An action of A on X is a map F: AxX to X such that F|_X = id: X to X. The restriction F|_A: A to X of an action is called a cyclic map. Special cases of these notions include group actions and the Gottlieb groups of a space, each of which has been studied extensively. We prove some general results about actions and their Eckmann-Hilton duals. For instance, we classify the actions on an H-space that are compatible with the H-structure. As a corollary, we prove that if any two actions F and F' of A on X have cyclic maps f and f' with Omega(f) = Omega(f'), then Omega(F) and Omega(F') give the same action of Omega(A) on Omega(X). We introduce a new notion of the category of a map g and prove that g is cocyclic if and only if the category is less than or equal to 1. From this we conclude that if g is cocyclic, then the Berstein-Ganea category of g is <= 1. We also briefly discuss the relationship between a map being cyclic and its cocategory being <= 1.
Collection of (equivariant) $\rm{PL}$-mappings admitting a relative abelian, cyclic, quaternionic, bicyclic, and quaternionic-cyclic structures are constructed.
The notion of a cyclic map g: A -> X is a natural generalization of a Gottlieb element in pi_n(X). We investigate cyclic maps from a rational homotopy theory point of view. We show a number of results for rationalized cyclic maps which generalize well-known results on the rationalized Gottlieb groups.
We introduce the notion of homotopy inner products for any cyclic quadratic Koszul operad $\mathcal O$, generalizing the construction already known for the associative operad. This is done by defining a colored operad $\widehat{\mathcal O}$, which describes modules over $\mathcal O$ with invariant inner products. We show that $\widehat{\mathcal O}$ satisfies Koszulness and identify algebras over a resolution of $\widehat{\mathcal O}$ in terms of derivations and module maps. As an application we construct a homotopy inner product over the commutative operad on the cochains of any Poincaré duali…
We introduce the notion of homotopy inner products for any cyclic quadratic Koszul operad $\mathcal O$, generalizing the construction already known for the associative operad. This is done by defining a colored operad $\hat{\mathcal O}$, which describes modules over $\mathcal O$ with invariant inner products. We show that $\hat{\mathcal O}$ satisfies Koszulness and identify algebras over a resolution of $\hat{\mathcal O}$ in terms of derivations and module maps. An application to Poincaré duality on the chain level of a suitable topological space is given.
We show that the tensor product of two cyclic $A_\infty$-algebras is, in general, not a cyclic $A_\infty$-algebra, but an $A_\infty$-algebra with homotopy inner product. More precisely, we construct an explicit combinatorial diagonal on the pairahedra, which are contractible polytopes controlling the combinatorial structure of an $A_\infty$-algebra with homotopy inner products, and use it to define a categorically closed tensor product. A cyclic $A_\infty$-algebra can be thought of as an $A_\infty$-algebra with homotopy inner products whose higher inner products are trivial. However, the highe…
In the present paper the cyclic homology functor from the category of $A_\infty$-algebras over any commutative unital ring $K$ to the category of graded $K$-modules is constructed. Further, it is showed that this functor sends homotopy equivalences of $A_\infty$-algebras into isomorphisms of graded modules. As a corollary, it is obtained that the cyclic homology of an $A_\infty$-algebra over any field is isomorphic to the cyclic homology of the $A_\infty$-algebra of homologies for the source $A_\infty$-algebra.
In this paper we apply homotopical localization to the framework of differential graded algebras over an operad. We get plus construction by performing nullification with respect to an universal acyclic algebra. This plus construction for Lie and Leibniz algebras up to homotopy yields two new homology theories for associative algebras. They are closely related to cyclic and Hochschild homologies. We also compute their first homology groups.
We generalize theorems of McGibbon-Roitberg, Iriye, and Meier on the relations between phantom maps and rational homotopy, and apply them to provide new calculational examples of the homotopy sets Ph(X, Y ) of phantom maps and the subsets SPh(X, Y ) of special phantom maps.
