The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in [Seg04] in order to formalize the concept of field theories. Our main result identifies the homotopy type of the classifying space of the embedded d-dimensional cobordism category for all d. For d = 2, our results lead to a new proof of the generalized Mumford conjecture, somewhat different in spirit from the original one, presented in [MW02]. Contents 1. Introduction and results 2. The cobordism category and its sheaves 2.1. The cobordism category 2.2. Recollection from [MW02] on sheaves 2.3. A sheaf model for the cobordism category 2.4. Cocycle sheaves 3. The Thom spectra and their sheaves 3.1. The spectrum MT (d) and its infinite loop space 3.2. Using Phillips' submersion theorem 4. Proof of the main theorem 5. Tangential structures 6. Connectedness issues 6.1. Discussion 6.2. Surgery 6.3. Connectivity 6.4. Parametrized surgery 7. Harer type stability and C 2 References
D. Mumford conjectured in [33] that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra generated by certain classes κ i of dimension 2i. For the purpose of calculating rational cohomology, one may replace the stable moduli space of Riemann surfaces by BΓ ∞ , where Γ ∞ is the group of isotopy classes of automorphisms of a smooth oriented connected surface of "large" genus. Tillmann's theorem [44]
Introduction 1. Topological Hochschild homology and localization 2. The homotopy groups of T (A|K) 3. The de Rham-Witt complex and TR • * (A|K; p) 4. Tate cohomology and the Tate spectrum 5. The Tate spectral sequence for T (A|K) 6. The pro-system TR • * (A|K; p, Z/p v ) Appendix A. Truncated polynomial algebras References * The first named author was supported in part by NSF Grant and the Alfred P. Sloan Foundation. The second named author was supported in part by The American Institute of Mathematics. i ! −→ TC(C b z (P A ); p) j −→ TC(C b q (P A ); p) ∂ −→ Σ TC(C b z (P A ) q ; p), *
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