© 1996, Topology Atlas

In the early 1940's it was shown that the arc, the square disk, the unit cube, and finally the n-ball, were incapable of supporting an exactly 2-to-1 map, by Harrold, Roberts, Civin and Chernavskii respectively. This was followed by Mioduszewski's in-depth study of 2-to-1 maps in Dissertationes Math. 1961. Since then, many other spaces and types of spaces have been shown either to support or not to support exactly k-to-1 maps (for various k) and have been shown to be the image (or not to be the image) of a k-to-1 map. But this subject is far from being well understood. A detailed survey of k-to-1 maps appears in Marcel Dekker's "Continua with the Houston Problem Book." Some of the most important questions being worked on currently are listed below. (A continuum is a compact, connected metric space.)

Question 1. (Sam Nadler and L.E. Ward, Jr.) I s it true that no tree-like continuum can be the 2-to-1 image of a continuum?

Regarding Question 1, we know that the following types of continua, if tree-like, cannot be the 2-to-1 image of any continuum: dendrites, hereditarily indecomposable continua, and indecomposable arc-continua. Furthermore, if a continuum (tree-like or not) has any of the following properties, then it cannot be the 2-to-1 image of a continuum:

(1) every subcontinuum has a cut point,

(2) every subcontinuum has a finite separating set and the continuum is hereditarily unicoherent, or

(3) every subcontinuum has an endpoint.

Question 2. For integers k > 2, which continua are k-to-1 images? All continua?

Question 3. (S. Miklos) Exactly which continua are k-to-1 images of dendrites?

For Question 3, it has been shown that each k-to-1 image of a dendrite is one- dimensional, it must contain a simple closed curve, and it cannot contain uncountably many disjoint arcs. And of course it must be a Peano continuum.

Question 4. Is there an indecomposable arc-like continuum that admits a 2-to-1 map? (No use trying the classic Knaster Buckethandle space; J. Mioduszewski proved in 1961 that it does not admit a 2-to-1 map.)

Question 5. (Mioduszewski) Does the pseudo-arc admit a 2-to-1 map?

Question 6. Is there an integer k > 2 and a continuum that admits no k-to-1 map?

Question 7. Given an integer k > 1 and two graphs, G and H, when does a k-to-1 map exist from G onto H?

For Question 7, there should be some way to look at the adjacency matrices of two given graphs and decide if a k-to-1 map exists from one onto the other. Oddly enough, a complete answer is known only for the class of finitely discontinuous k-to-1 functions between graphs.

Question 8. Is there a 2-to-1 finitely discontinuous function defined on the Knaster Buckethandle continuum ?

Some further questions and related information may be found in Jo Heath's recent publications concerning k-to-1 maps.