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$Z₂^{k}$-actions with a special fixed point set

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Pergher P., de Oliveira R.

Fundamenta Mathematicae

2005

tom 186

nr 2

97-109

Let Fⁿ be a connected, smooth and closed n-dimensional manifold satisfying the following property: if $N^{m}$ is any smooth and 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. We describe the equivariant cobordism classification of smooth actions $(M^{m};Φ)$ of the group $G = Z₂^{k}$ on closed smooth m-dimensional manifolds $M^{m}$ for which the fixed point set of the action is a submanifold Fⁿ with the above property. This generalizes a result of F. L. Capobianco, who obtained this classification for $Fⁿ = ℝP^{2r}$ (P. E. Conner and E. E. Floyd had previously shown that $ℝP^{2r}$ has the property in question). In addition, we establish some properties concerning these Fⁿ and give some new examples of these special manifolds.

Commuting involutions whose fixed point set consists of two special components

100%

Pergher P., de Oliveira R.

Fundamenta Mathematicae

2008

tom 201

nr 3

241-259

Let Fⁿ be a connected, smooth and closed n-dimensional manifold. We call Fⁿ a manifold with property 𝓗 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 𝓗, and where dim(K) < dim(L). The description is given in terms of the set of equivariant cobordism classes of involutions fixing K ∪ L.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Pergher P.

2 de Oliveira R.

1 2008

1 2005

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$Z₂^{k}$-actions with a special fixed point set

Commuting involutions whose fixed point set consists of two special components