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By generalizing the whisker topology on the $n$th homotopy group of pointed space $(X, x_0)$, denoted by $\pi_n^{wh}(X, x_0)$, we show that $\pi_n^{wh}(X, x_0)$ is a topological group if $n \ge 2$. Also, we present some necessary and sufficient conditions for $\pi_n^{wh}(X,x_0)$ to be discrete, Hausdorff and indiscrete. Then we prove that $L_n(X,x_0)$ the natural epimorphic image of the Hawaiian group $\mathcal{H}_n(X, x_0)$ is equal to the set of all classes of convergent sequences to the identity in $\pi_n^{wh}(X, x_0)$. As a consequence, we show that $L_n(X, x_0) \cong L_n(Y, y_0)$ if $\pi_n^{wh}(X, x_0) \cong \pi_n^{wh}(Y, y_0)$, but the converse does not hold in general, except for some conditions. Also, we show that on some classes of spaces such as semilocally $n$-simply connected spaces and $n$-Hawaiian like spaces, the whisker topology and the topology induced by the compact-open topology of $n$-loop space coincide. Finally, we show that $n$-SLT paths can transfer $\pi_n^{wh}$ and hence $L_n$ isomorphically along its points.

For any pointed space X, the mod p Lannes-Zarati homomorphism φ s H ˜ ⁎ X is considered as a graded associated version of the mod p Hurewicz map h ⁎ : π ⁎ ( Q X ) → H ⁎ Q X in the E 2 -term of Adams spectral sequence. Therefore, the study of the mod p Lannes-Zarati homomorphism plays an important role in the determination of the image of the mod p Hurewicz map. In this paper, we investigate the behavior of φ s = φ s F p = φ s H ˜ ⁎ S 0 , s ≤ 2 , for an odd prime p.

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.

Topological complexity $\TC{B}$ of a space $B$ is introduced by M. Farber to measure how much complex the space is, which is first considered on a configuration space of a motion planning of a robot arm. We also consider a stronger version $\TCM{B}$ of topological complexity with an additional condition: in a robot motion planning, a motion must be stasis if the initial and the terminal states are the same. Our main goal is to show the equalities $\TC{B} = \catBb{\double{B}}+1$ and $\TCM{B} = \catBB{\double{B}}+1$, where $\double{B}=B{\times}B$ is a fibrewise pointed space over $B$ whose projection and section are given by $p_{\double{B}}=\proj_{2} : B{\times}B \to B$ the canonical projection to the second factor and $s_{\double{B}}=\Delta_{B} : B \to B{\times}B the diagonal. In addition, our method in studying fibrewise L-S category is able to treat a fibrewise space with singular fibres. Recently, we found a problem with the proof of Theorem 1.13 which states that for a fibrewise well-pointed space $X$ over $B$, we have $\catBB{X}$ = \catBb{X}$ and that for a locally finite simplicial complex $B$, we have $\TC{B} = \TCM{B}$. While we still conjecture that Theorem 1.13 is true, this problem means that, at present, no proof is given to exist. Alternatively, we show the difference between two invariants $\catBb{X}$ and $\catBB{X}$ is at most 1 and the conjecture is true for some cases. We give further corrections mainly in the proof of Theorem 1.12., Comment: 12pages original + 5pages errata

Homotopy categorical groups of any pointed space are defined via the fundamental groupoid of iterated loop spaces. This notion allows, paralleling the group case, to introduce the notion of K-categorical groups $\mathbb{K}_iR$ of any ring R. We also show the existence of a fundamental categorical crossed module associated to any fibre homotopy sequence and then, $\mathbb{K}_1R$ and $\mathbb{K}_2R$ are characterized, respectively, as the homotopy cokernel and kernel of the fundamental categorical crossed module associated to the fibre homotopy sequence $FR\xrightarrow{{d_{R} }}BGLR\xrightarrow{{q_{R} }}BGLR^{ + } $ As consequence, the 3th level of the Postnikov tower of the K-theory spectrum of R is classified by this categorical crossed module.

For a pointed space X, the subgroups of self-homotopy equivalences Aut]N (X), AutΩ(X), Aut∗(X) and AutΣ(X) are considered, where Aut]N (X) is the group of all self-homotopy classes f of X such that f] = id : πi(X) → πi(X) for all i ≤ N ≤ ∞, AutΩ(X) is the group of all the above f such that Ωf = id; Aut∗(X) is the group of all selfhomotopy classes g of X such that g∗ = id : Hi(X) → Hi(X) for all i ≤ ∞, AutΣ(X) is the group of all the above g such that Σg = id. We will prove that AutΩ(X1×· · ·×Xn) has two factorizations similar to those of Aut]N (X1×· · ·×Xn) in reference [10], and that AutΣ(X1∨· · ·∨Xn), Aut∗(X1∨· · ·∨Xn) also have factorizations being dual to the former two cases respectively.

