using the unstable Novikov spectral sequence. This should be very close to the largest orders of v1-periodic elements, and we conjecture that the elements of largest order are v1-periodic. 6. /pub/Kuhn/loopspaces Let $T(j)$ be the dual of the $j^{th}$ Brown-Gitler spectrum (at the prime 2) with top class in dimension $j$. Then it is known that $T(j)$ is a retract of a suspension spectrum, is dual to a stable summand of $\Omega^2 S^3$, and that the homotopy colimit of a certain sequence $T(j) \rightarrow T(2j) \rightarrow \ldots$ is a 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 $K(Z/2,1) = RP^{\infty}$ as one of the summands. Refining a question posed by Doug Ravenel, I discuss a generalization of this picture. I consider certain finite spectra $T(n,j)$ for $n,j \geq 0$ (with $T(1,j) = T(j)$), dual to summands of $\Omega^{n+1}S^{N}$, conjecture generalizations of all of the above, and prove that all these conjectures are correct in cohomology. So, for example, $T(n,j)$ has unstable cohomology, and the cohomology of the colimit of a certain sequence $T(n,j) \rightarrow T(n,2j) \rightarrow \dots$ agrees with the cohomology of the wedge of stable summands of $K(V,n)$'s corresponding to the wedge occurring in the $n=1$ case above. One can also map the $T(n,j)$ to each other as $n$ varies, and the cohomological calculations suggest conjectures related to symmetric products of spheres. 7. /pub/Kuhn/symmetricpowers If $bF_q$ is the finite field of order $q$ and characteristic $p$, 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. Important families of objects in $F(q)$ include the families $S_n, S^n, \Lambda^n, \Bar{S}^n$, and $cT^n$, with $c \in F_q[\Sigma_n]$, defined by $S_n(V) = (V^{\otimes n})^{\Sigma_n}$, $ S^n(V) = V^{\otimes n}/\Sigma_n$, $\Lambda^n(V) = n^{th} \text{ exterior power of } V$, $\Bar{S}^*(V) = S^*(V)/(p^{th} \text{ powers})$, and $cT^n(V) = c(V^{\otimes n})$. Fixing $F$, we discuss the problem of computing $Hom_{F(q)}(S_m, F \circ G)$, for all $m$, given knowledge of $Hom_{F(q)}(S_m, G)$ for all $m$. When $q = p$, we 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. 8. /pub/Levi/comp Ran Levi ran---.math.nwu.edu Northwestern University (previous, University of Heidelberg) A comparison Criterion for certain loop spaces We study a comparison criterion for loop spaces on $p$-localized classifying spaces of certain finite $p$-perfect groups $G$. In particular we show that, under certain hypotheses, the homotopy type of those spaces is determined by the mod-$p$ cohomology of $G$ together with a finite Postnikov system. Appeared in Contemporary Math. 181, (1995) 9. /pub/Levi/conj Ran Levi ran---.math.nwu.edu Northwestern University (previous, University of Heidelberg) A counterexample to a conjecture of Cohen Let G be a finite p-superperfect group. A conjecture of F. Cohen suggests that \Omega BG^p is resolvable by finitely many fibrations over spheres and iterated loop spaces on spheres, where (-)^p denotes the p-completion functor of Bousfield and Kan. We produce a counter-example to this conjecture and discuss some related aspects of the homotopy type of \Omega BG^p. Appeared in Progress in Math. Vol 136, (1996) Birkhauser Verlag. 10. /pub/Levi/fin Ran Levi ran---math.nwu.edu Northwestern University On p-completed classifying spaces of discrete groups and finite complexes We show that for certain discrete $p$-perfect groups $G$, in particular for all $p$-perfect groups of finite cohomological dimension, the loop space on the $p$-completed classifying space $\lbgp$ is a retract of the loop spaces on a certain finite complex. For finite $vcd$ groups we provide a bound on the the dimension of such a complex. Submitted 11. /pub/Levi/gr Ran Levi ran---.math.nwu.edu Northwestern University (previous, University of Heidelberg) On Homological rate of Growth and the Homotopy type of \Omega BG^p Let G be a finite p-perfect group. We show that the mod-p homology of \Omega BG^p grows either polynomially or semi-exponentially. A conjecture due to F. Cohen states that \Omega BG^p for such groups G is spherically resolvable of finite weight. We show that any space X, which satisfies the conclusion of Cohen's conjecture has the property that its