We develop methods for proving that certain extensions of polynomial functors
do not split naturally. As an application we give a functorial description of
the third and the fourth stable homotopy groups of the classifying spaces of
free abelian groups.

In this paper, we compute the homology coalgebra and cohomology algebra over
a field of all generalized moment-angle complexes and give a duality theorem on
complementary moment-angle complexes.

It is a classical result that groupoids model homotopy 1-types, in the sense
that there is an equivalence between the homotopy categories, via the
classifying space and fundamental groupoid functors. We extend this to stable
homotopy 1-types and Picard groupoids.

The paper is devoted to show that topological homotopy groups commute with
inverse limits under certain circumstances. As a consequence, we present some
conditions under which the topological homotopy group of an inverse limit space
is a topological group. We also give some conditions for countability of
homotopy groups.

This paper confirms the following suggestion of Kontsevich. In the
appropriate derived sense, an action of the framed little disks operad and a
trivialization of the circle action is the same information as an action of the
Deligne-Mumford-Knudsen operad. This improves an earlier result of the author
and Bruno Vallette.

This paper shows that the kernel of the Witten genus tensor Z[1/6] is
generated by total spaces of Cayley plane bundles, but only after restricting
the Witten genus to string bordism. It does so by showing that the divisibility
properties of Cayley plane bundle characteristic numbers arising in
Borel-Hirzebruch Lie-group-theoretic calculations correspond precisely to the
divisibility properties arising in the Hovey-Ravenel-Wilson
BP-Hopf-ring-theoretic calculation of string bordism at primes >3.

The aim of this paper is to investigate the homology groups of mathematical
models of concurrency. We study the Baues-Wirsching homology groups of a small
category associated with a partial monoid action on a set. We prove that these
groups can be reduced to the Leech homology groups of the monoid. For a trace
monoid with an action on a set, we will build a cubical complex of free Abelian
groups with homology groups isomorphic to the integral homology groups of the
action category.

In "Generalized Group Characters and Complex Oriented Cohomology Theories",
Hopkins, Kuhn, and Ravenel develop a way to study cohomology rings of the form
E^*(BG) in terms of a character map. The character map can be interpreted as a
map of cohomology theories beginning with a height n cohomology theory E and
landing in a height 0 cohomology theory with a rational algebra of coefficients
that is constructed out of E.

Classical homological algebra considers chain complexes, resolutions, and
derived functors in additive categories. We describe ``track algebras in
dimension n'', which generalize additive categories, and we define higher order
chain complexes, resolutions, and derived functors. We show that higher order
resolutions exist in higher track categories, and that they determine higher
order Ext-groups. In particular, the E_m-term of the Adams spectral sequence
(m<n+3) is a higher order Ext-group, which is determined by the track algebra
of higher cohomology operations.

Let $B$ be a finite CW complex and $G$ a compact connected Lie group. We show
that the number of gauge groups of principal $G$-bundles over $B$ is finite up
to $A_n$-equivalence for $n<\infty$. As an example, we give a lower bound of
the number of $A_n$-equivalence types of gauge groups of principal
$\SU(2)$-bundles over $S^4$.

In this paper, we develop a diamond graph theory and apply the theory to the
(co)homology of the Lie algebra generated by positive systems of the classical
semi-simple Lie algebras over the field of complex numbers. As an application,
we give the weight decomposition of the diamond Lie algebra with Dynkin graph
$A_{n+1}$ and compute the rank of every weight subgraph of it.

Given a compact Riemannian manifold $(M g)$ and Morse function $f:m\to
\mathbb{R}$ whose gradient flow satisfies the Morse-Smale condition, (i.e. the
stable and unstable manifolds of f intersect transversely) we construct a chain
complex called the Morse-Witten Complex. Our goal on this paper is show that
the homology of the Morse-Witten complex is isomorphic to the singular homology
of $M$.

Let $E_{d}(\ell)$ denote the space of all closed $n$-gons in $\R^{d}$ (where
$d\ge 2$) with sides of length $\ell_1,..., \ell_n$, viewed up to translations.
The spaces $E_d(\ell)$ are parameterized by their length vectors
$\ell=(\ell_1,..., \ell_n)\in \R^n_{>}$ encoding the length parameters.
Generically, $E_{d}(\ell)$ is a closed smooth manifold of dimension
$(n-1)(d-1)-1$ supporting an obvious action of the orthogonal group ${O}(d)$.
However, the quotient space $E_{d}(\ell)/{O}(d)$ (the moduli space of shapes of
$n$-gons) has singularities for a generic $\ell$, assuming that $d>3$;

The relations of Barratt and Miller are shown to include all relations among
the elements $P^i\chi P^{n-i}$ in the mod $p$ Steenrod algebra, and a minimal
set of relations is given.

We prove that if a finite group $G$ acts smoothly on a manifold $M$ so that
all the isotropy subgroups are abelian groups with rank $\leq k$, then $G$ acts
freely and smoothly on $M \times \bbS^{n_1} \times ...\times \bbS^{n_k}$ for
some positive integers $n_1,..., n_k$. We construct these actions using a
recursive method, introduced in an earlier paper, that involves abstract fusion
systems on finite groups. As another application of this method, we prove that
every finite solvable group acts freely and smoothly on some product of spheres
with trivial action on homology.

Among the classical variants of the Pr\"ufer surface, some are homotopy
equivalent to a CW-complex (namely, a point or a wedge of a continuum of
circles) and some are not. The obstruction comes from the existence of
uncountably many `infinitesimal bridges' linking two metrizable open
subsurfaces inside the surface. We show that any non-metrizable surface that
possesses such a system of infinitesimal bridges cannot be homotopy equivalent
to a CW-complex.

The question of the existence of Universal homotopy commutative and homotopy
associative H-spaces (called Abelian H-spaces) is studied. Such a space T(X)
would prolong a map from X into an Abelian H-space to a unique H-map from T
into X. Examples of such pairs (X,T) are given and conditions are discussed
which limit the possible spaces X for which such a T can exist. The Anick
spaces are shown not to be universal Abelian H-spaces for the corresponding
Moore spaces, but conditions are discussed which could lead to a universal
property with respect to a more limited range of targets.

