Title: A Fundamental Group for Disconnected Spaces:
Applying an Old Idea to Symbolic Dynamics
Abstract: The fundamental group for a connected topological space is often described visually. It is based on paths along the space which start and end at the same point. The actual group comes from equivalence classes of such paths. However, this is just perspective. In this lecture, the speaker will review a different approach to fundamental groups, and indicate how this abstract alternative leads to analogous groups for other classes of topological objects. The development of axioms for Algebraic Geometry required a new formulation for the fundamental group. The new context required an algebraic version. Currently, there is an complete theory in the language of categories: If ${\cal C}$ is a category with certain diagrammatic properties, then there is an intrinsic notion of ``connected ${\cal C}$-object'', and there is a (almost-functorial) assignment of a group to the subcategory of all connected ${\cal C}$-objects. The categorical construction applies to symbolic dynamics. Symbolic dynamics considers contexts in which an object consists of a topological space paired with a group action. These categories include objects which are connected in the universal sense even though the topological component is not. The speaker will report on some preliminary investigations of the fundamental groups that emerge. Return to the seminar page. Please send comments about this page to Maurice Rojas at rojas@math.tamu.edu. Last Modified on 25/Feb/02 |