homology grows at most hyper-polynomially of finite degree. Thus we conclude that if a group $G$ satisfies the Cohen conjecture then the homology of \Omega BG^p grows polynomially. This enable us to produce counter examples to the conjecture. We study some further homotopy properties of our examples. We also show that the mod p homology of \Omega BG^p is a finitely generated, Lie nilpotent algebra provided it grows polynomially. Preprint 12. /pub/Levi/lsht Ran Levi Torsion in Loop Space Homology of Rationally Contractible Spaces Abstract Let $\R$ be a torsion free principal ideal domain. We study the growth of torsion in loop space homology of simply-connected $\dg\R$-coalgebras $C$, whose homology admits an exponent $r$ in $R$. Here by loop space homology we mean the homology of the loop algebra construction on $C$. We compute a bound on the growth of torsion in such objects and show that in general this bound is best possible. Our methods are applied to certain simply-connected spaces associated with classifying spaces of finite groups, where we are able to deduce the existence of global exponents in loop space homology. ----------------- Clarence has switched http daemons, and would appreciate hearing from users about any changes you notice. The new software is supposed to make downloading quicker and more reliable. We have just one new paper this time. Mark Hovey Papers uploaded to Hopf between Aug 9,1996 and Aug 18, 1996: 1. /pub/Gorbunov-Mahowald-Symons/peterme Infinite subgroups of the Morava stabilizer groups V. Gorbounov M. Mahowald P. Symonds We discuss certain infinite subgroups of the Morava stabilizer groups and outline some applications in homotopy theory. Consider a cyclic algebra ${\Bbb D}$ over ${\Bbb Q_p}$ of index $p-1$ with Hasse invariant ${1\over {p-1}}$. Let ${\Bbb S}l$ be the group of strict units of ${\Bbb D}$ of reduced norm one. The main result is the following: \begin{thm}\label{one} There is a p-th root of unity $\alpha\in {\Bbb S}l$ and a (p-1)-st root of $-p$ in ${Bbb D}$ such that \begin{enumerate} \item \{$\displaystyle \alpha^{\omega^i}$, $1\leq i\leq p-1$\} generate a subgroup $G$ of ${\Bbb S}l$, which is isomorphic to a free product of $p-1$ copies of ${\Bbb Z/p}$: ${\Bbb Z/p}*\cdots *{\Bbb Z/p}$ \item $G$ is dense in ${\Bbb S}l$. \end{enumerate} \end{thm} ------------------ A few words from Clarence. Hopf is now running new software and hardware. It is possible that the "gzip-on-the-fly" and other ftp tricks will be broken for a short while. Any problems should be reported to Clarence. It would be very helpful for Clarence if all abstracts had author and title lines that look like the following example: Great facts about Lie groups A. Borel, S. Lie, and H. Weyl Notice there are no \author or \title commands, all the authors are on the same line, and it is lower case. So please submit your abstracts in this form from now on. Note that if you want to use TeX (which I don't recommend), you can still make life easier for Clarence, I think, by using \title{ Fermat's Last Theorem } \author{ P. Fermat } There are 8 new papers this time. Mark Hovey Papers uploaded to Hopf between Aug 18,1996 and Sep 17, 1996: 1. /pub/Arkowitz-Lupton/ark_lup7 Title: Rational Obstruction Theory and Applications to Homotopy Sets. Authors: Martin Arkowitz and Gregory Lupton. Abstract: We develop an obstruction theory for homotopy of homomorphisms $f,g : {\Cal M }\to{\Cal N }$ between minimal differential graded algebras. We assume that ${\Cal M }=\Lambda V$ has an obstruction decomposition given by $V=V_0\oplus V_1$ and that $f$ and $g$ are homotopic on $\Lambda V_0$. An obstruction is then obtained as a vector space homomorphism $V_1\to H^*({\Cal N})$. We investigate the relationship between the condition that $f$ and $g$ are homotopic and the condition that the obstruction is zero. The obstruction theory is then applied to study various questions about the set of homotopy classes of maps $[{\Cal M },{\Cal N }]$. We study cohomologically trivial homotopy classes of maps from ${\Cal M }$ to ${\Cal N }$. We investigate a conjecture of Copeland-Shar on the homotopy set $[{\Cal M },{\Cal N }]$. We give examples of minimal algebras that have few homotopy classes of self-maps. Because of the equivalence of the homotopy category of minimal algebras and the homotopy category of rational spaces, this study yields analogous results for rational spaces. By exploiting basic properties of rationalization, we de-localize some of the results about rational spaces to obtain information on the set of homotopy classes of maps between two finite complexes. 