This paper gives a partial description of the homotopy type of K, the space of long knots in 3-dimensional Euclidean space. The primary result is the construction of a homotopy equivalence between K and the free little 2-cubes object over the space of prime knots. In proving the freeness result, a close correspondence is discovered between the Jaco-Shalen-Johannson decomposition of knot complements and the little cubes action on K. Beyond studying long knots in 3-space, we show that for any compact manifold M the space of embeddings of R^n x M in R^n x M with support in I^n x M admits an action of the operad of little (n+1)-cubes. If M=D^k this embedding space is the space of framed long n-knots in R^{n+k}, and the action of the little cubes operad is an enrichment of the monoid structure given by the connected-sum operation., Comment: 35 pages, 17 figues. Revision history: V2 corrected mis-statement of a theorem of Hatcher's. V3 corrected statement and proof of the satellite decomposition of knots. V4 notational changes, extra figures, added some detail to proof of the freeness result. V5 formatting changes, references updated. v6 fixed the statement on when the space EC(j,D^n) is a j+1-fold loop space to take into account Haefliger's work

We introduce and study various properties of the secondary cohomology of a space. Certain Steenrod squares are shown to be related to the action of the symmetric groups on secondary cohomology. To Jan–Erik Roos on his sixty–Þfth birthday For a Þeld k we choose the Eilenberg–MacLane space Z n = K(k;n) by the realization of the simplicial k–vector space generated by the non–basepoint singular simplices of the n–sphere S n = S 1 ^:::^S 1 . The permutation of the smash product factors S 1 yields an action of the symmetric group on on S n and hence on Z n . Moreover the quotient map S n C S m ! S n+m induces a cup product map n : Z n C Z m ! Z m+n with n;m > 1; see the Appendix below. It is well known that the (reduced) cohomology e H n (X;k) of a path-connected pointed space X is the same as the set [X;Z n ] of homotopy classes fxg of pointed maps x : X ! Z n . Moreover the cup product of the cohomology algebra H E = H E (X;k) = e H E a k is induced by the map n, that is fxg [ fyg = fn(x;y)g. The

For a 1-connected space X Adams's bar construction B(C*(X)) describes H*(ΩX) only as a graded module and gives no information about the multiplicative structure. Thus it is not possible to iterate the bar construction in order to determine the cohomology of iterated loop spaces Ω iX. In this paper for an n-connected pointed space X a sequence of A(∞)-algebra structures , k = 1, 2, . . . , n, is constructed, such that for each k ≤ n there exists an isomorphism of graded algebras

We show that, given a fibrewise pointed space ( X , p , s ), the smash product functor ( − ) Λ B X : Top B ∗ → Top B ∗ has a right adjoint if and only if the product functor ( − ) × B X : Top B → Top B has a right adjoint.

The purpose of this paper is to discuss the application of the theory of System Health Management to a simple sun pointed space vehicle attitude control system architecture and subsequently to analyze how well it defines the need for Fault Management. This paper also compares the results of this method with the results of an application of commonly used Classical Fault Management techniques. Following a short introduction, an overview of the tools utilized by both methods of determining needed Fault Management and a review of the foundational principles of both methods is given, before the example and findings are discussed.

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.

The (pointed) coarse shape category Sh * ( Sh ⋆ * ), having (pointed) topological spaces as objects and having the (pointed) shape category as a subcategory, was recently constructed. Its isomorphisms classify (pointed) topological spaces strictly coarser than the (pointed) shape type classification. In this paper we introduce a new algebraic coarse shape invariant which is an invariant of shape and homotopy, as well. For every pointed space ( X , ⋆ ) and for every k ∈ N 0 , the coarse shape group π ˇ k * ( X , ⋆ ) , having the standard shape group π ˇ k ( X , ⋆ ) for its subgroup, is defined. Furthermore, a functor π ˇ k * : Sh ⋆ * → Grp is constructed. The coarse shape and shape groups already differ on the class of polyhedra. An explicit formula for computing coarse shape groups of polyhedra is given. The coarse shape groups give us more information than the shape groups. Generally, π ˇ k ( X , ⋆ ) = 0 does not imply π ˇ k * ( X , ⋆ ) = 0 (e.g. for solenoids), but from pro- π k ( X , ⋆ ) = 0 follows π ˇ k * ( X , ⋆ ) = 0 . Moreover, for pointed metric compacta ( X , ⋆ ) , the n -shape connectedness is characterized by π ˇ k * ( X , ⋆ ) = 0 , for every k ⩽ n .

Given a pair of an abelian partial monoid M and a pointed space X, let C M (R 1 ,X) denote the configuration space of finite distinct points in R 1 parametrized by the partial monoid X^ M. In this note we will show that if M is embedded in a topological abelian group then the natural map C M (R 1 ,X)! C ±M (R 1 ,X), induced by the inclusion M ± M ={a b| a, b 2 M}, is a group completion. This generalizes the result of Caruso (1) that the space of "positive and negative particles" in R 1 parametrized by X is weakly equivalent to 1 1 X.

Let X be a topological space. Often we associate with X an object that depends on X as well as on a point x of X. The point x is called a base point and the pair (X, x) is called a pointed space. If (X, x) and (Y, y) are two pointed spaces, then a continuous map f : X → Y such that f(x) = y is called a map between pointed spaces. Let f : X → Y be a homeomorphism and x be a point of X. Then f is a homeomorphism between pointed spaces (X, x) and (Y, f(x)). The composite of two maps between pointed spaces is again a map between pointed spaces and the identity map I(X, x) : (X, x) → (X, x) is always a homeomorphism of pointed spaces for each x ∈ X.

In Chapter 7 we presented cohomology theory, and in Chapter 9 we introduced K-theory. Both theories have some properties in common. In this chapter we unify these properties and define the generalized cohomology theories. Prom this point of view we shall be able to obtain several results that follow from the formal properties rather than from the specific characteristics of the theory in question. Further, we shall prove a theorem that shows that our approach to both theories is quite general. Namely, we prove the Brown representability theorem, which shows that in an adequate category of spaces every generalized cohomology theory is represented by some classifying spaces, such as the Eilenberg-Mac Lane spaces in the case of cohomology and the spaces BU × ℤ and BU in the case of K-theory. Thus cohomology can always be expressed in homotopical terms. Finally we see that the representability of the cohomology theories implies the existence of certain objects, called spectra, which topologically, or better, homotopically, encode all the information concerning their associated cohomology and homology theories.

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.