--------------------------------------
These are mostly papers that just made it out of Clarence's e-mail.  The
new policy at Hopf is that e-mail submssions are strongly deprecated.
Please use the web form if at all possible.  There is a significant and
unpredictable delay associated with e-mail submission and it is easier
for papers to get misplaced.

11 new papers this time, from Chernov-Rudyak, Dugger (3),
Dwyer-Wilkerson, Ibanez-Rudyak-Tralle, Ibanez-Rudyak-Tralle-Ugarte,
Oliver, Oprea-Rudyak, Rudyak, and Wilkerson.  

                  Mark Hovey

New papers appearing on hopf between 5/13/03 and 5/17/03

1.
http://hopf.math.purdue.edu/cgi-bin/generate?/Chernov-Rudyak/wavefronts

Title: Affine Linking Numbers and Causality Relations for Wave Fronts
Authors: Vladimir Chernov (Tshernov), Yuli Rudyak

Addresses: 
  V. Chernov, Department of Mathematics, 6199 Bradley Hall, 
  Dartmouth College, Hanover NH 03755, USA

  Yuli Rudyak, Department of Mathematics, University of Florida, 358 
  Little Hall, PO Box 118105 Gainesville,  FL 32611-8105 U.S.A 

e-mail: rudyak@math.ufl.edu
e-mail: Vladimir.Chernov@dartmouth.edu

4 figures (eps files)

Abstract: Let M be an oriented manifold. We study the causal relations
between the wave fronts W and W' that originated at some points of M. We
introduce a numerical topological invariant CRI(W, W') (the so-called
causality relation invariant) that, in particular, gives the algebraic
number of times the wave front W passed through the point that was the
W' before the front W' originated.  This invariant can be easily
calculated from the current picture of wave fronts on M without the
knowledge of the propagation law for the wave fronts. Moreover, in fact
we even do not need to know the topology of M outside of a part V of M
such that W and W' are null-homotopic in V.  

2.
http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger/krspecDD

Title: An Atiyah-Hirzebruch spectral sequence for KR-theory
Author: Daniel Dugger

Department of Mathematics, University of Oregon, Eugene, OR 97403
email: ddugger@math.uoregon.edu

Abstract:

We construct a spectral sequence for KR-theory which is analagous to 
the spectral sequence relating motivic cohomlogy to algebraic K-theory.

3.
http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger/mult1DD

Title: Multiplicative structures on spectral sequences I
Authors: Daniel Dugger

Department of Mathematics, University of Oregon, Eugene, OR 97403
email: ddugger@math.uoregon.edu

Abstract:

This is mostly an expository paper, recording basic facts about towers
of homotopy fiber sequences.  We show that a pairing of towers induces
an associated pairing of spectral sequences, for towers of spaces and
towers of spectra.  In the hope that this might eventually be a
useful reference for people, feel free to send me suggestions for
things that should be improved (with the understanding that it might
be a while before I get around to implementing them).

4.
http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger/mult2DD

Title: Multiplicative structures on spectral sequences II
Authors: Daniel Dugger

Department of Mathematics, University of Oregon, Eugene, OR 97403
email: ddugger@math.uoregon.edu

Abstract:

This paper summarizes the constructions of pairings for some of the
standard spectral sequences in algebraic topology.  

5.
http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Wilkerson/NT

Title: Normalizers of Tori
Authors: W. G. Dwyer AND C. W. Wilkerson, Notre Dame and Purdue

  Suppose that G is a connected compact Lie group and that T   G
is a maximal torus, or in other words a maximal connected abelian
subgroup. The normalizer NT of T lies in a short exact sequence
(1.1)                 1 -> T -> NT -> W -> 1
in which W is a finite group called the Weyl group of G.  In this pa-
per we reformulate some ideas of Tits [27 ] in order to describe exactly
which groups appear as such an NT . This leads to an analogous deter-
mination of which groups appear as the normalizer NT~ of a maximal
2-discrete torus in a connected 2-compact group (1.16).

  In the compact Lie group case, NT determines G up to isomorphism
[3], and so in listing the possible NT 's we are giving an alternative
approach to the classification of connected compact Lie groups them-
selves.  In contrast, it is not known that the normalizer of a maximal
2-discrete torus in a connected 2-compact group X determines X up
to equivalence.  However, this seems likely to be true [23 ] [19 ], and
we hope that the results of this paper will eventually contribute to a
classification of connected 2-compact groups.

6.
http://hopf.math.purdue.edu/cgi-bin/generate?/Ibanez-Rudyak-Tralle/aspherical

Title: On the fundamental groups of  symplectically aspherical manifolds
Authors: R. Ibanez, Yu. Rudyak, A. Tralle

Adresses of Authors:
R. Ibanez, Departamento de Matematicas, Facultad de
Ciencias, Universidad del Pais Vasco, Apdo. 644, 48080 Bilbao, Spain

Yu. Rudyak, Department of Mathematics, Universoty of Florida, 
358 Little Hall, Gainesville, FL 32601, USA

A. Tralle, Department of Mathematics, University of Warmia and Mazura,
10561 Olsztyn, Poland

email: mtpibtor@lg.ehu.es
       rudyak@math.ufl.edu
       tralle@matman.uwm.edu.pl   
         
In this paper we are interested in the fundamental groups of closed
symplectically aspherical manifolds; i.e., of symplectic manifolds whose 
symplectic form vanishes on 2-dimensional spherical homology classes. 
Motivated by some results of Gompf, we  consider two classes of fundamental 
groups of symplectically aspherical  manifolds: with trivial and-non-trivial 
second homotopy group. Relations between  these classes are discussed. We show 
that several important classes of groups can be realized in both classes. Also, 
we notice that there are some dimensional phenomena in the realization problem.


