Splitting Theorems for Certain Equivariant Spectra (Memoirs of the American Mathematical Society)

Splitting Theorems for Certain Equivariant Spectra (Memoirs of the American Mathematical Society)

Paperback

Up to 2 WeeksUsually despatched within 2 weeks

Description

Let $G$ be a compact Lie group, $\Pi$ be a normal subgroup of $G$, $\mathcal G=G[LAMBDA]Pi$, $X$ 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 $(E\mathfrak F_+\wedge C)^\Pi \rarrow (E\mathfrak F'_+\wedge C)^\Pi \rarrow (E(\mathfrak F',\mathfrak F)\wedge C)^\Pi$ 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 Wirthmuller 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 $U$.

Contents

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

Product Details

  • ISBN13: 9780821820469
  • Format: Paperback
  • ID: 9780821820469
  • ISBN10: 082182046X

Delivery Information

  • Saver Delivery: Yes
  • 1st Class Delivery: Yes
  • Courier Delivery: Yes
  • Store Delivery: Yes

Prices are for internet purchases only. Prices and availability in WHSmith Stores may vary significantly

Close