# $Z{₂}^{k}$-actions with a special fixed point set

Fundamenta Mathematicae (2005)

• Volume: 186, Issue: 2, page 97-109
• ISSN: 0016-2736

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## Abstract

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

## How to cite

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Pedro L. Q. Pergher, and Rogério de Oliveira. "$Z₂^{k}$-actions with a special fixed point set." Fundamenta Mathematicae 186.2 (2005): 97-109. <http://eudml.org/doc/283374>.

@article{PedroL2005,
abstract = {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.},
author = {Pedro L. Q. Pergher, Rogério de Oliveira},
journal = {Fundamenta Mathematicae},
keywords = {; fixed data; characteristic number, Wu class; property ; Stiefel-Whitney},
language = {eng},
number = {2},
pages = {97-109},
title = {$Z₂^\{k\}$-actions with a special fixed point set},
url = {http://eudml.org/doc/283374},
volume = {186},
year = {2005},
}

TY - JOUR
AU - Pedro L. Q. Pergher
AU - Rogério de Oliveira
TI - $Z₂^{k}$-actions with a special fixed point set
JO - Fundamenta Mathematicae
PY - 2005
VL - 186
IS - 2
SP - 97
EP - 109
AB - 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.
LA - eng
KW - ; fixed data; characteristic number, Wu class; property ; Stiefel-Whitney
UR - http://eudml.org/doc/283374
ER -

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