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There are 7 new papers this time, from Arone-Lambrechts-Volic,
Broto-Levi-Oliver, Dwyer-Wilkerson, Gillespie, Naumann,
Ulrich-Wilkerson, and YauD.

                  Mark Hovey

New papers appearing on hopf between 7/8/06 and 8/4/06

1.
http://hopf.math.purdue.edu/cgi-bin/generate?/Arone-Lambrechts-Volic/CalculusFormalityEmbeddings

Title: Calculus of functors, operad formality, and rational homology of
embedding spaces

Authors:
Gregory Arone, 
Department of Mathematics, University of Virginia, Charlottesville, VA, USA. 

Pascal Lambrechts
Institut Math\'{e}matique, 2 Chemin du Cyclotron, B-1348
Louvain-la-Neuve, Belgium 

Ismar Voli\'c
Department of Mathematics, University of Virginia, Charlottesville, VA, USA

Abstract:

Let M be a smooth manifold and V a Euclidean space. Let Ebar(M,V) be the
homotopy fiber of the map from Emb(M,V) to Imm(M,V). This paper is about
the rational homology of Ebar(M,V). We study it by applying embedding
calculus and orthogonal calculus to the bi-functor (M,V) |--> HQ
/\Ebar(M,V)_+. Our main theorem states that if the dimension of V is
more than twice the embedding dimension of M, the Taylor tower in the
sense of orthogonal calculus (henceforward called ``the orthogonal
tower'') of this functor splits as a product of its
layers. Equivalently, the rational homology spectral sequence associated
with the tower collapses at E^1. In the case of knot embeddings, this
spectral sequence coincides with the Vassiliev spectral sequence. The
main ingredients in the proof are embedding calculus and Kontsevich's
theorem on the formality of the little balls operad.  We write explicit
formulas for the layers in the orthogonal tower of the functor HQ
/\Ebar(M,V)_+. The formulas show, in particular, that the (rational)
homotopy type of the layers of the orthogonal tower is determined by the
(rational) homotopy type of M. This, together with our rational
splitting theorem, implies that under the above assumption on
codimension, the rational homology groups of Ebar(M,V) are determined by
the rational homotopy type of M.
 
2.
http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Levi-Oliver/blo3

Title of paper:  Discrete models for the $p$-local homotopy theory of 
compact Lie groups and $p$-compact groups

Authors:  Carles Broto, Ran Levi, Bob Oliver

AMS Classification:  Primary 55R35. Secondary 55R40, 57T10

Addresses of authors:  
Departament de Matem\`atiques 
Universitat Aut\`onoma de Barcelona 
E--08193 Bellaterra, Spain

Department of Mathematical Sciences
University of Aberdeen, Meston Building 339 
Aberdeen AB24 3UE, U.K.

LAGA, Institut Galil\'ee 
Av. J-B Cl\'ement 
93430 Villetaneuse, France

Abstract: 
We define and study a certain class of spaces which includes $p$-completed 
classifying spaces of compact Lie groups, classifying spaces of $p$-compact 
groups, and $p$-completed classifying spaces of certain locally finite 
discrete groups.  These spaces are determined by fusion and linking systems 
over ``discrete $p$-toral groups'' --- extensions of $(\Z/p^\infty)^r$ by 
finite $p$-groups --- in the same way that classifying spaces of $p$-local 
finite groups as defined in \cite{BLO2} are determined by fusion and 
linking systems over finite $p$-groups.  We call these structures 
``$p$-local compact groups''.

3.
http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Wilkerson/PiOneHopf

The fundamental group of a $p$-compact group

W. G. Dwyer and C. W. Wilkerson

The notion of a $p$-compact group is a homotopy theoretic version of the
geometric or analytic notion of compact Lie group, although the homotopy
theory differs from the geometry is that there are parallel theories of
$p$-compact groups, one for each prime number~$p$. A key feature of the
theory of compact Lie groups is the relationship between centers and
fundamental groups; these play off against one another, at least in the
semisimple case, in that the center of the simply connected form is the
fundamental group of the adjoint form. There are explicit ways to
compute the center or fundamental group of a compact Lie group in terms
of the normalizer of the maximal torus.  For some time there has in fact
been a corresponding formula for the center of $p$-compact groups, but
in general the fundamental group has eluded analysis.  The purpose of
the present paper is to remedy this deficit.

