By P.J. Hilton

Because the advent of homotopy teams through Hurewicz in 1935, homotopy idea has occupied a in demand position within the improvement of algebraic topology. This monograph presents an account of the topic which bridges the distance among the basic techniques of topology and the extra advanced therapy to be present in unique papers. the 1st six chapters describe the basic rules of homotopy idea: homotopy teams, the classical theorems, the precise homotopy series, fibre-spaces, the Hopf invariant, and the Freudenthal suspension. the ultimate chapters talk about J. H. C. Whitehead's cell-complexes and their program to homotopy teams of complexes.

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**Example text**

138. Let Ko be a C00 function in Rn with n : \y\ < c}, Ko (0) ± 0, (11) for some c > 0, and K(x) = ANK0(x) for some N € N, where A is the Laplacian in R". The local means are given by K(t9g)(x) = f K(y)g(x + ty)dy9 0 < t < 1, g G ^ ( R " ) , (12) with its obvious counterpart Ko(l,g). If 2iV > max(s,<7p) then Lp\\q j (13) as equivalent quasi-norms. The proof in [Triy] shows that c > 0 in (11) may assumed to be small. We insert (9) in (12) and obtain, by the support properties of f(x — 2/c), -2/c)] {x) = YJ"kK{tJ){x-2k\ K ) \LP =(J2\ak\p)kp\\K(tJ)\Lp\\ U^2akf('-2k)\ \ k J (14) k (15) k and, by (13), finally (10) with Aspq = Bspq.

3/lx, and bo may be considered as an (s, p)K>L-atom located at go,o- Again we assume that (21) is an optimal atomic decomposition, U I fpq\ € N,/c e Z"}. (27) We put (28) and apply Step 6 to / 2 : (29) in analogy to (26). As for / / we may use Step 5 and obtain f( | Bspq\\ < (30) c uniformly for all admissible atoms bo, the inclusion Since ||ft0 I Bs B F*w (27) and (30) yield pq (31) The left-hand inequality in (4) now follows from (28), (29) and (31). The proof of part (i) of the theorem is complete.

Let — m — 1 < s < —m for some m G No. Then L~ = m in (7). Hence we have to find a function f e Aspr where 0 < q < oo, with [ (32) for which there is no c > 0 such that (6) holds for all j e N and all ak. Let n = 1 and let % be the characteristic function of the interval [0,1]. 23 Special properties 41 Then x € Asp+m, since Lp a Ap+m, and / = -£^x € 4 ^ . We thus have (32) interpreted as in (8). Let x^ = 2~jk; then —x —z M J\X) w*^/ /c=0 essentially coincides with (2) if a* = 1. Assume that (6) holds for some c > 0; then (34) for all j e N.