Previously we constructed operations in the mod 2 homology spectral sequence
associated to a cosimplicial E_\infty-space X. According to Bousfield, the
correct target for this spectral sequence is the homology of Tot X. Noting that
in this setting Tot X is an E_\infty-space, we show that our operations agree
with the usual Araki-Kudo operations in the target. We also prove that the
multiplication in the spectral sequence agrees with the multiplication in
H_*(Tot X).

Cohomology of a topological space with coefficients in stacks of abelian
2-groups is considered. A 2-categorical analog of the theorem of Grothendieck
is proved, relating cohomology of the space with coefficients in a 2-stage
spectrum and the Ext groups of appropriate stacks.

Such modern applications of topology as data analysis and digital image
analysis have to deal with noise and other uncertainty. In this environment,
topological spaces often appear equipped with a real valued function.
Persistence is a measure of robustness of the homology classes of the
filtration of the lower level sets of this function. In this paper we introduce
the homology group of filtration as the product of the kernels of the homology
maps of the inclusions.

The concept of topological persistence, introduced recently in computational
topology, finds applications in studying a map in relation to the topology of
its domain. Since its introduction, it has been extended and generalized in
various directions. However, no attempt has been made so far to extend the
concept of topological persistence to a generalization of `maps' such as
cocycles which are discrete analogs of closed differential forms, a well known
concept in differential geometry.

We prove that, for p an odd prime, every finite p-group of rank 3 acts freely
on a finite complex X homotopy equivalent to a product of three spheres.

We construct an example of an $A_{\infty}$ algebra structure defined over a
finite dimensional graded vector space.

It is known that viewing the fundamental group $\pi_{1}(X)$ as the quotient
space of the loop space $\Omega X$ with the compact-open topology does not
always give rise to a topological group. In this paper, free topological groups
are used to introduce a new group topology on the fundamental group. The
resulting invariant $\pi_{1}^{\tau}$ takes values in the category of
topological groups and is useful for studying homotopy in spaces that lack
universal covers.

We survey research on the homotopy theory of the space map(X, Y) consisting
of all continuous functions between two topological spaces. We summarize
progress on various classification problems for the homotopy types represented
by the path-components of map(X, Y). We also discuss work on the homotopy
theory of the monoid of self-equivalences aut(X) and of the free loop space LX.
We consider these topics in both ordinary homotopy theory as well as after
localization. In the latter case, we discuss algebraic models for the
localization of function spaces and their applications.

In this note, we give a simple proof of the Borsuk-Ulam theorem for
$Z_p$-actions. We prove that, if $S^n$ and $S^m$ are equipped with free
$Z_p$-actions (p prime) and $f: S^n \to S^m$ is a $Z_p$-equivariant map, then
$n \leq m$.

R. Thom and M. Adachi proved that the oriented cobordism class of $\CC
P^{2k-1}$ and $ \RR P^{2m-1}$ to be zero if $k=1,2 ; m= 1, 2,3, 4$ and $ k=3 ;
m= 5, 6$ respectively. We construct oriented manifolds having the boundary
either $\CC P^{2k-1}$ or $ \RR P^{4k+1}$ for each $k > 0$. The main tool is the
theory of quasitoric manifolds and small covers.

Let M be a homogeneous space admitting a left translation by a connected Lie
group G. The adjoint to the action gives rise to a map from G to the monoid of
self-homotopy equivalences of M.The purpose of this paper is to investigate the
injectivity of the homomorphism which is induced by the adjoint map on the
rational homotopy. In particular, the visible degrees are determined explicitly
for all the cases of simple Lie groups and their associated homogeneous spaces
of rank one which are classified by Oniscik.

We study exceptional torsion in the integral cohomology of a family of
p-groups associated to p-adic Lie algebras. A spectral sequence E_r^{*,*}[g] is
defined for any Lie algebra g which models the Bockstein spectral sequence of
the corresponding group in characteristic p. This spectral sequence is then
studied for complex semisimple Lie algebras like sl_n(C), and the results there
are transferred to the corresponding p-group via the intermediary arithmetic
Lie algebra defined over Z.

We start from any small strict monoidal braided Ab-category and extend it to
a monoidal nonstrict braided Ab-category which contains braided bialgebras. The
objects of the original category turn out to be modules for these bialgebras

A Bott tower is an iterated $\CP ^1$-bundle over a point, where each $\CP
^1$-bundle is the projectivization of a rank $2$ decomposable complex vector
bundle. For a Bott tower, the filtered cohomology is naturally defined. We show
that isomorphism classes of Bott towers are distinguished by their filtered
cohomology rings. We even show that any filtered cohomology ring isomorphism
between two Bott towers is induced by an isomorphism of the Bott towers.

We study the Linial--Meshulam model of random two-dimensional simplicial
complexes. One of our main results states that for $p\ll n^{-1}$ a random
2-complex $Y$ collapses simplicially to a graph and, in particular, the
fundamental group $\pi_1(Y)$ is free and $H_2(Y)=0$, a.a.s. We also prove that,
if the probability parameter $p$ satisfies $p\gg n^{-1/2+\epsilon}$, where
$\epsilon>0$, then an arbitrary finite two-dimensional simplicial complex
admits a topological embedding into a random 2-complex, with probability
tending to one as $n\to \infty$.

For a topological space $X$ we study continuous maps $f : X\to \mathbb R^m$
such that images of every pairwise distinct $k$ points are affinely (linearly)
independent. Such maps are called affinely (linearly) $k$-regular embeddings.
We investigate the cohomology obstructions to existence of regular embeddings
and give some new lower bounds on the dimension $m$ as function of $X$ and $k$,
for the cases $X$ is $\mathbb R^n$ or $X$ is an $n$-dimensional manifold. In
the latter case, some nonzero Stiefel-Whitney classes of $X$ help to improve
the bound.

We show in this text how the most important homology equivalences of
fundamental Algebraic Topology can be obtained as reductions associated to
discrete vector fields. Mainly the homology equivalences whose existence --
most often non-constructive -- is proved by the main spectral sequences, the
Serre and Eilenberg-Moore spectral sequences. On the contrary, the constructive
existence is here systematically looked for and obtained.

Let $G=SU(2)$ and $\Omega G$ the space of based loops in SU(2). Motivated by
the theory of Hamiltonian $LG$-spaces, we explicitly compute the topological
equivariant $K$-theory $K_G^*(\Omega G)$ as an $R(G)$-module.

On d\'emontre une conjecture due \'a N. Kuhn concernant la cohomologie
singuli\'ere \'a coefficients mod p des espaces, comme module instable sur
l'alg\'ebre de Steenrod. Notre d\'emonstration de ce r\'esultat, d\'ej\'a connu
en caract\'eristique 2, fait appel \'a une m'ethode nouvelle, qui fonctionne en
toute caracteristique. De cette mani\'ere on r\'etablit un r'esultat de [S98]
dont la preuve est incompl\'ete dans le cas d'un nombre premier impair.

----

We prove some basic facts about moduli prestacks of one-dimensional formal
A-module laws, moduli stacks of one-dimensional formal A-modules, and the flat
cohomology groups of each, when A is a p-adic number ring. This is the first
paper in a series about moduli stacks of one-dimensional formal A-modules,
methods for computing their flat cohomology, and applications to stable
homotopy of spheres.

We compute the integral homology and cohomology groups of configuration
spaces of two distinct points on a given real projective space. The explicit
answer is related to the (known multiplicative structure in the) integral
cohomology---with simple and twisted coefficients---of the dihedral group of
order 8 (in the case of unordered configurations) and the elementary abelian
2-group of rank 2 (in the case of ordered configurations).

We prove that the Halperin-Carlsson conjecture holds for any closed manifold
with a free (Z_2)^m action whose orbit space is a small cover.

We correct the proof of the main theorem of our paper of the same title
(Topology, 34, no.3, 633-649, 1995).

I provide a complete analysis of the motivic Adams spectral sequences
converging to the bigraded coefficients of the 2-completions of the motivic
spectra BPGL and kgl over p-adic fields, p>2. The former spectrum is the
algebraic Brown-Peterson spectrum at the prime 2 (and hence is part of the
study of algebraic cobordism), and the latter is a certain BPGL-module that
plays the role of a "connective" motivic algebraic K-theory spectrum.

Masuda (2008) provided the characterization of real Bott manifolds in terms
of three operations on upper triangular matrices. We provide a combinatorial
characterization of real Bott manifolds up to diffeomorphism in terms of
operations on directed acyclic graphs. Our observation leads to several new
invariants of real Bott manifolds.

In this article, we introduce the notion of a functor on coarse spaces being
coarsely excisive- a coarse analogue of the notion of a functor on topological
spaces being excisive. Further, taking cones, a coarsely excisive functor
yields a topologically excisive functor, and for coarse topological spaces
there is an associated coarse assembly map from the topologically exicisive
functor to the coarsely excisive functor.

We describe the projectives in the category of functors from a graded poset
to abelian groups. Based on this description we define a related condition,
pseudo-projectivity, and we prove that this condition is enough for the
vanishing of the derived direct limits. We apply this result to deduce a
generalized version of a theorem of Whitehead for the pushout. The dual results
for inverse limits are also considered.

Generalized \'etale homotopy pro-groups $\pi_1^{\ets}(\mc{C}, x)$ associated
to pointed connected small Grothendieck sites $(\mc{C}, x)$ are defined and
their relationship to Galois theory and the theory of pointed torsors for
discrete groups is explained.

In this first paper of a series we study various operads of natural
operations on Hochschild cochains and relationships between them.

We compute the cohomology with group ring coefficients of the complement of a
finite collection of affine hyperplanes in a finite dimensional complex vector
space. It is nonzero in exactly one degree, namely the degree equal to the rank
of the hyperplane arrangement.

We develop a stable analogue to the theory of cosimplicial frames in model
cagegories; this is used to enrich all homotopy categories of stable model
categories over the usual stable homotopy category and to give a different
description of the smash product of spectra which is compared with the known
descriptions; in particular, the original smash product of Boardman is
identified with the newer smash products coming from a symmetric monoidal model
of the stable homotopy category.

$A_\infty$ categories are a mathematical structure that appears in
topological field theory, string topology, and symplectic topology. This paper
studies the cyclic homology of a Calabi-Yau $A_\infty$ category, and shows that
it is naturally an equivariant topological conformal field theory, and in
particular, contains an involutive Lie bialgebra structure. Applications of the
Lie bialgebra to string topology, Fukaya category and symplectic field theory
are given.

We make some computations in stable motivic homotopy theory over Spec
\mathbb{C}, completed at 2. Using homotopy fixed points and the algebraic
K-theory spectrum, we construct a motivic analogue of the real K-theory
spectrum KO. We also establish a theory of connective covers to obtain a
motivic version of ko. We establish an Adams spectral sequence for computing
motivic ko-homology. The E_2-term of this spectral sequence involves Ext groups
over the subalgebra A(1) of the motivic Steenrod algebra. We make several
explicit computations of these E_2-terms in interesting special cases.

Minimum numbers of fixed points or of coincidence components (realized by
maps in given homotopy classes) are the principal objects of study in
topological fixed point and coincidence theory. In this paper we investigate
fiberwise analoga and represent a general approach e.g. to the question when
two maps can be deformed until they are coincidence free. Our method involves
normal bordism theory, a certain pathspace EB and a natural generalization of
Nielsen numbers.

Looking at the cartesian product of a topological space with itself, a
natural map to be considered on that object is the involution that interchanges
the coordinates, i.e. that maps (x,y) to (y,x). The so-called halfsquaring
construction, now also called "symmetric squaring construction", in Cech
homology with Z/2-coefficients was introduced in [arXiv:0709.1774] as a map
from the k-th Cech homology group of a space X to the 2k-th Cech homology group
of X \times X divided by the above mentioned involution.

We present a new technique for analyzing the v_0-Bockstein spectral sequence
studied by Shimomura and Yabe. Employing this technique, we derive a
conceptually simpler presentation of the homotopy groups of the E(2)-local
sphere for p > 3. We identify and correct some errors in the original
Shimomura-Yabe calculation. We deduce the related K(2)-local homotopy groups,
and discuss their manifestation of Gross-Hopkins duality.

In this article, we investigate the functors from modules to modules that
occur as the summands of tensor powers and the functors from modules to Hopf
algebras that occur as natural coalgebra summands of tensor algebras. The main
results provide some explicit natural coalgebra summands of tensor algebras. As
a consequence, we obtain some decompositions of Lie powers over the general
linear groups.

We introduce characteristic classes for the spectral sequence associated to a
split short exact sequence of Hopf algebras. We show that these characteristic
classes can be seen as obstructions for the vanishing of differentials in the
spectral sequence and prove a decomposition theorem. We also interpret our
results in the settings of group and Lie algebra extensions and prove some
interesting corollaries concerning the collapse of the
(Lyndon-)Hochschild-Serre spectral sequence and the order of characteristic
classes.

In this paper, we recall the definition of twisted K-theory in various
settings. We prove that for a twist $\tau$ corresponding to a three dimensional
integral cohomology class of a space X, there exist a "universal coefficient"
isomorphism K_{*}^{\tau}(X)\cong
K_{*}(P_{\tau})\otimes_{K_{*}(\mathbb{C}P^{\infty})} \hat{K}_{*} where $P_\tau$
is the total space of the principal $\mathbb{C}P^{\infty}$-bundle induced over
X by $\tau$ and $\hat K_*$ is obtained form the action of
$\mathbb{C}P^{\infty}$ on K-theory.

A convenient 2-category of topological stacks is constructed which is both
complete and Cartesian closed. This 2-category, called the 2-category of
compactly generated stacks, is the analogue of classical topological stacks,
but for a different Grothendieck topology. In fact, there is an equivalence of
2-categories between compactly generated stacks and those classical topological
stacks which admit locally compact atlases. Compactly generated stacks are also
equivalent to a bicategory of topological groupoids and principal bundles, just
as in the classical case.

We prove that the operad B of natural operations on the Hochschild cohomology
has the homotopy type of the operad of singular chains on the little disks
operad. To achieve this goal, we introduce crossed interval groups and show
that B is a certain crossed interval extension of an operad T whose homotopy
type is known. This completes the investigation of the algebraic structure on
the Hochschild cochain complex that has lasted for several decades.

Given a family of based CW-pairs
$(\underline{X},\underline{A})=\{(X;A)\}^m_{i=1}$ together with an abstract
simplicial complex $K$ with $m$ vertices, there is an associated based
CW-complex $Z(K;(\underline{X},\underline{A}))$ known as a generalized
moment-angle complex.

The goal of this paper is to prove a Koszul duality result for E_n-operads in
differential graded modules over a ring. The case of an E_1-operad, which is
equivalent to the associative operad, is classical. For n>1, the homology of an
E_n-operad is identified with the n-Gerstenhaber operad and forms another well
known Koszul operad. Our main theorem asserts that an operadic cobar
construction on the dual cooperad of an E_n-operad defines a cofibrant model of
E_n.

We gather conditions on a class H of continuous maps of topological spaces
that allow a reasonable theory of fibrations up to an equivalence (a map from
this class) which we call H-fibrations. The weak homotopy equivalences recover
quasifibrations and homology equivalences yield homology fibrations. We study
local H-fibrations that behave nicely with respect to homotopy colimits
together with universal H-fibrations that behave nicely with respect to
pullbacks. We then proceed to classify H-fibrations up to a natural notion of
equivalence.

By studying the representation theory of a certain infinite $p$-group and
using the generalised characters of Hopkins, Kuhn and Ravenel we find useful
ways of understanding the rational Morava $E$-theory of the classifying spaces
of general linear groups over finite fields. Making use of the well understood
theory of formal group laws we establish more subtle results integrally,
building on relevant work of Tanabe. In particular, we study in detail the
cases where the group has dimension less than or equal to the prime $p$ at
which the $E$-theory is localised.

Motivated by the problem of dealing with incomplete or imprecise acquisition
of data in computer vision and computer graphics, we extend results concerning
the stability of persistent homology with respect to function perturbations to
results concerning the stability with respect to domain perturbations. More
precisely, by encoding sets using their distance functions, we prove that the
multidimensional matching distance between rank invariants of persistent
homology groups is always upperly bounded by the Hausdorff distance between
sets.

We prove structural theorems for computing the completion of a G-spectrum at
the augmentation ideal of the Burnside ring of a finite group G. First we show
that a G-spectrum can be replaced by a spectrum obtained by allowing only
isotropy groups of prime power order without changing the homotopy type of the
completion. We then show that this completion can be computed as a homotopy
colimit of completions of spectra obtained by further restricting isotropy to
one prime at a time, and that these completions can be computed in terms of
completion at a prime.

Given a simplicial complex with weights on its simplices, and a nontrivial
cycle on it, we are interested in finding the cycle with minimal weight which
is homologous to the given one. Assuming that the homology is defined with
integer coefficients, we show the following : For a finite simplicial complex
$K$ of dimension greater than $p$, the boundary matrix $[\partial_{p+1}]$ is
totally unimodular if and only if $H_p(L, L_0)$ is torsion-free, for all pure
subcomplexes $L_0, L$ in $K$ of dimensions $p$ and $p+1$ respectively, where
$L_0$ is a subset of $L$.

Let $X$ be a finitistic space having the mod 2 cohomology algebra of the
product of two projective spaces. We study free involutions on $X$ and
determine the possible mod 2 cohomology algebra of orbit space of any free
involution, using the Leray spectral sequence associated to the Borel fibration
$X \hookrightarrow X_{\mathbb{Z}_2} \longrightarrow B_{\mathbb{Z}_2}$.

We describe a coordinate-free notion of conformal nets as a mathematical
model of conformal field theory. We define defects between conformal nets and
introduce composition of defects, thereby providing a notion of morphism
between conformal field theories. Altogether we characterize the algebraic
structure of the collection of conformal nets as a symmetric monoidal
tricategory. Dualizable objects of this tricategory correspond to
conformal-net-valued 3-dimensional local topological quantum field theories.

A Bott manifold is a closed smooth manifold obtained as the total space of an
iterated $\C P^1$-bundle starting with a point, where each $\C P^1$-bundle is
the projectivization of a Whitney sum of two complex line bundles. A
\emph{$\Q$-trivial Bott manifold} of dimension $2n$ is a Bott manifold whose
cohomology ring is isomorphic to that of $(\CP^1)^n$ with $\Q$-coefficients. We
find all diffeomorphism types of $\Q$-trivial Bott manifolds and show that they
are distinguished by their cohomology rings with $\Z$-coefficients.

We prove that for a toric manifold $M$, any graded ring isomorphism
$H^\ast(M) \to H^\ast(\prod_{i=1}^{m}\CP^{n_i})$ is induced by a diffeomorphism
$\prod_{i=1}^m \CP^{n_i} \to M$.

In this note we present examples of localization functors (in the category of
spaces) whose composition with certain cellularization functors is not
idempotent, and vice versa.

The paper is devoted to the problem of finding explicit combinatorial
formulae for the Pontryagin classes. We discuss two formulae, the classical
Gabrielov-Gelfand-Losik formula based on investigation of configuration spaces
and the local combinatorial formula obtained by the author in 2004. The latter
formula is based on the notion of a universal local formula introduced by the
author and on the usage of bistellar moves. We give a brief sketch for the
first formula and a rather detailed exposition for the second one.

We prove that the Goodwillie tower of a weak equivalence preserving functor
from spaces to spectra can be expressed in terms of the tower for stable
mapping spaces. Our proof is motivated by interpreting the functors P_n and D_n
as pseudo-differential operators which suggests certain `integral'
presentations based on a derived Yoneda embedding. These models allow one to
extend computational tools available for the tower of stable mapping spaces. As
an application we give a classical expression for the derivative over the
basepoint.

We use the orientation underlying the Hirzebruch genus of level three to map
the beta family at the prime p=2 into the ring of divided congruences. This
procedure, which may be thought of as the elliptic greek letter beta
construction, yields the f-invariants of this family.

Divided difference operators are degree-reducing operators on the cohomology
of flag varieties that are used to compute algebraic invariants of the ring
(for instance, structure constants). We identify divided difference operators
on the equivariant cohomology of G/P for arbitrary partial flag varieties of
arbitrary Lie type, and show how to use them in the ordinary cohomology of G/P.
We provide three applications. The first shows that all Schubert classes of
partial flag varieties can be generated from a sequence of divided difference
operators on the highest-degree Schubert class.

The approach we present here is a modification of the Morse theory for unital
C*-algebras.It helps us to study the geometry of the noncommutative CW
complexes introduced in[1] and [2]. A geometric condition for a unital
C*-algebra to admit a noncommutative CW complex decomposition is studied. Some
examples to illustrate these geometric information in practice are given.

We compute the set of naive pointed homotopy classes of endomorphisms of the
projective line P^1 over the spectrum of a field. Our computation compares well
with Fabien Morel's one of the motivic pointed homotopy classes of
endomorphisms of P^1: there is an a priori monoid structure on the set of naive
homotopy classes and the group completion of this monoid is isomorphic to the
group of motivic homotopy classes.

We study the twisted cohomology groups of $A_\infty$-algebras defined by
twisting elements and their behavior under morphisms and homotopies using the
bar construction. We define higher Massey products on the cohomology groups of
general $A_\infty$-algebras and establish the naturality under morphisms and
their dependency on defining systems. We construct a spectral sequence
converging to the twisted cohomology groups an show that the higher
differentials are given by the $A_\infty$-algebraic Massey products.

We give a construction of an L-infinity map from any L-infinity algebra into
its truncated Chevalley-Eilenberg complex as well as its cyclic and A-infinity
analogues. This map fits with the inclusion into the full Chevalley-Eilenberg
complex (or its respective analogues) to form a homotopy fiber sequence of
L-infinity-algebras. Application to deformation theory and graph homology are
given. We employ the machinery of Maurer-Cartan functors in L-infinity and
A-infinity algebras and associated twistings which should be of independent
interest.

This paper gives a short summary of the central role played by Ed Brown's
``twisting cochains'' in higher Franz-Reidemeister (FR) torsion and higher
analytic torsion. Briefly, any fiber bundle gives a twisting cochain which is
unique up to fiberwise homotopy equivalence. However, when they are based, the
difference between two of them is a higher algebraic K-theory class measured by
higher FR torsion. Flat superconnections are also equivalent to twisting
cochains.

We give a complete description of the integral cohomology ring of the flag
manifold E_8/T, where E_8 denotes the compact exceptional Lie group of rank 8
and T its maximal torus, by the method due to Borel and Toda. This completes
the computation of the integral cohomology rings of the flag manifolds for all
compact connected simple Lie groups.

We prove surjectivity of the comparison map from continuous bounded
cohomology to continuous cohomology for Hermitian Lie groups with finite
center. For general semisimple Lie groups with finite center, the same argument
shows that the image of the comparison map contains all the even generators.
Our proof uses a Hirzebruch type proportionality principle in combination with
Gromov's results on boundedness of primary characteristic classes and classical
results of Cartan and Borel on the cohomology of compact homogeneous spaces.

In this paper we complete the description of the $B\mathbb{Z}
/p$-cellularization of the classifying spaces of all finite groups, for all
primes $p$. The techniques are based in a careful analysis of the $p$-fusion
structure of the groups involved -with special attention to their strongly
closed subgroups- and Chach\'olski's description of the $A$-cellular
approximation.

