${ Z_2^k}$-actions with a special fixed point set

Pedro L. Q. Pergher¹ ; Rogério de Oliveira²

¹ Centro de Ciências Exatas e Tecnologia Departamento de Matemática Universidade Federal de São Carlos Caixa Postal 676; CEP 13.565-905 São Carlos, SP, Brazil
² Departamento de Ciências Exatas Campus Universitário de Três Lagoas Universidade Federal de Mato Grosso do Sul Caixa Postal 210; CEP 79603-011 Três Lagoas, MS, Brazil

Fundamenta Mathematicae, Tome 186 (2005) no. 2, pp. 97-109.

Résumé

Let $F^n$ 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 $T:N^m \to N^m$ is a smooth involution whose fixed point set is $F^n$, then $m=2n$. We describe the equivariant cobordism classification of smooth actions $(M^m; {\mit \Phi })$ of the group $G=Z_2^k$ on closed smooth $m$-dimensional manifolds $M^m$ for which the fixed point set of the action is a submanifold $F^n$ with the above property. This generalizes a result of F.~L. Capobianco, who obtained this classification for $F^n={\mathbb R}P^{2r}$ (P. E. Conner and E. E. Floyd had previously shown that ${\mathbb R}P^{2r}$ has the property in question). In addition, we establish some properties concerning these $F^n$ and give some new examples of these special manifolds.

Détail

DOI : 10.4064/fm186-2-1

Keywords: connected smooth closed n dimensional manifold satisfying following property smooth closed m dimensional manifold smooth involution whose fixed point set describe equivariant cobordism classification smooth actions mit phi group closed smooth m dimensional manifolds which fixed point set action submanifold above property generalizes result capobianco who obtained classification mathbb conner floyd had previously shown mathbb has property question addition establish properties concerning these examples these special manifolds