Vol. 90, No. 2, 1980

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Ends of fundamental groups in shape and proper homotopy

Michael L. Mihalik

Vol. 90 (1980), No. 2, 431–458

The number of topological ends of the universal cover of a finite complex, K, is either 0, 1, 2, or and only depends upon the fundamental group of K. Call this number e(K). We wish to define numbers e(X) for compact metric spaces analogous to e(K). To accomplish this we extend the theory of ends for finitely generated groups to certain inverse sequences of finitely generated groups and their inverse limits. Classifications for these inverse sequences and their inverse limits analogous to those for finitely generated groups are derived. Whenever the fundamental pro-group of a compact metric space, X, satisfies certain properties, we obtain a shape invariant number e(X) (either 0, 1, 2 or ) and analyze what e(X) describes geometrically.

Mathematical Subject Classification 2000
Primary: 55Q07
Secondary: 55Q70
Received: 4 May 1979
Revised: 19 September 1979
Published: 1 October 1980
Michael L. Mihalik
Mathematics Department
Vanderbilt University
Nashville TN 37240
United States