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

Tom 186 / 2005

Pedro L. Q. Pergher, Rogério de Oliveira Fundamenta Mathematicae 186 (2005), 97-109 MSC: Primary 57R85; Secondary 57R75. DOI: 10.4064/fm186-2-1


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.


  • Pedro L. Q. PergherCentro de Ciências Exatas e Tecnologia
    Departamento de Matemática
    Universidade Federal de São Carlos
    Caixa Postal 676; CEP 13.565-905
    São Carlos, SP, Brazil
  • Rogério de OliveiraDepartamento 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

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