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Bounds on the dimension of manifolds with involution fixing \(F^n \cup F^2\). (English) Zbl 1154.57033
Let \(M^m\) be an \(m\)-dimensional, smooth and closed manifold, equipped with a smooth involution \(T:M^m \rightarrow M^m\). The fixed point set of \(T\), \(F\), is a finite and disjoint union of closed submanifolds, \(F = \cup_{j=0}^{n}F^j\), where \(F^j\) denotes the union of those components of \(F\) having dimension \(j\). The \(5/2\)-Theorem of J. M. Boardman [Bull. Am. Math. Soc. 73, 136-138 (1967; Zbl 0153.25403)] states that, if \(M^m\) is nonbounding, then \(m \leq 5/2n\). Further, this estimate for \(m\) is best possible. C. Kosniowski and R. E. Stong obtained in [Topology 17, 309-330 (1978; Zbl 0402.57005)] the following strengthened version of the Boardman theorem: if the normal bundle of \(F\) in \(M^m\) is not a boundary, then \(m \leq 5/2n.\) The generality of this result allows the possibility that fixed components of all dimensions \(j\), \(0 \leq j \leq n\), occur. Therefore this raises the question of improving this estimate for more specific fixed point sets \(F\). In this direction, the authors deal with the case in which \(F = F^n \cup F^2\), where \(n > 2 \). For each natural number \(n\), write \(n=2^pq\), where \(p \geq 0\) and \(q\) is odd, and set \( m(n) = 2n+p-q+1,p \leq q + 1\) and \( m(n) = 2n + 2^{p-q}, p \geq q.\)
Denote by \(\beta\) the stable cobordism class of the normal bundle of \(F^2\) in \(M^m\). In a previous paper by the authors [Topology Appl. 153 (14), 2499–2507 (2006; Zbl 1102.57019)], it was proved that, if \(\beta\) is nonzero, then \(m \leq m(n-2) + 4\). Further, it was shown that this bound is best possible. This estimate is valid for any \(n > 2\) and any \(\beta\), and the authors had additionally shown that there are seven such classes \(\beta\). This suggests the question of improving this estimate for specific values of \(n\) and \(\beta\). Inspired in this setting the authors defined the number \(\varphi(n, \beta)\) as being the greatest number \(m\) satisfying the fact that there exists an involution \((M^m,T)\) having fixed point set of the form \(F = F^n \cup F^2\) and such that the normal bundle over \(F^2\) represents \(\beta\). In their previous paper [Arch. Math. 87, No. 3, 280–288 (2006; Zbl 1101.57016)], the authors calculated \(\varphi(n, \beta)\) for every \(\beta\) and \(n\) odd.
The paper in question concerns the calculation of \(\varphi(n, \beta)\) for \(n\) even. The authors provide an enumeration of the seven classes \(\beta\) as \(\beta_i\), \(i=1,2,\dots,7\), in terms of the characteristic numbers of these classes. The case in which \( n \equiv 0\) mod \(4\) is completely solved. Also, \(\varphi(n, \beta_i)\) is calculated in the cases: 7mm
(i)
\(i=3,5\) or \(7\) and \(n\) satisfies the fact that \(n-2 = 2^p q\) where \(q\) is odd and \( p \leq q.\)
(ii)
\( i= 1\) or \(4\) and \( n>2\) is any even.
The cases where \( n \equiv 2\) mod \(4\) and \( i = 2\) or \(6\), and where \(n-2 = 2^p q\) with \( p > q\) and \( i = 3, 5\) or \(7\), are left open. The difficulty in this cases consists in finding suitable maximal examples.

MSC:
57R85 Equivariant cobordism
57R75 \(\mathrm{O}\)- and \(\mathrm{SO}\)-cobordism
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