Homotopy Type Of CW-complexes Research Articles

On the axiomatic systems of singular cohomology theory

On the category of pairs of topological spaces having a homotopy type of CW complexes the singular (co)homology theory was axiomatically studied by J. Milnor [8]. In particular, Milnor gave additivity axiom for a (co)homology theory and proved that any additive (co)homology theory on the given category is isomophic to the singular (co)homology. On the other hand, the singular homology is a homology with compact support [3]. In the paper [6], L. Mdzinarishvili proposed partially compact support property for a cohomology theory and gave another axiomatic characterization of the singular cohomology theory. In this paper, we will give additional different axiomatic characterizations of the singular cohomology theory. Moreover, we will study connections of the mentioned axiomatic systems (cf. [2]).

CARDINALITY OF RATIONAL HOMOTOPY TYPES OF SIMPLY CONNECTED CW COMPLEXES

The aim of this paper is to consider the following problem: given a simply connected CW complex X of finite type, find the cardinal of the set of rational homotopy types of CW complexes Y such that H*(Y, ℚ) is isomorphic, as a commutative graded algebra, to H*(X, ℚ). For this purpose we introduce a long exact sequence, called the Whitehead exact sequence, which we associated with the minimal Sullivan model of X and which allows us to define the notion of the adapted couple associated with X. Thus we show that the rational homotopy type of X is completely determined by the equivalence class of its adapted couple.

On the homotopy type of CW-complexes with aspherical fundamental group

This paper is concerned with the homotopy type distinction of finite CW-complexes. A ( G , n ) -complex is a finite n-dimensional CW-complex with fundamental-group G and vanishing higher homotopy-groups up to dimension n − 1 . In case G is an n-dimensional group there is a unique (up to homotopy) ( G , n ) -complex on the minimal Euler-characteristic level χ min ( G , n ) . For every n we give examples of n-dimensional groups G for which there exist homotopically distinct ( G , n ) -complexes on the level χ min ( G , n ) + 1 . In the case where n = 2 these examples are algebraic.

ADJUNCTION OF mod P EQUIVALENCES IN HOMOTOPY THEORY

We formalize the adjunction of maps of localized nilpotent spaces of the homotopy type of CW-complexes. Using the semilocalization theory of M. Bendersky, in which only the higher homotopy groups are localized, we obtain an adjunction theorem with fewer nilpotency conditions on the spaces. The two main theorems are in the form of a theorem on maps of triads by J. P. May.

Power-cancellation of CW-complexes with few cells

In this paper we use the fact that the rings of integer matrices have the power-substitution property in order to obtain a power-cancellation property for homotopy types of CW-complexes with one cell in dimensions 0 and 4n and a finite number of cells in dimension 2n.

On the Mapping Theorem for Lusternik-Schnirelmann Category II

Let X and Y be 1-connected spaces having the homotopy type of cw-complexes.Definition 0.1. A continuous map f:X → Y is Ω-split if Ωf:ΩX → ΩY admits a retraction up to homotopy.In [6] we prove the following “mapping theorem”:THEOREM 0.1. (a) If f is Ω-split, then cat(X) ≦ cat(Y);(b) If π*(f) is split injective and ΩY has the homotopy type of a product of Eilenberg-MacLane spaces, then f is Ω-split.

The simplicial bundle of a CW fibration

Suppose we are given a fibration p : E → B p:E \to B over a connected base, with both base and fibre having the homotopy type of CW complexes. We construct a fibre bundle over B B fibre homotopy equivalent to the given fibration and with fibre a simplicial complex. Further, the transformations of the fibre arising from the transition functions of this bundle are simplicial maps. From this, we deduce that the weak spectral sequence constructor axioms are sufficient to determine the Serre spectral sequence of a CW fibration.

Corrections: ``On the homotopy type of CW complexes with the form $S^{2}\cup e^{4}\cup e^{6}$''

Corrections: ``On the homotopy type of CW complexes with the form $S^{2}\cup e^{4}\cup e^{6}$''

On the homotopy type of CW complexes with the form $S^{2}\cup e^{4}\cup e^{6}$

On the homotopy type of CW complexes with the form $S^{2}\cup e^{4}\cup e^{6}$

Rational BP operations and the Chern character

This paper will study certain rational BP operations. We will work in the category of spaces having the homotopy type of CW complexes. Recall that for − 1 ≤ n ≤ ∞ there is defined in this category the homology theory BP(n)*(X). (See (7).) In particular, BP(∞)*(X) = BP*(X) is the usual Brown Peterson homology of X, BP(1)*(X) = l*(X) is the canonical direct summand of connective K-theory with Qp coefficients (Qp are the integers localized at the prime p), BP(0)*(X) = Hℤ*(X) is ordinary homology with Qp coefficients, and BP(−1)*(X) = Hℤ/p*(X) is ordinary homology with ℤp coefficients (ℤp are the integers reduced mod p). For 1 ≤ n ≤ ∞, BP(n)*(X) is a module over BP(n)*(pt) = Qp[v1, …, vn] (deg vs = 2ps − 2). The BP(n) theories are all related via canonical maps p(m, n):BP(m)*(X) → BP(n)*(X) whose performance is described in (7).

A remark on ?homotofp fibrations?

We define a category TopF of “homotopy fibrations with fibre F” (or rather “maps with homotopy fibre F”) and show that this category is closed under certain colimits and “homotopy colimits”. It follows that the geometric realization of a semisimplicial object in TopF is again in TopF. As a corollary we show that for a homotopy everything H-space A*(i.e. a (special) Γ-space in the sense of G. Segal (s.[9],[10])) with homotopy inverse the loop space of the classifying space of A* is homotopy equivalent (not only weakly s. [9],[iO])to A1 even without assuming that all spaces involved have the homotopy type of CW-complexes (compare [8]).

Suspension of the Lusternik-Schnirelmann Category

Let cat be the Lusternik-Schnirelmann category structure as defined by Whitehead [6] and let be the category structure as defined by Ganea [2],We prove thatandIt is known that w ∑ cat X = conil X for connected X. Dually, if X is simply connected,1. We work in the category of based topological spaces with the based homotopy type of CW-complexes and based homotopy classes of maps. We do not distinguish between a map and its homotopy class. Constant maps are denoted by 0 and identity maps by 1.We recall some notions from Peterson's theory of structures [5; 1] which unify the definitions of the numerical homotopy invariants akin to the Lusternik-Schnirelmann category.