2. /pub/Arone-Kankaanrinta/Liehomology The homology of certain subgroups of the symmetric group with coefficients in Lie(n). Greg Arone and Marja Kankaanrinta Abstract: The authors compute the mod p homology of groups of the form \Sigma_{n_1}\times\cdots\times\Sigma_{n_k} with twisted coefficients in the module Lie(n), where n_1+\cdots+n_k=n. 3. /pub/Gorbunov-Siegel-Symonds/mor-stab-at2 The Cohomology of the Morava Stabilizer Group $\Bbb S_2$ at the Prime $3$ Vassily Gorbounov Stephen F. Siegel Peter Symonds We compute the cohomology of the Morava stabilizer group {\small $\Bbb S_2$} at the prime $3$ by resolving it by a free product ${\Bbb Z}/3*{\Bbb Z}/3$ and analyzing the ``relation module.'' 4. /pub/Gorbunov-Symonds/higherEO2 (This file has associated with it three .ps files which contain pictures. These are called fign.ps, wher n=1,2,3. If you get the .ps file, instead of the .dvi file, these will already be included. Mark) Toward the homotopy groups of the higher real $K$-theory $EO_2$ Vassily Gorbounov Peter Symonds The higher real $K$-theories $EO_n$ have been constructed by Hopkins and Miller recently. When $n=2$ this construction suggests a way of defining an "integral elliptic cohomology", which yields the usual elliptic cohomology when 6 is inverted. Our main result is the calculation of the initial term of the spectral sequence $H^*(G,E)^{\mbox{\rm \small Gal}} \Rightarrow \pi _*EO_2$, which converges to the homotopy groups of the "elliptic cohomology" spectrum localized at the prime 3. Here $G$ is the group $\Bbb Z/3\rtimes\Bbb Z/4$ and $E$ is an infinitely generated module for $G$ over $\Bbb Z_3$, which arises from the theory of formal groups. We also show how the integral modular forms of Deligne appear naturally in this initial term. This calculation was originally sketched by Hopkins and Miller, but the details were never published, so we have proceeded by our own methods. 5. /pub/Greenlees/augmentation_ideals ``Augmentation ideals of equivariant cohomology rings.'' J.P.C.Greenlees School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK. j.greenlees------sheffield.ac.uk The purpose of this note is to establish a number of useful results about the augmentation ideal $J$ for the coefficient ring $F_G^*$ of a Noetherian complex orientable equivariant cohomology theory. The results show that various naturally occurring substitutes for the ideal have the same radical, and can therefore be used instead of the augmentation ideal in all geometric constructions. 6. /pub/Greenlees/Rational_O2_equivariant J.P.C.Greenlees ``Rational O(2)-equivariant cohomology theories.'' 8 pp A complete algebraic model is given for rational O(2)-spectra. The main input is the model for SO(2)-spectra described in ``Rational S1-equivariant stable homotopy theory'', and the special case of ``Rational Mackey functors for compact Lie groups I'' applying to O(2). It turns out that any O(2)-spectrum is described by an SO(2)-spectrum with O(2)/SO(2)-action, together with an equivariant sheaf over the space of dihedral subgroups of O(2). ---------------------------------------------------------------------- | John Greenlees | Tel : (0114)-282 4437 direct | | Mathematics and Statistics | (0114)-276 8555 central | | University of Sheffield | | | Hicks Building | Fax : (0114)-282 4292 | | Sheffield S3 7RH | Email: j.greenlees---sheffield.ac.uk | ---------------------------------------------------------------------- | WWW Homepage | |http://www.shef.ac.uk/uni/academic/I-M/ms/staff/greenlees/index.html| ---------------------------------------------------------------------- 7. /pub/Greenlees/Rational_Mackey_functors_I (This is a new version of a paper that was already on the archive.) J.P.C.Greenlees ``Rational Mackey functors for compact Lie groups I.'' 32pp The category of rational Mackey functors is