Commuting involutions whose fixed point set consists of two special components

Pedro L. Q. Pergher ; Rogério de Oliveira

Fundamenta Mathematicae, Tome 201 (2008), p. 241-259 / Harvested from The Polish Digital Mathematics Library

Résumé

Let Fⁿ be a connected, smooth and closed n-dimensional manifold. We call Fⁿ a manifold with property when it has the following property: if $N^m$ is any smooth closed m-dimensional manifold with m > n and $T:N^m → N^m$ is a smooth involution whose fixed point set is Fⁿ, then m = 2n. Examples of manifolds with this property are: the real, complex and quaternionic even-dimensional projective spaces $RP^{2n}$, $CP^{2n}$ and $HP^{2n}$, and the connected sum of $RP^{2n}$ and any number of copies of Sⁿ × Sⁿ, where Sⁿ is the n-sphere and n is not a power of 2. In this paper we describe the equivariant cobordism classification of smooth actions $(M^m; Φ)$ of the group $Z₂^k$ on closed smooth m-dimensional manifolds $M^m$ for which the fixed point set of the action consists of two components K and L with property , and where dim(K) < dim(L). The description is given in terms of the set of equivariant cobordism classes of involutions fixing K ∪ L.

Détail

Publié le : 2008-01-01

Zbl 1160.57030

EUDML-ID : urn:eudml:doc:282845