##
**Duality for smooth families in equivariant stable homotopy theory.**
*(English)*
Zbl 1029.55011

Astérisque. 285. Paris: Société Mathématique de France. 108 p. EUR 25.00; $ 36.00 (2003).

This article is an original work concerning a duality theorem for the equivariant stable homotopy category, though it is written in book form. In chapter 1, consisting of a single page, it is explained what made the idea used here occur to the author. The motivating classical duality theorem presented there is the following: Let \({\mathcal A}\) be an abelian category and let \(\text{Sh}(X)\) denote the category of sheaves on a scheme \(X\) into \({\mathcal A}\). For a proper map \(f: X\to Y\) of schemes, one has two sorts of direct image functors \(f_*,f_!: \text{Sh}(X)\to \text{Sh}(Y)\). Then there is a natural equivalence \(f_!\simeq f_*\) in the derived category of chain complexes of sheaves on \(Y\) into \({\mathcal A}\). The purpose of this paper is to prove (Theorem 4.9) an analogue of this equivalence in the equivariant topological context under certain conditions.

Assume that a space always means a compactly generated weak Hausdorff \(G\)-space, where \(G\) is a compact Lie group. Let \(G\text{Top}/X\) be the category of \(G\)-spaces over \(X\), equipped with a Grothendieck topology which is given by defining the covering to be given by all colimits. A based \(G\)-space over \(X\) is a \(G\)-space \(Z\) with maps \(p_Z: Z\to X\) and \(i_Z: X\to Z\) such that \(p_Z\cdot i_Z= \text{Id}_X\). Then one finds that the category \(G\text{Top}_*/X\) of based \(G\)-spaces over \(X\) becomes canonically equivalent to that of sheaves \(F\) of sets over \(G\text{Top}/X\), together with a morphism of sheaves \(C_X\to F\). Here \(C_X\) denotes the constant sheaf of sets on \(G\text{Top}/X\), given by \(C_X(Z)=*\) for \(Z\in G\text{Top}/X\). This implies, for example, that a sheaf of spectra of \(G\text{Top}/X\) can be regarded as a spectrum parametrized over \(X\). So it follows that one may work in the parametrized or fiberwise homotopy category of based \(G\)-spaces over \(X\) instead of the category of sheaves of sets on \(G\text{Top}/X\). This is the key point of the author’s approach.

The main theorem is given in Chapter 4. If one quotes it without worrying about notations, then it is sketched as follows. One considers the \(G\)-map \(f: X\to Y\) which is an equivariant bundle whose fibre is a smooth compact manifold with actions of subgroups of \(G\) varying smoothly over \(Y\). Let \(f_{\sharp}(Z)\) denote the pushout of \(i_Z\) and \(f\) for a based \(G\)-space \(Z\) over \(X\), and let \(f^*(Z)\) denote the pullback of \(p_Z\) and \(f\) for a closed \(G\)-space \(Z\) over \(Y\). Also put \(C_f= X\times_Y X/(X\times_Y X-\Delta)\), called the dualizing object of \(f\), where \(\Delta\) is the diagonal subspace of \(X\times_Y X\) which is viewed as a \(G\)-space over \(X\) by the projection to the first factor then there exists its inverse \(C^{-1}_f\) in the stable homotopy category over \(X\). One defines the functor \(f_!\) by putting \(f_!(Z)= f_{\sharp}(Z\wedge_X C^{-1}_f)\) for a based \(G\)-space \(Z\) over \(X\) and writes \(f_*\) for the right adjoint of \(f^*\). Then the main theorem (Theorem 4.9) states that for a certain spectrum \(E\) there is a natural weak equivalence \(f_*(E)\simeq f_!(E)\) in the category of spectra over \(Y\). In particular, when \(Y\) is a point and \(X\) is a smooth \(G\)-manifold, this implies just Poincaré duality as given in [L. G. Lewis and M. Steinberger, Equivariant stable homotopy theory, Lectures Notes in Mathematics 1213, Springer-Verlag (1986; Zbl 0611.55001)]. A detailed account of the foundations of \(G\)-spaces and \(G\)-spectra over a base space needed for describing this theorem exactly is given in two chapters. In fact, three lemmas are stated in Chapter 3 with proofs deferred to Chapter 7 which is devoted only to the proofs of these results.

The main part of the proof of the theorem is given in Chapter 5. The author begins with the case where \(Y\) is compact. This is outlined as follows. Take an embedding of \(X\) into \(Y\times S^V\) for some \(G\)-representation \(V\) of \(G\) and denote by \(S(\nu_X)\) the sphere bundle of the normal bundle \(\nu_X\) of \(X\) in \(Y\times S^V\). And let \(S(\tau_X)\) denote the sphere bundle of the tangent bundle \(\tau_X\) of \(X\). Then the dualizing object \(C_f\) is naturally equivalent to \(S(\tau_X)\) in the category of based \(G\)-spaces over \(X\), namely \(C_f\simeq S(\tau_X)\), and also clearly it holds that \(S(\tau_X)\wedge_X S(\nu_X)\simeq X\times S^V\). It follows that \(C^{-1}_f\simeq \Sigma^{-V}_X S(\nu_X)\). Also, since \(f_{\sharp}(S(\nu_X))\) becomes the Thom space of \(\nu_X\) in the category of based \(G\)-spaces over \(Y\), one finds that the Pontryagin-Thom construction of the above embedding yields a map \(\varphi: f_*(T)\to f_{\sharp}(T\wedge_X C^{-1}_f)\) of \(G\)-spaces over \(Y\). By actually constructing its homotopy inverse, it is shown that \(\varphi\) is an equivalence. The general case is obtained by a colimit argument over the compact skeleta of \(Y\).

