Cohomotopy sets of 4-manifolds

Abstract

Elementary geometric arguments are used to compute the group of homotopy classes of maps from a 4-manifold X to the 3-sphere, and to enumerate the homotopy classes of maps from X to the 2-sphere. The former completes a project initiated by Steenrod in the 1940's, and the latter provides geometric arguments for and extensions of recent homotopy theoretic results of Larry Taylor. These two results complete the computation of all the cohomotopy sets of closed oriented 4-manifolds and provide a framework for the study of Morse 2-functions on 4-manifolds, a subject that has garnered considerable recent attention.

20 pages, 6 figures; this version to appear in the FreedmanFest (G&T Monographs, Volume 18)

References in corpus (1)

• Fiber connected, indefinite Morse 2-functions on connected n-manifolds

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• On fundamental groups of spaces of framed embeddings of a circle in a 4-manifold
• A geometric computation of cohomotopy groups in co-degree one