Homology, Homotopy and Applications Articles (Project Euclid)
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The latest articles from Homology, Homotopy and Applications on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTFri, 28 Jan 2011 09:11 ESThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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A universal property for $Sp(2)$ at the prime $3$
http://projecteuclid.org/euclid.hha/1251832557
<strong>Jelena Grbić</strong>, <strong>Stephen Theriault</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 11, Number 1, 1--15.</p><p><strong>Abstract:</strong><br/>
We study a universal property of $Sp(2)$ in the category of $3$-local homotopy associative, homotopy commutative
$H$-spaces. We show that while $Sp(2)$ fails to be universal in the full category, there is a subcategory in which
it is universal for its $7$-skeleton.
</p>projecteuclid.org/euclid.hha/1251832557_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTWeight structure on Kontsevich's noncommutative mixed motiveshttp://projecteuclid.org/euclid.hha/1355321484<strong>Gonçalo Tabuada</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 14, Number 2, 129--142.</p><p><strong>Abstract:</strong><br/>
In this article we endow Kontsevich’s triangulated category
$\mathrm{KMM}_k$ of noncommutative mixed motives with a nondegenerate
weight structure in the sense of Bondarko. As an
application we obtain: (1) a convergent weight spectral sequence
for every additive invariant (e.g., algebraic $K$-theory, cyclic
homology, topological Hochschild homology, etc.); (2) a ring
isomorphism between $K_0(\mathrm{KMM}_k)$ and the Grothendieck ring
of the category of noncommutative Chow motives; (3) a precise
relationship between Voevodsky’s (virtual) mixed motives and
Kontsevich’s noncommutative (virtual) mixed motives.
</p>projecteuclid.org/euclid.hha/1355321484_Wed, 12 Dec 2012 09:11 ESTWed, 12 Dec 2012 09:11 ESTThe equvariant slice filtration: A primerhttp://projecteuclid.org/euclid.hha/1355321485<strong>Michael A. Hill</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 14, Number 2, 143--166.</p><p><strong>Abstract:</strong><br/>
We present an introduction to the equivariant slice filtration.
After reviewing the definitions and basic properties, we
determine the slice-connectivity of various families of naturally
arising spectra. This leads to an analysis of pullbacks of slices
defined on quotient groups, producing new collections of slices.
Building on this, we determine the slice tower for the Eilenberg-
Mac Lane spectrum associated to a Mackey functor for a cyclic
p-group. We then relate the Postnikov tower to the slice tower
for various spectra. Finally, we pose a few conjectures about the
nature of slices and pullbacks.
</p>projecteuclid.org/euclid.hha/1355321485_Wed, 12 Dec 2012 09:11 ESTWed, 12 Dec 2012 09:11 ESTHomology decompositions and groups inducing fusion systemshttp://projecteuclid.org/euclid.hha/1355321486<strong>Assaf Libman</strong>, <strong>Nora Seeliger</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 14, Number 2, 167--187.</p><p><strong>Abstract:</strong><br/>
We relate the construction of groups which realize saturated
fusion systems and signaliser functors with homology decompositions
of $p$-local finite groups. We prove that the cohomology
ring of Robinson’s construction is in some precise sense very
close to the cohomology ring of the fusion system it realizes.
</p>projecteuclid.org/euclid.hha/1355321486_Wed, 12 Dec 2012 09:11 ESTWed, 12 Dec 2012 09:11 ESTEvery binary self-dual code arises from Hilbert symbolshttp://projecteuclid.org/euclid.hha/1355321487<strong>Ted Chinburg</strong>, <strong>Ying Zhang</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 14, Number 2, 189--196.</p><p><strong>Abstract:</strong><br/>
In this paper we construct binary self-dual codes using the
étale cohomology of $\mu_2$ on the spectra of rings of $S$-integers
of global fields. We will show that up to equivalence, all selfdual
codes of length at least 4 arise from Hilbert pairings on
rings of $S$-integers of $\mathbb{Q}$. This is an arithmetic counterpart of a
result of Kreck and Puppe, who used cobordism theory to show
that all self-dual codes arise from Poincaré duality on real three
manifolds.
</p>projecteuclid.org/euclid.hha/1355321487_Wed, 12 Dec 2012 09:11 ESTWed, 12 Dec 2012 09:11 ESTChromatic subdivision of a simplicial complexhttp://projecteuclid.org/euclid.hha/1355321488<strong>Dmitry N. Kozlov</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 14, Number 2, 197--209.</p><p><strong>Abstract:</strong><br/>
We prove that the protocol complex of the immediate snapshot
read/write complex for $n + 1$ processors is a simplicial subdivision
of the input complex. Our proof is purely geometric,
using the Schlegel diagram construction.
</p>projecteuclid.org/euclid.hha/1355321488_Wed, 12 Dec 2012 09:11 ESTWed, 12 Dec 2012 09:11 EST$K$-motives of algebraic varietieshttp://projecteuclid.org/euclid.hha/1355321489<strong>Grigory Garkusha</strong>, <strong>Ivan Panin</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 14, Number 2, 211--264.</p><p><strong>Abstract:</strong><br/>
A kind of motivic algebra of spectral categories and modules
over them is developed to introduce $K$-motives of algebraic varieties.
As an application, bivariant algebraic $K$-theory $K(X; Y)$
as well as bivariant motivic cohomology groups $H^{p;q}(X; Y; \mathbb{Z})$
are defined and studied. We use Grayson’s machinery
to produce the Grayson motivic spectral sequence connecting
bivariant $K$-theory to bivariant motivic cohomology. It is shown
that the spectral sequence is naturally realized in the triangulated
category of $K$-motives constructed in the paper. It is also
shown that ordinary algebraic $K$-theory is represented by the
$K$-motive of the point.
</p>projecteuclid.org/euclid.hha/1355321489_Wed, 12 Dec 2012 09:11 ESTWed, 12 Dec 2012 09:11 ESTQuillen model structions on the category of graphshttp://projecteuclid.org/euclid.hha/1355321490<strong>Jean-Marie Droz</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 14, Number 2, 265--284.</p><p><strong>Abstract:</strong><br/>
We present different ways of endowing a particular category
of graphs with Quillen model structures. We show, among other
things, that the core of a graph can be seen as its homotopy type
in an appropriate Quillen model structure, and that an infinity
of Quillen model structures exist for our particular category of
graphs.
</p>projecteuclid.org/euclid.hha/1355321490_Wed, 12 Dec 2012 09:11 ESTWed, 12 Dec 2012 09:11 ESTIntegral excision for $K$-theoryhttp://projecteuclid.org/euclid.hha/1383943665<strong>Bjørn Dundas</strong>, <strong>Harald Øyen Kittang</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 1, 1--26.</p><p><strong>Abstract:</strong><br/>
If A is a homotopy cartesian square of ring spectra satisfying connectivity hypotheses, then the cube induced by Goodwillie's integral cyclotomic trace $K(\mathcal{A}) \to TC(\mathcal{A})$ is homotopy
cartesian. In other words, the homotopy fiber of the cyclotomic trace satisfies excision.
The method of proof gives as a spin-off new proofs of some old results, as well as some new results, about periodic cyclic homology, and - more relevantly for our current application - the $\mathbf{T}$-Tate
spectrum of topological Hochschild homology, where $\mathbf{T}$ is the circle group.
</p>projecteuclid.org/euclid.hha/1383943665_Fri, 08 Nov 2013 15:47 ESTFri, 08 Nov 2013 15:47 ESTÉtale homotopy types and bisimplicial hypercovershttp://projecteuclid.org/euclid.hha/1383943666<strong>Michael D. Misamore</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 1, 27--49.</p><p><strong>Abstract:</strong><br/>
Suppose $(C,x)$ is a pointed locally connected small
Grothendieck site, and let $(X,z)$ denote any connected
locally fibrant simplicial sheaf $X$ equipped with a "geometric"
point $z$. Following Artin-Mazur, an étale homotopy type of
$X$ may then be defined via the geometrically pointed hypercovers
of $X$ to yield a pro-object of the homotopy category, but this
is not the only possible definition. In Étale homotopy of
simplicial schemes , Friedlander defined another étale
homotopy type of a simplicial scheme $X$ by taking diagonals of
geometrically pointed bisimplicial hypercovers.
In this paper, these two types are shown to be
pro-isomorphic by means of a direct comparison of the associated
cocycle categories. Friedlander's construction of étale homotopy
types as actual pro-simplicial sets relies on a rigidity property of
the étale topology that may not always be available for arbitrary
sites; the cocycle methods employed here do not have this limitation.
By consequence, the associated homotopy types constructed from
hypercovers and bisimplicial hypercovers are shown to be
pro-isomorphic on any locally connected small Grothendieck site, and
the comparison at the level of cocycles shows, in particular, that
both abelian and non-abelian sheaf cohomology may be computed via
bisimplicial hypercovers on arbitrary small Grothendieck sites.
</p>projecteuclid.org/euclid.hha/1383943666_Fri, 08 Nov 2013 15:47 ESTFri, 08 Nov 2013 15:47 ESTHomology and robustness of level and interlevel setshttp://projecteuclid.org/euclid.hha/1383943667<strong>Paul Bendich</strong>, <strong>Herbert Edelsbrunner</strong>, <strong>Dmitriy Morozov</strong>, <strong>Amit Patel</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 1, 51--72.</p><p><strong>Abstract:</strong><br/>
Given a continuous function $f\colon \mathbb{X} \to \mathbb{R}$ on a topological space,
we consider the preimages of intervals and their homology groups
and show how to read the ranks of these groups from the extended
persistence diagram of $f$. In addition, we quantify the robustness of the homology classes
under perturbations of $f$ using well groups, and we show
how to read the ranks of these groups from the same extended
persistence diagram. The special case $\mathbb{X} = \mathbb{R}^3$ has ramifications
in the fields of medical imaging and scientific visualization.
</p>projecteuclid.org/euclid.hha/1383943667_Fri, 08 Nov 2013 15:47 ESTFri, 08 Nov 2013 15:47 ESTOn semisimplicial sets satisfying the Kan conditionhttp://projecteuclid.org/euclid.hha/1383943668<strong>James E. McClure</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 1, 73--82.</p><p><strong>Abstract:</strong><br/>
A semisimplicial set has face maps but not degeneracies. A
basic fact, due to Rourke and Sanderson, is that a semisimplicial
set satisfying the Kan condition can be given a simplicial
structure. The present paper gives a combinatorial proof of this
fact and a generalization to multisemisimplicial sets.
</p>projecteuclid.org/euclid.hha/1383943668_Fri, 08 Nov 2013 15:47 ESTFri, 08 Nov 2013 15:47 ESTChevalley cohomology for aerial Kontsevich graphshttp://projecteuclid.org/euclid.hha/1383943669<strong>Walid Aloulou</strong>, <strong>Didier Arnal</strong>, <strong>Ridha Chatbouri</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 1, 83--100.</p><p><strong>Abstract:</strong><br/>
Let $T_{\operatorname{poly}}(\mathbb{R}^d)$ denote the space of
skew-symmetric polyvector fields on $\mathbb{R}^d$, turned into a
graded Lie algebra by means of the Schouten bracket. Our aim is to
explore the cohomology of this Lie algebra, with coefficients in the
adjoint representation, arising from cochains defined by linear
combination of aerial Kontsevich graphs. We prove that this cohomology
is localized at the space of graphs without any isolated vertex, any
"hand" or any "foot". As an application, we explicitly compute the
cohomology of the "ascending graphs" quotient complex.
</p>projecteuclid.org/euclid.hha/1383943669_Fri, 08 Nov 2013 15:47 ESTFri, 08 Nov 2013 15:47 ESTOn the homology of the space of singular knotshttp://projecteuclid.org/euclid.hha/1383943670<strong>Hossein Abbaspour</strong>, <strong>David Chataur</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 1, 101--126.</p><p><strong>Abstract:</strong><br/>
In this paper we introduce various associative products on the homology of the space of knots and singular knots in $S^n$. We prove that these products are related through a desingularization map.
