Involutions whose top dimensional component of the fixed point set is indecomposable
In 1967, \textit{J. Boardman} proved the famous Five Halves Theorem [Bull. Am. Math. Soc. 73, 136--138 (1967; Zbl 0153.25403)] which states: if \(M^m\) is a smooth closed \(n\)-dimensional manifold, and if \(T:M^m \to M^m\) is a smooth involution on \(M^m\) for which the fixed point set \(F = \displaystyle {\bigcup_{j =0}^{n}} F^j \) does not bound, then \(m \leq\frac{5}{2}n\) (here, \(F^j\) denotes the union of those components of \(F\) having dimension \(j\)). Further, in this generality, this estimative is best possible. In recent years, the author of the present paper obtained several results in the following relevant direction: can we obtain improvements for the Boardman bounds if we impose additional conditions on \(F\)? In the present paper, the author deals with this question when the top dimensional component of \(F\), \(F^n\), is indecomposable; we recall that a closed manifold is indecomposable if its unoriented cobordism class cannot be expressed as a sum of products of lower dimensional cobordism classes. In this case, the author proves that the Boardman bound \(m \leq\frac{5}{2}n\) can be improved to \(m \leq 2n+1\). To do that, the author discovered a very special characteristic class associated to the total space of projective bundles. The author also constructs, for each \( n \geq 2\) not of the form \(2^t - 1\), a special involution \((M^{2n+1},T)\) so that the dimension of the top dimensional component of the fixed point set is \(n\) and with this top dimensional component being indecomposable, thus showing that the bound in question is best possible.
scientific article; zbMATH DE number 5721540
scientific article