Homotopy classes of maps between Knaster continua.pdf

a r X i v : m a t h / 9 9 0 4 1 6 2 v 1 [ m a t h .G N ] 2 8 A p r 1 9 9 9 HOMOTOPY CLASSES OF MAPS BETWEEN KNASTER CONTINUA Piotr Minc Abstract. By a Knaster continuum we understand the inverse limit of copies of [0, 1] with open bonding maps. We prove that for any two Knaster continua K1 and K2, there are 2?0 distinct homotopy types of maps of K1 onto K2 that map the endpoint of K1 to the endpoint of K2. 1. Introduction Let R denote the set of real numbers and let I denote the interval [0, 1]. For any real number t, let [t] denote the greatest integer less than or equal to t. Let v : R → I be defined by the formula v (t) = { t? [t] , if [t] is even [t] + 1? t, if [t] is odd. For each positive integer n, let gn : I → I be defined by the formula gn (t) = v (nt). Observe that gn stretches n times and then folds the resulting interval [0, n] onto [0, 1]. The map g2 is the very well known “roof-top” map. For any two positive integers m and n, gm ? gn = gmn. Consequently, gn and gm commute (see, for example, [8, Proposition 2.2]). Let N = {n1, n2, . . . } be a sequence of integers 1. Consider the inverse sequence (*) I gn1←?? I gn2←?? I gn3←?? I gn4←?? . . . By the Knaster continuum associated with the sequence N we will understand the inverse limit of (*). Observe that the same Knaster continuum can be associated with two different sequences. For example the inverse limit does not change if we replace N by {n1n2 . . . nj1 , nj1+1 . . . nj2 , . . . }. However, it should be noted here that there are 2?0 topologically distinct Knaster continua [2](see also [14]). For a Knaster continuum K, let e denote the endpoint (0, 0, . . . ). By πi we will understand the projection of K onto the i-th component in the inverse system (i = 0, 1, . . . ). 1991 Mathematics Subject Classification. Primary 54F15, 54F50. Key words and phrases. Knaster continua, homotopy. This research was supported in part by NSF grant # DMS-9505108. Typeset by AMS-TEX 1 2 PIOTR MINC Let S1 denote t