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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.

It is proved that a dendroid is smooth if and only if it can be represented as Ihe inverse limit of an inverse sequence of finite dendrites with bonding mappings which are monotone relatiye to points forming a thread. As a consequence another proof of the existence of a universal smooth dendroid [4) is obtained. § 1. Preliminaria. All spaces considered in this paper are assumed to be metric and all mappings are continuous. A dendroid means a hereditarily unicoherent and arcwise connected continuum. If, moreover, it is locally connected, it is called a dendrite. By a ramification point of a dendroid X we understand a point which is the centre of a simple triod C9ntained in X. A dendroid having at most one ramification point t is called afan, and t is called its top. A fan with at most n endpoints is called an n -fan. A dendroid X is said to be smooth at a point p E X provided that for each sequence of points all E X which is convergent to a point a E X the sequence of arcs pa" converges to the arc pa. A mapping f: X -. Y of a continuum X onto Y is said to be monotone relative to a point p E X if for each continuum Q in Y such thatf(p) E Q the setf-l(Q) is connected (see [6], p. 720). The author is very grateful to Professor Henryk Torunczyk for his important sugestions, which have contributed to the preparation of the present version of the paper. § 2. The main result and corollaries. The following result is a particular case (for dendrites) of Corollary 4 of [1]. THEOREM A. Let an im:erse sequence {Xi,P} be gillen of dendrites Xi containing poiflts pi such that 1 P(pi+!) = pi and 20 the bonding mappings p: X H1 -. Xi are monotone relatil'e to points pH!. Then the inverse limit X = ~im{Xi,p} is a dendroid which is smooth at the thread p = {pi}. The aim of this paper is to prove the inverse theorem, so that the characterization can be obtained of smooth dendroids as inverse limits of finite dendrites with bonding mappings which are monotone relative to some points forming a thread of the inverse sequence. Namely we shall prove the following ~ Fundamenta Mathematicae CXXIV/2 164 W. J. Charatonik MAIN THEOREM. Let a dendroid X be given which is smooth at a point p E X. Then for each i E {I, 2, ... } there exist finite dendrites Xi, mappings fi: Xi+l -+ Xi and points pi E Xi such that 10 P(pH 1) = pi and 20 the mappings P are monotone relative to pHi and the inverse limit !:im{Xi,ji} is homeomorphic to X in such a wey that the thread {pi} corresponds to the point p. Before proving the theorem we pose some problems and give corollaries. PROBLEM. What continua X can be obtained as inverse limits of locally COllnected continua Xi with bonding mappings fi satisfying conditions 10 and 20 of Theorem A? In a proposition below we give an answer to this question for the class of COlltinua which are hereditarily unicoherent at a point. Recall that a continuum X is said to be hereditarily unicoherent at a point p e X if the intersection of any two subcontinua of X each of which contains p is connected~ PROPOSITION 1. Let an inverse sequence {Xi,fi} be given of locally connected continua Xi containing points pi such that conditions 1 and 2 of Theorem A hold. If the inverse limit X = !:im{Xi,ji} is hereditarily unicoherent at the thread p = {/}, then each Xi is a dendrite and hence X is a dendroid which is smooth at p. In fact, by Corollary 1 of [2] each natural projection from X onto' XI is monotone relative to p. Hence, by Theorem 2.5 of [6], p. 721, each Xi is hereditarily unicoherent at pi. So Xi is a dendrite by Theorem 2.2 of [3], p. 63. Now we are interested in the universal smooth dendroid. Its existence has been proved in [4]. We show that the standard methods of McCord [7] applied to the class of pointed finite dendrites with mappings monotone relative to distinguished points, together with the Main Theorem, give another proof of the existence of a universal smooth dendroid. For this purpose we need some auxiliary concepts. A pair (X, x) where x e X is called a pointed space. Let :/C be a class of mappings of