Subject: new Hopf listings From: Mark Hovey Date: 31 Aug 1999 15:34:57 -0400 5 new papers this time. Mark Hovey New papers uploaded to hopf between 8/19/99 and 8/31/99. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Crossley-Whitehouse/hiconj Title: Higher Conjugation Cohomology in Commutative Hopf Algebras Authors: M. D. Crossley and Sarah Whitehouse Abstract Text: The dual Steenrod algebra can be expressed as the homotopy of a smash product of two copies of the Eilenberg-MacLane spectrum, and the conjugation arises by permutation of the two factors. This can be generalized to an action of the symmetric group $\Sigma_n$ acting on an n-1-fold tensor product of copies of the dual Steenrod algebra; this action was described by the second author in [6], purely in terms of the Hopf algebra structure. So, formally, one has a similar action for any commutative Hopf algebra. In this paper, we study the cohomology ring $H^*(\Sigma_n; A^{\otimes n-1})$, where $A$ is a graded commutative Hopf algebra. We show that for a certain class of Hopf algebras the cohomology ring is independent of the coproduct provided $n$ and $(n-2)!$ are invertible in the ground ring. Then, by choosing a sufficiently simple coproduct, we are able to deduce significant information about the $\Sigma_n$ invariants of $A^{\otimes n-1}$, including dimensions and algebra structure. In particular, we give a complete solution to the "conjugation invariants" problem for the mod $p$ dual Steenrod algebra when $p>2$. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Gorbounov-Malikov-Schechtman/g erbes Title: Gerbes of chiral differential operators Authors: Vassily Gorbounov, Fyodor Malikov, and Vadim Schechtman This is an improved version of the paper we have submitted earlier. In particular we have computed the "conformal anomaly": the obstruction to the existence of a globally defined Virasoro field. In this note we compute the cohomological obstruction to the existence of certain sheaves of vertex algebras on smooth varieties. These sheaves have been introduced and studied in the previous work by Malikov, Schechtman and Vaintrob, and are canonically defined for an arbitrary $X$. One can try to define a purely even counterpart of $\Omega^{ch}_X$, a sheaf of graded vertex algebras $\CO^{ch}_X$, called a {\it chiral structure sheaf}. The obstraction to its existence turns out to admit a very simple expression in terms of characteristic classes of $X$, namely it is expressed in terms of the second component of Chern character of the tangent bundle of $X$. The obstruction to the existence of a globally defined Virasoro field $L(z)$; it is given by the first Chern class $c_1(X)/2$, cf. Theorem 9.1. In particular, Theorem 3, provides a geometric criterion for a manifold to admit a $BU\langle 6\rangle$-structure: those are precisely the manifolds which admit the above mentioned sheaf $\CO^{ch}_X$ and for which the conformal anomaly vanishes. If such a manifold is Calabi-Yau (i.e. has the trivial canonical bundle) then $\CO^{ch}_X$ is a sheaf of {\it conformal} vertex algebras, cf. Corollary 9.3. From a different viewpoint, one can regard the above result as a geometric interpretation of the second component of the Chern character. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/GrayB/associativity Associativity in two-cell complexes Brayton Gray Let P be the mapping cone of an element in an even stem in the homotopy groups of spheres localized at an odd prime. Generalizing the case of a mod p^r Moore space, we show that the smash square of P splits as a wedge of two iterated suspensions of P. Furthermore, this can be done in a unique way satisfying certain identities, and if p>3, one of these identities is an associativity condition. There are two consequences: 1) If E is a (commutative) associative ring spectrum, then E^P is as well when localized at p>3. 2) A Samelson product can be defined in homotopy with coefficients in P which will satisfy all the usual identities including the Jacobi identity if p>3. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/GrayB/twocells On the homotopy groups of 2-cell complexes Brayton Gray In 1978, Cohen, Moore, and Neisendorfer gave a decomposition of the loops on a mod p^r Moore space when p>2. This decomposition involved an atomic factor T^(2n+1) which was encompassed in a fibration sequence with other terms whose homotopy was better understood. This paper considers the case when the mod p^r Moore space is replaced by the mapping cone P of an element in an even stem. Exactly the same results are obtained when the attaching map is divisible by p, or the dimension of P is even. The first obstruction to such a result is displayed in general, and the example of beta_1 is presented. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Jianzhong-Woo/forget Title: Phantom maps and Forgetable maps Authors: Pan Jianzhong and Moo Ha Woo In this note, we attack a question posed ten years ago by Tsukiyama about the injectivity of the so-called Forgetable map. We show that we can insert the Forgetable map in an exact sequence and that the problem can be reduced to the computation of the sequence which turns out unexpectedly to be related to the phantom map problem and the famous Halperin conjecture in rational homotopy theory. ---------------------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.wesleyan.edu/~mhovey/archive/ If this doesn't work or is missing a few issues, try http://www.lehigh.edu/~dmd1/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 the URL listed. The general Hopf archive URL is http://hopf.math.purdue.edu There are links to conference announcements, Purdue seminars, and other math related things on this page as well. The largest archive of math preprints is at http://xxx.lanl.gov There is an algebraic topology section in this archive. The most useful way to browse it or submit papers to it is via the front end developed by Greg Kuperberg: http://front.math.ucdavis.edu To get the announcements of new papers in the algebraic topology section at xxx, send e-mail to math@xxx.lanl.gov with subject line "subscribe" (without quotes), and with the body of the message "add AT" (without quotes). You can also access Hopf through ftp. 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. Clarence has explicit instructions for the form of this abstract: see http://hopf.math.purdue.edu/pub/submissions.html In particular, your abstract is meant to be read by humans, so should be as readable as possible. I reserve the right to edit unreadable abstracts. You should then e-mail Clarence at wilker@math.purdue.edu telling him what you have uploaded. I am solely responsible for these messages---don't send complaints about them to Clarence. Thanks to Clarence for creating and maintaining the archive.