PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 130, Number 6, Pages 1867–1873 S 0002-9939(01)06266-9 Article electronically published on November 15, 2001 COVERING MAPS THAT ARE NOT COMPOSITIONS OF COVERING MAPS OF LESSER ORDER JERZY KRZEMPEK (Communicated by Alan Dow) Abstract. In 1995 J.W. Heath asked which exactly n-to-one maps are com- positions of exactly k-to-one maps with 1 <k<n. This paper deals with compositions of covering maps. Exactly n-to-one covering maps on locally arcwise connected continua that are not factorable into covering maps of or- der ≤ n - 1 are constructed for all n’s, and characterized in algebraic terms (fundamental groups). They are not proper compositions of exactly k-to-one maps, open maps, or locally one-to-one maps. A (continuous) map is said to be of order ≤ n when each of its point-inverses has at most n points. Maps of order ≤ 2 are called simple. Let us recall the following three problems from the literature—each of them concerns maps between compact metric spaces. Problem A (K. Borsuk, R. Molski [2]). Is every locally one-to-one map a compo- sition of a finite number of simple locally one-to-one maps ? 1 Problem B (J.D. Baildon [1]). Is every open map of order ≤ n defined on a finite- dimensional space necessarily a finite composition of simple open maps ? Problem C (J.W. Heath [6]). Under what circumstances are exactly n-to-one maps finite compositions of maps of lesser order ? 2 J. Dydak [3] showed that if p is prime, then the map z → z p of the unit complex circle is not factorable into locally one-to-one maps of order ≤ p − 1. These same maps z → z p are not compositions of simple open maps (Baildon [1]). On the other hand, both Problems A and B have affirmative answers in the case of maps from zero-dimensional spaces 3 . Problem C was considered by L.R. Griffus [4]. Lastly, let us mention that exactly four-to-one maps from the interval [0, 1] are not factorable Received by the editors November 6, 2000 and, in revised form, January 9, 2001. 2000 Mathematics Subject Classification. Primary 54C10; Secondary 05C25. Key words and phrases. Composition, covering map, locally one-to-one, open, exactly k-to-one map, group action, fundamental group. 1 A map f is locally one-to-one if each point in its domain has a neighborhood U such that the restriction f |U is one-to-one. Such maps were called elementary in [2, 3]. Borsuk and Molski [2] also asked another question: Does there exist a map of order ≤ n which is not a finite composition of simple maps ? K. Sieklucki [10] proved that every map of order ≤ n defined on a finite-dimensional compact metric space is such a composition, and constructed an infinite-dimensional counter-example. For recent results on this problem see Krzempek [7]. 2 A function is called exactly n-to-one if each of its point-inverses consists of n points. 3 More precisely: Every clopen map of order ≤ n defined on a zero-dimensional metric space is a composition of n - 1 simple clopen maps whose domains are metrizable (cf. Krzempek [7], c 2001 American Mathematical Society 1867 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use