Subject: new Hopf listings
Date: 21 Jan 2003 09:54:34 -0500
From: Mark Hovey
Reply-To: mhovey@wesleyan.edu
To: dmd1@lehigh.edu
The dvipdf and dviselect programs don't seem to be working quite right
on Hopf. Only a very few papers are affected, but if you have any
trouble with pdf files, use the dvi file instead.
13 new papers this time, from BrownR, BrownR-Higgins,
Chataur-Rodriguez-Scherer, Hovey, Hovey-Strickland (2 papers), Hung (4),
Hung-Nam (2), and Marzantowicz-Prieto.
Mark Hovey
New papers appearing on hopf between 1/08/03 and 01/21/03
1.
http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/fields-artxx
Title of Paper: Crossed complexes and homotopy groupoids as non
commutative tools for higher dimensional local-to-global problems
Author(s): Ronald Brown
AMS Classification numbers: 01-01,16E05,18D05,18D35,55P15,55Q05
Already submitted to the xxx LANL archive, include the id. no.,
math.AT/0212271
Addresses of Authors:
Ronald Brown
Mathematics Division,
School of Informatics,
University of Wales, Bangor
Gwynedd LL57 1UT, U.K.
Email address of Authors
r.brown@bangor.ac.uk
Abstract: We outline the main features of the definitions and
applications of crossed complexes and cubical $\omega$-groupoids with
connections. These give forms of higher homotopy groupoids, and new
views of basic algebraic topology and the cohomology of groups, with the
ability to obtain some non commutative results and compute some homotopy
types.
2.
http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR-Higgins/orbitgpdxx
Title of Paper: The fundamental groupoid of the quotient of a Hausdorff
space by a discontinuous action of a discrete group is the orbit
groupoid of the induced action
Author(s): Ronald Brown and Philip J. Higgins
AMS Classification numbers: 0F34, 20L13, 20L15, 57S30
Already submitted to the xxx LANL archive, include the id. no.,
math.AT/0212271
Addresses of Authors:
Ronald Brown
Mathematics Division,
School of Informatics,
University of Wales, Bangor
Gwynedd LL57 1UT, U.K.
Philip J. Higgins
Department of Mathematical Sciences,
Science Laboratories,
South Rd.,
Durham, DH1 3LE, U.K.
Email address of Authors
r.brown@bangor.ac.uk
p.j.higgins@durham.ac.uk
Text of Abstract (try for 20 lines or less)
The main result is that the fundamental groupoid of the orbit space of a
discontinuous action of a discrete group on a Hausdorff space which
admits a universal cover is the orbit groupoid of the fundamental
groupoid of the space. We also describe work of Higgins and of Taylor
which makes this result usable for calculations. As an example, we
compute the fundamental group of the symmetric square of a space.
The main result, which is related to work of Armstrong, is due to Brown
and Higgins in 1985 and was published in sections 9 and 10 of Chapter 9
of the first author's book on Topology (1988 edition). This is a
somewhat edited, and in one point (on normal closures) corrected,
version of those sections.
Because of its provenance, this should be read as a graduate text rather
than an article. The Exercises should be regarded as further
propositions for which we leave the proofs to the reader. It is
expected that this material will be part of a new edition of the book.
3.
http://hopf.math.purdue.edu/cgi-bin/generate?/Chataur-Rodriguez-Scherer/operadplus
Plus-construction of algebras over an operad, cyclic and
Hochschild homologies up to homotopy
David Chataur, Jose L. Rodriguez, and Jerome Scherer
math.AT/0301130
CRM Barcelona, dchataur@crm.es
Universidad de Almeria, jlrodri@ual.es
Universidad Autonoma de Barcelona, jscherer@mat.uab.es
The aim of this paper is to show how to apply the machinery of
homotopical localization to the framework of differential graded
algebras over an operad. By performing nullification with respect to a
universal acyclic algebra one obtains a plus-construction, which doesn't
affect Quillen homology and quotients out the maximal perfect ideal of
$\pi_0$. For any associative algebra the general linear Lie
(resp. Leibniz) algebra is a Lie (resp. Leibniz) algebra up to
homotopy. The plus-construction yields then two new homology theories,
closely related to cyclic and Hochschild homology (they coincide with
the classical cyclic and Hochschild homology over the rational). We also
compute the first homology groups of these theories, in analogy with the
computation of the first $K$-theory groups of a ring.
4.
http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/barcelona
Chromatic phenomena in the algebra of BP_{*}BP-comodules
Mark Hovey
Wesleyan University
mhovey@wesleyan.edu
This paper begins with an exposition of the author's research on the
category of BP_*BP-comodules, much of which is joint with Neil
Strickland. We give an overview of the results obtained in the papers
Hovey/comodule, Hovey-Strickland/torsion-comod, and
Hovey-Strickland/derived-ln. The main result of that work is that the
category of E(n)_*E(n)-comodules is equivalent to a localization of the
category of BP_*BP-comodules (the localization is L_n, analogous to the
topological L_n).
The main new result in this paper is that, analogously, the stable
homotopy category of E(n)_*E(n)-comodules is equivalent to a
localization (the finite localization L_n^f this time, not L_n) of the
stable homotopy category of BP_*BP-comodules. These stable homotopy
categories were constructed in Hovey/comodule, and are supposed to model
stable homotopy theory; it is like stable homotopy theory where there
are no differentials in the Adams-Novikov spectral sequence. Our result
embeds the Miller-Ravenel and Hovey-Sadofsky change of rings theorems as
special cases of isomorphisms like
[X,Y]=[L_n^f X, Y]
for L_n^f-local objects Y.
5.
http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Strickland/torsion-comod
Comodules and Landweber exact homology theories
Mark Hovey and Neil Strickland
Wesleyan University University of Sheffield
mhovey@wesleyan.edu N.P. Strickland@sheffield.ac.uk
We show that, if E is a Landweber exact ring spectrum, then the category
of E_*E-comodules is equivalent to the localization of the category of
BP_*BP-comodules with respect to the hereditary torsion theory of
v_n-torsion comodules, where n is the height of E. In particular, the
category of E(n)_*E(n)-comodules is equivalent to the category of
(v_n^{-1}BP)_*(v_n^{-1}BP)-comodules. We also prove structure theorems
for E_*E-comodules; we show every E_*E-comodule has a primitive, we
classify the invariant radical ideals, and we prove a version of the
Landweber filtration theorem.
6.
http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Strickland/derived-ln
Local cohomology of BP_*BP-comodules
Mark Hovey and Neil Strickland
Wesleyan University University of Sheffield
mhovey@wesleyan.edu N.P. Strickland@sheffield.ac.uk
In the paper torsion-comod (announced above) on this archive, we showed
that the category of E(n)_*E(n)-comodules is a localization of the
category of BP_*BP-comodules. In this paper, we study the resulting
localization functor L_n on the category of BP_*BP-comodules. It is an
algebraic analogue of the usual topological localization L_n. It is
left exact, so has right derived functors L_n^i. We show that these
derived functors are closely related to the local cohomology groups of
BP_*-modules studied by Greenlees and May; in fact, they coincide with
Cech cohomology with respect to I_{n+1}. We also construct a spectral
sequence of comodules analogous to the Greenlees-May spectral sequence
(of modules) converging to BP_*(L_n X) whose E_2-term involves
L_n^i(BP_*X). The proofs require getting a partial understanding of
injective objects in the category of BP_*BP-comodules.
7.
http://hopf.math.purdue.edu/cgi-bin/generate?/Hung/2001
Title of Paper: On triviality of Dickson invariants in the homology of
the Steenrod algebra
Author: Nguy\^{e}n H. V. Hung
2000 Mathematics Subject Classification: Primary 55P47, 55Q45,
55S10, 55T15.
