CW Complex Of Finite Type Research Articles

ON TOPOLOGICAL COMPLEXITY OF GORENSTEIN SPACES

In this paper, using Sullivan's approach to rational homotopy theory of simply-connected finite type CW complexes, we endow the $\mathbb{Q}$-vector space $\mathcal{E}xt_{C^{\ast}(X;\mathbb{Q})}(\mathbb{Q}, C^{\ast}(X;\mathbb{Q}))$ with a graded commutative algebra structure. This leads us to introduce the $\mathcal{E}xt$-version of higher (resp. module, homology) topological complexity of $X_0$, the rationalization of $X$ (resp. of $X$ over $\mathbb{Q}$). We then compare these invariants and their respective ordinary ones for Gorenstein spaces.
 We also highlight, in this context, the benefit of Adams-Hilton models over a field of odd characteristics especially through two cases, the first one when the space is a $2$-cell CW-complex and the second one when it is a suspension.

Rational homotopy via Sullivan models and enriched Lie algebras

Rational homotopy theory originated in the late 1960s and the early 1970s with the simultaneous but distinct approaches of Quillen (1969), Sullivan (1977) and Bousfield–Kan (1972). Each approach associated to a path connected space X an “algebraic object” A which is then used to construct a rational completion of X , X \to X_{\mathbb Q} . These constructions are homotopy equivalent for simply connected CW complexes of finite type, in which case H_*(X_{\mathbb Q})\cong H_*(X)\otimes \mathbb Q and \pi_*(X_{\mathbb Q}) \cong \pi_*(X)\otimes \mathbb Q . Otherwise, they may be different; in fact, Quillen’s construction is only available for simply connected spaces. In this review, discussion is limited to Sullivan’s completions, and the notation X\to X_{\mathbb Q} is reserved for these. We briefly review the construction, and follow that with a review of developments and examples over the subsequent decades, but often without the proofs. Since the explicit form of Sullivan’s completion has lent itself to a wide variety of applications in a range of fields, this survey will necessarily be modest in scope.

The group of self-homotopy equivalences of a rational space cannot be a free abelian group

In this paper, we prove that a free abelian group cannot occur as the group of self-homotopy equivalences of a rational CW-complex of finite type. Thus, we generalize a result due to Sullivan–Wilkerson showing that if $X$ is a rational CW-complex of finite type such that $\mathrm{dim}\,H^{*}(X, \mathbb{Z}) < \infty$ or $\mathrm{dim}\,\pi_{*}(X) < \infty$, then the group of self-homotopy equivalences of $X$ is isomorphic to a linear algebraic group defined over $\mathbb{Q}$.

Relative Gottlieb groups of mapping spaces and their rational cohomology

Let f:X →Y be a map of simply connected CW-complexes of finite type. Put maxπ★(Y)⊗Q = max{ i | πi(Y)⊗Q≠0 }. In this paper we compute the relative Gottlieb groups of f when X is an F0-space and Y is a product of odd spheres. Also, under reasonable hypothesis, we determine these groups when X is a product of odd spheres and Y is an F0-space. As a consequence, we show that the rationalized G-sequence associated to f splits into a short exact sequence. Finally, we prove that the rational cohomology of map(X,Y;f) is infinite dimensional whenever maxπ★(Y)⊗Q is even.

The depth and LS category of a topological space

The depth of an augmented ring $\varepsilon \colon A\to k $ is the least $p$, or ∞, such that \begin {equation*} \Ext _A^p(k , A)\neq 0. \end {equation*} When $X$ is a simply connected finite type CW complex, $H_*(\Omega X;\mathbb {Q})$ is a Hopf algebra and the universal enveloping algebra of the Lie algebra $L_X$ of primitive elements. It is known that $\depth H_*(\Omega X;\mathbb {Q}) \leq \cat X$, the Lusternik-Schnirelmann category of $X$.&#x0D; For any connected CW complex we construct a completion $\widehat {H}(\Omega X)$ of $H_*(\Omega X;\mathbb {Q})$ as a complete Hopf algebra with primitive sub Lie algebra $L_X$, and define $\depth X$ to be the least $p$ or ∞ such that \[ \Ext ^p_{UL_X}(\mathbb {Q}, \widehat {H}(\Omega X))\neq 0. \] Theorem: for any connected CW complex, $\depth X\leq \cat X$.

