Abstract

We define an extension of the nth homotopy group which can distinguish a larger class of spaces. (E.g., a converging sequence of disjoint circles and the disjoint union of countably many circles, which have isomorphic fundamental groups, regardless of choice of basepoint.) We do this by introducing a generalization of homotopies, called component-homotopies, and defining the nth extended homotopy group to be the set of component-homotopy classes of maps from compact subsets of (0,1)ⁿ into a space, with a concatenation operation. We also introduce a method of tree-adjoinment for "connecting" disconnected metric spaces and show how this method can be used to calculate the extended homotopy groups of an arbitrary metric space.

Degree

College and Department

Physical and Mathematical Sciences; Mathematics

Rights

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