Memoirs of the American Mathematical Society 2000; 89 pp; softcover Volume: 144 ISBN10: 082182046X ISBN13: 9780821820469 List Price: US$48 Individual Members: US$28.80 Institutional Members: US$38.40 Order Code: MEMO/144/686
 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 fixedpoint 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 \(U\). Readership 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
