| This thesis describes three distinct projects. First, we construct a discrete model of the homotopy theory of S1-spaces. We define a category P with objects composed of a simplicial set and a cyclic set along with compatibility data. P inherits a model structure from the model structures on the constituent categories. We show that there is a Quillen equivalence between P and the model category of S1-spaces where weak equivalences and fibrations are maps inducing weak equivalences and fibrations on passage to fixed point sets.; Next, we study the category of equivariant diagram spectra indexed on the category WG of based G-spaces homeomorphic to finite G-CW-complexes (G a compact Lie group). We show that there is a model structure on this category which admits a monoidal Quillen equivalence to the category of orthogonal G-spectra. There is a model-theoretic identification of the fibrant objects as functors Z such that for A ∈ WG the collection {lcub}Z(A ∧ SW){rcub} forms an O-G-prespectrum as W varies over the universe U. We show that a functor is fibrant if and only if it takes G-homotopy pushouts to G-homotopy pullbacks and is compatible with equivariant Atiyah duality for orbit spaces G/H+.; Finally, we compute the topological Hochschild homology of Thom spectra. For a cofibrant S-algebra R, THH (R) is equivalent to the cyclic bar construction NcycR. When R is an Einfinity-ring spectrum, N cycR ≅ R ⊗ S1. We prove that the Thom spectrum functor M from Einfinity-maps f : X → BG to commutative S-algebras commutes with tensors. Therefore M(f ⊗ S1) ≅ M f ⊗ S1. We construct a symmetric monoidal product ⋆L on based L (1)-spaces over BG by mimicking the construction of the smash product ∧L of S-modules. Ainfinity-spaces and Einfinity-spaces are monoids and commutative monoids for ⋆L . Thus, an Ainfinity-map f : X → BG induces a composite Ncyc⋆f:&vbm0;N cyc⋆X&vbm0;→&vbm0;N cyc⋆BG&vbm0;→BG where the last map is levelwise multiplication. We show that M is a strong symmetric monoidal functor from based L (1)-spaces to S-modules, which implies that M( Ncyc⋆f ) ≅ NcycMf. There is a comparison between NcycBX and the free loop space ΛBX. We show that under certain hypotheses there is a splitting Ncyc Mf ≃ Mf ∧ BX+, allowing us to recover calculations of THH for bordism spectra and H Z /p. |