We construct a discrete model of the homotopy theory of $S^1$-spaces. We define a category $\sP$ with objects composed of a simplicial set and a cyclic set along with suitable compatibility data. $\sP$ inherits a model structure from the model structures on the categories of simplicial sets and cyclic sets. We then show that there is a Quillen equivalence between $\sP$ and the model category of $S^1$-spaces in which weak equivalences and fibrations are maps inducing weak equivalences and fibrations on passage to all fixed point sets.
We generalize some results of Gray and McGibbon-Roitberg on relations between phantom maps and rational homotopy to relative phantom maps. Since the $\lim^1$ and the profinite completion techniques do not apply to relative phantom maps, we develop new techniques.
We compute the homotopy groups of the spaces of self maps of Lie groups of rank 2, SU(3), Sp(2), and G_2. We use the cell structures of these Lie groups and the standard methods of homotopy theory.
We show that Connes' B-operator on a cyclic differential graded k-module M is a model for the canonical circle action on the geometric realization of M. This implies that the negative cyclic homology and the periodic cyclic homology of a differential graded category can be identified with the homotopy fixed points and the Tate fixed points of the circle action on its Hochschild complex.
In this article we study thick ideals defined by periodic self maps in the stable motivic homotopy category over $\mathbb{C}$. In addition, we extend some results of Ruth Joachimi about the relation between thick ideals defined by motivic Morava K-theories and the preimages of the thick ideals in the stable homotopy category under Betti realization.
Let $\mathcal{E}(X)$ be the group of homotopy classes of self homotopy equivalences for a connected CW complex $X$. We observe two classes of maps $\mathcal{E}$-maps and co-$\mathcal{E}$-maps. They are defined as the maps $X\to Y$ that induce the homomorphisms $\mathcal{E}(X)\to \mathcal{E}( Y)$ and $\mathcal{E}(Y)\to \mathcal{E}(X)$, respectively. We give some rationalized examples related to spheres, Lie groups and homogeneous spaces by using Sullivan models. Furthermore, we introduce an $\mathcal{E}$-equivalence relation between rationalized spaces $X_{\mathbb{Q}}$ and $Y_{\mathbb{Q}}$ as a…
For a pointed topological space $X$, we use an inductive construction of a simplicial resolution of $X$ by wedges of spheres to construct a "higher homotopy structure" for $X$ (in terms of chain complexes of spaces). This structure is then used to define a collection of higher homotopy invariants which suffice to recover $X$ up to weak equivalence. It can also be used to distinguish between different maps $f$ from $X$ to $Y$ which induce the same morphism on homotopy groups $f_*$ from $π_* X$ to $π_* Y$.
In general the processes of taking a homotopy inverse limit of a diagram of spectra and smashing spectra with a fixed space do not commute. In this paper we investigate under what additional assumptions these two processes do commute. In fact we deal with an equivariant generalization which involves spectra and smash products over the orbit category of a discrete group. Such a situation naturally occurs if one studies the equivariant homology theory associated to topological cyclic homology. The main theorem of this paper will play a role in the generalization of the results obtained by Boekst…
We introduce a strong homotopy notion of a cyclic symmetric inner product of an A-infinity algebra and prove a characterization theorem in the formalism of the infinity inner products by Tradler. We also show that it is equivalent to the notion of a non-constant symplectic structure on the corresponding formal non-commutative supermanifold. We show that (open Gromov-Witten type) potential for a cyclic filtered A-infinity algebra is invariant under the cyclic filtered A-infinity homomorphism up to reparametrization, cyclization and a constant addition, generalizing the work of Kajiura.
We establish certain conditions which imply that a map $f:X\to Y$ of topological spaces is null homotopic when the induced integral cohomology homomorphism is trivial; one of them is: $H^*(X)$ and $π_*(Y)$ have no torsion and $H^*(Y)$ is polynomial.
Normal maps between discrete groups $N\rightarrow G$ were characterized [FS] as those which induce a compatible topological group structure on the homotopy quotient $EN\times_N G$. Here we deal with topological group (or loop) maps $N\rightarrow G$ being normal in the same sense as above and hence forming a homotopical analogue to the inclusion of a topological normal subgroup in a reasonable way. We characterize these maps by a compatible simplicial loop space structure on $Bar_\bullet(N,G)$, invariant under homotopy monoidal functors, e.g. Localizations and Completions. In the course of char…
We raise the problem of realisability of rings as $\{X,X\}$ the ring of stable homotopy classes of self-maps of a space $X$. By focusing on $A_n^2$-polyhedra, we show that the direct sum of three endomorphism rings of abelian groups, one of which must be free, is realisable as $\{X,X\}$ modulo the acyclic maps. We also show that $\mathbb{F}_p^3$ is not realisable in the setting of finite type $A_n^2$-polyhedra, for $p$ any prime.