One has two examples of the main theorem. In Chapter 6 the author shows that the Wirthmüller and the Adams isomorphisms [loc. cit.] are special cases of this duality theorem.

Finally, the reviewer wants to add that this paper is very readable and self-contained, and moreover almost every result presented here is proven in full detail.

Assume that a space always means a compactly generated weak Hausdorff \(G\)-space, where \(G\) is a compact Lie group. Let \(G\text{Top}/X\) be the category of \(G\)-spaces over \(X\), equipped with a Grothendieck topology which is given by defining the covering to be given by all colimits. A based \(G\)-space over \(X\) is a \(G\)-space \(Z\) with maps \(p_Z: Z\to X\) and \(i_Z: X\to Z\) such that \(p_Z\cdot i_Z= \text{Id}_X\). Then one finds that the category \(G\text{Top}_*/X\) of based \(G\)-spaces over \(X\) becomes canonically equivalent to that of sheaves \(F\) of sets over \(G\text{Top}/X\), together with a morphism of sheaves \(C_X\to F\). Here \(C_X\) denotes the constant sheaf of sets on \(G\text{Top}/X\), given by \(C_X(Z)=*\) for \(Z\in G\text{Top}/X\). This implies, for example, that a sheaf of spectra of \(G\text{Top}/X\) can be regarded as a spectrum parametrized over \(X\). So it follows that one may work in the parametrized or fiberwise homotopy category of based \(G\)-spaces over \(X\) instead of the category of sheaves of sets on \(G\text{Top}/X\). This is the key point of the author’s approach.

The main theorem is given in Chapter 4. If one quotes it without worrying about notations, then it is sketched as follows. One considers the \(G\)-map \(f: X\to Y\) which is an equivariant bundle whose fibre is a smooth compact manifold with actions of subgroups of \(G\) varying smoothly over \(Y\). Let \(f_{\sharp}(Z)\) denote the pushout of \(i_Z\) and \(f\) for a based \(G\)-space \(Z\) over \(X\), and let \(f^*(Z)\) denote the pullback of \(p_Z\) and \(f\) for a closed \(G\)-space \(Z\) over \(Y\). Also put \(C_f= X\times_Y X/(X\times_Y X-\Delta)\), called the dualizing object of \(f\), where \(\Delta\) is the diagonal subspace of \(X\times_Y X\) which is viewed as a \(G\)-space over \(X\) by the projection to the first factor then there exists its inverse \(C^{-1}_f\) in the stable homotopy category over \(X\). One defines the functor \(f_!\) by putting \(f_!(Z)= f_{\sharp}(Z\wedge_X C^{-1}_f)\) for a based \(G\)-space \(Z\) over \(X\) and writes \(f_*\) for the right adjoint of \(f^*\). Then the main theorem (Theorem 4.9) states that for a certain spectrum \(E\) there is a natural weak equivalence \(f_*(E)\simeq f_!(E)\) in the category of spectra over \(Y\). In particular, when \(Y\) is a point and \(X\) is a smooth \(G\)-manifold, this implies just Poincaré duality as given in [L. G. Lewis and M. Steinberger, Equivariant stable homotopy theory, Lectures Notes in Mathematics 1213, Springer-Verlag (1986; Zbl 0611.55001)]. A detailed account of the foundations of \(G\)-spaces and \(G\)-spectra over a base space needed for describing this theorem exactly is given in two chapters. In fact, three lemmas are stated in Chapter 3 with proofs deferred to Chapter 7 which is devoted only to the proofs of these results.

The main part of the proof of the theorem is given in Chapter 5. The author begins with the case where \(Y\) is compact. This is outlined as follows. Take an embedding of \(X\) into \(Y\times S^V\) for some \(G\)-representation \(V\) of \(G\) and denote by \(S(\nu_X)\) the sphere bundle of the normal bundle \(\nu_X\) of \(X\) in \(Y\times S^V\). And let \(S(\tau_X)\) denote the sphere bundle of the tangent bundle \(\tau_X\) of \(X\). Then the dualizing object \(C_f\) is naturally equivalent to \(S(\tau_X)\) in the category of based \(G\)-spaces over \(X\), namely \(C_f\simeq S(\tau_X)\), and also clearly it holds that \(S(\tau_X)\wedge_X S(\nu_X)\simeq X\times S^V\). It follows that \(C^{-1}_f\simeq \Sigma^{-V}_X S(\nu_X)\). Also, since \(f_{\sharp}(S(\nu_X))\) becomes the Thom space of \(\nu_X\) in the category of based \(G\)-spaces over \(Y\), one finds that the Pontryagin-Thom construction of the above embedding yields a map \(\varphi: f_*(T)\to f_{\sharp}(T\wedge_X C^{-1}_f)\) of \(G\)-spaces over \(Y\). By actually constructing its homotopy inverse, it is shown that \(\varphi\) is an equivalence. The general case is obtained by a colimit argument over the compact skeleta of \(Y\).

One has two examples of the main theorem. In Chapter 6 the author shows that the Wirthmüller and the Adams isomorphisms [loc. cit.] are special cases of this duality theorem.

Finally, the reviewer wants to add that this paper is very readable and self-contained, and moreover almost every result presented here is proven in full detail.

Reviewer: Haruo Minami (Nara)

### MSC:

55P91 | Equivariant homotopy theory in algebraic topology |

55P42 | Stable homotopy theory, spectra |

55R10 | Fiber bundles in algebraic topology |

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |