Principal bundles on two-dimensional CW-complexes with disconnected structure group

Given any topological group G, the topological classification of principal G-bundles over a finite CW-complex X is long known to be given by the set of free homotopy classes of maps from X to the corresponding classifying space BG. This classical result has been long-used to provide such classification in terms of explicit characteristic classes. However, even when X has dimension 2, there is a case in which such explicit classification has not been explicitly considered. This is the case where G is a Lie group, whose group of components acts nontrivially on its fundamental group $\pi_1G$ . Here, we deal with this case and obtain the classification, in terms of characteristic classes, of principal G-bundles over a finite CW-complex of dimension 2, with G is a Lie group such that $\pi_0G$ is abelian.

Coronas for properly combable spaces

This paper is a systematic approach to the construction of coronas (i.e. Higson dominated boundaries at infinity) of combable spaces. We introduce three additional properties for combings: properness, coherence and expandingness. Properness is the condition under which our construction of the corona works. Under the assumption of coherence and expandingness, attaching our corona to a Rips complex construction yields a contractible [Formula: see text]-compact space in which the corona sits as a [Formula: see text]-set. This results in bijectivity of transgression maps, injectivity of the coarse assembly map and surjectivity of the coarse co-assembly map. For groups we get an estimate on the cohomological dimension of the corona in terms of the asymptotic dimension. Furthermore, if the group admits a finite model for its classifying space [Formula: see text], then our constructions yield a [Formula: see text]-structure for the group.

Hecke operators in Bredon (co)homology, K-(co)homology and Bianchi groups

In this paper, we provide a framework for the study of Hecke operators acting on the Bredon (co)homology of an arithmetic discrete group. Our main interest lies in the study of Hecke operators for Bianchi groups. Using the Baum–Connes conjecture, we can transfer computations in Bredon homology to obtain a Hecke action on the [Formula: see text]-theory of the reduced [Formula: see text]-algebra of the group. We show the power of this method giving explicit computations for the group [Formula: see text]. In order to carry out these computations we use an Atiyah–Segal type spectral sequence together with the Bredon homology of the classifying space for proper actions.

Subtle characteristic classes for Spin-torsors

Extending (Smirnov and Vishik, Subtle Characteristic Classes, arXiv:1401.6661), we obtain a complete description of the motivic cohomology with $${{\,\mathrm{\mathbb {Z}}\,}}/2$$ Z / 2 -coefficients of the Nisnevich classifying space of the spin group $$Spin_n$$ S p i n n associated to the standard split quadratic form. This provides us with very simple relations among subtle Stiefel–Whitney classes in the motivic cohomology of Čech simplicial schemes associated to quadratic forms from $$I^3$$ I 3 , which are closely related to $$Spin_n$$ S p i n n -torsors over the point. These relations come from the action of the motivic Steenrod algebra on the second subtle Stiefel–Whitney class. Moreover, exploiting the relation between $$Spin_7$$ S p i n 7 and $$G_2$$ G 2 , we describe completely the motivic cohomology ring of the Nisnevich classifying space of $$G_2$$ G 2 . The result in topology was obtained by Quillen (Math Ann 194:197–212, 1971).

Massey products in the homology of the loop space of a p-completed classifying space: finite groups with cyclic Sylow p-subgroups

Let $G$ be a finite group with cyclic Sylow $p$ -subgroups, and let $k$ be a field of characteristic $p$ . Then $H^{*}(BG;k)$ and $H_*(\Omega BG{{}^{{}^{\wedge }}_p};k)$ are $A_\infty$ algebras whose structure we determine up to quasi-isomorphism.

A note on the modular representation on the $\mathbb Z/2$-homology groups of the fourth power of real projective space and its application

We write $BV_h$ for the classifying space of the elementary Abelian 2-group $V_h$ of rank $h,$ which is homotopy equivalent to the cartesian product of $h$ copies of $\mathbb RP^{\infty}.$ Its cohomology with $\mathbb Z/2$-coefficients can be identified with the graded unstable algebra $P^{\otimes h} = \mathbb Z/2[t_1, \ldots, t_h]= \bigoplus_{n\geq 0}P^{\otimes h}_n$ over the Steenrod ring $\mathcal A$, where grading is by the degree of the homogeneous terms $P^{\otimes h}_n$ of degree $n$ in $h$ generators with the degree of each $t_i$ being one. Let $GL_h$ be the usual general linear group of rank $h$ over $\mathbb Z/2.$ The algebra $P^{\otimes h}$ admits a left action of $\mathcal A$ as well as a right action of $GL_h.$ A central problem of homotopy theory is to determine the structure of the space of $GL_h$-coinvariants, $\mathbb Z/2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}H_n(BV_h; \mathbb Z/2) ,$ where ${\rm Ann}_{\overline{\mathcal A}}H_n(BV_h; \mathbb Z/2) ={\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}_n]^{*}$ denotes the space of primitive homology classes, considered as a representation of $GL_h$ for all $n.$ Solving this problem is very difficult and still unresolved for $h\geq 4.$ The aim of this Note is of studying the dimension of $\mathbb Z/2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}_n]^{*}$ for the case $h = 4$ and the "generic" degrees $n$ of the form $k(2^{s} - 1) + r.2^{s},$ where $k,\, r,\, s$ are positive integers. Applying the results, we investigate the behaviour of the Singer cohomological "transfer" of rank $4$, which is a homomorphism from a certain subquotient of the divided power algebra $\Gamma(a_1^{(1)}, \ldots, a_4^{(1)})$ to mod-2 cohomology groups ${\rm Ext}_{\mathcal A}^{4, 4+n}(\mathbb Z/2, \mathbb Z/2)$ of the algebra $\mathcal A.$ Singer's algebraic transfer is one of the relatively efficient tools in determining the cohomology of the Steenrod algebra.