We also compute some of these products and prove the non-triviality of the desingularization morphism. Using direct computations, we prove that some of these products are indeed commutative.
</p>projecteuclid.org/euclid.hha/1383943670_Fri, 08 Nov 2013 15:47 ESTFri, 08 Nov 2013 15:47 ESTCohomology of loop spaces of the symmetric space $EI$http://projecteuclid.org/euclid.hha/1383943671<strong>Younggi Choi</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 1, 127--136.</p><p><strong>Abstract:</strong><br/>
We determine the mod $p$ cohomology of the loop space and the double loop space of the symmetric space of exceptional type $EI$ exploiting the Serre spectral sequence and the Eilenberg-Moore spectral sequence.
</p>projecteuclid.org/euclid.hha/1383943671_Fri, 08 Nov 2013 15:47 ESTFri, 08 Nov 2013 15:47 ESTHomotopy type of space of maps into a $K(G,n)$http://projecteuclid.org/euclid.hha/1383943672<strong>Jaka Smrekar</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 1, 137--149.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a connected CW complex and let $K(G,n)$ be an Eilenberg-Mac Lane CW complex where $G$ is abelian.
As $K(G,n)$ may be taken to be an abelian monoid, the weak homotopy type of the
space of continuous functions $X \to K(G,n)$ depends only upon the homology groups of $X$.
The purpose of this note is to prove that this is true for the actual homotopy type.
Precisely, the space $\mathrm{map}_* \big(X, K(G,n)\big)$ of pointed continuous maps $X \to K(G,n)$ is shown to be
homotopy equivalent to the Cartesian product
\[ \prod_{i \leq n} \mathrm{map}_* \big(M_i, K(G,n)\big). \]
Here, $M_i$ is a Moore complex of type $M\big(H_i(X), i\big)$. The spaces of functions are equipped with the compact
open topology.
</p>projecteuclid.org/euclid.hha/1383943672_Fri, 08 Nov 2013 15:47 ESTFri, 08 Nov 2013 15:47 ESTProfinite $G$-spectrahttp://projecteuclid.org/euclid.hha/1383943673<strong>Gereon Quick</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 1, 151--189.</p><p><strong>Abstract:</strong><br/>
We construct a stable model structure on profinite spectra
with a continuous action of an arbitrary profinite group. The
motivation is to provide a natural framework in a subsequent
paper for a new and conceptually simplified construction of continuous
homotopy fixed point spectra and of continuous homotopy
fixed point spectral sequences for Lubin-Tate spectra under
the action of the extended Morava stabilizer group.
</p>projecteuclid.org/euclid.hha/1383943673_Fri, 08 Nov 2013 15:47 ESTFri, 08 Nov 2013 15:47 ESTContinuous homotopy fixed points for Lubin-Tate spectrahttp://projecteuclid.org/euclid.hha/1383943674<strong>Gereon Quick</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 1, 191--222.</p><p><strong>Abstract:</strong><br/>
We provide a new and conceptually simplified construction of
continuous homotopy fixed point spectra for Lubin-Tate spectra
under the action of the extended Morava stabilizer group. Moreover,
our new construction of a homotopy fixed point spectral
sequence converging to the homotopy groups of the homotopy
fixed points of Lubin-Tate spectra is isomorphic to an Adams
spectral sequence converging to the homotopy groups of the
spectra constructed by Devinatz and Hopkins. The new idea is
built on the theory of profinite spectra with a continuous action
by a profinite group.
</p>projecteuclid.org/euclid.hha/1383943674_Fri, 08 Nov 2013 15:47 ESTFri, 08 Nov 2013 15:47 ESTOin trivialities of Steifel-Whitney classes of vector bundles over iterated suspensions of Dold manifoldshttp://projecteuclid.org/euclid.hha/1383943675<strong>Ajay Singh Thakur</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 1, 223--233.</p><p><strong>Abstract:</strong><br/>
A space $X$ is called $W$-trivial if for every vector bundle $\xi$ over $X$, the total Stiefel-Whitney class $W(\xi) = 1$. In this article we shall investigate whether the
suspensions $\Sigma^k D(m,n)$ of Dold manifolds are $W$-trivial or not.
</p>projecteuclid.org/euclid.hha/1383943675_Fri, 08 Nov 2013 15:47 ESTFri, 08 Nov 2013 15:47 ESTReal equivariant bordism for elementary abelian 2-groupshttp://projecteuclid.org/euclid.hha/1383943676<strong>Moritz Firsching</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 1, 235--251.</p><p><strong>Abstract:</strong><br/>
We give a description of real equivariant bordism for the group $G = Z/2 \times \cdots \times Z/2$, which is similar to the description of complex equivariant bordism for the
group $S^1 \times \cdots \times S^1$ given by Hanke.
</p>projecteuclid.org/euclid.hha/1383943676_Fri, 08 Nov 2013 15:47 ESTFri, 08 Nov 2013 15:47 ESTCellular decomposition and free resolution for split metacyclic spherical space formshttp://projecteuclid.org/euclid.hha/1383943677<strong>L. L. Fêmina</strong>, <strong>A. P. T. Galves</strong>, <strong>O. Manzoli Neto</strong>, <strong>M. Spreafico</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 1, 253--278.</p><p><strong>Abstract:</strong><br/>
Given a free isometric action of a split metacyclic group
on odd dimensional sphere, we obtain an explicit finite cellular
decomposition of the sphere equivariant with respect to the
group action. A cell decomposition of the factor space and an
explicit description of the associated cellular chain complex of
modules over the integral group ring of the fundamental group
follow. In particular, the construction provides a simple explicit
4-periodic free resolution for the split metacyclic groups.
</p>projecteuclid.org/euclid.hha/1383943677_Fri, 08 Nov 2013 15:47 ESTFri, 08 Nov 2013 15:47 ESTSymmetric continuous cohomology of topological groupshttp://projecteuclid.org/euclid.hha/1383943678<strong>Mahender Singh</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 1, 279--302.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce a symmetric continuous cohomology
of topological groups. This is obtained by topologizing
a recent construction due to Staic, where a symmetric
cohomology of abstract groups is constructed. We give a characterization
of topological group extensions that correspond to
elements of the second symmetric continuous cohomology. We
also show that the symmetric continuous cohomology of a profinite
group with coefficients in a discrete module is equal to the
direct limit of the symmetric cohomology of finite groups. In
the end, we also define symmetric smooth cohomology of Lie
groups and prove similar results.
</p>projecteuclid.org/euclid.hha/1383943678_Fri, 08 Nov 2013 15:47 ESTFri, 08 Nov 2013 15:47 ESTConnectivity at infinity for braid groups on complete graphshttp://projecteuclid.org/euclid.hha/1383943679<strong>John Meier</strong>, <strong>Liang Zhang</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 1, 303--311.</p><p><strong>Abstract:</strong><br/>
We show that the connectivity at infinity for configuration
spaces on complete graphs is determined by the connectivity
of chessboard complexes.
</p>projecteuclid.org/euclid.hha/1383943679_Fri, 08 Nov 2013 15:47 ESTFri, 08 Nov 2013 15:47 ESTPower operations in orbifold Tate $K$-theoryhttp://projecteuclid.org/euclid.hha/1383943680<strong>Nora Ganter</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 1, 313--342.</p><p><strong>Abstract:</strong><br/>
We formulate the axioms of an orbifold theory with power
operations. We define orbifold Tate $K$-theory, by adjusting
Devoto’s definition of the equivariant theory, and proceed to
construct its power operations. We calculate the resulting sym-
metric powers, exterior powers and Hecke operators and put our
work into context with orbifold loop spaces, level structures on
the Tate curve and generalized Moonshine.
</p>projecteuclid.org/euclid.hha/1383943680_Fri, 08 Nov 2013 15:47 ESTFri, 08 Nov 2013 15:47 ESTLimit the theorems for Betti numbers of random simplicial complexeshttp://projecteuclid.org/euclid.hha/1383943681<strong>Matthew Kahle</strong>, <strong>Elizabeth Meckes</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 1, 343--374.</p><p><strong>Abstract:</strong><br/>
There have been several recent articles studying homology of
various types of random simplicial complexes. Several theorems
have concerned thresholds for vanishing of homology groups,
and in some cases expectations of the Betti numbers; however,
little seems known so far about limiting distributions of random
Betti numbers.
In this article we establish Poisson and normal approximation
theorems for Betti numbers of different kinds of random simplicial
complexes: Erd˝os–R´enyi random clique complexes, random
Vietoris-Rips complexes, and random ˇCech complexes. These
results may be of practical interest in topological data analysis.
</p>projecteuclid.org/euclid.hha/1383943681_Fri, 08 Nov 2013 15:47 ESTFri, 08 Nov 2013 15:47 ESTThe geometric realization of monomial ideal rings and a theorem of Trevisanhttp://projecteuclid.org/euclid.hha/1383945273<strong>A. Bahri</strong>, <strong>M. Bendersky</strong>, <strong>F. R. Cohen</strong>, <strong>S. Gitler</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 2, 1--7.</p><p><strong>Abstract:</strong><br/>
A direct proof is presented of a form of Alvise Trevisan’s
theorem, that every monomial ideal ring is represented by
the cohomology of a topological space. Certain of these rings
are shown to be realized by polyhedral products indexed by
simplicial complexes.
</p>projecteuclid.org/euclid.hha/1383945273_Fri, 08 Nov 2013 16:14 ESTFri, 08 Nov 2013 16:14 ESTHigher $K$-theory of Koszul cubeshttp://projecteuclid.org/euclid.hha/1383945274<strong>Satoshi Mochizuki</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 2, 9--51.</p><p><strong>Abstract:</strong><br/>
The main objective of this paper is to determine generators
of the topological filtrations on the higher $K$-theory of a noetherian
commutative ring with unit A. We introduce the concept
of Koszul cubes and give a comparison theorem between the
$K$-theory of Koszul cubes with that of topological filtrations.
</p>projecteuclid.org/euclid.hha/1383945274_Fri, 08 Nov 2013 16:14 ESTFri, 08 Nov 2013 16:14 ESTHomological dimensions of ring spectrahttp://projecteuclid.org/euclid.hha/1383945275<strong>Mark Hovey</strong>, <strong>Keir Lockridge</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 2, 53--71.</p><p><strong>Abstract:</strong><br/>
We define homological dimensions for $S$-algebras, the generalized
rings that arise in algebraic topology. We compute the
homological dimensions of a number of examples, and establish
some basic properties. The most difficult computation is the
global dimension of real $K$-theory $KO$ and its connective version
$ko$ at the prime 2. We show that the global dimension of
$KO$ is 2 or 3, and the global dimension of $ko$ is 4 or 5.
</p>projecteuclid.org/euclid.hha/1383945275_Fri, 08 Nov 2013 16:14 ESTFri, 08 Nov 2013 16:14 ESTSpaces of topological complexity onehttp://projecteuclid.org/euclid.hha/1383945276<strong>Mark Grant</strong>, <strong>Gregory Lupton</strong>, <strong>John Oprea</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 2, 73--81.</p><p><strong>Abstract:</strong><br/>
We prove that a space whose topological complexity equals 1 is homotopy equivalent to some odd-dimensional sphere. We prove a similar result, although not in complete generality, for
spaces $X$ whose higher topological complexity ${\sf TC}_n(X)$ is as low as possible, namely $n-1$.
</p>projecteuclid.org/euclid.hha/1383945276_Fri, 08 Nov 2013 16:14 ESTFri, 08 Nov 2013 16:14 ESTPower maps on $p$-regular Lie groupshttp://projecteuclid.org/euclid.hha/1383945277<strong>Stephen Theriault</strong><p><strong>Source: </strong>Homology Homotopy Appl., Volume 15, Number 2, 83--102.</p><p><strong>Abstract:</strong><br/>
A simple, simply-connected, compact Lie group $G$ is $p$-regular if it is homotopy equivalent to a product of spheres when localized at $p$. If $A$ is the corresponding wedge of spheres, then it is well known
that there is a $p$-local retraction of $G$ off $\Omega\Sigma A$. We show that that complementary factor is very well behaved, and this allows us to deduce properties of $G$ from those of $\Omega\Sigma A$.
We apply this to show that, localized at $p$, the $p$ th -power map on $G$ is an $H$-map. This is a significant step forward in Arkowitz-Curjel and McGibbon's programme for identifying
which power maps between finite $H$-spaces are $H$-maps.
</p>projecteuclid.org/euclid.hha/1383945277_Fri, 08 Nov 2013 16:14 ESTFri, 08 Nov 2013 16:14 ESTHomology operations and cosimplicial iterated loop spaceshttp://projecteuclid.org/euclid.hha/1401800069<strong>Philip Hackney</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 1--25.</p><p><strong>Abstract:</strong><br/>
If $X$ is a cosimplical $E_{n+1}$ space then $\operatorname{Tot}(X)$ is an $E_{n+1}$ space and its mod 2 homology $H_*(\operatorname{Tot}(X))$ has Dyer-Lashof and Browder operations.
It's natural to ask if the spectral sequence converging to $H_*(\operatorname{Tot}(X))$ admits compatible operations. In this paper we give a positive answer to this question.
</p>projecteuclid.org/euclid.hha/1401800069_20140603085432Tue, 03 Jun 2014 08:54 EDTSecondy multiplication in Tate cohomology of generalized quaternion groupshttp://projecteuclid.org/euclid.hha/1401800070<strong>Martin Langer</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 27--47.</p><p><strong>Abstract:</strong><br/>
Let $k$ be a field, and let $G$ be a finite group. By a theorem of D. Benson, H. Krause, and S. Schwede, there is a canonical element in the Hochschild cohomology of the Tate cohomology
$\gamma_G\in H\! H^{3,-1} \hat{H}^*(G)$ with the following property: Given any graded $\hat{H}^*(G)$-module $X$, the image of $\gamma_G$ in $\mathrm{Ext}^{3,-1}_{\hat{H}^*(G)} (X,X)$ is zero
if and only if $X$ is isomorphic to a direct summand of $\smash{\hat{H}^*(G,M)}$ for some $kG$-module $M$. In particular, if $\gamma_G=0$ then every module is a direct summand of a realizable $\hat{H}^*(G)$-module.
We prove that the converse of that last statement is not true by studying in detail the case of generalized quaternion groups. Suppose that $k$ is a field of characteristic $2$ and $G$ is generalized
quaternion of order $2^n$ with $n\geq 3$. We show that $\gamma_G$ is non-trivial for all $n$, but there is an $\hat{H}^*(G)$-module detecting this non-triviality if and only if $n=3$.
</p>projecteuclid.org/euclid.hha/1401800070_20140603085432Tue, 03 Jun 2014 08:54 EDTSimplification of complexes of persistent homology computationshttp://projecteuclid.org/euclid.hha/1401800071<strong>Paweł Dłotko</strong>, <strong>Hubert Wagner</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 49--63.</p><p><strong>Abstract:</strong><br/>
In this paper we focus on preprocessing for persistent homology
computations. We adapt some techniques that were successfully
used for standard homology computations. The main
idea is to reduce the complex prior to generating its boundary
matrix, which is costly to store and process. We discuss the following
reduction methods: elementary collapses, coreductions
(as defined by Mrozek and Batko), and acyclic subspace methods
(introduced by Mrozek, Pilarczyk, and Żelazna).
</p>projecteuclid.org/euclid.hha/1401800071_20140603085432Tue, 03 Jun 2014 08:54 EDTFree 2-rank of symmetry of products of Milnor manifoldshttp://projecteuclid.org/euclid.hha/1401800072<strong>Mahender Singh</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 65--81.</p><p><strong>Abstract:</strong><br/>
A real Milnor manifold is the non-singular hypersurface of
degree $(1; 1)$ in the product of two real projective spaces. These
manifolds were introduced by Milnor to give generators for the
unoriented cobordism algebra, and they admit free actions by
elementary abelian 2-groups. In this paper, we obtain some
results on the free 2-rank of symmetry of products of finitely
many real Milnor manifolds under the assumption that the
induced action on mod 2 cohomology is trivial. Similar results
are obtained for complex Milnor manifolds that are defined analogously.
Here the free 2-rank of symmetry of a topological space
is the maximal rank of an elementary abelian 2-group that acts
freely on that space.
</p>projecteuclid.org/euclid.hha/1401800072_20140603085432Tue, 03 Jun 2014 08:54 EDTNote on the homotopy groups of a bouquet $S^1\vee Y$, $Y$ 1-connectedhttp://projecteuclid.org/euclid.hha/1401800073<strong>Joseph Roitberg</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 83--87.</p><p><strong>Abstract:</strong><br/>
A study is made of the action of the fundamental group of a
bouquet of a circle and a 1-connected space on the higher homotopy
groups. If the 1-connected space is a suspension space, it
is shown, with the aid of a theorem of Hartley on wreath products
of groups and the Hilton-Milnor theorem, that the action
is residually nilpotent. An unsuccessful approach in the case of
a general 1-connected space is discussed, as it has some interesting
features.
</p>projecteuclid.org/euclid.hha/1401800073_20140603085432Tue, 03 Jun 2014 08:54 EDTHolohonies for connections with values in $L_\infty$-algebrashttp://projecteuclid.org/euclid.hha/1401800074<strong>Camilo Arias Abad</strong>, <strong>Florian Schätz</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 89--118.</p><p><strong>Abstract:</strong><br/>
Given a flat connection $\alpha$ on a manifold $M$ with values in a filtered $L_\infty$-algebra $\mathfrak{g}$, we construct a morphism
$\mathsf{hol}^{\infty}_\alpha \colon C_\bullet(M) \rightarrow \mathsf{B} \hat{\mathbb{U}}_\infty(\mathfrak{g})$, which generalizes the holonomy map associated to a flat connection with values in a Lie algebra.
The construction is based on Gugenheim's $\mathsf{A}_{\infty}$-version of de Rham's theorem, which in turn is based on Chen's iterated integrals. Finally, we discuss examples related to the geometry of
configuration spaces of points in Euclidean space $\mathbb{R}^d$, and to generalizations of the holonomy representations of braid groups.
</p>projecteuclid.org/euclid.hha/1401800074_20140603085432Tue, 03 Jun 2014 08:54 EDTThe homology graph of a precubical sethttp://projecteuclid.org/euclid.hha/1401800075<strong>Thomas Kahl</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 119--138.</p><p><strong>Abstract:</strong><br/>
Precubical sets are used to model concurrent systems. We
introduce the homology graph of a precubical set, which is a
directed graph whose nodes are the homology classes of the
precubical set. We show that the homology graph is invariant
under weak morphisms that are homeomorphisms.
</p>projecteuclid.org/euclid.hha/1401800075_20140603085432Tue, 03 Jun 2014 08:54 EDTPostnikov towers with fibers generalized Eilenberg-Mac Lane spaceshttp://projecteuclid.org/euclid.hha/1401800076<strong>Kouyemon Iriye</strong>, <strong>Daisuke Kishimoto</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 139--157.</p><p><strong>Abstract:</strong><br/>
A generalized Postnikov tower (GPT) is defined as a tower of
principal fibrations with the classifying maps into generalized
Eilenberg–Mac Lane spaces. We study fundamental properties
of GPT’s such as their existence, localization and length. We
further consider the distribution of torsion in a GPT of a finite
complex, motivated by the result of McGibbon and Neisendorfer. We also give an algebraic description of the length of a
GPT of a rational space.
</p>projecteuclid.org/euclid.hha/1401800076_20140603085432Tue, 03 Jun 2014 08:54 EDTComplexification and homotopyhttp://projecteuclid.org/euclid.hha/1401800077<strong>Wojciech Kucharz</strong>, <strong>Łukasz Maciejewski</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 159--165.</p><p><strong>Abstract:</strong><br/>
Let $Y$ be a real algebraic variety. We are interested in determining the supremum, $\beta(Y)$, of all nonnegative integers $n$ with the following property: For every $n$-dimensional compact connected
nonsingular real algebraic variety $X$, every continuous map from $X$ into $Y$ is homotopic to a regular map. We give an upper bound for $\beta(Y)$, based on a construction involving complexification
of real algebraic varieties. In some cases, we obtain the exact value of $\beta(Y)$.
</p>projecteuclid.org/euclid.hha/1401800077_20140603085432Tue, 03 Jun 2014 08:54 EDTKei modules and unoriented link invariantshttp://projecteuclid.org/euclid.hha/1401800078<strong>Michael Grier</strong>, <strong>Sam Nelson</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 167--177.</p><p><strong>Abstract:</strong><br/>
We define invariants of unoriented knots and links by enhancing the integral kei counting invariant $\Phi_X^{\mathbb{Z}}(K)$ for a finite kei $X$ using representations of the kei algebra ,
$\mathbb{Z}_K[X]$, a quotient of the quandle algebra $\mathbb{Z}[X]$ defined by Andruskiewitsch and Graña. We give an example that demonstrates that the enhanced invariant is stronger
than the unenhanced kei counting invariant. As an application, we use a quandle module over the Takasaki kei on $\mathbb{Z}_3$ which is not a $\mathbb{Z}_K[X]$-module to detect the non-invertibility of a virtual knot.
</p>projecteuclid.org/euclid.hha/1401800078_20140603085432Tue, 03 Jun 2014 08:54 EDTVersal deformation theory of algebras over a quadratic operadhttp://projecteuclid.org/euclid.hha/1401800079<strong>Alice Fialowski</strong>, <strong>Goutam Mukherjee</strong>, <strong>Anita Naolekar</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 179--198.</p><p><strong>Abstract:</strong><br/>
We develop deformation theory of algebras over quadratic
operads where the parameter space is a commutative local algebra.
We also give a construction of a distinguised deformation of
an algebra over a quadratic operad with a complete local algebra
as its base—the so-called versal deformation—which induces all
other deformations of the given algebra.
</p>projecteuclid.org/euclid.hha/1401800079_20140603085432Tue, 03 Jun 2014 08:54 EDTExact sequences of commutative monoids and semimoduleshttp://projecteuclid.org/euclid.hha/1401800080<strong>Jawad Y. Abuhlail</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 199--214.</p><p><strong>Abstract:</strong><br/>
Basic homological lemmas well known for modules over rings
and, more generally, in the context of abelian categories, have
been extended to many other concrete and abstract-categorical
contexts by various authors. We propose a new such extension,
specifically for commutative monoids and semimodules; these
two contexts are equivalent since the forgetful functors from
varieties of semimodules to the variety of commutative monoids
preserve all limits and colimits.
</p>projecteuclid.org/euclid.hha/1401800080_20140603085432Tue, 03 Jun 2014 08:54 EDTOn connective $K$-theory of elementary abelian 2-groups and local dualityhttp://projecteuclid.org/euclid.hha/1401800081<strong>Geoffrey M. L. Powell</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 215--243.</p><p><strong>Abstract:</strong><br/>
The connective $ku$-(co)homology of elementary abelian $2$-groups is determined as a functor of the elementary abelian $2$-group, using the action of the Milnor operations $Q_0, Q_1$ on mod $2$ group
cohomology, the Atiyah-Segal theorem for $KU$-cohomology, together with an analysis of the functorial structure of the integral group ring; the functorial structure then reduces calculations to the rank 1 case.
These results are used to analyse the local cohomology spectral sequence calculating $ku$-homology, via a functorial version of local duality for Koszul complexes, giving a conceptual explanation of results of Bruner and
Greenlees.
</p>projecteuclid.org/euclid.hha/1401800081_20140603085432Tue, 03 Jun 2014 08:54 EDTDescribing high-order statistical dependence using "concurrence topology," with application to functional MRI brain datahttp://projecteuclid.org/euclid.hha/1401800082<strong>Steven P. Ellis</strong>, <strong>Arno Klein</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 245--264.</p><p><strong>Abstract:</strong><br/>
In multivariate data analysis dependence beyond pair-wise can
be important. With many variables, however, the number of simple
summaries of even third-order dependence can be unmanageably
large.
“Concurrence topology” is an apparently new method for describing
high-order dependence among up to dozens of dichotomous (i.e.,
binary) variables (e.g., seventh-order dependence in 32 variables).
This method generally produces summaries of dependence of manageable
size. (But computing time can be lengthy.) For time series,
this method can be applied in both the time and Fourier domains.
Write each observation as a vector of 0’s and 1’s. A “concurrence”
is a group of variables all labeled “1” in the same observation. The
collection of concurrences can be represented as a filtered simplicial
complex. Holes in the filtration indicate relatively weak or negative
association among the variables. The pattern of the holes in the
filtration can be analyzed using persistent homology.
We applied concurrence topology on binarized, resting-state,
functional MRI data acquired from patients diagnosed with
attention-deficit hyperactivity disorder and from healthy controls.
An exploratory analysis finds a number of differences between
patients and controls in the topologies of their filtrations, demonstrating
that concurrence topology can find in data high-order structure
of real-world relevance.
</p>projecteuclid.org/euclid.hha/1401800082_20140603085432Tue, 03 Jun 2014 08:54 EDTMayer-Vietoris sequences in stable derivatorshttp://projecteuclid.org/euclid.hha/1401800083<strong>Moritz Groth</strong>, <strong>Kate Ponto</strong>, <strong>Michael Shulman</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 265--294.</p><p><strong>Abstract:</strong><br/>
We show that stable derivators, like stable model cate-
gories, admit Mayer-Vietoris sequences arising from cocartesian
squares. Along the way we characterize homotopy exact squares
and give a detection result for colimiting diagrams in derivators.
As an application, we show that a derivator is stable if and only
if its suspension functor is an equivalence.
</p>projecteuclid.org/euclid.hha/1401800083_20140603085432Tue, 03 Jun 2014 08:54 EDTGraphs associated with simplicial complexeshttp://projecteuclid.org/euclid.hha/1401800084<strong>A. Grigor'yan</strong>, <strong>Yu. V. Muranov</strong>, <strong>Shing-Tung Yau</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 295--311.</p><p><strong>Abstract:</strong><br/>
The cohomology of digraphs was introduced for the first time by Dimakis and Müller-Hoissen. Their algebraic definition is based on a differential calculus on an algebra of functions on the set of vertices
with relations that follow naturally from the structure of the set of edges. A dual notion of homology of digraphs, based on the notion of path complex, was introduced by the authors, and the first methods for computing
the (co)homology groups were developed. The interest in homology on digraphs is motivated by physical applications and relations between algebraic and geometrical properties of quivers. The digraph
$G_B$ of the partially ordered set $B_{S}$ of simplexes of a simplicial complex $S$ has graph homology that is isomorphic to the simplicial homology of $S$. In this paper, we introduce the concept of cubical
digraphs and describe their homology properties. In particular, we define a cubical subgraph $G_{S}$ of $G_B$, whose homologies are isomorphic to the simplicial homologies of $S$.
</p>projecteuclid.org/euclid.hha/1401800084_20140603085432Tue, 03 Jun 2014 08:54 EDTGlobal orthogonal spectrahttp://projecteuclid.org/euclid.hha/1401800085<strong>Anna Marie Bohmann</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 313--332.</p><p><strong>Abstract:</strong><br/>
For any compact Lie group $G$, there are several well-established definitions of a $G$-equivariant spectrum. In this paper, we develop the definition of a global orthogonal spectrum. Loosely speaking,
this is a coherent choice of orthogonal $G$-spectrum for each compact Lie group $G$. We use the framework of enriched indexed categories to make this precise. We also consider equivariant $K$-theory
and $\operatorname{Spin}^c$-cobordism from this perspective, and we show that the Atiyah-Bott-Shapiro orientation extends to the global context.
</p>projecteuclid.org/euclid.hha/1401800085_20140603085432Tue, 03 Jun 2014 08:54 EDTA purely homotopy-theoretci proof of the Blakers-Massey theorem for $n$-cubeshttp://projecteuclid.org/euclid.hha/1401800086<strong>Brian A. Munson</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 333--339.</p><p><strong>Abstract:</strong><br/>
Goodwillie’s proof of the Blakers-Massey Theorem for n- cubes relies on a lemma whose proof invokes transversality. The
rest of his proof follows from general facts about cubes of spaces
and connectivities of maps. We present a purely homotopytheoretic
proof of this lemma. The methods are elementary,
using a generalization and modification of an argument originally
due to Puppe used to prove the Blakers-Massey Theorem
for squares.
</p>projecteuclid.org/euclid.hha/1401800086_20140603085432Tue, 03 Jun 2014 08:54 EDTCohomology of algebras over weak Hopf algebrashttp://projecteuclid.org/euclid.hha/1401800087<strong>J. N. Alonso Álvarez</strong>, <strong>J. M. Fernández Vilaboa</strong>, <strong>R. González Rodríguez</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 1, 341--369.</p><p><strong>Abstract:</strong><br/>
In this paper we present the Sweedler cohomology for a cocommutative weak Hopf algebra $H$. We show that the second cohomology group classifies completely weak crossed products, having a common preunit,
of $H$ with a commutative left $H$-module algebra $A$.
</p>projecteuclid.org/euclid.hha/1401800087_20140603085432Tue, 03 Jun 2014 08:54 EDTHigher Morse moduli spaces and $n$-categorieshttp://projecteuclid.org/euclid.hha/1408712332<strong>Sonja Hohloch</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 2, 1--32.</p><p><strong>Abstract:</strong><br/>
We generalize Cohen & Jones & Segal's flow category, whose objects are the critical points of a Morse function and whose morphisms are the Morse moduli spaces between the critical points
to an $n$-category. The $n$-category construction involves repeatedly doing Morse theory on Morse moduli spaces for which we have to construct a class of suitable Morse functions. It turns out to be
an 'almost strict' $n$-category, i.e. it is a strict $n$-category 'up to canonical isomorphisms'.
</p>projecteuclid.org/euclid.hha/1408712332_20140822085854Fri, 22 Aug 2014 08:58 EDTHomological descent for motivic homology theorieshttp://projecteuclid.org/euclid.hha/1408712333<strong>Thomas Geisser</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 2, 33--43.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to give homological descent theorems
for motivic homology theories (for example, Suslin homology)
and motivic Borel-Moore homology theories (for example,
higher Chow groups) for certain hypercoverings.
</p>projecteuclid.org/euclid.hha/1408712333_20140822085854Fri, 22 Aug 2014 08:58 EDTDerived categories of absolutely flat ringshttp://projecteuclid.org/euclid.hha/1408712334<strong>Greg Stevenson</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 2, 45--64.</p><p><strong>Abstract:</strong><br/>
Let $S$ be a commutative ring with topologically noetherian spectrum, and let $R$ be the absolutely flat approximation of $S$. We prove that subsets of the spectrum of $R$ parametrise the localising subcategories
of $\mathsf{D}(R)$. Moreover, we prove the telescope conjecture holds for $\mathsf{D}(R)$. We also consider unbounded derived categories of absolutely flat rings that are not semi-artinian and exhibit a localising
subcategory that is not a Bousfield class and a cohomological Bousfield class that is not a Bousfield class.
</p>projecteuclid.org/euclid.hha/1408712334_20140822085854Fri, 22 Aug 2014 08:58 EDTWeak Lefschetz for Chow groups: Infinitesimal liftinghttp://projecteuclid.org/euclid.hha/1408712335<strong>D. Patel</strong>, <strong>G. V. Ravindra</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 2, 65--84.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic zero, and let $Y \subset X$ be a smooth ample hyperplane section. The Weak Lefschetz conjecture for Chow groups states
that the natural restriction map $\mathrm{CH}^p (X)_{\mathbb{Q}} \to \mathrm{CH}^p (Y)_{\mathbb{Q}}$ is an isomorphism for all $p \lt \dim (Y) / 2$. In this note, we revisit a strategy introduced by Grothendieck
to attack this problem by using the Bloch-Quillen formula to factor this morphism through a continuous $\mathrm{K}$-cohomology group on the formal completion of $X$ along $Y$. This splits the conjecture
into two smaller conjectures: one consisting of an algebraization problem and the other dealing with infinitesimal liftings of algebraic cycles. We give a complete proof of the infinitesimal part of the conjecture.
</p>projecteuclid.org/euclid.hha/1408712335_20140822085854Fri, 22 Aug 2014 08:58 EDTCrossed modules of rackshttp://projecteuclid.org/euclid.hha/1408712336<strong>Alissa S. Crans</strong>, <strong>Friedrich Wagemann</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 2, 85--106.</p><p><strong>Abstract:</strong><br/>
We generalize the notion of a crossed module of groups to
that of a crossed module of racks. We investigate the relation
to categorified racks, namely strict 2-racks, and trunk-like objects
in the category of racks, generalizing the relation between
crossed modules of groups and strict 2-groups. Then we explore
topological applications. We show that by applying the rackspace
functor, a crossed module of racks gives rise to a covering.
Our main result shows how the fundamental racks associated to
links upstairs and downstairs in a covering fit together to form
a crossed module of racks.
</p>projecteuclid.org/euclid.hha/1408712336_20140822085854Fri, 22 Aug 2014 08:58 EDT$L_{\infty}$-algebras of local observables from higher prequantum bundleshttp://projecteuclid.org/euclid.hha/1408712337<strong>Domenico Fiorenza</strong>, <strong>Christopher L. Rogers</strong>, <strong>Urs Schreiber</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 2, 107--142.</p><p><strong>Abstract:</strong><br/>
To any manifold equipped with a higher degree closed form, one can associate an $L_\infty$-algebra of local observables that generalizes the Poisson algebra of a symplectic manifold. Here, by means of an explicit
homotopy equivalence, we interpret this $L_\infty$-algebra in terms of infinitesimal autoequivalences of higher prequantum bundles. By truncating the connection data on the prequantum bundle, we produce
analogues of the (higher) Lie algebras of sections of the Atiyah Lie algebroid and of the Courant Lie 2-algebroid. We also exhibit the $L_\infty$-cocycle that realizes the $L_\infty$-algebra of local observables
as a Kirillov-Kostant-Souriau-type $L_\infty$-extension of the Hamiltonian vector fields. When restricted along a Lie algebra action, this yields Heisenberg-like $L_\infty$-algebras such as the string Lie 2-algebra
of a semisimple Lie algebra.
</p>projecteuclid.org/euclid.hha/1408712337_20140822085854Fri, 22 Aug 2014 08:58 EDTUniversal enveloping crossed module of a Lie crossed modulehttp://projecteuclid.org/euclid.hha/1408712338<strong>José Manuel Casas</strong>, <strong>Rafael F. Casado</strong>, <strong>Emzar Khmaladze</strong>, <strong>Manuel Ladra</strong>. <p><strong>Source: </strong>Homology, Homotopy and Applications, Volume 16, Number 2, 143--158.</p><p><strong>Abstract:</strong><br/>
We construct a pair of adjoint functors between the categories
of crossed modules of Lie and associative algebras, which
extends the classical one between the categories of Lie and associative
algebras. This result is used to establish an equivalence
of categories of modules over a Lie crossed module and its universal
enveloping crossed module.
</p>projecteuclid.org/euclid.hha/1408712338_20140822085854Fri, 22 Aug 2014 08:58 EDT