Address of Author:
Current Address: Department of Mathematics,
Johns Hopkins University, 3400 N. Charles Street, Baltimore MD
21218 - 2689 E-mail address: nhvhung@math.jhu.edu
Permanent Address: Department of Mathematics, Vietnam National
University, Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam
E-mail address: nhvhung@vnu.edu.vn
Abstract: Let ${\cal A}$ be the mod 2 Steenrod algebra and $D_k$
the Dickson algebra of $k$ variables. We study the Lannes-Zarati
homomorphisms
$$
\varphi_k: Ext_{\cal A}^{k,k+i}(F_2,F_2)\to (F_2\otimes_{\cal A}
D_k)_i^*,
$$
which correspond to an associated graded of the Hurewicz map $
H:\pi_*^s(S^0)\cong \pi_*(Q_0S^0)\to H_*(Q_0S^0)$. An algebraic
version of the long-standing conjecture on spherical classes
predicts that $\varphi_k=0$ in positive stems, for $k>2$. That the
conjecture is no longer valid for $k=1$ and $2$ is respectively an
exposition of the existence of Hopf invariant one classes and
Kervaire invariant one classes.
This conjecture has been proved for $k=3$ by Hung [Trans AMS 349
(1997), 3893-3910]. It has been shown that $\varphi_k$ vanishes on
decomposable elements for $k>2$ [Hung and Peterson, Math. Proc.
Camb. Phil. Soc. 124 (1998), 253-264] and on the image of Singer's
algebraic transfer for $k>2$ [Hung, 1997; Hung and Nam, Trans AMS
353 (2001), 5029-5040]. In this paper, we establish the conjecture
for $k=4$. To this end, our main tools include (1) an explicit
chain-level representation of $\varphi_k$ and (2) a squaring
operation $Sq^0$ on $(F_2\otimes_{\cal A} D_k)^*$, which commutes
with the classical $Sq^0$ on $Ext_{\cal A}^k(F_2,F_2)$ through the
Lannes-Zarati homomorphism.
(To appear in Math. Proc. Camb. Phil. Soc. 134 (2003).)
8.
http://hopf.math.purdue.edu/cgi-bin/generate?/Hung/2002h
Title of Paper: The cohomology of the Steenrod algebra and
representations of the general linear groups
Author: Nguy\^{e}n H. V. Hung
2000 Mathematics Subject Classification:
Primary 55P47, 55Q45,
55S10, 55T15.
Address of Author:
Current Address: Department of Mathematics,
Wayne State University 656 W. Kirby Street, Detroit, MI 48202
(USA) E-mail address: nhvhung@@math.wayne.edu
Permanent Address: Department of Mathematics, Vietnam National
University, Hanoi 334 Nguyen Trai Street, Hanoi, Vietnam E-mail
address: nhvhung@@vnu.edu.vn
Abstract:
Let $Tr_k$ be the algebraic transfer that maps from the
coinvariants of certain $GL_k$-representation to the cohomology of
the Steenrod algebra. This transfer was defined by W. Singer as an
algebraic version of the geometrical transfer $tr_k: \pi_*^S((B\V
_k)_+) \to \pi_*^S(S^0)$. It has been shown that the algebraic
transfer is highly nontrivial, more precisely, that $Tr_k$ is an
isomorphism for $k=1, 2, 3$ and that $Tr= \oplus_k Tr_k$ is a
homomorphism of algebras.
In this paper, we first recognize the phenomenon that if we start
from any degree $d$, and apply $Sq^0$ repeatedly at most $(k-2)$
times, then we get into the region, in which all the iterated
squaring operations are isomorphisms on the coinvariants of the
$GL_k$-representation. As a consequence, every finite
$Sq^0$-family in the coinvariants has at most $(k-2)$ non zero
elements. Two applications are exploited.