Let Y = Hom(Z^n, SU(2)) denote the space of commuting n-tuples in SU(2). We
determine the homotopy type of the suspension of Y and compute the integral
cohomology groups of Y for all positive integers n.

In this paper the spaces of $q$-tuples of points in a Euclidean space,
without $k$-wise coincidences are studied (configuration-like spaces). A
transitive group action by permuting these points is considered, and some new
upper bounds on the genus (in the sense of Krasnosel'skii-Schwarz and
Clapp-Puppe) for this action are given. Some theorems of Cohen-Lusk type for
coincidence points of continuous maps to Euclidean spaces are deduced.

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 HH^{3,-1} H*(G) with the following property:
Given any graded H*(G)-module X, the image of the canonical element in
Ext^{3,-1}(X,X) is zero if and only if X is isomorphic to a direct summand of
H*(G,M) for some kG-module M. In particular, if the canonical element vanishes,
then every module is a direct summand of a realizable H*(G)-module.

We construct a cobordism group for embedded graphs in two different ways,
first by using sequences of two basic operations, called "fusion" and
"fission", which in terms of cobordisms correspond to the basic cobordisms
obtained by attaching or removing a 1-handle, and the other one by using the
concept of a 2-complex surface with boundary is the union of these knots.

We provide a model of the String group as a finite-dimensional 2-group in the
bicategory of Lie groupoids, left-principal bibundles, and bibundle maps. This
bicategory is a generalization of the 2-category of Lie groupoids, smooth
functors, and smooth natural transformations and so our notion of smooth
2-group subsumes the notion of Lie 2-group introduced by Baez-Lauda. More
precisely we classify certain central extensions of these 2-groups in terms of
a topological group cohomology introduced by G. Segal the late 60's, and our
String 2-group is a special case of such extensions.

In 1971, Kunio Murasugi proved a necessary condition on a knot's Alexander
polynomial for that knot to be periodic of prime power order. In this paper I
present an alternate proof of Murasugi's condition which is subsequently used
to extend his result to the twisted Alexander polynomial.

Manfred Stelzer has pointed out that part of Corollary 4.5 of our paper
"Topological nonrealization results via the Goodwillie tower approach to
iterated loopspace homology" [Alg. Geo. Top. 8 (2008), 2109--2129] was not
sufficiently proved, and, indeed, is likely incorrect as stated. This
necessitates a little more argument to finish the proof of the main theorem of
the original paper. The statement of this theorem, and all the examples, remain
unchanged.

This paper explains why Goodwillie-Weiss calculus of embeddings can offer new
information about the Euclidean embedding dimension of P^m only for m < 16.
Concrete scenarios are described in these low-dimensional cases, pinpointing
where to look for potential, but critical, high-order obstructions in the
corresponding Taylor towers. For m > 15, the relation TC^S(P^m) > n-1 is
translated into the triviality of a certain cohomotopy Euler class which, in
turn, becomes the only Taylor obstruction to producing an n-dimensional
Euclidean embedding of P^m.

We study the spaces of string links and homotopy string links in an arbitrary
manifold using multivariable manifold calculus of functors. We construct
multi-cosimplicial models for both spaces and deduce certain convergence
properties of the associated Bousfield-Kan homotopy and cohomology spectral
sequences when the ambient manifold is a Euclidean space of dimension four or
more.

Using fixed-point-free group actions, we set up a scheme to define nested
classes of groups indexed over ordinals. Restricting to cellular actions on
CW-complexes, we find new classes as well as new characterizations for some
well-known classes, such as virtually polycyclic groups. We generalize
properties of the virtual cohomological dimension of a group to groups with
jump (co)homology and prove that a core subclass of a new class of groups has
jump (co)homology.

In this paper, we study algebras over the little cubes operads introduced by
Boardman and Vogt, using the formalism of higher category theory.

In this paper we extend our correlation functions to the open/closed case.
This gives rise to actions of an open/closed version of the Sullivan PROP as
well as an action of the relevant moduli space. There are several unexpected
structures and conditions that arise in this extension which are forced upon us
by considering the open sector.

We propose a homology theory for locally compact spaces with ends in which
the ends play a special role. The approach is motivated by results for graphs
with ends, where it has been highly successful. But it was unclear how the
original graph-theoretical definition could be captured in the usual language
for homology theories, so as to make it applicable to more general spaces.

Colocalization is a right adjoint to the inclusion of a subcategory. Given a
ring-spectrum R, one would like a spectral sequence which connects a given
colocalization in the derived category of R-modules and an appropriate
colocalization in the derived category of graded modules over the graded ring
of homotopy groups of R. We show that, under suitable conditions, such a
spectral sequence exists. This generalizes the local-cohomology spectral
sequence of Dwyer, Greenlees and Iyengar. Some applications are presented.

We introduce a new cubical model for homotopy types. More precisely, we'll
define a category Qs with the following features: Qs is a PROP containing the
classical box category as a subcategory, the category Qs-Set of presheaves of
sets on Qs models the homotopy category, and combinatorial symmetric monoidal
model categories with cofibrant unit all have homotopically well behaved Qs-Set
enrichments.

We discuss properties of the complete Euler characteristic of a group G of
type FP over the complex numbers and we relate it to the L2-Euler
characteristic of the centralizers of the elements of G.

The topological fundamental group $\pi_{1}^{top}$ is a topological invariant
that assigns to each space a quasi-topological group and is discrete on spaces
which are well behaved locally. For a totally path-disconnected, Hausdorff,
unbased space $X$, we compute the topological fundamental group of the "hoop
earring" space of $X$, which is the reduced suspension of $X$ with disjoint
basepoint.

The object of this paper is to prove that the standard categories in which
homotopy theory is done, such as topological spaces, simplicial sets, chain
complexes of abelian groups, and any of the various good models for spectra,
are all homotopically self-contained. The left half of this statement
essentially means that any functor that looks like it could be a tensor product
(or product, or smash product) with a fixed object is in fact such a tensor
product, up to homotopy. The right half says any functor that looks like it
could be Hom into a fixed object is so, up to homotopy.

