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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
##### Keywords:
involution; fixed point set; stable cobordism class
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##### References:
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