The first main theorem is that $Tr_k$ is not an isomorphism for
$k\geq 5$. Furthermore, $Tr_k$ is not an isomorphism in infinitely
many degrees for each $k > 5$. We also show that if $Tr_{\ell}$
detects a nonzero element in certain degrees of
$\text{Ker}(Sq^0)$, then it is not a monomorphism and further,
$Tr_k$ is not a monomorphism in infinitely many degrees for each
$k>\ell$.
The second main theorem is that the elements of any $Sq^0$-family
in the cohomology of the Steenrod algebra, except at most its
first $(k-2)$ elements, are either all detected or all not
detected by $Tr_k$, for every $k$. Applications of this study to
the cases $k=4$ and $5$ show that $Tr_4$ does not detect the three
families $g$, $D_3$, $p'$ and $Tr_5$ does not detect the family
$\{h_{n+1}g_n |\; n\geq 1\}$.
9.
http://hopf.math.purdue.edu/cgi-bin/generate?/Hung/HungTAMS01
Title of Paper: Spherical classes and the Lambda algebra
Author: Nguy\^{e}n H. V. Hung
2000 Mathematics Subject Classification: Primary 55P47, 55Q45,
55S10, 55T15.
Address: Department of Mathematics, Vietnam National University,
Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam
E-mail address: nhvhung@vnu.edu.vn
Abstract: Let $\Gamma^{\wedge}= \oplus_k \Gamma_k^{\wedge}$ be
Singer's invariant-theoretic model of the dual of the Lambda
algebra with $H_k(\Gamma^{\wedge})\cong Tor_k^{\cal A}(F_2, F_2)$,
where ${\cal A}$ denotes the mod 2 Steenrod algebra. We prove that
the inclusion of the Dickson algebra, $D_k$, into
$\Gamma_k^{\wedge}$ is a chain-level representation of the
Lannes--Zarati dual homomorphism
$$
\varphi_k^*: F_2\otimes_{\cal A} D_k \to Tor^{\cal A}_k(F_2,F_2)
\cong H_k(\Gamma^{\wedge}).
$$
The Lannes--Zarati homomorphisms themself, $\varphi_k$, correspond
to an associated graded of the Hurewicz map
$$
H:\pi_*^s(S^0)\cong \pi_*(Q_0S^0)\to H_*(Q_0S^0)\,.
$$
Based on this result, we discuss some algebraic versions of the
classical conjecture on spherical classes, which states that {\it
Only Hopf invariant one and Kervaire invariant one classes are
detected by the Hurewicz homomorphism.} One of these algebraic
conjectures predicts that every Dickson element, i. e. element in
$D_k$, of positive degree represents the homology class $0$ in
$Tor^{\cal A}_k(F_2, F_2)$ for $k>2$.
10.
http://hopf.math.purdue.edu/cgi-bin/generate?/Hung/HungTAMS97
Title of Paper: Spherical classes and the algebraic transfer
Author: Nguy\^{e}n H. V. Hung
1991 Mathematics Subject Classification: Primary 55P47, 55Q45,
55S10, 55T15.
Address of Author: Department of Mathematics, Vietnam National
University, Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam
E-mail address: nhvhung@vnu.edu.vn
Abstract: We study a weak form of the classical conjecture which
predicts that there are no spherical classes in $Q_0S^0$ except
the elements of Hopf invariant one and those of Kervaire invariant
one. The weak conjecture is obtained by restricting the Hurewicz
homomorphism to the homotopy classes which are detected by the
algebraic transfer.
We prove that the weak conjecture is equivalent to the following
one: Every positive degree Dickson invariant of at least 3
variables belongs to the image of the Steenrod algebra acting on
the corresponding polynomial algebra. This conjecture is proved
for the case of 3 variables in two different ways.
11.
http://hopf.math.purdue.edu/cgi-bin/generate?/Hung-Nam/HungNamJA01
Title of Paper: The hit problem for the modular invariants of linear
groups
Author: Nguy\^{e}n H. V. Hung and Tran Ngoc Nam
2000 Mathematics Subject Classification: Primary 55S10, Secondary
55Q45.
Address of authors: Department of Mathematics, Vietnam National
University, Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam
E-mail address: nhvhung@vnu.edu.vn
E-mail address: namtn@vnu.edu.vn
Abstract: Let the mod 2 Steenrod algebra, ${\cal A}$, and the
general linear group, $GL_k:=GL(k, F_2)$, act on
$P_{k}:=F_2[x_{1},...,x_{k}]$ with $\deg(x_{i})=1$ in the usual
manner. We prove that, for a family of some rather small subgroups
$G$ of $GL_k$, every element of positive degree in the invariant
algebra $P_{k}^G$ is hit by ${\cal A}$ in $P_{k}$. In other words,
$(P_{k}^G)^+ \subset {\cal A}^+\cdot P_{k}$, where $(P_{k}^G)^+$
and ${\cal A}^+$ denote respectively the submodules of $P_{k}^G$
and ${\cal A}$ consisting of all elements of positive degree. This
family contains most of the parabolic subgroups of $GL_k$. It
should be noted that the smaller the group G is the harder the
problem turns out to be. Remarkably, when $G$ is the smallest
group of the family, the invariant algebra $P_{k}^G$ is a
polynomial algebra in $k$ variables, whose degrees are $\leq 8$
and fixed while $k$ increases.
It has been shown by Hung [Trans AMS 349 (1997), 3893-3910] that,
for $G=GL_k$, the inclusion $(P_{k}^{GL_k})^+\subset {\cal
A}^+\cdot P_{k}$ is equivalent to a week algebraic version of the
long-standing conjecture stating that the only spherical classes
in $Q_0S^0$ are the elements of Hopf invariant one and those of
Kervaire invariant one.
12.
http://hopf.math.purdue.edu/cgi-bin/generate?/Hung-Nam/HungNamTAMS01
Title of Paper: The hit problem for the Dickson algebra
Author: Nguy\^{e}n H. V. Hung and Tran Ngoc Nam
2000 Mathematics Subject Classification: Primary 55S10, Secondary
55P47, 55Q45, 55T15.
Address of authors: Department of Mathematics, Vietnam National
University, Hanoi 334 Nguyen Trai Street, Hanoi, Vietnam
E-mail address: nhvhung@vnu.edu.vn
E-mail address: namtn@vnu.edu.vn
Abstract: Let the mod 2 Steenrod algebra, ${\cal A}$, and the
general linear group, $GL(k, F_2)$, act on $P_{k}:=
F_2[x_{1},...,x_{k}]$ with $|x_{i}|=1$ in the usual manner. We
prove the conjecture of the first-named author in {\it Spherical
classes and the algebraic transfer}, (Trans. AMS 349 (1997),
3893-3910) stating that every element of positive degree in the
Dickson algebra $D_{k}:=(P_{k})^{GL(k,F_2)}$ is ${\cal
A}$-decomposable in $P_{k}$ for arbitrary $k>2$. This conjecture
was shown to be equivalent to a weak algebraic version of the
classical conjecture on spherical classes, which states that the
only spherical classes in $Q_0S^0$ are the elements of Hopf
invariant one and those of Kervaire invariant one.
13.
http://hopf.math.purdue.edu/cgi-bin/generate?/Marzantowicz-Prieto/Marprieto
The unstable equivariant fixed point index and the equivariant degree
by Waclaw Marzantowicz and Carlos Prieto
A correspondence between the equivariant degree introduced by Ize,
Massab\'o, and Vignoli and an unstable version of the equivariant fixed
point index defined by the second author and Ulrich is shown. With the
help of conormal maps and properties of the unstable index, we prove a
sum decomposition formula for the index and consequently also for the
degree. As an application, we decompose equivariant homotopy groups as
direct sums of smaller groups of fixed orbit types, and we give a
geometric interpretation of each summand in terms of conormal maps.
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put
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