Nilpotent CW-complex Research Articles

Spaces of functions and sections with paracompact domain

We study spaces of continuous functions and sections with domain a paracompact Hausdorff k-space $X$ and range a nilpotent CW complex $Y$ , with emphasis on localization at a set of primes. For $\mathop {\rm map}\nolimits _\phi (X,\,Y)$ , the space of maps with prescribed restriction $\phi$ on a suitable subspace $A\subset X$ , we construct a natural spectral sequence of groups that converges to $\pi _*(\mathop {\rm map}\nolimits _\phi (X,\,Y))$ and allows for detection of localization on the level of $E^2$ . Our applications extend and unify the previously known results.

Homotopy transfer and rational models for mapping spaces

By using homotopy transfer techniques in the context of rational homotopy theory, we show that if $C$ is a coalgebra model of a space $X$, then the $A_\infty$-coalgebra structure in $H_*(X;\mathbb{Q})\cong H_*(C)$ induced by the higher Massey coproducts provides the construction of the Quillen minimal model of $X$. We also describe an explicit $L_\infty$-structure on the complex of linear maps ${\rm Hom}(H_*(X; \mathbb{Q}), \pi_*(\Omega Y)\otimes\mathbb{Q})$, where $X$ is a finite nilpotent CW-complex and $Y$ is a nilpotent CW-complex of finite type, modeling the rational homotopy type of the mapping space ${\rm map}(X, Y)$. As an application we give conditions on the source and target in order to detect rational $H$-space structures on the components.

$$L_{\infty }$$ rational homotopy of mapping spaces

In this paper we describe explicit \(L_\infty \) algebras modeling the rational homotopy type of any component of the spaces \(\text{ map}(X,Y)\) and \(\text{ map}^*(X,Y)\) of free and pointed maps between the finite nilpotent CW-complex \(X\) and the finite type nilpotent CW-complex \(Y.\) When \(X\) is of finite type, non necessarily finite, we also show that the algebraic covers of these \(L_\infty \) algebras model the universal covers of the corresponding mapping spaces.

Inductive LS cocategory and localisation

In this paper we prove that the inductive cocategory of a nilpotent CW-complex of finite type X, indcocat X, is bounded above by an expression involving the inductive cocategory of the p-localisations of X. Also, we show that the inductive cocategory is generic for 1-connected H 0-spaces of finite type. Our arguments are the dualisation of classical results due to Ganea that allow us to improve previous results by Cornea and Stanley on LS category.

L∞ models of based mapping spaces

In this paper, for any pointed map f: X ? Y between finite type nilpotent CW-complexes, we obtain L ∞ and Lie models of map psi f (X,Y), the pointed space of based maps homotopic to f, in terms of Lie algebras constructed from the Quillen models of X and Y. The main advantage of our approach is to allow X to be an infinite dimensional CW-complex, in which case map * f (X,Y) has no longer the homotopy type of a finite type CW-complex. © 2011 The Mathematical Society of Japan.

Hurewicz–Serre theorem in extension theory

The paper is devoted to generalizations of the Cencelj&#8211;Dranishnikov theorems relating extension properties of nilpotent CW complexes to their homology groups. Here are the main results of the paper:<br /> Theorem 0.1. <i> Let $L$ be a nilpotent CW

Rank of the fundamental group of any component of a function space

We compute the rank of the fundamental group of any connected component of the space map(X, Y) for X and Y connected, nilpotent CW complexes of finite type with X finite. For the component corresponding to a general homotopy class f : X → Y, we give a formula directly computable from the Sullivan model for f. For the component of the constant map, our formula retrieves a known expression for the rank in terms of classical invariants of X and Y. When both X and Y are rationally elliptic spaces with positive Euler characteristic, we use our formula to determine the rank of the fundamental group of any component of map(X, Y) explicitly in terms of the homomorphism induced by f on rational cohomology.

Phantom Maps from a Strong Homology Viewpoint

Let X and Y be connected nilpotent CW–complexes of finite type. In this paper we deal with another complete analysis of a set of phantom maps in terms of strong homology by using the derived functors. We also prove that the set of homotopy classes of maps is independent of the choice of both rational homotopy equivalent classes and completion genera.

Extension of maps into nilpotent spaces III

Let M be a nilpotent CW-complex. We give necessary and sufficient cohomological dimension theory conditions for a finite-dimensional metric compactum X so that every map A → M , where A is a closed subset of X, can be extended to a map X → M . This is a generalization of a result by Dranishnikov [Mat. Sb. (1991)] where such conditions were found for simply-connected CW-complexes M, and Cencelj and Dranishnikov [Canad. Bull. Math. (2001) and Topology Appl. (2002)] where a condition of finitely generatedness was imposed on the nilpotent CW-complex M.