For any space $Y$, we let $\HZp_i(Y)$ denotes $\lim{}_n\HH_i(Y;\Z/p^n)$.
Suppose that $X$ is a connected $p$-compact group, with maximal torus
$T$ and torus normalizer $\NT$. It is known that the map 
$\pi_1(T)\to\pi_1(X)$ is surjective or equivalently
that the map $\HZp_2(\BB T)\to\HZp_2(\BB X)$ is surjective.
We prove the following statement.

Main Theorem: If $X$ is a connected $p$-compact group, then the kernel
  of the map $\HZp_2 \BB T\to \HZp_2\BB\NT$ is the same as the
  kernel of the map $\HZp_2 \BB T\to \HZp_2\BB X$. Equivalently, the
  image of the map $\HZp_2\BB T\to \HZp_2(\BB\NT)$ is (naturally)
  isomorphic to $\pi_1X$.

  There is a proof of the corresponding statement for compact Lie groups
  which relies on the Feshbach double coset formula Our proof of the
  MainTheorem uses a transfer calculation that in practice amounts to a
  weak homological reflection of the double coset formula; we can get
  away with this because we have a splitting of $\HZp_2(\BB\NT)$.

  It is possible to derive from the MainTheorem a more explicit formula
  for $\pi_1X$; this formula is known for $p$~odd as a consequence of
  the classification theorem for $p$ odd. Our demonstration does not use
  the classification theorem.

  Let $W$ denote the Weyl group of $X$. If $p$ is odd, then $\pi_1X$ is
  naturally isomorphic to the module of coinvariants $\HH_0(W;\HZp_2(\BB
  T))$ .  If $p=2$, then up to factors which do not contribute to
  $\pi_1X$, the normalizer of the torus in $X$ is derived by
  $\Ftwo$-completion from the normalizer $\NT_G$ of a maximal torus
  $T_G$ in a connected compact Lie group~$G$ . The image of the map
  $\HH_2(\BB T_G;\Z)\to\HH_2(\BB\NT_G;\Z)$ is isomorphic to $\pi_1G$ ,
  and so by the MainTheorem the tensor product of this image with
  $\Ztwo$ is $\pi_1X$. This image can be computed from the marked
  reflection lattice $(\pi_1T_G, \{b_\sigma,\beta_\sigma\})$
  corresponding to the root system of $G$ or, after tensoring with
  $\Ztwo$, from the marked complete reflection lattice
  $(\pi_1T,\{b_\sigma,\beta_\sigma\})$ associated to $X$ The upshot is
  that $\pi_1X$ is the quotient of $\pi_1T=\pi_2\BB T=\HZtwo_2\BB T$ by
  the $\Ztwo$--submodule generated by the elements $\{b_\sigma\}$.

  Another way to describe this calculation is the following. For each
  reflection $s_\alpha$ in the Weyl group~$W$, let $u_\alpha$ be a
  generator over $\Zp$ of the rank~1 submodule of $\pi_1T$ given by the
  image of $(1-s_\alpha)$. If $p$ is odd let $v_\alpha=u_\alpha$; if
  $p=2$, let $v_\alpha=u_\alpha$ or $u_\alpha/2$, according to whether
  the marking of $s_\alpha$ is trivial or non-trivial. Then $\pi_1X$ is
  the quotient of $\pi_1T$ by the $\Zp$-span of the
  elements~$v_\alpha$. See the upcoming even classification by Andersen
  and Grodal for more details.

4.
http://hopf.math.purdue.edu/cgi-bin/generate?/Gillespie/quasi-coherent

Title: A Quillen Approach to Derived Categories and Tensor Products

Author: James Gillespie 

AMS Classification numbers: 55U35, 18G15, 18E30  

4000 University Drive
Penn State McKeesport
McKeesport, PA 15132

Abstract: We put a monoidal model category structure on the category 
of chain complexes of quasi-coherent sheaves over a quasi-compact 
and semi-separated scheme X. The approach generalizes and simplifies 
methods used by the author to build monoidal model structures  on 
the category of chain complexes of modules over a ring and chain 
complexes of sheaves over a ringed space. Indeed, much of the paper
is dedicated to showing that in any Grothendieck category G, a nice 
enough class of objects, which we call a Kaplansky class, induces a 
model structure on the category Ch(G) of chain complexes. We also 
find simple conditions to put on the Kaplansky class which will 
guarantee that our model structure in monoidal. We see that the 
common model structures used in practice are all induced by such 
Kaplansky classes.  

