The homotopy dimension of codiscrete subsets of the 2-sphere 𝕊²

J. W. Cannon; G. R. Conner

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Let X and Y be one-dimensional Peano continua. If the fundamental groups of X and Y are isomorphic, then X and Y are homotopy equivalent. Every homomorphism from the fundamental group of X to that of Y is a composition of a homomorphism induced from a continuous map and a base point change isomorphism.

A characterization of smoothness in dendroids

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Whitney continua of graphs admit all homotopy types of compact connected ANRs

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Conditions are investigated that imply noncontractibility of curves. In particular, a plane noncontractible dendroid is constructed which contains no homotopically fixed subset. A new concept of a homotopically steady subset of a space is introduced and its connections with other related concepts are studied.

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Non-separating subcontinua of planar continua

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We revisit an old question of Knaster by demonstrating that each non-degenerate plane hereditarily unicoherent continuum X contains a proper, non-degenerate subcontinuum which does not separate X.