Methods and Applications of Analysis

Volume 26 (2019)

Number 1

In Memory of Professor John N. Mather: Part 3 of 3

Guest Editors: Sen Hu, University of Science and Technology, China; Stanisław Janeczko, Polish Academy of Sciences, Poland; Stephen S.-T. Yau, Tsinghua University, China; and Huaiqing Zuo, Tsinghua University, China.

Equivariant control data and neighborhood deformation retractions

Pages: 13 – 36

DOI: https://dx.doi.org/10.4310/MAA.2019.v26.n1.a2

Authors

Markus J. Pflaum (Department of Mathematics, University of Colorado, Boulder, Co., U.S.A.)

Graeme Wilkin (Department of Mathematics, University of York, Heslington, York, United Kingdom)

Abstract

In this article we study Whitney (B) regular stratified spaces with the action of a compact Lie group $G$ which preserves the strata. We prove an equivariant submersion theorem and use it to show that such a $G$-stratified space carries a system of $G$-equivariant control data. As an application, we show that if $A \subset X$ is a closed $G$-stratified subspace which is a union of strata of $X$, then the inclusion $i : A \hookrightarrow X$ is a $G$-equivariant cofibration. In particular, this theorem applies whenever $X$ is a $G$-invariant analytic subspace of an analytic $G$-manifold $M$ and $A \hookrightarrow X$ is a closed $G$-invariant analytic subspace of $X$.

Keywords

equivariant control data, equivariant Whitney stratification, equivariant tubular neighbourhood, equivariant cofibration

2010 Mathematics Subject Classification

32C25, 57N80, 57R99

Full Text (PDF format)

Received 8 August 2018

Accepted 29 January 2019

Published 14 November 2019