shown to be equivalent to the category of equivariant sheaves on the Weyl-toral category. The advantage is that the latter category only has morphisms in one direction, and may thus be seen to have global dimension equal to the rank of the group. This version is a simplified and corrected version of an earlier preprint. ---------------------------------------------------------------------- | John Greenlees | Tel : (0114)-282 4437 direct | | Mathematics and Statistics | (0114)-276 8555 central | | University of Sheffield | | | Hicks Building | Fax : (0114)-282 4292 | | Sheffield S3 7RH | Email: j.greenlees---sheffield.ac.uk | ---------------------------------------------------------------------- | WWW Homepage | |http://www.shef.ac.uk/uni/academic/I-M/ms/staff/greenlees/index.html| ---------------------------------------------------------------------- 8. /pub/Ishiguro/pairing Pairings of p-compact groups and H-structures on the classifying spaces of finite loop spaces Kenshi Ishiguro Fukuoka University, Fukuoka 814-80, Japan We consider the maps between classifying spaces of p-compact groups of the form $BX \times BY --->>> BZ$. The main theorem shows that if the restriction map on BY is a weak epimorphism, then the restriction on BX should factor through the classifying spaces of the center of the p-compact group Z. An application implies that, for a finite loop space X, its classifying space BX is an H-space (Hopf space) if and only if X is the product of a torus and a finite abelian group. It is also shown, for a compact Lie group G, exactly when the p-completed space $(BG)\p$ has an H-structure. ---------------- There are 2 new papers this time. Mark Hovey Papers uploaded to Hopf between Sep 18,1996 and Oct 2, 1996: 1. /pub/Arkowitz-Gutierrez/ark_gut1 \title COMULTIPLICATIONS ON FREE GROUPS AND WEDGES OF CIRCLES \endtitle \author Martin Arkowitz and Mauricio Gutierrez \endauthor \address Mathematics Department, Dartmouth College, Hanover, NH 03755\endaddress \email Martin.Arkowitz\---Dartmouth.EDU\endemail \address Mathematics Department, Tufts University, Medford, MA 02155 \endaddress \email mgutierr\---tufts.edu\endemail \abstract By means of the fundamental group functor, a co-H-space structure or a co-H-group structure on a wedge of circles is seen to be equivalent to a comultiplication or a cogroup structure on a free group $F$. We consider individual comultiplications on $F$ and their properties such as associativity, coloop structure, existence of inverses, etc. as well as the set of all comultiplications of $F$. For a comultiplication $m$ of $F$ we define a subset $\Delta_m \subseteq F$ of quasi-diagonal elements which is basic to our investigation of associativity. The subset $\Delta _m$ can be determined algorithmically and contains the set of diagonal elements $D_m$. We show that $D_m$ is a basis for the largest subgroup $A_m$ of $F$ on which $m$ is associative and that $A_m$ is a free factor of $F$. We also give necessary and sufficient conditions for a comultiplication $m$ on $F$ to be a coloop in terms of the Fox derivatives of $m$ with respect to a basis of $F$. In addition, we consider inverses of a comultiplication, the collection of cohomomorphisms between two free groups with comultiplication and the action of the group $\aut F$ on the set of comultiplications of $F$. We give many examples to illustrate these notions. We conclude by translating these results from comultiplications on free groups to co-H-space structures on wedges of circles. \endabstract 2. /pub/Costenoble-Waner/Surgery_for_compact_G The equivariant Spivak normal bundle and equivariant surgery for compact Lie groups Steven R. Costenoble and Stefan Waner Dept. of Mathematics 103 Hofstra University Hempstead, NY 11550 We generalize the results of [Costenoble and Waner, The equivariant Spivak normal bundle and equivariant surgery, Mich. Math. J. 39 (1992), 415-424] to compact Lie groups. Using a suitable ordinary equivariant homology and cohomology, we define equivariant Poincar\'e complexes with the properties that (1) every compact $G$-manifold is an equivariant Poincar\'e complex, (2) every finite equivariant Poincar\'e complex (with some mild additional hypotheses) has an