Hochschild cohomology of Sullivan algebras and mapping spaces

Let f:X→Y be a map between simply connected spaces having the homotopy of finite type CW-complexes, where H∗(Y,Q) is finite dimensional and ϕ:(∧V,d)→(B,d) a Sullivan model of f. We consider (B,d) as a module over ∧V via the mapping ϕ. Let map(X,Y;f) denote the component of f in the space of mappings from X to Y. In this paper we show that there is a canonical injection π∗(Ωmap(X,Y;f))⊗Q→HH∗(∧V;B).

Spaces Realized and Non-Realized as Dold-Lashof Classifying Spaces

Let $X$ be a simply connected CW-complex of finite type. Denote by ${\rm Baut}_{1}(X)$ the Dold-Lashof classifying space of fibrations with fiber $X$. This paper is a survey about the problem of realizing Dold-Lashof classifying spaces. We will also present some new results: we show that not all rank-two rational $H$-spaces can be realized as ${\rm Baut}_{1}(X)$ for simply connected, rational elliptic space $X$. Moreover, we construct an infinite family of rational spaces $X,$ such that ${\rm Baut}_{1}(X)$ is rationally a finite $H$-space of rank-two (up to rational homotopy type).

On the Theriault conjecture for self homotopy equivalences

Our main purpose in this paper is to resolve, in a rational homotopy theory context, the following open question asked by S. Theriaul: given a topological space X , what one may say about the nilpotency of aut _1(X) when the cocategory of its classifying space Baut _1(X) is finite? Here aut _1(X) denotes the path component of the identity map in the set of self homotopy equivalences of X . More precisely, we prove that \mathrm {HNil}_\mathbb Q(\mathrm {aut}_1(X))\leqslant\mathrm {cocat}_\mathbb Q(\mathrm {Baut}_1(X)), when X is a simply connected CW-complex of finite type and that the equality holds when Baut _1(X) is coformal. Many intersections with other popular open questions will be discussed.

A new lower bound for LS-category

Let X be a simply connected CW-complex of finite type and {mathbb {K}} an arbitrary field. In this paper, we use the Eilenberg–Moore spectral sequence of C_*(Omega (X), mathbb K) to introduce a new homotopical invariant textsc {r}(X, {mathbb {K}}). If X is a Gorenstein space with nonzero evaluation map, then textsc {r}(X, {mathbb {K}}) turns out to interpolate mathrm {depth}(H_*(Omega (X), {mathbb {K}})) and mathrm {e}_{{mathbb {K}}}(X). We also define for any minimal Sullivan algebra (Lambda V,d) a new spectral sequence and make use of it to associate to any 1-connected commutative differential graded algebra (A, d) a similar invariant textsc {r}(A,d). When (Lambda V,d) is a minimal Sullivan model of X, this invariant fulfills the relation textsc {r}(X, {mathbb {K}}) = textsc {r}(Lambda V,d).

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.

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.

Realization of 2-solvable nilpotent groups as groups of classes of homotopy self-equivalences

Let X be a 1-connected CW-complex of finite type and ε ♯ ( X ) be the group of homotopy classes of self-equivalences of X which induce the identity on homotopy groups. In this paper, we prove that every finitely generated 2-solvable rational nilpotent group is realizable as ε ♯ ( X ) where X is the rationalization of a 1-connected CW-complex of finite type.

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.

Genus sets and SNT sets of certain connective covering spaces

We study the genus and SNT sets of connective covering spaces of familiar finite CW-complexes, both of rationally elliptic type (e.g. quaternionic projective spaces) and of rationally hyperbolic type (e.g. one-point union of a pair of spheres). In connection with the latter situation, we are led to an independently interesting question in group theory: if f is a homomorphism from Gl(�,A) to Gl(n,A), � < n, A = Z, resp. Zp, does the image of f have infinite, resp. uncountably infinite, index in Gl(n,A)? 1. Introduction and statement of results. In this paper, we study the genus sets and SNT sets of certain m-connective covering spaces Xh mi , following the work initiated by McGibbon and Moller ((16)), and continued by McGibbon and Roitberg ((17)). Before stating our main results, we recall the basic notions; in the following definitions, X and Y are assumed to be spaces of the homotopy type of nilpotent, finite type CW-complexes.

The evaluation subgroup of a fibre inclusion

Let ξ : X → j E → p B be a fibration of simply connected CW complexes of finite type with classifying map h : B → B aut 1 ( X ) . We study the evaluation subgroup G n ( E , X ; j ) of the fibre inclusion as an invariant of the fibre-homotopy type of ξ. For spherical fibrations, we show the evaluation subgroup may be expressed as an extension of the Gottlieb group of the fibre sphere provided the classifying map h induces the trivial map on homotopy groups. We extend this result after rationalization: We show that the decomposition G ∗ ( E , X ; j ) ⊗ Q = ( G ∗ ( X ) ⊗ Q ) ⊕ ( π ∗ ( B ) ⊗ Q ) is equivalent to the condition ( h ♯ ) Q = 0 .

H–space structure on pointed mapping spaces

We investigate the existence of an H-space structure on the function space, F_*(X,Y,*), of based maps in the component of the trivial map between two pointed connected CW-complexes X and Y. For that, we introduce the notion of H(n)-space and prove that we have an H-space structure on F_*(X,Y,*) if Y is an H(n)-space and X is of Lusternik-Schnirelmann category less than or equal to n. When we consider the rational homotopy type of nilpotent finite type CW-complexes, the existence of an H(n)-space structure can be easily detected on the minimal model and coincides with the differential length considered by Y. Kotani. When X is finite, using the Haefliger model for function spaces, we can prove that the rational cohomology of F_*(X,Y,*) is free commutative if the rational cup length of X is strictly less than the differential length of Y, generalizing a recent result of Y. Kotani.

Free and semi-inert cell attachments

Let Y Y be the space obtained by attaching a finite-type wedge of cells to a simply-connected, finite-type CW-complex. We introduce the free and semi-inert conditions on the attaching map which broadly generalize the previously-studied inert condition. Under these conditions we determine H ∗ ( Ω Y ; R ) H_*(\Omega Y;R) as an R R -module and as an R R -algebra, respectively. Under a further condition we show that H ∗ ( Ω Y ; R ) H_*(\Omega Y;R) is generated by Hurewicz images. As an example we study an infinite family of spaces constructed using only semi-inert cell attachments.

Homotopy Lie algebras, lower central series and the Koszul property

Let X and Y be finite-type CW-complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k-rescaling of the rational cohomology ring of X. Assume H^*(X,Q) is a Koszul algebra. Then, the homotopy Lie algebra pi_*(Omega Y) tensor Q equals, up to k-rescaling, the graded rational Lie algebra associated to the lower central series of pi_1(X). If Y is a formal space, this equality is actually equivalent to the Koszulness of H^*(X,Q). If X is formal (and only then), the equality lifts to a filtered isomorphism between the Malcev completion of pi_1(X) and the completion of [Omega S^{2k+1}, Omega Y]. Among spaces that admit naturally defined homological rescalings are complements of complex hyperplane arrangements, and complements of classical links. The Rescaling Formula holds for supersolvable arrangements, as well as for links with connected linking graph.

The product formula for Lusternik–Schnirelmann category

If C=C_\phi denotes the mapping cone of an essential phantom map \phi from the suspension of the Eilenberg-Mac Lane complex K=K(Z,5) to the 4-sphere S=S^4 we derive the following properties: (1) The LS category of the product of C with any n-sphere S^n is equal to 3; (2) The LS category of the product of C with itself is equal to 3, hence is strictly less than twice the LS category of C. These properties came to light in the course of an unsuccessful attempt to find, for each positive integer m, an example of a pair of 1-connected CW-complexes of finite type in the same Mislin (localization) genus with LS categories m and 2m. If \phi is such that its p-localizations are inessential for all primes p, then by the main result of [J. Roitberg, The Lusternik-Schnirelmann category of certain infinite CW-complexes, Topology 39 (2000), 95-101], the pair C_*, C where C_*= S wedge \Sigma ^2 K, provides such an example in the case m=1.