The group of self-homotopy equivalences of \(S^ 2\)-bundles over \(S^ 4\). II: applications
This is the second part of an earlier work of the same title [ibid. 9, 308-326 (1986; Zbl 0604.55004)]. The results of the first part are used here to classify the homotopy types of complexes of the form \(X=S^ 2\cup e^ 4\cup e^ 6\). Such a complex X is said to be of type (m, n, \(\epsilon)\) where \(x_ 2\cdot x_ 2=\pm mx_ 4\), \(x_ 2\cdot x_ 4=\pm nx_ 6\) (m, \(n\geq 0)\) and \(Sq^ 2(x_ 4')=\epsilon x_ 6'\) \((\epsilon =0\), 1) where the \(x_ i\) (resp. \(x_ i')\) are the generators in \({\mathbb{Z}}-\) (resp. \({\mathbb{Z}}_ 2-)\) cohomology. Let \(m\equiv 1 mod 2\), then \(\epsilon =0\), and the homotopy type is determined by (m, n, \(\epsilon)\). Similarly for \(m\equiv 0 mod 2\) and \(\epsilon =1\). However for \(m\equiv 0 mod 2\) and \(\epsilon =0\), there are two homotopy types corresponding to (m, n, \(\epsilon)\). In the special case where X is a smooth manifold of type (m, 1, 0), the Pontryagin class can be used to distinguish these two homotopy types.
CW-complexes with a cell in dimension 2, 4, and 6
homotopy types of complexes of the form \(S^ 2\cup e^ 4\cup e^ 6\)
scientific article; zbMATH DE number 4011425
scientific article