Let $f:G\rightarrow H$ be a homomorphism of groups, we construct a topological space $X_f$ such that its group of homeomorphisms is isomorphic to $G$, its group of homotopy classes of self-homotopy equivalences is isomorphic to $H$ and the natural map between the group of homeomorphisms of $X_f$ and the group of homotopy classes of self-homotopy equivalences of $X_f$ is precisely $f$. In addition, realization problems involving homology, homotopy groups and groups of automorphisms are considered.
In this article topologies on metagroups are studied. They are related with generalized $C^*$-algebras over ${\bf R}$ or ${\bf C}$. Homomorphisms and quotient maps on them are investigated. Structure of topological metagroups is scrutinized. In particular, topologies on smashed products and smashed twisted wreath products of metagroups are scrutinized, which are making them topological metagroups. Moreover, their inverse homomorphism systems are studied.
Let $φ\colonΓ\to G$ be a homomorphism of groups. We consider factorizations $Γ\xrightarrow{f} M\xrightarrow{g} G$ of $φ$ such that either $g$ or $f$ are universal normal maps (namely, crossed modules). These two factorizations are natural generalizations of the usual normal closure and normalizer of a subgroup. Iterating these universal factorizations yield towers related, on the one hand, to hypercentral group extensions, Bousfield's localizations, and relative Schur multipliers. Dually, they extend to a relative situation, the automorphisms tower of a centerless group. Our constructions have…
Let $G$ be a real linear algebraic group and $L$ a finitely generated cosimplicial group. We prove that the space of homomorphisms $Hom(L_n,G)$ has a homotopy stable decomposition for each $n\geq 1$. When $G$ is a compact Lie group, we show that the decomposition is $G$-equivariant with respect to the induced action of conjugation by elements of $G$. The spaces $Hom(L_n,G)$ assemble into a simplicial space $Hom(L,G)$. When $G=U$ we show that its geometric realization $B(L,U)$, has a non-unital $E_\infty$-ring space structure whenever $Hom(L_0,U(m))$ is path connected for all $m\geq1$.
We prove that the topological connectivity of a graph homomorphism complex Hom($G,K_m$) is at least $m-D(G)-2$, where $\displaystyle D(G)=\max_{H\subseteq G}δ(H)$. This is a strong generalization of a theorem of Cukić and Kozlov, in which $D(G)$ is replaced by the maximum degree $Δ(G)$. It also generalizes the graph theoretic bound for chromatic number, $\displaystyleχ(G)\leq D(G)+1$, as $\displaystyleχ(G)=\min\{ m:\text{Hom}(G,K_m)\neq\varnothing\}$. Furthermore, we use this result to examine homological phase transitions in the random polyhedral complexes Hom$(G(n,p),K_m)$ when $p=c/n$ for a…
The purpose of this article is to give an exposition of topological properties of spaces of homomorphisms from certain finitely generated discrete groups to Lie groups $G$, and to describe their connections to classical representation theory, as well as other structures. Various properties are given when $G$ is replaced by a small category, or the discrete group is given by a right-angled Artin group.
Let X be a topological space. The homology of the iterated loop space $H_*Ω^n X$ is an algebra over the homology of the framed n-disks operad $H_*f\mathcal{D}_n$ \cite{Getzler:BVAlg,Salvatore-Wahl:FrameddoBVa}. We determine completely this $H_*f\mathcal{D}_n$-algebra structure on $H_*(Ω^n X;\mathbb{Q})$. We show that the action of $H_*(SO(n))$ on the iterated loop space $H_*Ω^n X$ is related to the J-homomorphism and that the BV-operator vanishes on spherical classes only in characteristic other than 2.
It is well-known that a homomorphism p between topological groups K, G is a covering homomorphism if and only if p is an open epimorphism with discrete kernel. In this paper we generalize this fact, in precisely, we show that for a connected locally path connected topological group G, a continuous map p is a generalized covering if and only if K is a topological group and p is an open epimorphism with prodiscrete (i.e, product of discrete groups) kernel. To do this we first show that if G is a topological group and H is any generalized covering subgroup of fundamental group of G, then H is as …
We prove that every group can be realized as the homeomorphism group and as the group of (pointed) homotopy classes of (pointed) self-homotopy equivalences of infinitely many non-homotopy-equivalent Alexandroff spaces.
This article records basic topological, as well as homological properties of the space of homomorphisms Hom(L,G) where L is a finitely generated discrete group, and G is a Lie group, possibly non-compact. If L is a free abelian group of rank equal to n, then Hom(L,G) is the space of ordered n-tuples of commuting elements in G. If G=SU(2), a complete calculation of the cohomology of these spaces is given for n=2, 3. An explicit stable splitting of these spaces is also obtained, as a special case of a more general splitting.
The purpose of this note is to observe that a homomorphism of discrete groups $f:Γ\to G$ arises as the induced map $π_0(\mathfrak{M})\to π_0(\mathfrak{X})$ on path components of some closed normal inclusion of topological groups $\mathfrak{M}\subseteq \mathfrak{X},$ if and only if the map $f$ can be equipped with a crossed module structure. In that case an essentially unique realization $\mathfrak{M}\subseteq \mathfrak{X}$ exists by homotopically discrete topological groups.