pointed spaces which is closed with respect to taking compositions. The class & of pointed polyhedra is called :/C -amalgamable if for each finite sequence (P1,Pi)' (P2,P2), ... , (Pn,Pn) of members of & and mappings fi: (PitPi) (Q, q) where (Q, q) e & and each fi e :/C there exist a member (P, p) of & with embeddings gi: (Pi,Pj)-(P,p) and a mappingfe:/C of (P,p) onto (Q,q) such that fi = fgj for each i e {I, 2, ... , n}. Let & be a class of pointed polyhedra. We say that a pointed continuum (X, x) is (&, :/C)-like if there is an inverse sequence of members of & with bonding mappings belonging to :/C such that (X, x) is the inverse limit of that sequence. Using exactly the same arguments as McCord uses in his proof of Theorem 1 of [7], Part 3, p. 72-77 and considering the concepts introduced above, we get PROPOSITION 2. If a class & of pointed polyhedra is :/C -amalgamable, then there exists a universal (&, f)-like continuum. Smooth dendroids as inverse limits 0/ dendrites 165 Denote by !!} the class of pointed finite dendrites and by.,/{ the class of mappings which are monotone relative to distinguished points. Now we are able to reformulate the Main Theorem and Theorem A as follows: COROLLARY 1. A pair (X,p) is a pointed dendroid which is smooth at p if and only if (X,p) is (f!),~It)-like. PROPOSITION 3. The class f!) is .,/{-amalgamable. Proof. Let (PbPl)' ... , (P",P,J be pointed finite dendrites and let f;: (P;,p;) -+ (Q, q) 'be monotone relative to PI with (Q-, q) e!!J. Let P be the one-point union of P; with points Pl>P2' ... ,p" identified to a point peP. Then (P,p) is a pointed finite dendrite. We consider g; as a natural embedding of (Pi,p;) into (P,p) and define f: (P,p) -+ (Q, q) by flP; =f;, or more exactly f(x) =fi(g,l(X») for x e g;(P;). One can observe (simply by definitions) thatf is monotone relative to p. So all conditions of the definition are satisfied. Corollary I and Propositions 2 and 3 lead to COROLLARY :2 ([4], Theorem 3.1, p. 992). There exists a universal smooth dendroid. § 3. Proof of the Main Theorem. The following result of Mackowiak will be used in the sequel. THEOREM B ([6], Corollary 2.10, p. 722). Let a continuous mapping f map a dendroid X onto a dendroid Y, and let p eX. Then f is monotone relative to p if and only if f!px is monotone for each x e X. For each natural number i let pi be the cone over the set Ai = {O, I}' and let P be the cone over the Cantor set C = {O, I}OO; i.e., pi = Aix[O, l]jAix{O} and P = Cx [0, l]/Cx {O} are the 2i-fan and the Cantor fan respectively. Denote by t i the top of the fan pi and by t the top of the fan P. The projections C -+ Ai and Ai + 1 -+ Ai induce maps pi: P -+ pi and J: pi+1 -+ pi. We shall employ the following result of Grispolakis and Tymchatyn. THEOREM C ([5], Theorem 2.3, p. 132). Each smooth dendroid X can be embedded into a smooth dendroid Dx such that there exists a mapping g: P -+ Dx satisfying the conditions: 1° if g(X1 ,Yl) = g(X2,Y2), then Yl = Y2 for each (X1oYl), (x2,Y2)eP; 2° if g(X1 ,J'1) = g(X2,J'l) and O

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?

In any theoretical investigation of a real physical system, one is always forced to make simplifying assumptions concerning the true nature of the system. Since such idealizations are inevitable, it is reasonable to inquire as to how far one can go in this direction and still obtain satisfactory results. In certain cases, for example the motion of the planets, the equations of celestial mechanics provide a quite accurate model of the real physical system. In other situations, such as ecological or chemical interactions, or the study of large scale atmospheric phenomena, one either writes down certain reasonable relations between the quantities involved and their rates of change, or one tremendously reduces the number of actual equations involved. If such “leaps of faith” are to be of any use, it is necessary to study “rough” equations in “rough” terms. This in a nutshell, is our aim in this chapter and the next one. In other words, we want to fit these vague notions into a precise mathematical framework.

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.