On fibering four-and five-manifolds
Put very crudely the topology of n-dimensional manifolds can be described in terms of general theorems for \(n\geq 5\), and also for \(n=4\), provided that the fundamental group is not too large, and one is prepared to neglect smoothing. However in dimension 3 the situation is completely different, largely because one lacks the ``room'' necessary to carry out general ce Chern character associated with W. Examples arising from the Lie group \(G_ 2\) are investigated in detail. If \(G_ 2\) is given a standard CW-structure, \(X^ 3=G_ 2^{[6]}\), \(Y^{11}=G_ 2^{[11]}/G_ 2^{[6]}\) and \(S^{14}=G_ 2/G_ 2^{[11]}\) are K- theory spheres. Denoting suspension by increased superscripts, it is shown that \(\{Y^{n+7},X^ n\}\cong {\mathbb{Z}}/60{\mathbb{Z}}\) and a generator has e-invariant \((120)^{-1} mod 2^{-1}{\mathbb{Z}}\) and so e is monic. Further analysis of the homotopy of \(G_ 2\) and of the other simply connected H-spaces \(G_{2,b}\) with the same mod 2 homotopy type leads to complete information on the action of self-maps in integral homology modulo torsion both unstably and stably; the K-theoretic restrictions are best possible. It is shown also that the only H-equivalences of such spaces for any multiplications are the identity maps, except possibly for \(G_{2,-2}\).
Farrell's fibration theorem for 5-manifolds
topological fibrations of manifolds over the circle
scientific article; zbMATH DE number 4040855
scientific article