The Conner Conjecture for Compact Group Actions on Equivariant CW Complexes

Year 2024, Volume: 9 Issue: 2, 534 - 550, 29.12.2024

Mehmet Onat

https://doi.org/10.33484/sinopfbd.1526629

Abstract

Let $X$ be a paracompact space of finite cohomological dimension over $\Lambda $ ($\mathbb{Z}$, $\mathbb{Z}_{p}$ or $\mathbb{Q}$) and $G$ be a compact Lie group acting on $X$ with finite many orbit types. It has been proven by R. Oliver that if $X$ is acyclic over $\Lambda $, then the orbit space $X/G$ is also acyclic over $\Lambda $. In this study, for rational coefficients, this result will be proven for finite-dimensional compact (non-Lie) group actions on finite dimensional equivariant CW complexes with finitely many connective orbit types. Furthermore, it will be showen that the fixed point space of an acyclic space over rationals is acyclic for compact connected (non-Lie) group actions on paracompact spaces under certain conditions.

Keywords

Conner conjecture, Acyclic spaces, Compact group actions

Ekivaryant CW Kompleksler Üzerine Kompakt Grup Etkileri İçin Conner'in Sanısı

Abstract

$X$, $\Lambda $ ($\mathbb{Z}$, $\mathbb{Z}_{p}$ or $\mathbb{Q}$) üzerinde sonlu kohomolojik boyuta sahip bir parakompakt uzay ve $G$, $X$ üzerine sonlu sayıda orbit tipi ile etki eden kompakt bir Lie grubu olsun. R. Oliver tarafIndan kanıtlandı ki eğer $X$, $\Lambda $ üzerinde asiklik bir uzay ise $X/G$ orbit uzayı da $\Lambda $ üzerinde asiklik bir uzaydır. Bu çalışmada, sonlu sayıda bağlantılı orbit tipli, sonlu boyutlu ekivaryant CW kompleksler üzerine sonlu boyutlu kompakt grup (Lie grubu olmayabilir) etkileri için bu sonuş kanıtlanacaktır. Ayrıca, bazı koşullar altında parakompakt uzaylar üzerine etki eden kompakt bağlantılı grup (Lie grubu olmayabilir) etkileri için rasyonel sayılar üzerinde bir asiklik uzayın sabit nokta uzayının asiklik olduğu gösterilecektir.

Keywords

Conner'in sanısı, Asiklik uzaylar, Kompakt grup etkileri

Thanks

Yazar, yorumları ve önerileri ile bu makalenin geli¸stirilmesine ve netle¸stirilmesine yardımcı olan hakemlere te¸sekkür etmeyi bir borç bilir.

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