Homotopy Decompositions of Spaces
نویسنده
چکیده
0.1. Why Do We Want to Decompose a Space? Basically the goal of mathematics is to classify certain objects. For instance, we are able to classify 2-dimensional manifolds. Then we are trying to classify 3-manifolds while Poincaré conjecture sounds difficult to be solved. In homotopy theory, a general question is how to classify spaces (up to homotopy). The general idea for classifying spaces is: (1). Decompose spaces (up to homotopy) into “smaller spaces”; and (2). Study “indecomposable factors” (atomic spaces). A decomposition of a space X means X ' Y ∨ Z or X ' Y × Z. Usually we want to decompose a co-H-space as a wedge of smaller spaces and an H-space as a product of smaller spaces. Exercise. A path-connected co-H-space does not have a proper product decomposition. A path-connected H-space does not have a proper wedge decomposition. (A proper decomposition here means that each factor has non-trivial mod p homology.)
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