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In this paper, we compute ${\rm Ext}_{A}^{s}(\widetilde{H}^*(B\mathbb{Z}/p),\mathbb{F}_p)$ for $s\leq 1$. Using this result, we investigate the behavior of $\varphi_3^{\mathbb{F}_p}$ and $\varphi_s^{\widetilde{H}^*(B\mathbb{Z}/p)}\ (s\leq1)$ for an odd prime $p$., Comment: 36 pages

The normalized singular chains of a path connected pointed space $X$ may be considered as a connected $E_{\infty}$-coalgebra $\mathbf{C}_*(X)$ with the property that the $0^{\text{th}}$ homology of its cobar construction, which is naturally a cocommutative bialgebra, has an antipode, i.e. it is a cocommutative Hopf algebra. We prove that a continuous map of path connected pointed spaces $f: X\to Y$ is a weak homotopy equivalence if and only if $\mathbf{C}_*(f): \mathbf{C}_*(X)\to \mathbf{C}_*(Y)$ is an $\mathbf{\Omega}$-quasi-isomorphism, i.e. a quasi-isomorphism of dg algebras after applying the cobar functor $\mathbf{\Omega}$ to the underlying dg coassociative coalgebras. The proof is based on combining a classical theorem of Whitehead together with the observation that the fundamental group functor and the data of a local system over a space may be described functorially from the algebraic structure of the singular chains., Comment: This version contains minor changes based on a referee report mostly concerning presentation and typos. We have also added new acknowledgements

Using the strong shift equivalence of pointed space maps associated to filtration pairs for isolated invariant sets of a discrete dynamical system we define a strong Conley index independent of the choice of filtration pairs and prove that it is invariant under continuation.

Our aim in this section is to compute (or describe), for any integer n0, the free strongly \({\mathbb{A}}^{1}\)-invariant sheaf generated by the n-th smash power of \({\mathbb{G}}_{m}\), in other words the free strongly \({\mathbb{A}}^{1}\)-invariant sheaf on nunits.

Nous comparons et calculons trois sous-espaces vectoriels remaquables de la cohomologie d'un CW-complexe.We compare three natural subvectorspaces of the cohomology of a simply connected pointed space. These spaces are related to, respectively, the “last cells” of the space, the “dual Gottlieb groups” and the “evaluation map” defined by the Spivak construction.

Let /' be a fibrewise co-Hopf space over a topological space B. A /^-suspension space rBX is a generalization of a fibrewise suspension space ZBX for any fibrewise pointed space X over B. Making use of /'^-suspensio n space, we define /"s-Hopf construction and prove a /"^-suspension formula which generalizes the suspension formula of C. S. Hoo. The dual formula is also proved.

In this paper, for any pointed map f: X → Y between finite type nilpotent CW-complexes, we obtain L∞ and Lie models of mapf*(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 mapf*(X,Y) has no longer the homotopy type of a finite type CW-complex.

In this chapter we shall introduce the concepts of path connectedness and of homotopy of continuous maps between two spaces. We shall study the sets of homotopy classes of maps and relate this with path connectedness. Finally, we shall define the homotopy groups of a topological space, which are important algebraic invariants for such spaces.

In this paper we introduce filtration pairs for isolated invariant sets of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated invariant set. Moreover, the maps defining the shift equivalence can be chosen canonically. Lastly, we define partially ordered Morse decompositions and prove the existence of Morse set filtrations for such decompositions., Comment: 23 pages

In this chapter we work over a pointed base space B. A fibrewise pointed space over B consists of a space X together with maps $$ B\xrightarrow{s}X\xrightarrow{p}B $$ such that p ∘ s = 1 B . In other words, X is a fibrewise space over B with section s. The alternative terminology sectioned fibrewise space is also widely used. Note that the projection is necessarily a quotient map and the section is necessarily an embedding. It is often convenient to regard B as a subspace of X so that the projection retracts X onto B. To simplify the exposition in what follows let us assume, once and for all, that the embedding is closed, as is necessarily the case when X is a Hausdorff space. We regard any subspace of X containing B as a fibrewise pointed space in the obvious way; no other subspaces will be admitted. When s is a cofibration we describe X as cofibrant.

The fiberwise homotopy theory is quite a recent development. Ideally, any exposition of the fiberwise homotopy theory is based on a fiberwise version of topology, but for limited purposes ordinary topology is sufficient. Fiberwise homotopy is an equivalence relation between fiberwise maps. Examples can easily be given of the fibrewise maps that are homotopic as ordinary maps but are not fibrewise homotopic. A fibrewise pointed space reduces to a pointed space or space with basepoint when the base space is just a point. It is not customary to require that subspaces must contain the basepoint but from our point of view this is an essential condition. It is also not customary to require the basepoint to be closed. A fibrewise pointed homotopy into the fibrewise constant is called a fibrewise pointed nulhomotopy. A fibrewise pointed space is said to be fibrewise pointed contractible if it has the same fibrewise pointed homotopy type as the base space, in other words if the identity is fibrewise pointed nulhomotopic. Finally, the chapter briefly explains fiberwise cofibrations and fibrations.

A loop space X is by definition a triple (X, BX, e) in which X is a space, BX is a connected pointed space, and e: X -QBX is a homotopy equivalence from X to the space QBX of based loops in BX. We will say that a loop space X is finite if the integral homology H*(X, Z) is finitely generated as a graded abelian group, i.e., if X appears at least homologically to be a finite complex. In this paper we prove the following theorem.

In this note, we determine the action of the first differential in the Atiyah–Hirzebruch spectral sequence for the nth Morava K-theory of the p-adic construction Dp(ℓ)(X)=Cℓ(p)∧ΣpX∧p of a pointed space X for n⩾1. Our formula gives us a way to compute the first differentials in the Atiyah–Hirzebruch spectral sequence for the Morava K-theory of iterated loop spaces.Analogous formulas are also proved for the Steenrod operations.

A central feature in the study of many partial differential equations is the existence of special solutions. In this paper we consider two such examples. These are the existence of traveling wave solutions of reaction diffusion systems and the existence of radial solutions of semilinear elliptic equations. Both of these problems can be reduced to finding a particular solution of a system of ordinary differential equations. As we shall see, the particular solution corresponds to a trajectory in phase space which “connects” two equilibrium points. That is, the particular solution satisfies a system of ordinary differential equations of the form $$U'\left( z \right) = F\left( U \right),\quad U \in {R^{n}}$$ (1.1a) , together with boundary conditions $$\underline {\mathop{{\lim }}\limits_{{z \to - \infty }} } \;\,U\left( z \right) = A\;and\;\underline {\mathop{{\lim }}\limits_{{z \to + \infty }} } \;\,U\left( z \right) = B $$ (1.1b) where A and B are equilibria of (1.1a). We refer to a solution of (1.1) as an “A → B connection” or just “connection”.

Problems concerning the extension of continuous functions are central to topology. One is given a space X and a subspace A of X. One is also given a space E and a map f: A → E. The question is: does there exist an extension of f over X, i.e. a map g: X → E such that gA = f?

Following the programme outlined in the first chapter we now turn to the category T (2) of pairs associated with the category T of spaces and maps. We begin by discussing the category of spaces under a given space, then turn to the category of spaces over a given space, and finally consider the category of spaces over and under a given space.