AMS Bookstore LOGO amslogo
Return to List

AMS TextbooksAMS Applications-related Books

Splitting Theorems for Certain Equivariant Spectra
L. Gaunce Lewis, Jr., Syracuse University, NY

Memoirs of the American Mathematical Society
2000; 89 pp; softcover
Volume: 144
ISBN-10: 0-8218-2046-X
ISBN-13: 978-0-8218-2046-9
List Price: US$51
Individual Members: US$30.60
Institutional Members: US$40.80
Order Code: MEMO/144/686
[Add Item]

Request Permissions

Let \(G\) be a compact Lie group, \(\Pi\) be a normal subgroup of \(G\), \(\mathcal G=G/\Pi\), \(X\) be a \(\mathcal G\)-space and \(Y\) be a \(G\)-space. There are a number of results in the literature giving a direct sum decomposition of the group \([\Sigma^\infty X,\Sigma^\infty Y]_G\) of equivariant stable homotopy classes of maps from \(X\) to \(Y\). Here, these results are extended to a decomposition of the group \([B,C]_G\) of equivariant stable homotopy classes of maps from an arbitrary finite \(\mathcal G\)-CW sptrum \(B\) to any \(G\)-spectrum \(C\) carrying a geometric splitting (a new type of structure introduced here). Any naive \(G\)-spectrum, and any spectrum derived from such by a change of universe functor, carries a geometric splitting. Our decomposition of \([B,C]_G\) is a consequence of the fact that, if \(C\) is geometrically split and \((\mathfrak F',\mathfrak F)\) is any reasonable pair of families of subgroups of \(G\), then there is a splitting of the cofibre sequence \((E\mathfrak F_+ \wedge C)^\Pi \longrightarrow (E\mathfrak F'_+ \wedge C)^\Pi \longrightarrow (E(\mathfrak F', \mathfrak F) \wedge C)^\Pi\) constructed from the universal spaces for the families. Both the decomposition of the group \([B,C]_G\) and the splitting of the cofibre sequence are proven here not just for complete \(G\)-universes, but for arbitrary \(G\)-universes.

Various technical results about incomplete \(G\)-universes that should be of independent interest are also included in this paper. These include versions of the Adams and Wirthmüller isomorphisms for incomplete universes. Also included is a vanishing theorem for the fixed-point spectrum \((E(\mathfrak F',\mathfrak F) \wedge C)^\Pi\) which gives computational force to the intuition that what really matters about a \(G\)-universe \(U\) is which orbits \(G/H\) embed as \(G\)-spaces in \(U\).


Graduate students and research mathematicians interested in algebraic topology.

Table of Contents

  • Introduction
  • Notational conventions
Part 1. Geometrically Split Spectra
Part 2. A Toolkit for Incomplete Universes
Part 3. The Longer Proofs
  • Acknowledgments
  • Bibliography
Powered by MathJax

  AMS Home | Comments:
© Copyright 2014, American Mathematical Society
Privacy Statement

AMS Social

AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia