Two natural questions are answered in the negative: (1) If a space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic? (2) Can adding arcs to a space cause an essential curve to become nulhomotopic? The answer to the first question clarifies the relationship between the notions of a space being homotopically Hausdorff and $\pi_1$-shape injective.Comment: 12 pages, 5 figure
We discuss the recoverable and irrecoverable energy densities associated with a pulse at a point in the propagation medium and derive easily computed expressions to calculate these quantities. Specific types of fields are required to retrieve the recoverable portion of the energy density from the point in the medium, and we discuss the properties that these fields must have. Several examples are given to illustrate these concepts.
The classical archipelago is a non-contractible subset of 3 which is homeomorphic to a disk except at one non-manifold point. Its fundamental group, , is the quotient of the topologist's product of , the fundamental group of the shrinking wedge of countably many copies of the circle (the Hawaiian earring), modulo the corresponding free product. We show is locally free, not indicable, and has the rationals both as a subgroup and a quotient group. Replacing with arbitrary groups yields the notion of archipelago groups.Surprisingly, every archipelago of countable groups is isomorphic to either ( ) or ( 2 ), the cases where the archipelago is built from circles or projective planes respectively. We conjecture that these two groups are isomorphic and prove that for large enough cardinalities of G i , (G i ) is not isomorphic to either.
Abstract. A solenoid is an inverse limit of circles. When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give different groups; in particular, the nicest embeddings are unknotted at each level, and give an Abelian fundamental group, while other embeddings have non-Abelian groups. We show using geometry that every solenoid has uncountably many embeddings with non-homeomorphic complements.In this paper we study 3-manifolds which are complements of solenoids in S 3 . This theory is a natural extension of the study of knot complements in S 3 ; many of the tools that we use are the same as those used in knot theory and braid theory.We will mainly be concerned with studying the geometry and fundamental groups of 3-manifolds which are solenoid complements. We review basic information about solenoids in section 1. In section 2 we discuss the calculation of the fundamental group of solenoid complements. In section 3 we show that every solenoid has an embedding in S 3 so that the complementary 3-manifold has an Abelian fundamental group, which is in fact a subgroup of Q (Theorem 3.5). In section 4 we show that each solenoid has an embedding whose complement has a non-Abelian fundamental group (Theorem 4.3). In section 5 we take a more geometric approach, and show that each solenoid admits uncountably many embeddings in S 3 with non-homeomorphic complements (Theorem 5.4). We achieve this by showing that these complements have distinct geometries using JSJ theory, and thus by Mostow-Prasad rigidity are distinct manifolds.
Every Peano continuum has a strong deformation retract to a deforested continuum, that is, one with no strongly contractible subsets attached at a single point. In a deforested continuum, each point with a one-dimensional neighborhood is either fixed by every self-homotopy of the space, or has a neighborhood which is a locally finite graph. A minimal deformation retract of a continuum (if it exists) is called its core. Every one-dimensional Peano continuum has a unique core, which can be obtained by deforestation. We give examples of planar Peano continua that contain no core but are deforested.
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