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Daniel Davis
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Let G be a profinite group, {X_alpha}_alpha a cofiltered diagram of discrete G-spectra, and Z a spectrum with trivial G-action. We show how to define the homotopy fixed point spectrum F(Z, holim_alpha X_alpha)^{hG} and that when G has finite virtual cohomological dimension (vcd), it is equivalent to F(Z, holim_alpha (X_alpha)^{hG}). With these tools, we show that the K(n)-local Spanier-Whitehead dual is always a homotopy fixed point spectrum, a well-known Adams-type spectral sequence is actually a descent spectral sequence, and, for a sufficiently nice k-local profinite G-Galois extension E, with K a closed normal subgroup of G, the equivalence (E^{h_kK})^{h_kG/K} simeq E^{h_kG} (due to Behrens and the author), where (-)^{h_k(-)} denotes k-local homotopy fixed points, can be upgraded to an equivalence that just uses ordinary (non-local) homotopy fixed points, when G/K has finite vcd.
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In this thesis we will investigate rational G-spectra for a profinite group G. We will provide an algebraic model for this model category whose injective dimension can be calculated in terms of the Cantor-Bendixson rank of the space of closed subgroups of G, denoted SG. The algebraic model we consider is chain complexes of Weyl-G-sheaves of rational vector spaces over the spaces. The key step in proving that this is an algebraic model for G-spectra is in proving that the category of rational G-Mackey functors is equivalent to Weyl-G-sheaves. In addition to the fact that this sheaf description utilises the topology of G and the closed subgroups of G in a more explicit way than Mackey functors do, we can also calculate the injective dimension. In the final part of the thesis we will see that the injective dimension of the category of Weyl-G-sheaves can be calculated in terms of the Cantor-Bendixson rank of SG, hence giving the injective dimension of the category of Mackey functors via the earlier equivalence.
If C is the model category of simplicial presheaves on a site with enough points, with fibrations equal to the global fibrations, then it is well-known that the fibrant objects are, in general, mysterious. Thus, it is not surprising that, when G is a profinite group, the fibrant objects in the model category of discrete G-spectra are also difficult to get a handle on. However, with simplicial presheaves, it is possible to construct an explicit fibrant model for an object in C, under certain finiteness conditions. Similarly, in this paper, we show that if G has finite virtual cohomological dimension and X is a discrete G-spectrum, then there is an explicit fibrant model for X. Also, we give several applications of this concrete model related to closed subgroups of G.
For a profinite group $G$, let $(text{-})^{hG}$, $(text{-})^{h_dG}$, and $(text{-})^{hG}$ denote continuous homotopy fixed points for profinite $G$-spectra, discrete $G$-spectra, and continuous $G$-spectra (coming from towers of discrete $G$-spectra), respectively. We establish some connections between the first two notions, and by using Postnikov towers, for $K vartriangleleft_c G$ (a closed normal subgroup), give various conditions for when the iterated homotopy fixed points $(X^{hK})^{hG/K}$ exist and are $X^{hG}$. For the Lubin-Tate spectrum $E_n$ and $G <_c G_n$, the extended Morava stabilizer group, our results show that $E_n^{hK}$ is a profinite $G/K$-spectrum with $(E_n^{hK})^{hG/K} simeq E_n^{hG}$, by an argument that possesses a certain technical simplicity not enjoyed by either the proof that $(E_n^{hK})^{hG/K} simeq E_n^{hG}$ or the Devinatz-Hopkins proof (which requires $|G/K| < infty$) of $(E_n^{dhK})^{h_dG/K} simeq E_n^{dhG}$, where $E_n^{dhK}$ is a construction that behaves like continuous homotopy fixed points. Also, we prove that (in general) the $G/K$-homotopy fixed point spectral sequence for $pi_ast((E_n^{hK})^{hG/K})$, with $E_2^{s,t} = H^s_c(G/K; pi_t(E_n^{hK}))$ (continuous cohomology), is isomorphic to both the strongly convergent Lyndon-Hochschild-Serre spectral sequence of Devinatz for $pi_ast(E_n^{dhG})$, with $E_2^{s,t} = H^s_c(G/K; pi_t(E_n^{dhK}))$, and the descent spectral sequence for $pi_ast((E_n^{hK})^{hG/K})$.
The project of Greenlees et al. on understanding rational G-spectra in terms of algebraic categories has had many successes, classifying rational G-spectra for finite groups, SO(2), O(2), SO(3), free and cofree G-spectra as well as rational toral G-spectra for arbitrary compact Lie groups. This paper provides an introduction to the subject in two parts. The first discusses rational G-Mackey functors, the action of the Burnside ring and change of group functors. It gives a complete proof of the well-known classification of rational Mackey functors for finite G. The second part discusses the methods and tools from equivariant stable homotopy theory needed to obtain algebraic models for rational G-spectra. It gives a summary of the key steps in the classification of rational G-spectrain terms of a symmetric monoidal algebraic category. Having these two parts in the same place allows one to clearly see the analogy between the algebraic and topological classifications.
We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. We recall (from May, Quinn, and Ray) that a commutative ring spectrum A has a spectrum of units gl(A). To a map of spectra f: b -> bgl(A), we associate a commutative A-algebra Thom spectrum Mf, which admits a commutative A-algebra map to R if and only if b -> bgl(A) -> bgl(R) is null. If A is an associative ring spectrum, then to a map of spaces f: B -> BGL(A) we associate an A-module Thom spectrum Mf, which admits an R-orientation if and only if B -> BGL(A) -> BGL(R) is null. We also note that BGL(A) classifies the twists of A-theory. We develop and compare two approaches to the theory of Thom spectra. The first involves a rigidified model of A-infinity and E-infinity spaces. Our second approach is via infinity categories. In order to compare these approaches to one another and to the classical theory, we characterize the Thom spectrum functor from the perspective of Morita theory.
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