IJMMS 32:2 (2002) 81–92 PII. S0161171202109136 http://ijmms.hindawi.com © Hindawi Publishing Corp. SPACES WHOSE ONLY FINITE-SHEETED COVERS ARE THEMSELVES. PART I MATHEW TIMM Received 15 September 2001 This is a survey of results and open questions related to the topology of spaces that have no nontrivial finite-sheeted covers. 2000 Mathematics Subject Classification: 57M10, 54C10, 54H25, 20E34. 1. Introduction. This paper focuses on those connected topological spaces that have the property that all of their finite-sheeted connected covering spaces have total space homeomorphic to the base space. Note that there are two ways that a space M can satisfy this property: either (1) M has no nontrivial finite-sheeted covers or (2) M has a k-fold connected cover p : X → M for some k ≥ 2, and the total space of every connected finite-sheeted cover p : X → M is such that X is homeomorphic to M. In this paper, we consider those spaces that satisfy the first condition, that is, those that have no nontrivial finite-sheeted covers. In the sequel we will survey items related to spaces satisfying the second condition. A reader with a current interest in the second type of space can consult [34, 39]. Section 2 of this paper presents basic terminology and some elementary examples. Section 3 presents what is known about such metric continua. Section 4 presents what is known in the context of spaces with more structure such as low-dimensional man- ifolds and cell complexes. The development includes: examples, means to construct additional examples, and statements of interesting problems that, to the knowledge of the author, are unsolved as of this date. It is of interest to note that spaces with no nontrivial finite-sheeted covers are related to two problems of significant histori- cal interest: the question of whether every nonseparating planar continuum has the fixed point property and the question of whether every compact 3-manifold can be decomposed into finitely many geometric pieces. See Scott [35]. It has been attempted to make the paper as self-contained as possible. Unfortu- nately, omissions have no doubt occurred. For more complete treatments of the gen- eral topology, including the topology of inverse limit spaces, consult Engelking [8]. A good reference for the algebraic topology is Spanier [36]. The books by Hempel [16] and Jaco [21] and the survey paper by Scott [35] are good references on 3-manifold topology. For the more specialized group theory, refer to the books by Robinson [33] and Magnus et al. [27] while the more elementary group theory can be found in Hunger- ford [20]. Also refer to Kirby [25] for the most recent version of his problem list. Kirby’s paper, (which contains an extensive bibliography) can also be thought of as a crash course in the topology of low-dimensional manifolds.