equivariant spherical Spivak normal fibration, and (3) the $\pi$-$\pi$ Theorem holds for equivariant Poincar\'e pairs under suitable gap hypotheses. ----------------- There is one new paper this time. Mark Hovey Papers uploaded to Hopf between Oct 2,1996 and Oct 10, 1996: 1. /pub/Palmieri/palmieri-f-iso Quillen stratification for the Steenrod algebra John H. Palmieri Let A be the mod 2 Steenrod algebra, and let Q denote the category of exterior sub-Hopf algebras of A, where the morphisms are given by inclusions. The restriction maps Ext_A (Z/2,Z/2) --> Ext_E (Z/2,Z/2), for E in Q, can be assembled into a map i : Ext_A (Z/2, Z/2) --> lim_Q Ext_E (Z/2,Z/2). There is an action of A on this inverse limit, and i factors through the invariants under this action, giving a map g : Ext_A (Z/2, Z/2) --> ( lim_Q Ext_E (Z/2,Z/2) )^A. We show that g is an F-isomorphism. ---------------- We have five new papers this time. Mark Hovey Papers uploaded to Hopf between Oct 10,1996 and Nov 1, 1996: 1./pub/DDavis/emb3 Embeddings of real projective spaces Donald M. Davis Abstract We tabulate known results on embeddings of real projective spaces, and prove one family of new results: P^n can be embedded in R^{2n-4} if n = 2^i + 3. 2. /pub/Feldman-Wilce/fibdegen Title: Degenerate fibres in the Stone-Cech compactification of the universal bundle of a finite group: An application of homotopy theory to general topology Authors: David Feldman and Alexander Wilce Abstract: If p: E ----->B is a continuous surjection between completely regular spaces E and B , we may apply the Stone-Cech compactification functor \beta to obtain a surjection \beta p: \beta E ----->\beta B . It is well-known that if E = B x F where F is a finite set and p is projection on the first factor, then \beta E = \beta B x \beta F , and \beta p is again projection on the first factor. In this paper, we apply \beta to an n-fold covering map, that is, a local homeomorphism p: E ----->B such that p^-1 (b) has cardinality n for any b \in B . We show that the fibres of \beta p , while never exceeding n points, may degenerate to sets whose cardinality properly divides n (in contrast with the more usual, explosive sort of Stone-Cech ``pathology''). What is particularly striking about this phenomenon is that it depends on a homotopy invariant, the sectional category, of the map p . In particular, we show that if p: E ----->B is an H-bundle where H is a finite group, then \beta p has degenerate fibres iff p has infinite sectional category. In the special case where G is a p-group and p:EG ----->BG is the universal G-bundle, we can show more precisely that every possible G-orbit occurs somewhere as a fibre of \beta p . The proof of this theorem uses a weak form of the so-called generalized Sullivan conjecture, which is now a theorem of H.~Miller. It is interesting to see the structure group G manifesting itself in this way even though it is not explicitly part of the data fed to the Stone-Cech functor. Algebraic and general topology have grown far appart in recent years. Accordingly, we have tried to include enough detail to make the paper essentially self contained. Regarding the Stone-Cech compactification, we use few facts beyond the basic definitions. Readers unfamiliar with universal G -bundles should bear in mind the simplest non-trivial example, G= Z/2Z . The double cover of the infinite real projective space RP^\infty is a universal Z/2Z -bundle. No other finite group has a universal bundle which is so easily pictured; it is this case which motivated some of our terminology. 3. /pub/Hovey-Sadofsky/lnpic Invertible spectra in the $E(n)$-local stable homotopy category Mark Hovey and Hal Sadofsky Recall that the Picard group, first introduced into stable homotopy theory by Hopkins, is the group of isomorphism classes of smash-invertible spectra. For the ordinary stable homotopy category, this group is just Z on the spheres S^n. For localized categories,however the situation may be more complex. Hopkins, Mahowald, and Sadofsky, and also Strickland, have studied the Picard group of the K(n)-local category. In this paper we study the Picard group of the L_n-local, or E(n)-local, stable homotopy category. We find that if n is large relative to the prime p, the answer is just Z again. The proof involves a few general results that should be of independant interest. In particular, we give two proofs of a generalized Miller-Ravenel change of rings theorem, one of which depends on a general (unfortuately unpublished, but not tremendously difficult) change of rings theorem due to Hopkins. We also give an E(n) version of the Landweber filtration theorem, and show that the E(n) Adams spectral sequence always converges. We conclude the paper with the simplest example where n is not large relative to p, when n=1 and p=2. Here the Picard group is Z plus Z/2. 