A note on the non-trivial elements in the cohomology groups of the Steenrod algebra

Let $F_2$ be the prime field of two elements and let $GL_s:= GL(s, F_2)$ be the general linear group of rank $s.$ Denote by $\mathscr A$ the Steenrod algebra over $F_2.$ The (mod-2) Lambda algebra, $\Lambda,$ is one of the tools to describe those mysterious "Ext-groups". In addition, the $s$-th algebraic transfer of William Singer \cite{Singer} is also expected to be a useful tool in the study of them. This transfer is a homomorphism $Tr_s: F_2 \otimes_{GL_s}P_{\mathscr A}(H_{*}(B\mathbb V_s))\to {\rm Ext}_{\mathscr {A}}^{s,s+*}(F_2, F_2),$ where $\mathbb V_s$ denotes an elementary abelian $2$-group of rank $s$, and $H_*(B\mathbb V_s)$ is the (mod-2) homology of a classifying space of $\mathbb V_s,$ while $P_{\mathscr A}(H_{*}(B\mathbb V_s))$ means the primitive part of $H_*(B\mathbb V_s)$ under the action of $\mathscr A.$ It has been shown that $Tr_s$ is highly non-trivial and, more precisely, that $Tr_s$ is an isomorphism for $s\leq 3.$ In addition, Singer proved that $Tr_4$ is an isomorphism in some internal degrees. He also investigated the image of the fifth transfer by using the invariant theory. In this note, we use another method to study the image of $Tr_5.$ More precisely, by direct computations using a representation of $Tr_5$ over the algebra $\Lambda,$ we show that $Tr_5$ detects the non-zero elements $h_0d_0\in {\rm Ext}_{\mathscr A}^{5, 5+14}(F_2, F_2),\ h_2e_0 = h_0g\in {\rm Ext}_{\mathscr A}^{5, 5+20}(F_2, F_2)$ and $h_3e_0 = h_4h_1c_0\in {\rm Ext}_{\mathscr A}^{5, 5+24}(F_2, F_2).$ The same argument can be used for homological degrees $s\geq 6$ under certain conditions.

On the hit problem for the algebra $H^{*}(B\mathbb F_2^{6}, \mathbb F_2)$ as a module over the mod two Steenrod algebra

Let $\mathbb F_2^{s}$ be the elementary Abelian 2-group of rank $s$ and let $\mathcal A_2$ be the mod two Steenrod algebra. We are interested in determining the dimension of the $\mathbb F_2$- graded vector space $\{\mathbb F_2\otimes_{\mathcal A_2} H^{*}(B\mathbb F_2^{s}, \mathbb F_2)\}_{n\geq 0},$ where $B\mathbb F_2^{s}$ is the classifying space of $\mathbb F_2^{s}$ and $$H^{*}(B\mathbb F_2^{s}, \mathbb F_2) \cong P_s:= \mathbb F_2[x_1, \ldots, x_s],$$ the algebra polynomial on $s$ variables of degree $1,$ viewed as a module over $\mathcal A_2.$ This problem is called \textit{the "hit" problem} for Steenrod algebra. It is solved for the cases $s\leq 4$ and in some generic degrees for $s = 5.$ However, the problem is not yet known for $s\geq 6.$ The hit problem has been proved surprisingly difficult, even with the help of the computer. In this note, based on Kameko's homomorphism and an algorithm on the MAGMA computer algebra system, we explicitly compute the dimension of $\mathbb F_2\otimes_{\mathcal A_2} H^{*}(B\mathbb F_2^{s}, \mathbb F_2)$ for $s = 6$ and in the generic degree $n = 5(2^{d+4} - 1) + 49.2^{d+4}$ with $d > 0.$ This study is a continuation of our recent work in \cite{D.P4}.

UNITARY FUNCTOR CALCULUS WITH REALITY

We construct a calculus of functors in the spirit of orthogonal calculus, which is designed to study ‘functors with reality’ such as the Real classifying space functor, $\BU_\Bbb{R}(-)$ . The calculus produces a Taylor tower, the n-th layer of which is classified by a spectrum with an action of $C_2 \ltimes \U(n)$ . We further give model categorical considerations, producing a zigzag of Quillen equivalences between spectra with an action of $C_2 \ltimes \U(n)$ and a model structure on the category of input functors which captures the homotopy theory of the n-th layer of the Taylor tower.