Implementing an idea due to John Baez and James Dolan we define new
invariants of Whitney stratified manifolds by considering the homotopy theory
of smooth transversal maps. To each Whitney stratified manifold we assign
transversal homotopy monoids, one for each natural number. The assignment is
functorial for a natural class of maps which we call stratified normal
submersions. When the stratification is trivial the transversal homotopy
monoids are isomorphic to the usual homotopy groups. We compute some simple
examples and explore the elementary properties of these invariants.

We show, at the primes 2, 3, and 5, that no map from MU to BP defining a
universal p-typical formal group law on BP preserves power operations. In
particular, such a map cannot define a commutative MU-algebra structure on BP.
Our results apply more generally to show that the p-typical complex
orientations on a number of standard spectra are not commutative MU-algebra
maps.

We develop a functorial approach to the study of the homotopy groups of
spheres and Moore spaces $M(A,n)$, based on the Curtis spectral sequence and
the decomposition of Lie functors as iterates of simpler functors such as the
symmetric or exterior algebra functors. The discussion takes place over the
integers, and includes a functorial description of the derived functors of
certain Lie algebra functors, as well as of all the main cubical functors (such
as the degree 3 component $SP^3$ of the symmetric algebra functor).

An $n$-FC ring is a left and right coherent ring whose left and right self
FP-injective dimension is $n$. The work of Ding and Chen in \cite{ding and chen
93} and \cite{ding and chen 96} shows that these rings possess properties which
generalize those of $n$-Gorenstein rings. In this paper we call a (left and
right) coherent ring with finite (left and right) self FP-injective dimension a
Ding-Chen ring. In case the ring is Noetherian these are exactly the Gorenstein
rings.

Let $p:E -> B$ be a principal fibration with classifying map $w:B -> C$. It
is well-known that the group $[X,\Omega C]$ acts on $[X,E]$ with orbit space
the image of $p_#$, where $p_#: [X,E] -> [X,B]$. The isotropy subgroup of the
map of $X$ to the base point of $E$ is also well-known to be the image of $[X,
\Omega B]$. The isotropy subgroups for other maps $e:X -> E$ can definitely
change as $e$ does.

We construct an example of a Peano continuum $X$ such that: (i) $X$ is a
one-point compactification of a polyhedron; (ii) $X$ is weakly homotopy
equivalent to a point (i.e.

$\pi_n(X)$ is trivial for all $n \geq 0$); (iii) $X$ is noncontractible; and
(iv) $X$ is homologically and cohomologically locally connected (i.e. $X$ is a
$HLC$ and $clc$ space). We also prove that all classical homology groups
(singular, \v{C}ech, and Borel-Moore), all classical cohomology groups
(singular and \v{C}ech), and all finite-dimensional Hawaiian groups of $X$ are
trivial.

Given a maximal finite subgroup G of the nth Morava stabilizer group at a
prime p, we address the question: is the associated higher real K-theory EO_n a
summand of the K(n)-localization of a TAF-spectrum associated to a unitary
similitude group of type U(1,n-1)? We answer this question in the affirmative
for p in {2, 3, 5, 7} and n = (p-1)p^{r-1} for a maximal finite subgroup
containing an element of order p^r. We answer the question in the negative for
all other odd primary cases.

In this paper, we show that the abelianization of the kernel of the Magnus
representation of the automorphism group of a free group is not finitely
generated.

Given a finitely presented group $G$, Hopf's formula expresses the second
integral homology of $G$ in terms of generators and relators. We give an
algorithm that exploits Hopf's formula to estimate $H_2(G;k)$, with
coefficients in a finite field k, and give examples using $G=SL_2$ over
specific rings of integers. These examples are related to a conjecture of
Quillen.

In work by Ausoni, Dundas and Rognes a half magnetic monopole is discovered
and describes an obstruction to creating a determinant K(ku) \to ku*. In fact
it is an obstruction to creating a determinant gerbe map from K(ku) to K(Z,3).
We describe this obstruction precisely using monoidal categories and define the
notion of oriented 2-vector bundles, which removes this obstruction so that we
can define a determinant gerbe. We also generalize Brylinskis notion of a
connective structure to 2-vector bundles, in a way compatible with the
determinant gerbe.

In a preceding article the authors and Tran Ngoc Nam constructed a minimal
injective resolution of the mod 2 cohomology of a Thom spectrum. A Segal
conjecture type theorem for this spectrum was proved. In this paper one shows
that the above mentioned resolutions can be realized topologically. In fact
there exists a family of co?brations inducing short exact sequences in mod 2
cohomology. The resolutions above are obtained by splicing together these short
exact sequences. Thus the injective resolutions are realizable in the best
possible sense.

We explore connections between our earlier work, in which we constructed
spectra that interpolate between bu and HZ, and earlier work of Kuhn and Priddy
on the Whitehead conjecture and of Rognes on the stable rank filtration in
algebraic K-theory. We construct a "chain complex of spectra" that is a
bu-analogue of an auxiliary complex used by Kuhn-Priddy; we conjecture that
this chain complex is "exact"; and we give some supporting evidence. We tie
this to work of Rognes by showing that our auxiliary complex can be constructed
in terms of the stable rank filtration.

Let $\H_g$ be a genus $g$ handlebody and $\MCG_{2n}(\T_g)$ be the
$2n$-punctured mapping class group of $\T_g=\partial\H_g$. In this paper we
study two particular subgroups of $\MCG_{2n}(\T_g)$ which generalize Hilden
groups. As well as Hilden groups are related to plate closures of braids, these
generalizations are related to Heegaard splittings of manifolds and to bridge
decompositions of links. Connections between these subgroups and motion groups
of links in closed 3-manifolds are also provided.

The focus of this paper is the comparison of two unstable homotopy spectral
sequences-- the unstable mod p Adams spectral sequence that computes the
unstable homotopy of a $p-$complete space, and the Goerss--Hopkins spectral
sequence, which computes the unstable homotopy of the space of E-infinity maps
between Hk-algebras, where k is the algebraic closer of the field with p
elements and p is an odd prime.