5.
http://hopf.math.purdue.edu/cgi-bin/generate?/Naumann/qisoneu

Niko Naumann
Quasi-isogenies and Morava stabilizer groups

For every prime $p$ and integer $n\ge 3$ we explicitly construct an
abelian variety $A/\F_{p^n}$ of dimension $n$ such that for a suitable
prime $l$ the group of quasi-isogenies of $A/\F_{p^n}$ of $l$-power
degree is canonically a dense subgroup of the $n$-th Morava stabilizer
group at $p$.  We also give a variant of this result taking into account
a polarization.  This is motivated by a perceivable generalization of
topological modular forms to more general topological automorphic forms.

For this, we prove some results about approximation of local units in
maximal orders which is of independent interest. For example, it gives a
precise solution to the problem of extending automorphisms of the
$p$-divisible group of a simple abelian variety over a finite field to
quasi-isogenies of the abelian variety of degree divisible by as few
primes as possible.

6.
http://hopf.math.purdue.edu/cgi-bin/generate?/Ulrich-Wilkerson/uw06rev1

Field degrees and multiplicities for non-integral extensions

Bernd Ulrich
Clarence W. Wilkerson
Department of Mathematics, Purdue University, West Lafayette, IN 47907
Department of Mathematics, Purdue University, West Lafayette, IN 47907
ulrich@math.purdue.edu
cwilkers@purdue.edu

Let $k$ be a field and $S = k[t_1,\hdots,t_d]$ a polynomial ring with
variables $t_i$ of degree one. Consider a $k$-subalgebra $R$ generated
by $m$ homogeneous elements $\{x_1,\hdots,x_m\}$.  In general, if $x$ is
a homogeneous element in a graded object, we denote its degree by $|x|$.

{\bf Problem.} {\it Let $[S:R]$ denote the degree of the underlying
fraction field extension. If $S$ is algebraic over $R$, calculate
$[S:R]$ from the $\{|x_i|\}$ }.

First, one has a form of Bezout's Theorem:

\begin{thm}\label{BezoutsThm} If $S$ is integral over $R$, the following hold:
\begin{enumerate}
\item $[S:R]$ divides $\prod{|x_i|}$.
\item If $m=d$, then $[S:R] = \prod{|x_i|}$.
\end{enumerate}
\end{thm}

In this paper, we consider the case that $m = d$ and obtain a converse to
part (b) above:

\begin{thm}\label{MainTheorem} If $S$ is algebraic over $R$, $m=d$, and 
$[S:R] = \prod{|x_i|}$, then $S$ is integral over $R$ $($equivalently, $S$
is finitely generated as an $R$-module$)$.
\end{thm}

We also note that if $S$ is not integral over $R$, then $[S:R]$ need not
even divide $\prod{|x_i|}$.

Our proofs rely on reduction to the case of standard graded $k$-algebras. 

An interesting application of Theorem 1.2 is in the study of rings of
invariants of finite groups acting on a polynomial ring:

\begin{thm}\label{Invariants}
Let $V$ be a $d$-dimensional vector space over the field $k$, $V^\#$ its
$k$-dual, and $S = S[V^\#] = k[t_1,\hdots,t_d]$ the algebra of
polynomial functions on $V$. Let $W \subset GL(V)$ be a finite group.
There is an induced action on $S$. Then $S^W = R$ is a polynomial
algebra over $k$ if and only if there exist homogeneous elements $\{x_1,
\hdots,x_d\}$ of $R$ such that
\begin{enumerate}
\item $S$ is algebraic over $k[x_1,\hdots,x_d]$, and
\item $|W| = \prod{|x_i|}$.
\end{enumerate}
\end{thm}

7.
http://hopf.math.purdue.edu/cgi-bin/generate?/YauD/module_alg

Title: Cohomology and deformation of module-algebras

Author: Donald Yau

Email: dyau@math.ohio-state.edu

Abstract: An algebraic deformation theory of module-algebras over a
bialgebra is constructed.  The cases of module-coalgebras,
comodule-algebras, and comodule-coalgebras are also considered.

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