We study degree one maps between closed orientable 4-manifolds with cyclic $π_1$, and obtain some results on the existence and finiteness, as well as some relation of $1$-domination and Euler Characteristics.
We study the homotopy types of certain spaces closely related to the spaces of algebraic (rational) maps from the $m$ dimensional real projective space into the $n$ dimensional complex projective space for $2\leq m\leq 2n$ (we conjecture this relation to be a homotopy equivalence). In an earlier article we proved that the homotopy types of the terms of the natural degree filtration approximate closer and closer the homotopy type of the space of continuous maps and obtained bounds that describe the closeness of the approximation in terms of the degrees of the maps. Here we improve the estimates…
We construct an abelian category A(G) of sheaves over a category of closed subgroups of the r-torus G and show it is of finite injective dimension. It can be used as a model for rational $G$-spectra in the sense that there is a homology theory \piA_*: G-spectra/Q --> A(G) on rational G-spectra with values in A(G), and the associated Adams spectral sequence converges for all rational $G$-spectra and collapses at a finite stage.
Let E be an H-space acting on a based space X. Then we refer to ev: E -> X, the map obtained by acting on the base point of X, as a ``generalized evaluation map." We establish several fundamental results about the rational homotopy behaviour of a generalized evaluation map, all of which apply to the usual evaluation map Map(X,X;1)->X. With mild hypotheses on X, we show that a generalized evaluation map ev factors, up to rational homotopy, through a map Gamma_ev: S_ev -> X where S_ev is a (relatively small) finite product of odd-dimensional spheres and the map induced by Gamma_ev on rational ho…
Primary cohomology operations, i.e., elements of the Steenrod algebra, are given by homotopy classes of maps between Eilenberg--MacLane spectra. Such maps (before taking homotopy classes) form the topological version of the Steenrod algebra. Composition of such maps is strictly linear in one variable and linear up to coherent homotopy in the other variable. To describe this structure, we introduce a hierarchy of higher distributivity laws, and prove that the topological Steenrod algebra satisfies all of them. We show that the higher distributivity laws are homotopy invariant in a suitable sens…
In this paper we classify the homotopy classes of proper maps $E\rightarrow \mathbb R^k$, where $E$ is a vector bundle over a compact Hausdorff space. As a corollary we compute the homotopy classes of proper maps $\mathbb R^n\rightarrow \mathbb R^k$. We find a stability range of such maps. We conclude with some remarks on framed submanifolds of non-compact manifolds, the relationship with proper homotopy classes of maps and the Pontryagin-Thom construction.
We compute the $G$-equivariant homotopy type of the real topological cyclic homology of spherical group rings with anti-involution induced by taking inverses in the group, where $G$ denotes the group $Gal(\mathbb{C}/\mathbb{R})$. The real topological Hochschild homology of a spherical group ring $\mathbb{S}[Γ]$, with anti-involution as described, is an $O(2)$-cyclotomic spectrum and we construct a map commuting with the cyclotomic structures from the $O(2)$-equivariant suspension spectrum of the dihedral bar construction on $Γ$ to the real topological Hochschild homology of $\mathbb{S}[Γ]$, wh…
We prove that homotopy invariants of finite degree distinguish homotopy classes of maps of a connected compact CW-complex to a nilpotent connected CW-complex with finitely generated homotopy groups.
In this paper, we study the equivariant homotopy type of a connected sum of linear actions on complex projective planes defined by Hambleton and Tanase. These actions are constructed for cyclic groups of odd order. We construct cellular filtrations on the connected sum using spheres inside unitary representations. A judicious choice of filtration implies a splitting on equivariant homology for general cyclic groups under a divisibility hypothesis, and in all cases for those of prime power order.