We show that K_{2i}(Z[x,y]/(xy),(x,y)) is free abelian of rank 1 and that
K_{2i+1}(Z[x,y]/(xy),(x,y)) is finite of order (i!)^2. We also compute
K_{2i+1}(Z[x,y]/(xy),(x,y)) in low degrees.

We describe partial semi-simplicial resolutions of several moduli spaces of
interest in topology, geometry and algebra. These are: moduli spaces of finite
sets, labelled configuration spaces of points in an open manifold, and moduli
spaces of surfaces with tangential structure.

This allows us to prove homology stability results for all these moduli
spaces, which often improve the known stability ranges and give explicit
stability ranges in many new cases.

We prove a Dold-Kan type correspondence between the category of dendroidal
abelian groups and a suitably constructed category of dendroidal complexes. Our
result naturally extends the classical Dold-Kan correspondence between the
category of simplicial abelian groups and the category of non-negatively graded
chain complexes.

This note is about spherical classes in $H_*Q_0S^0$. A conjecture, due to Ed.
Curtis, predicts that only Hopf invariant one and Kervaire invariant one
elements will give rise to spherical classes in $H_*Q_0S^0$. Yet, there has
been no proof of this conjecture around. Assuming that this conjecture fails,
there must exist some other spherical classes in $H_*Q_0S^0$.

We consider the problem of calculating the Hurewicz image of Mahowald's
family $\eta_i\in{_2\pi_{2^i}^S}$. This allows us to identify specific
spherical classes in $H_*\Omega_0^{2^{i+1}-8+k}S^{2^i-2}$ for $0\leqslant
k\leqslant 6$. We then identify the type of the subalgebras that these classes
give rise to, and calculate the $A$-module and $R$-module structure of these
subalgebras. We shall the discuss the relation of these calculations to the
Curtis conjecture on spherical classes in $H_*Q_0S^0$, and relations with
spherical classes in $H_*Q_0S^{-n}$.

Over a monoidal model category, under some mild assumptions, we equip the
categories of colored PROPs and their algebras with projective model category
structures. A Boardman-Vogt style homotopy invariance result about algebras
over cofibrant colored PROPs is proved. As an example, we define homotopy
topological conformal field theories and observe that such structures are
homotopy invariant.

This paper investigates if a differential graded algebra can have more than
one $A_\infty$-structure extending the given differential graded algebra
structure. We give a sufficient condition for uniqueness of such an
$A_\infty$-structure up to quasi-isomorphism using Hochschild cohomology. We
then extend this condition to Sagave's notion of derived $A_\infty$-algebras
after introducing a notion of Hochschild cohomology that applies to this.

We compute the mod-2 cohomology of the collection of all symmetric groups as
a Hopf ring, using the transfer product of Strickland and Turner, which sheds
considerable light on the cup product structure of the cohomology of an
individual symmetric group. The main ingredient is a primitivity result for the
coproduct on homology dual to the transfer product. We also briefly develop
related Hopf ring structures on rings of symmetric invariants.

We study topology of configuration spaces of planar linkages having one leg
of variable length. Such telescopic legs are common in modern robotics where
they are used for shock absorbtion and serve a variety of other purposes. Using
a Morse theoretic technique, we compute explicitly, in terms of the metric
data, the Betti numbers of configuration spaces of these mechanisms.

The category of differential graded operads is a cofibrantly generated model
category and as such inherits simplicial mapping spaces. The vertices of an
operad mapping space are just operad morphisms. The 1-simplices represent
homotopies between morphisms in the category of operads.

We synthesize work of U. Koschorke on link maps and work of B. Johnson on the
derivatives of the identity functor in homotopy theory. The result can be
viewed in two ways: (1) As a generalization of Koschorke's "higher Hopf
invariants", which themselves can be viewed as a generalization of Milnor's
invariants of link maps in Euclidean space; and (2) As a stable range
description, in terms of bordism, of the cross effects of the identity functor
in homotopy theory evaluated at spheres.

We define an interesting sub-category of the category of simplicial sets,
$\Sr$, whose objects are called regular. Both it and the subcategory ${\cal
S}_{f-{\rm reg}}$ of finite regular simplicial sets have good stability
properties under limits and union. The category ${\cal S}_{f-{\rm reg}}$ is
cartesian closed, in contrast to the category of finite simplicial sets which
is not cartesian closed.

Farber introduced a notion of topological complexity $\TC(X)$ that is related
to robotics. Here we introduce a series of numerical invariants $\TC_n(X),
n=0,1, ...$ such that $\TC_2(X)=\TC(X)$ and $\TC_n(X)\le \TC_{n+1}(X)$. For
these higher complexities, we also define their symmetric version in spirit of
Gonz\'alez-Landweber.

In the study of stratified spaces it is useful to examine spaces of popaths
(paths which travel from lower strata to higher strata) and holinks (those
spaces of popaths which immediately leave a lower stratum for their final
stratum destination). It is not immediately clear that for adjacent strata
these two path spaces are homotopically equivalent, and even less clear that
this equivalence can be constructed in a useful way (with a deformation of the
space of popaths which fixes start and end points and where popaths instantly
become members of the holink).

In this paper we consider an operation on the set of simplicial complexes,
which we call "doubling operation". We show that the moment-angle complex Z_K
is the real moment-angle complex RZ_L(K) for simplicial complex L(K) obtained
from K by applying "doubling operation". As an application of this operation we
prove the toral rank conjecture for Z_K by estimating the lower bound of the
cohomology rank (with rational coefficients) of the real moment-angle complexes
RZ_K.

In this paper we study topological invariants of a class of random groups.
Namely, we study right angled Artin groups associated to random graphs and
investigate their Betti numbers, cohomological dimension and topological
complexity. The latter is a numerical homotopy invariant reflecting complexity
of motion planning algorithms in robotics. We show that the topological
complexity of a random right angled Artin group assumes, with probability
tending to one, at most three values.