For a given pair of maps f,g:X->M from an arbitrary topological space to an n-manifold, the Lefschetz homomorphism is a certain graded homomorphism L:H(X)->H(M) of degree (-n). We prove a Lefschetz-type coincidence theorem: if the Lefschetz homomorphism is nontrivial then there is an x in X such that f(x)=g(x).
In this paper, we show that every topological group is a strong small loop transfer space at the identity element. This implies that the quasitopological fundamental group of a connected locally path connected topological group is a topological group. Also, we show that every covering space of a connected locally path connected topological group is a topological group and its covering map is homomorphism.
We present a characterization of cotorsion-free abelian groups in terms of homomorphisms from fundamental groups of Peano continua, which aligns naturally with the generalization of slenderness to non-abelian groups. In the process, we calculate the first homology group of the Griffiths twin cone.
Let $K$ be the K3 manifold. In this note, we discuss two methods to prove that certain generalized Miller--Morita--Mumford classes for smooth bundles with fiber $K$ are non-zero. As a consequence, we fill a gap in a paper of the first author, and prove that the homomorphism $Diff(K)\to π_0 Diff(K)$ does not split. One of the two methods of proof uses a result of Franke on the stable cohomology of arithmetic groups that strengthens work of Borel, and may be of independent interest.
We employ the Goodwillie spectral sequence for the iterated loop space functor in order to provide realizability conditions on certain unstable modules over the Steenrod algebra at an odd prime.
In this paper we propose and partially carry out a program to use $K$-theory to refine the topological realization problem of unstable algebras over the Steenrod algebra. In particular, we establish a suitable form of algebraic models for $K$-theory of spaces, called $ψ^p$-algebras, which give rise to unstable algebras by taking associated graded algebras mod $p$. The aforementioned problem is then split into (i) the \emph{algebraic} problem of realizing unstable algebras as mod $p$ associated graded of $ψ^p$-algebras and (ii) the \emph{topological} problem of realizing $ψ^p$-algebras as $K$-t…
Hexagon relations are algebraic realizations of four-dimensional Pachner moves, and there are hexagon relations admitting nontrivial cohomologies and leading thus to piecewise linear (PL) 4-manifold invariants. We show that some - but not all! - of the known nontrivial cohomologies can be obtained from a single integral bilinear form corresponding to a PL 4-manifold by using a Frobenius homomorphism for a half of `color' variables (or different Frobenius homomorphisms for both halves). This form can be regarded as a sophisticated analogue of the manifold's intersection form.
Given a finitely generated and projective Lie-Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an homeomorphism are also discussed. These results, in particular, apply to the s…
It is shown that most possibly truncated power series rings admit uncountably many filtered lambda-ring structures. The question of how many of these filtered lambda-ring structures are topologically realizable by the K-theory of torsionfree spaces is also considered for truncated polynomial rings.
We show that the classifying space functor $B: Mon \to Top*$ from the category of topological monoids to the category of based spaces is left adjoint to the Moore loop space functor $Ω': Top*\to Mon$ after we have localized $Mon$ with respect to all homomorphisms whose underlying maps are homotopy equivalences and $Top*$ with respect to all based maps which are (not necessarily based) homotopy equivalences. It is well-known that this localization of $Top*$ exists, and we show that the localization of $Mon$ is the category of monoids and homotopy classes of homotopy homomorphisms. To make this …
Let M denote the mapping class group of S, a compact connected oriented surface with one boundary component. The action of M on the nilpotent quotients of the fundamental group of S allows to define the so-called Johnson filtration and the Johnson homomorphisms. J. Levine introduced a new filtration of M, called the Lagrangian filtration. He also introduced a version of the Johnson homomorphisms for this new filtration. The first term of the Lagrangian filtration is the Lagrangian mapping class group, whose definition involves a handlebody bounded by S, and which contains the Torelli group. Th…
We discuss an infinite class of metabelian Von Neumann rho-invariants. Each one is a homomorphism from the monoid of knots to the real line. In general they are not well defined on the concordance group. Nonetheless, we show that they pass to well defined homomorphisms from the subgroup of the concordance group generated by anisotropic knots. Thus, the computation of even one of these invariants can be used to conclude that a knot is of infinite order. We introduce a method to give a computable bound on these invariants. Finally we compute this bound to get a new and explicit infinite set of t…
We define and develop a homotopy invariant notion for the topological complexity of a map $f:X \to Y$, denoted TC($f$), that interacts with TC($X$) and TC($Y$) in the same way cat($f$) interacts with cat($X$) and cat($Y$). Furthermore, TC($f$) and cat($f$) satisfy the same inequalities as TC($X$) and cat($X$). We compare it to other invariants defined in the papers [15,16,17,18,20]. We apply TC($f$) to studying group homomorphisms $f:H\to G$.
A topological groupoid G is K-pointed, if it is equipped with a homomorphism from a topological group K to G. We describe the homotopy groups of such K-pointed topological groupoids and relate these groups to the ordinary homotopy groups in terms of a long exact sequence. As an application, we give an obstruction to presentability of proper regular Lie groupoids.
Let S be a compact connected oriented surface with one boundary component, and let P be the fundamental group of S. The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of S, whose k-th term consists of the self-homeomorphisms of S that act trivially at the level of the k-th nilpotent quotient of P. Morita defined a homomorphism from the k-th term of the Johnson filtration to the third homology group of the k-th nilpotent quotient of P. In this paper, we replace groups by their Malcev Lie algebras and we study the "infinitesimal" version of the k-th Morita homomorp…
