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Memoirs of the American Mathematical Society
2000; 89 pp; softcover
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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 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.
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