BABYL OPTIONS: -*- rmail -*- Version: 5 Labels: Note: This is the header of an rmail file. Note: If you are seeing it in rmail, Note: it means the file has no messages in it.  1, filed, edited, forwarded,, Mail-from: From hovey@math.mit.edu Fri Feb 3 09:53:22 1995 Return-Path: Received: from nevanlinna.mit.edu by math.mit.edu (4.1/Math-2.0) id AA23567; Fri, 3 Feb 95 09:51:15 EST From: Mark Hovey Received: by nevanlinna.mit.edu; Fri, 3 Feb 95 09:51:11 EST Date: Fri, 3 Feb 95 09:51:11 EST Message-Id: <9502031451.AA21474@nevanlinna.mit.edu> To: hovey@math.mit.edu Subject: Hopf mailing list Reply-To: hovey@math.mit.edu *** EOOH *** Return-Path: From: Mark Hovey Date: Fri, 9 Aug 96 09:51:11 EST To: hovey@math.mit.edu Subject: Hopf mailing list Reply-To: hovey@math.mit.edu 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. ---------------------Instructions----------------------------- To subscribe or unsubscribe to this list, send a message to Don Davis at dmd1@lehigh.edu with your e-mail address and name. Please make sure he is using the correct e-mail address for you. To see past issues of this mailing list, point your WWW browser to http://www-math.mit.edu/~hovey/ If this doesn't work or is missing a few issues, try http://www.lehigh.edu/~dmd1/public/www-data/algtop.html , which also has the other messages sent to Don's list. To get the papers listed above, point your WWW client (Mosaic, Netscape) to http://hopf.math.purdue.edu/pub/hopf.html There are links to conference announcements, Purdue seminars, and other math related things on this page as well. You can also use ftp to hopf.math.purdue.edu, and login as ftp. Then cd to pub. Files are organized by author name, so papers by me are in pub/Hovey. If you want to download a file using ftp, you must type binary before you type get . To put a paper of yours on the archive, cd to /pub/incoming. Transfer the dvi file using binary, by first typing binary then put You should also transfer an abstract as well. To do this take the TeX file and save the abstract to a different file, without any \begin{document} commands or anything, and transfer that file. You can use ascii instead of binary for this. I am solely responsible for this mailing list---don't send complaints about it to Clarence. Thanks to Clarence for creating and maintaining the archive.