Two-to-one maps on solenoids and Knaster continua

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Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland

It is shown that 2-to-1 maps cannot be defined on certain solenoids, in particular on the dyadic solenoid, and on Knaster continua.

P. Civin, Two-to-one mappings of manifolds, Duke Math. J. 10 (1943), 49-57.
A. V. Chernavskiĭ, The impossibility of a strictly double continuous partition of the homologous cube, Dokl. Akad. Nauk SSSR 144 (1962), 286-289 (in Russian).
S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton Univ. Press, 1952.
V. B. Fugate and T. B. McLean, Compact groups of homeomorphisms on tree-like continua, Trans. Amer. Math. Soc. 267 (1981), 609-620.
O. G. Harrold, The non-existence of a certain type of continuous transformation, Duke Math. J. 5 (1939), 789-793.
J. Heath, Tree-like continua and exactly k-to-1 functions, Proc. Amer. Math. Soc. 105 (1989), 765-772.
E. Hewitt and K. Ross, Abstract Harmonic Analysis, Vol. I, Springer, 1963.
J. Mioduszewski, On two-to-one continuous functions, Dissertationes Math. (Rozprawy Mat.) 24 (1961).
L. Pontryagin, Topological Groups, Gordon and Breach, New York 1966.
I. Rosenholtz, Open maps of chainable continua, Proc. Amer. Math. Soc. 42 (1974), 258-264.
W. Scheffer, Maps between topological groups that are homotopic to homomorphisms, ibid. 33 (1972), 562-567.

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