Jo Heath, Auburn University, April 2004
A map is k-to-1 if every point inverse has exactly k many elements, and a map is at-most-k-to-1 if every point inverse has no more than k many elements. An at-most-2-to-1 map is also called “simple”. All spaces are assumed metric.
Some of the questions in the 1995 Survey [3] have been answered and this is reported in part 3 of this note. If the survey were being rewritten, the new material and questions from part 2 of this note would be incorporated. The two big questions in part 1, one for domains of 2-to-1 maps and one for images of 2-to-1 maps, were the major questions in the survey as well, and neither has been answered.
1. Two venerable questions.
(a) Mioduszewski, 1961, [10]: Is there a 2-to-1 map defined on the pseudoarc?
(b) Nadler and Ward, 1983, [11]: Is there a tree-like continuum that is the image of a 2-to-1 map from a continuum?
2. New results and questions.
(a) In 1979 J. Krasinkiewicz (see problem 5 in [1] or problem 130 in [7]) asked if there exists a finite-to-1 map from a hereditarily indecomposable continuum onto a hereditarily decomposable continuum. Piotr Minc [9] recently answered this question affirmatively by showing that the pseudoarc admits an at-most-3-to-1 onto a dendroid. In fact he has shown:
Piotr Minc (2004, [9]) Every chainable continuum maps at-most-3-to-1 onto some dendroid. Minc also asks:
Question. (Piotr Minc, [9]) Is it true that a chainable continuum is hereditarily decomposable if and only if it admits a simple map onto a dendroid?
(b) Related Question. (Van Nall, Jo Heath, 2003, [4]) Is there an indecomposable continuum that admits a simple map onto a dendroid?
Piotr Minc has come the closest to answering this question. In [8] he constructed an at-most-3-to-1 map from the Knaster bucket-handle space onto a dendroid.
3. Update for the 1995 survey paper on k-to-1 maps, [3].
Section 2.1, Question 2: Is it true that every continuum that is not tree-like is the 2-to-1 image of a continuum?
Answer: No, this is not true. In [5] it is shown that some of J. Rogers’ pseudo-solenoids are not 2-to-1 images.
Section 2.1. To the list of tree-like continua known not to be 2-to-1 images can now be added dendroids. In [4], Van Nall and Jo Heath show that no dendroid can be the 2-to-1 image of a continuum. Also, they ask:
Question. (Van Nall, Jo Heath, 2003, [4]) Is there a continuum that maps 2-to-1 onto a lambda-dendroid?
Section 2.2, Question 3. For k > 2, which continua are k-to-1 images?
Thomas Gonzales , [2], has shown that no hereditarily indecomposable tree-like continuum can be a k-to-1 image, for any k > 1.
Section 2.3, Question 7. Is there an indecomposable arc-like continuum that admits a 2-to-1 map?
Answer: Yes. Jerzy Krzempek, (2004, [6]), constructed an indecomposable chainable continuum that admits a 2-to-1 map. Also, he asked:
Question: Jerzy Krzempek (2004, [6]) Must the 2-to-1 image of a chainable indecomposable continuum be decomposable?
Section 2.4. Question 9. Is there a continuum X and an integer k > 2 such that there is no k-to-1 map defined on X?
Answer: Yes, and more than that. Thomas Gonzales , [2], has identified an continuum that cannot be the domain of a k-to-1 map for any k > 1
References
[1] Howard Cook, Andrew Lelek, W. Thomas Ingram, Continua, with the Houston Problem Book, Lecture Notes in Pure and Appl. Math. 170, Dekker, New York, (1995) 365 – 398.
[2]. Thomas Gonzalez, Exactly k-to-1 maps and hereditarily indecomposable tree-like continua, Proc. Amer. Math. Soc. 131 (2003), 3925-3927.
[3]. Jo Heath, Exactly k-to-1 maps: From pathological functions with finitely many discontinuites to well-behaved covering maps, Continua, with the Houston Problem Book, 89 – 102, Lecture Notes in Pure and Appl. Math. 170, Dekker, New York, 1995.
[4]. Jo Heath and Van Nall, No arc-connected treelike continuum is the 2-to-1 image of a continuum, Fundamenta Mathematicae, 180 (2003), 11 – 24.
[5]. Jo Heath, A non-treelike continuum that is not the 2-to-1 image of any continuum, Proc. Amer. Math. Soc. 124 (1996), 3571 – 3578.
[6] Jerzy Krzempek, An exactly two-to-one map from an indecomposable chainable continuum, (Gliwice, Poland), submitted for publication.
[7] Wayne Lewis, Continuum Theory Problems, Topology Proc. 8 (1983), 361 – 394.
[8] Piotr Minc, Bottlenecks in dendroids, Topology Appl. 129 (2003) 187-209.
[9] Piotr Minc, Mapping chainable continua onto dendroids, Topology Appl. 138 (2004) 287 – 298.
[10] Jerzy Mioduszewski, On two-to-one continuous functions, Dissertationes Math. (Rozprawy Mat.) 24 (1961), 43.
[11] Sam B. Nadler, Jr. and Lew E. Ward, Jr., Concerning exactly (n,1) images of continua, Proc. Amer. Math. Soc. 87 (1983), 351-354.