The homotopy type of the space of maps of a homology 3-sphere into the 2-sphere

Abstract

It is proved that if K is a compact, connected polyhedron such that H2(K; Z) = 0, then all the components in the space of maps of K into the 2-sphere are homeomorphic. For K a polyhedral homology 3-sphere the common homotopy type of the components is identified and shown to be independent of K. 1* Introduction and statements of results* Let K and X be a pair of compact, connected polyhedra and let M(K, X) denote the space of (continuous) maps of K into X. All mapping spaces will be equipped with the compact-open topology. Corresponding to each homotopy class of maps of K into X there is a (path-) component in M(K, X). For each pair of spaces K and X there arises then a natural classification problem, namely that of dividing the set of components in 'M(K, X) into homotopy types. The present paper is one in a series of papers, where we search through classical algebraic topology for methods, which are useful in the study of such classification problems. In [4], information on certain Whitehead products was used to tackle the classification problem for the set of components in the space of maps of the m-sphere Sm into the ^-sphere Sn, m ^ n ^ 1, and complete solutions were obtained in the cases m = n and m = n + 1. If the domain in the mapping space is not a suspension, the problem becomes more delicate, since normally, it is then difficult to construct nontrivial maps between the various components. For a mapping space with a manifold as domain it is sometimes possible to solve the classification problem for the components using information about a corresponding mapping space with a sphere as domain. As an example, knowledge of the fundamental group of the various components in M(S2, S2) was used in [5] to solve the classification problem for the countable number of components in the space of maps of an orientable closed surface into S2. In this paper, we shall investigate spaces of maps into the base space of a principal bundle. We will concentrate mainly on spaces of maps into S2, making use of the fact, that S2 is the base space in a principal S'-bundle, namely