#### Vol. 303, No. 1, 2019

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Degree-one, monotone self-maps of the Pontryagin surface are near-homeomorphisms

### Robert J. Daverman and Thomas L. Thickstun

Vol. 303 (2019), No. 1, 93–131
##### Abstract

We prove that a self-map of the closed Pontryagin surface can be approximated by homeomorphisms if and only if it is monotone and has degree $±1$. This adds to a body of theorems, each of which characterizes for some space or class of spaces those self-maps which are approximable by homeomorphisms.

##### Keywords
Pontryagin surface, Pontryagin disk, monotone map, near-homeomorphism, degree-one map, figure-eight, shrinkability criterion
##### Mathematical Subject Classification 2010
Primary: 57N05, 57P05
Secondary: 54B15, 54G99, 57N60