eBook ISBN:  9781470402778 
Product Code:  MEMO/144/686.E 
List Price:  $51.00 
MAA Member Price:  $45.90 
AMS Member Price:  $30.60 
eBook ISBN:  9781470402778 
Product Code:  MEMO/144/686.E 
List Price:  $51.00 
MAA Member Price:  $45.90 
AMS Member Price:  $30.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 144; 2000; 89 ppMSC: Primary 55; 57
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 \(U\).
ReadershipGraduate students and research mathematicians interested in algebraic topology.

Table of Contents

Chapters

Introduction

Notational conventions

Part 1. Geometrically split spectra

Section 1. The notion of a geometrically split $G$spectrum

Section 2. Geometrically split $G$spectra and $G$fixedpoint spectra

Section 3. Geometrically split $G$spectra and IIfixedpoint spectra

Section 4. Geometrically split spectra and finite groups

Section 5. The stable orbit category for an incomplete universe

Part 2. A toolkit for incomplete universes

Section 6. A vanishing theorem for fixedpoint spectra

Section 7. SpanierWhitehead duality and incomplete universes

Section 8. Change of group functors and families of subgroups

Section 9. Change of universe functors and families of subgroups

Section 10. The geometric fixedpoint functor $\Phi ^\Lambda $ for incomplete universes

Section 11. The Wirthmüller isomorphism for incomplete universes

Section 12. An introduction to the Adams isomorphism for incomplete universes

Part 3. The longer proofs

Section 13. The proof of Proposition 3.10 and its consequences

Section 14. The proofs of the main splitting theorems

Section 15. The proof of the sharp Wirthmüller isomorphism theorem

Section 16. The proof of the Adams isomorphism theorem for incomplete universes

Section 17. The Adams transfer for incomplete universes


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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 \(U\).
Graduate students and research mathematicians interested in algebraic topology.

Chapters

Introduction

Notational conventions

Part 1. Geometrically split spectra

Section 1. The notion of a geometrically split $G$spectrum

Section 2. Geometrically split $G$spectra and $G$fixedpoint spectra

Section 3. Geometrically split $G$spectra and IIfixedpoint spectra

Section 4. Geometrically split spectra and finite groups

Section 5. The stable orbit category for an incomplete universe

Part 2. A toolkit for incomplete universes

Section 6. A vanishing theorem for fixedpoint spectra

Section 7. SpanierWhitehead duality and incomplete universes

Section 8. Change of group functors and families of subgroups

Section 9. Change of universe functors and families of subgroups

Section 10. The geometric fixedpoint functor $\Phi ^\Lambda $ for incomplete universes

Section 11. The Wirthmüller isomorphism for incomplete universes

Section 12. An introduction to the Adams isomorphism for incomplete universes

Part 3. The longer proofs

Section 13. The proof of Proposition 3.10 and its consequences

Section 14. The proofs of the main splitting theorems

Section 15. The proof of the sharp Wirthmüller isomorphism theorem

Section 16. The proof of the Adams isomorphism theorem for incomplete universes

Section 17. The Adams transfer for incomplete universes