4. /pub/Mitchell/thomason Author: Stephen A. Mitchell Title: Hypercohomology spectra and Thomason's descent theorem Email: mitchell---math.washington.edu This is an expository paper on Thomason's etale descent theorem for Bott-periodic algebraic K-theory. It is mainly concerned with the proof of the theorem, and the machinery that goes into the proof. It includes an exposition of Jardine's closed model category structure on presheaves of spectra. Much background material is provided on Grothendieck sites, sheaf cohomology, etc. The paper is arranged so that readers who are not familiar with schemes or etale cohomology can still read the first and last chapters, which deal with Thomason's theorem for fields. This paper will appear in the proceedings of the 1996 Great Lakes K-theory conference. 5. /pub/Nassau/pnpnalg On the structure of $P(n)_\ast(P(n))$ for $p=2$ Christian Nassau We show that $P(n)_\ast(P(n))$ for $p=2$ with its geometrically induced structure maps is not an Hopf algebroid because neither the augmentation $\epsilon$ nor the coproduct $\Delta$ are multiplicative. As a consequence the algebra structure of $P(n)_\ast(P(n))$ is slightly different from what was supposed to be the case. We give formulas for $\epsilon(xy)$ and $\Delta(xy)$ and show that the inversion of the formal group of $P(n)$ is induced by an antimultiplicative involution $\Xi:P(n)\rightarrow P(n)$. Some consequences for multiplicative and antimultiplicative automorphisms of $K(n)$ for $p=2$ are also discussed. ------------------ We have five new papers again this time. Mark Hovey Papers uploaded to Hopf between Nov 1,1996 and Nov 18, 1996: 1. /pub/Buchstaber-Ray/decfmqd ABSTRACT FOR: DOUBLE COBORDISM, FLAG MANIFOLDS, AND QUANTUM DOUBLES Victor M Buchsaber and Nigel Ray Drinfeld's construction of quantum doubles is one of several recent advances in the theory of Hopf algebras (and their actions on rings) which may be attractively presented within the framework of complex cobordism; these developments were pioneered by S P Novikov and the first author. Here we extend their programme by discussing the geometric and homotopy theoretical interpretations of the quantum double of the Landweber-Novikov algebra, as represented by a subalgebra of operations in double complex cobordism. We base our study on certain families of bounded flag manifolds with double complex structure, originally introduced into cobordism theory by the second author. We give background information on double complex cobordism, and discuss the cell structure of the flag manifolds by analogy with the classic Schubert decomposition, allowing us to describe their complex oriented cohomological properties (already implicit in the Schubert calculus of Bressler and Evens). This yields a geometrical realization of the basic algebraic structures of the dual of the Landweber-Novikov algebra, as well as its quantum double. We work in the context of Boardman's eightfold way, which clarifies the relationship between the quantum double and the standard machinery of Hopf algebroids of homology cooperations. 2. /pub/Dwyer/Exotic.Cohomology.GLnZhalf Exotic cohomology for GL(n,Z[1/2]) by W. G. Dwyer We show that for n=32 the mod 2 group cohomology of GL(n,Z[1/2]) is not detected on the subgroup of diagonal matrices. This disproves an old conjecture, and suggests that the cohomology of these general linear groups may in general be difficult to understand. 3. /pub/Green-Leary/SpectrumChern Title: The spectrum of the Chern subring Authors: David J. Green, IEM Essen, Germany. david---exp-math.uni-essen.de Ian J. Leary, Univ. of Southampton, UK. ijl---maths.soton.ac.uk Status: Submitted for publication. Abstract: For certain subrings of the mod-$p$-cohomology of a compact Lie group, we give a description of the spectrum, analogous to Quillen's description of the spectrum of the whole cohomology ring. Subrings to which our theorem applies include the Chern subring. Corollaries include a characterization of those groups for which the Chern subring is F-isomorphic to the cohomology ring. 1991 Classification: Primary 20J06; Secondary 20D15, 55R40. 4. /pub/Green-Minh/gm_transfer Title: Transfer and Chern Classes for Extraspecial $p$-Groups Authors: David J. Green, IEM Essen, Germany. david---exp-math.uni-essen.de Pham Anh Minh, University of Hue, Vietnam. Status: Submitted for publication Abstract: In the cohomology ring of an extraspecial $p$-group, the subring generated by Chern classes and transfers is studied. This subring is strictly larger than the Chern subring, but still not the whole cohomology ring, even modulo nilradical. A formula is obtained relating Chern classes to transfers. 1991 Classification: Primary 20J06; Secondary 20D15, 55R40. 5. /pub/Hung-Peterson/spherical Spherical classes and the Dickson algebra Nguyen H. V. Hu'ng and Franklin P. Peterson We attack the conjecture that the only spherical classes in the homology of $Q_0S^0$ are Hopf invariant one and Kervaire invariant one elements. We do this by computing products in the $E^2$-term of the unstable Adams spectral sequence converging to $\pi_*(Q_0S^0)$ using results about the Dickson algebra and by studying the Lannes--Zarati homomorphism. --------------- We have one new paper this time, plus a whole lot of new pictures taken and scanned by Clarence. I'll give you Clarence's message about the pictures first, then the abstract of the new paper. Mark Hovey Clarence wrote: >In the directory /pub/pictures/95-96 on hopf, there are pictures >of people from the CMS Vancouver meeting, my stay at MIT, my stay in >Zurich, and the Fields Inst in May 96. >If anyone has requests for pictures of particular people/places >or if people would like to contribute pictures, just e-mail me. >Mine were taken on color print film, finished at the local >drug store, and 4x5 paper prints scanned in and eye corrected >for density and contrast, with some trimming also. >Here's the current list of the new crop: > aadem.jpg 21-Nov-96 14:39 28k > berrick.jpg 21-Nov-96 14:42 24k > billrich.gif 21-Nov-96 09:17 79k > campbell-goerss-van.jpg21-Nov-96 14:33 61k > campbell-peterson.gif 21-Nov-96 09:22 125k > carlos.jpg 21-Nov-96 14:42 23k > cww-as-cubscout.gif 21-Nov-96 14:35 163k > cww-mit.gif 21-Nov-96 09:17 123k > drav.jpg 21-Nov-96 14:42 20k > goerss.gif 21-Nov-96 09:24 73k > gottmit.gif 21-Nov-96 09:20 136k > gray-jardine.gif 21-Nov-96 09:20 94k > harper-moore.gif 21-Nov-96 09:20 98k > jardine.gif 21-Nov-96 09:20 62k > jeanner.jpg 21-Nov-96 14:42 39k > jhubb.gif 21-Nov-96 09:21 85k > jmcclure.gif 21-Nov-96 09:21 132k > klesh.jpg 21-Nov-96 14:46 36k > lin-kane.jpg 21-Nov-96 14:42 49k > miller-peterson.gif 21-Nov-96 09:22 388k > moller.jpg 21-Nov-96 14:50 28k > notbohm.jpg 21-Nov-96 14:54 36k > pengal.jpg 21-Nov-96 14:58 43k > rlevi.jpg 21-Nov-96 15:02 14k > sadofsky-minami.gif 21-Nov-96 09:22 70k > shipley-smith.gif 21-Nov-96 09:20 111k > suter.gif 21-Nov-96 09:25 97k > toronto1.jpg 21-Nov-96 15:05 20k > vincent.jpg 21-Nov-96 15:05 37k > wgd-lec.jpg 21-Nov-96 15:09 48k > wgd-van.gif 21-Nov-96 09:27 80k Papers uploaded to Hopf between Nov. 18,1996 and Dec. 19, 1996: 1. /pub/Silverman/hit_polys_and_conjugation Hit Polynomials and Conjugation in the Steenrod Algebra and its Dual Judith H. Silverman judith---iu-math.math.indiana.edu Let $A^*$ be the mod-2 Steenrod algebra of cohomology operations and $\chi$ its canonical antiautomorphism. For all positive integers $f$ and $k$, we show that the excess of the element $\chi[Sq(2^{k-1}f) \cdot Sq(2^{k-2}f) \cdots Sq(2f) \cdot Sq(f)]$ is $(2^k-1) \mu(f)$, where $\mu(f)$ denotes the minimal number of summands in any representation of $f$ as a sum of numbers of the form $2^i-1$. We also interpret this result in purely combinatorial terms. In so doing, we express the Milnor basis representation of the products $Sq(a_1) \ldots Sq(a_n)$ and $\chi[Sq(a_1) \ldots Sq(a_n)]$ in terms of the cardinalities of certain sets of matrices. For $s \geq 1$, let $P_s = F_2[x_1, \ldots, x_s]$ be the mod-2 cohomology of the $s$-fold product of $RP^{\infty}$ with itself, with its usual structure as an $A^*$-module. A polynomial $P \in P_s$ is {\em hit} if it is in the image of the action $\overline{A^*} \otimes P_s \longrightarrow P_s$, where $\overline{A^*}$ is the augmentation ideal of $A^*$. We prove that if the integers $e$, $f$, and $k$ satisfy $e<(2^k-1)\mu(f)$, then for any polynomials $E$ and $F$ of degrees $e$ and $f$ respectively, the product $E \cdot F^{2^k}$ is hit. This generalizes a result of Wood, conjectured by Peterson, and proves a conjecture of Singer and Silverman. ---------------- We have 6 new papers on hopf this time. Happy New Year! Mark Hovey New papers uploaded to Hopf between 12/20/96 and 12/30/96: 1. /pub/Dwyer/Exotic.Cohomology.GLnZhalf This is a new version of a paper already on the archive. I don't know how extensive the changes are. 2. /pub/Neumann-Neusel-Smith/ag1 Title: Rings of Generalized and Stable Invariants of Pseudoreflections and Pseudoreflection Groups Authors: Frank Neumann, Mara D. Neusel and Larry Smith Abstract: Let \rho: G --> GL(n, F) be a representation of a finite group G over the field F and F[V] the space of polynomial functions on V=F^n. We associate to G an ideal J_\infty(G) of F[V] called the ideal of stable invariants of \rho. If S is a set of pseudoreflections we associate to S the ideal I(S) of F[V] called the ideal of generalized invariants of S in the sense of Kac and Peterson. When G is a pseudoreflection group we investigate I(S) for various choices of S and the relation between J_\infty(G) and I(S). To a representation \rho, respectively to a set S of pseudoreflections, we also associate the rings gr_J_\infty(G) respectively gr_I(S) of stable and generalized invariants. We show that gr_I(S) is always a polynomial algebra over F and whenever \rho(G) is generated by semisimple pseudoreflections S that gr_J_\infty(G)=gr_I(S). This is the version which is published in J. of Algebra 182 (1996), 85-122. 3. /pub/Neumann-Neusel-Smith/ag4 (The fonts are weird in this file--I had trouble with my dvi viewer--Mark) Title: Rings of Generalized and Stable Invariants and Classifying Spaces of Compact Lie Groups Authors: Frank Neumann, Mara D. Neusel and Larry Smith Abstract: Let G be a compact connected Lie group with maximal torus T and Weyl group W(G). We show that the Eilenberg-Moore spectral sequence mod p (p odd prime) of the fibration G --> G/T --> BT collapses at the term E_2. This gives as corollary a different proof of the theorem of Kac, that the Serre Spectral sequence of the fibration T --> G --> G/T collapses at the term E_3. As an important step in the proof we show that the kernel of the induced map H^*(BT, F_p) --> H^*(G/T, F_p) can be identified with the ideal J_\infty (W(G)) in H^*(BT, F_p) of stable invariants of the Weyl group. The result can be applied to study torsion questions of H^*(BG, Z) in terms of the Weyl group action. This is the old version of the paper submitted to Inv. Math. 4. /pub/Shank/fmodsi Formal Modular Seminvariants R. James Shank Abstract: We construct a generating set for the ring of invariants for the four and five dimensional indecomposable modular representations of a cyclic group of prime order. We then observe that for the four dimensional representation the ring of invariants is generated in degrees less than or equal to 2p-3, and for the five dimensional representation the ring of invariants is generated in degrees less than or equal to 2p-2. 5. /pub/Strickland/mult Products on MU-modules by Neil Strickland We use the new categories of spectra and $MU$-modules constructed by Elmendorf, Kriz, Mandell and May to get improved results about multiplicative structures on spectra such as $P(n)$ and $E(n)$, particularly in the case $p=2$. 6. /pub/Strickland/poly Morava E-theory of symmetric groups by Neil Strickland We compute the completed $E(n)$ cohomology of the classifying spaces of the symmetric groups, and relate the answer to the theory of finite subgroups of formal groups. ----------------