The framed little 2-discs operad is homotopy equivalent to a cyclic operad. We show that the derived modular envelope of this cyclic operad (i.e., the modular operad freely generated in a homotopy invariant sense) is homotopy equivalent to the modular operad made from classifying spaces of diffeomorphism groups of 3-dimensional handlebodies with marked discs on their boundaries. A modification of the argument provides a new and elementary proof of K. Costello's theorem that the derived modular envelope of the associative operad is homotopy equivalent to the ``open string'' modular operad made …
We show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of induced maps of their chain algebras of based loop spaces. In the case of a universal acyclic map we obtain, for a wide class of spaces, an explicit algebraic description for these induced maps in terms of derived localization.
The incidence correspondence in the grassmannian which determines the tautological bundle defines a map between cycle spaces on grassmannians. These cycle spaces decompose canonically into a product of Eilenberg-MacLane spaces. These decompositions and the associated maps are calculated up to homotopy.
We give an alternative to Postnikov's homotopy classification of maps from 3-dimensional CW-complexes to homogeneous spaces G/H of Lie groups. It describes homotopy classes in terms of lifts to the group G and is suitable for extending the notion of homotopy to Sobolev maps. This is required for applications to variational problems of mathematical physics.
This is the second of a series of papers which are devoted to a comprehensive theory of maps between orbifolds. In this paper, we develop a basic machinery for studying homotopy classes of such maps. It contains two parts: (1) the construction of a set of algebraic invariants -- the homotopy groups, and (2) an analog of CW-complex theory. As a corollary of this machinery, the classical Whitehead theorem which asserts that a weak homotopy equivalence is a homotopy equivalence is extended to the orbifold category.
First we describe a class of homotopy Frobenius algebras via cyclic operads which we call cyclic $A_\infty$ algebras. We then define a suitable new combinatorial operad which acts on the Hochschild cochains of such an algebra in a manner which encodes the homotopy BV structure. Moreover we show that this operad is equivalent to the cellular chains of a certain topological (quasi)-operad of CW complexes whose constituent spaces form a homotopy associative version of the Cacti operad of Voronov. These cellular chains thus constitute a chain model for the framed little disks operad, proving a cyc…
We define a topological Hochschild (THH) and cyclic (TC) homology theory for differential graded (dg) categories and construct several non-trivial natural transformations from algebraic K-theory to THH(-). In an intermediate step, we prove that the homotopy theory of dg categories is Quillen equivalent, through a four step zig-zag of Quillen equivalences, to the homotopy theory of Eilenberg-Mac Lane spectral categories. Finally, we show that over the rationals two dg categories are topological equivalent if and only if they are quasi-equivalent.
Consider a push-out diagram of spaces C <-- A --> B, construct the homotopy push-out, and then the homotopy pull-back of the diagram one gets by forgetting the initial object A. We compare the difference between A and this homotopy pull-back. This difference is measured in terms of the homotopy fibers of the original maps. Restricting our attention to the connectivity of these maps, we recover the classical Blakers-Massey Theorem.
The purpose of this paper is to introduce $N_k(\ell)$-maps ($1\le k,\ell\le\infty$), which describe higher homotopy normalities, and to study their basic properties and examples. An $N_k(\ell)$-map is defined with higher homotopical conditions. It is shown that a homomorphism is an $N_k(\ell)$-map if and only if there exists fiberwise maps between fiberwise projective spaces with some properties. Also, the homotopy quotient of an $N_k(k)$-map is shown to be an $H$-space if its LS category is not greater than $k$. As an application, we investigate when the inclusions $\operatorname{SU}(m)\to\op…
In this paper we compute $RO(G)$-graded homotopy Mackey functors of $H\underline{\mathbb{Z}}$, the Eilenberg-Mac Lane spectrum of the constant Mackey functor of integers for cyclic p-groups and give a complete computation for $G = C_{p^2}$ . We also discuss homological algebra of $\underline{\mathbb{Z}}$-modules for cyclic $p$-groups, and interactions between these two. The goal of this paper is to understand various slice spectral sequences as $RO(G)$-graded spectral sequences of Mackey functors.