André–Quillen cohomology and rational homotopy of function spaces

We develop a simple theory of André–Quillen cohomology for commutative differential graded algebras over a field of characteristic zero. We then relate it to the homotopy groups of function spaces and spaces of homotopy self-equivalences of rational nilpotent CW-complexes. This puts certain results of Sullivan in a more conceptual framework.

Extension of maps to nilpotent spaces. II

Let M be a nilpotent CW-complex with finitely generated fundamental group. We give necessary and sufficient cohomological dimension theory conditions for a finite-dimensional metric compactum X so that every map A→ M, where A is a closed subset of X can be extended to a map X→ M. This is a generalization of a result by Dranishnikov [Mat. Sb. 182 (1991)] where such conditions were found for simply-connected CW-complexes M, and Cencelj and Dranishnikov forthcoming paper [Cannad. Bull. Math.] where such conditions were found for nilpotent CW-complexes M with finitely generated homotopy groups.

Extension of Maps to Nilpotent Spaces

AbstractWe show that every compactum has cohomological dimension 1 with respect to a finitely generated nilpotent group G whenever it has cohomological dimension 1 with respect to the abelianization of G. This is applied to the extension theory to obtain a cohomological dimension theory condition for a finite-dimensional compactum X for extendability of every map from a closed subset of X into a nilpotent CW-complex M with finitely generated homotopy groups over all of X.

Nilpotent subgroups of self equivalences of torsion spaces

LetX be a finitep-torsion based connected nilpotent CW-complex. We give a criterion of a subgroup of e(X), the group of self equivalences ofX, to be a nilpotent group, in terms of its action onE *(X), whereE is a CW-spectrum, satisfying some technical conditions.

A continuous localization and completion

The main goal of this paper is to construct a localization and completion of Bousfield-Kan type as a continuous functor for a virtually nilpotent CW-complex. Then the localization and completion of an A n {A_n} -space is given to be an A n {A_n} -homomorphism between A n {A_n} -spaces. For any general compact Lie group, this gives a continuous equivariant localization and completion for a virtually nilpotent G G -CW-complex. More generally, we have a continuous localization with respect to a system of core rings for a virtually nilpotent D \mathbf {D} -CW-complex for a polyhedral category D \mathbf {D} .

L′-Isolated Maps and Localizations

Let P be the set of primes, l ⊆ P a subset and l′ = P – l Recall that an H0-space is a space the rational cohomology of which is a free algebra.Cassidy and Hilton defined and investigated l′-isolated homomorphisms between locally nilpotent groups. Zabrodsky [8] showed that if X and Y are simply connected H0-spaces either with a finite number of homotopy groups or with a finite number of homology groups, then every rational equivalence f : X → Y can be decomposed into an l-equivalence and an l′-equivalence.In this paper we define and investigate l′-isolated maps between pointed spaces, which are of the homotopy type of path-connected nilpotent CW-complexes. Our definition of an l′-isolated map is analogous to the definition of an l′-isolated homomorphism. As every homomorphism can be decomposed into an l-isomorphism and an l′-isolated homomorphism, every map can be decomposed into an l-equivalence and an l′-isolated map.

Two results relating nilpotent spaces and cofibrations

We first prove a Blakers-Massey Theorem for nilpotent spaces: If (X, A) is an n-connected, n ⩾ 1 n \geqslant 1 , pair of nilpotent spaces, then under suitable conditions the map π ∗ ( X , A ) → π ∗ X / A {\pi _ \ast }(X,A) \to {\pi _ \ast }X/A is an isomorphism in dimension n + 1 n + 1 and an epimorphism in dimension n + 2 n + 2 . Next, we dualize the well-known fact that if the total space of a fibration is nilpotent, so is the fiber. Our dual theorem can be used to construct new examples of finite nilpotent CW complexes.

Genus and cancellation

IN HILTOK and ROITBERG [4]. an H-space E,, was constructed with the properties that E,,. x S3 ‘c Sp(2) x S3 and E,,, x E,w 5 Sp(2) x Sp(2), but yet E,w + Sp(2). That is, cancellation does not hold for the semi-group operation of Cartesian product on finite H-spaces, even up to homotopy type. It was clear from the construction however, that E,, and Sp(2) had homotopy equivalent p-localizations for all primes p. This prompted the definition by Mislin [7] that two nilpotent CW complexes X and Y are in the same genus (G(X) = G(Y)) if and only if their p-localizations X,, and Y, are homotopy equivalent for all primes p.