7.
http://hopf.math.purdue.edu/cgi-bin/generate?/Ibanez-Rudyak-Tralle-Ugarte/HomotopySymplecticKahler

Title: On certain geometric and  homotopy properties of closed symplectic 
manifolds
Authors: R. Ibanez, Yu. Rudyak, A. Tralle, L. Ugarte

Adresses of Authors:
R. Ibanez, Departamento de Matematicas, Facultad de
Ciencias, Universidad del Pais Vasco, Apdo. 644, 48080 Bilbao, Spain

Yu. Rudyak, Department of Mathematics, Universoty of Florida, 
358 Little Hall, Gainesville, FL 32601, USA

A. Tralle, Department of Mathematics, University of Warmia and Mazura,
10561 Olsztyn, Poland

L. Ugarte, Departamento de Matem\'aticas, Facultad de
Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain

email: mtpibtor@lg.ehu.es
       rudyak@math.ufl.edu
       tralle@matman.uwm.edu.pl
       ugarte\@posta.unizar.es  

The paper deals with relations between the Hard Lefschetz property, 
(non)vanishing of Massey products and the evenness of odd-degree Betti numbers 
of closed symplectic manifolds. It is known that closed symplectic manifolds can 
violate all these properties (in contrast with the case of Kaehler manifolds). 
However, the relations between such homotopy properties seem to be not analyzed. 
This analysis may shed a new light on topology of symplectic manifolds. In the 
paper, we summarize our knowledge in tables (different in the simply-connected 
and in symplectically aspherical cases). Also, we discuss the variation of 
symplectically harmonic Betti numbers on some 6-dimensional manifolds. 

8.
http://hopf.math.purdue.edu/cgi-bin/generate?/Oliver/limz

Equivalences of classifying spaces completed at the prime two
Bob Oliver

We prove here the Martino-Priddy conjecture for the prime $2$:  the 
$2$-completions of the classifying spaces of two groups $G$ and $G'$ are 
homotopy equivalent if and only if there is an isomorphism between their 
Sylow $2$-subgroups which preserves fusion.  This is a consequence of a 
technical algebraic result, which says that for a finite group $G$, the 
second higher derived functor of the inverse limit vanishes for a certain 
functor $\calz_G$ on the $2$-subgroup orbit category of $G$.  The proof of 
this result uses the classification theorem for finite simple groups.

9.
http://hopf.math.purdue.edu/cgi-bin/generate?/Oprea-Rudyak/cat3man

Title: Detecting Elements and Lusternik--Schnirelmann
Category of 3-Manifolds
Authors: John Oprea, Yuli Rudyak

Addresses: 
  John Oprea, Department of Mathematics, Cleveland State University, 
  Cleveland, Ohio 44115  U.S.A

  Yuli Rudyak, Department of Mathematics, University of Florida, 358 
  Little Hall, PO Box 118105 Gainesville,  FL 32611-8105 U.S.A 

e-mail: rudyak@math.ufl.edu

Abstract: In this paper, we give a new simplified calculation of the
Lusternik-Schnirelmann category of closed 3-manifolds. We also describe
when 3-manifolds have detecting elements and prove that 3-manifolds
satisfy the equality of the Ganea conjecture.

10.
http://hopf.math.purdue.edu/cgi-bin/generate?/Rudyak/PLstructures

(This is an updated version of a paper already on the archive)

Title: Piecewise linear structures on topological
 manifolds
Author: Yuli Rudyak

Address: Yuli Rudyak, Department of Mathematics, University of Florida, 358 
Little Hall, PO Box 118105 Gainesville,  FL 32611-8105, USA 

e-mail: rudyak@math.ufl.edu

Abstract: This is a survey paper where we expose the Kirby--Siebenmann
results on classification of PL structures on topological
manifolds and, in particular, the homotopy equivalence
TOP/PL=K(Z/2,3) and the Hauptvermutung for manifolds.


11.
http://hopf.math.purdue.edu/cgi-bin/generate?/Wilkerson/e8-lab

Lab Notes on the exceptional Lie group $E_8$ at the prime $2$
\author[C. W. Wilkerson]{Clarence W. Wilkerson, Jr.}
\dedicatory{Dedicated to Morton L. Curtis (1921-1989).}
\address{Department of Mathematics, Purdue University, West Lafayette, 
Indiana 47907-1395}
\thanks{Thanks to the National Science Foundation, Purdue University, 
Johns Hopkins University, and Fukuoka University for financial support 
during this research and the 2000 sabbatical of the author. Thanks to 
the Clay Foundation for travel support during this research.}
\email{wilker@math.purdue.edu}

 This is an account of the author's use 
of computer algebra tools to explore the structure 
of the maximal elementary abelian $2$-subgroups of the exceptional 
Lie group $E_8$. The principal result obtained thus far by these
methods is that any rank $8$ connected $2$-compact group $(BX,X)$ with
Weyl group isomorphic to that of the exceptional Lie group $E_8$  has
its normalizer of the maximal torus isomorphic to that of
$E_8$ at the prime $2$. Similar results hold for the comparison of 
possible exotic forms of $G_2$, $DI(4)$, $F_4$,
and $E_7/\Center(E_7)$ to the standard forms.\\

Corollaries of this result include that the Krull dimension of the
mod $2$ cohomology of such $BX$ is $9$ and that the cohomology
ring is not Cohen-Macaulay. \\ 


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