We compute the $\mathbb{A}^1$-localization of several invariants of schemes namely, topological Hochschild homology ($\mathrm{THH}$), topological cyclic homology ($\mathrm{TC}$) and topological periodic cyclic homology ($\mathrm{TP}$). This procedure is quite brutal and kills the completed versions of most of these invariants. The main ingredient for the vanishing statements is the vanishing of $\mathbb{A}^1$-localization of de Rham cohomology (and, eventually, crystalline cohomology) in positive characteristics.
Let M be a monoidal category endowed with a distinguished class of weak equivalences and with appropriately compatible classifying bundles for monoids and comonoids. We define and study homotopy-invariant notions of normality for maps of monoids and of conormality for maps of comonoids in M. These notions generalize both principal bundles and crossed modules and are preserved by nice enough monoidal functors, such as the normaliized chain complex functor. We provide several explicit classes of examples of homotopy-normal and of homotopy-conormal maps, when M is the category of simplicial sets …
We address the homotopy theory of 2-crossed modules of commutative algebras. In particular, we define the concept of a 2-fold homotopy between a pair of 1-fold homotopies connecting 2-crossed module maps $\A \to \A'$. We also prove that if the domain 2-crossed module $\A$ is free up to order one (i.e. if the bottom algebra is a polynomial algebra) then we have a 2-groupoid of 2-crossed module maps $\A \to \A'$ and their homotopies and 2-fold homotopies.
First we argue that many BV and homotopy BV structures, including both familiar and new examples, arise from a common underlying construction. The input of this construction is a cyclic operad along with a cyclically invariant Maurer-Cartan element in an associated Lie algebra. Using this result we introduce and study the operad of cyclically invariant operations, with instances arising in cyclic cohomology and $S^1$ equivariant homology. We compute the homology of the cyclically invariant operations; the result being the homology operad of $\mathcal{M}_{0,n+1}$, the uncompactified moduli spac…
The classifying space of a crossed complex generalises the construction of Eilenberg-Mac Lane spaces. We show how the theory of fibrations of crossed complexes allows the analysis of homotopy classes of maps from a free crossed complex to such a classifying space. This gives results on the homotopy classification of maps from a CW-complex to the classifying space of a crossed module and also, more generally, of a crossed complex whose homotopy groups vanish in dimensions between 1 and n. The results are analogous to those for the obstruction to an abstract kernel in group extension theory.
An algorithmic computation of the set of unpointed stable homotopy classes of equivariant fibrewise maps was described in a recent paper of the author and his collaborators. In the present paper, we describe a simplification of this computation that uses an abelian heap structure on this set that was observed in another paper of the author. A heap is essentially a group without a choice of its neutral element; in addition, we allow it to be empty.
We address the (pointed) homotopy theory of 2-crossed modules (of groups), which are known to faithfully represent Gray 3-groupoids, with a single object, and also connected homotopy 3-types. The homotopy relation between 2-crossed module maps will be defined in a similar way to Crans' 1-transfors between strict Gray functors, however being pointed, thus this corresponds to Baues' homotopy relation between quadratic module maps. Despite the fact that this homotopy relation between 2-crossed module morphisms is not, in general, an equivalence relation, we prove that if $A$ and $A'$ are 2-crosse…
We address the homotopy theory of 2-crossed modules of commutative algebras, which are equivalent to simplicial commutative algebras with Moore complex of length two. In particular, we construct for maps of 2-crossed modules a homotopy relation, and prove that it yields an equivalence relation in very unrestricted cases (freeness up to order one of the domain 2-crossed module). This latter condition strictly includes the case when the domain is cofibrant. Furthermore, we prove that this notion of homotopy yields a groupoid with objects being the 2-crossed module maps between two fixed 2-crosse…
Two locally generic maps f,g : M^n --> R^{2n-1} are regularly homotopic if they lie in the same path-component of the space of locally generic maps. Our main result is that if n is not 3 and M^n is a closed n-manifold then the regular homotopy class of every locally generic map f : M^n --> R^{2n-1} is completely determined by the number of its singular points provided that f is singular (i.e., f is not an immersion).
