Alpine Perspectives on Algebraic Topology: Third Arolla by Christian Ausoni, Kathryn Hess, Jerome Scherer

By Christian Ausoni, Kathryn Hess, Jerome Scherer

This quantity comprises the complaints of the 3rd Arolla convention on Algebraic Topology, which happened in Arolla, Switzerland, on August 18-24, 2008. This quantity comprises examine papers on reliable homotopy concept, the speculation of operads, localization and algebraic K-theory, in addition to survey papers at the Witten genus, on localization recommendations and on string topology - delivering a huge standpoint of contemporary algebraic topology

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However, when discussing reductions modulo a maximal ideal, it is sometimes more useful to regard the natural periodicity as having degree 2(pn − 1) with associated Z/2(pn − 1)-grading; more precisely, we will follow the ideas of [4] and consider gradings by Z/2(pn − 1) together with the non-trivial bilinear pairing ν : Z/2(pn − 1) × Z/2(pn − 1) −→ {1, −1}; ν(¯i, ¯j) = (−1)ij , where ¯i denotes the residue class i (mod 2(pn − 1)). 16 ANDREW BAKER We will denote by K = E∧E(n) K(n) the version of Morava K-theory associated to E, it is known that E is K-local in the category of E-modules and we can consider the localisation LK (E ∧ E) for which E∗∨ E = π∗ (LK (E ∧ E)).

Of course, if Gal(F/Fp ) is finite this interpretation is still valid but then all maps Gal(F/Fp ) −→ F are continuous. In each case, we obtain an isomorphism of Hopf algebroids ∼ Map(Gal(F/Fp ) F×n , F). 3) F[θk : k n 1]/(θkp − θk : k 1) ∼ = Mapc (S0n , F). 6, so this Hopf algebra over F is unipotent. Tensoring up with K• we have the following graded version. 4. The Hopf algebra (K• , K• ⊗F⊗Fp F(u,θ0 ) K• E) is unipotent. 5. 3) can be extended to all degrees of K• ⊗F⊗Fp F(u,θ0 ) K• E. To make this explicit, we consider Mapc (Sn , Fur ) with the nr action of Gal(F/Fp ) F× pn induced from the action on Sn used in defining Gn and r the F-semilinear action of Gal(F/Fp ) F× pn on Fu obtained by inducing up the r-th power of the natural 1-dimensional representation of F× pn .

2), endpoint-preserving functors φ : [n+1] → [m+1] are represented by constant integer-strings of length m+1, subdivided into n + 1 substrings. Such an integer-string determines, and is determined by, a map of ordinals ψ : [m] → [n]. More precisely, φ and ψ determine each other by the formulas: ψ(i)+1 = min{j | φ(j) > i} and φ(j)−1 = max{i | ψ(i) < j}. This duality is often referred to as Joyal-duality [29]. 5. The underlying category of the lattice path operad is ∆. Proof. Lu (m, n) = Cat∗,∗ ([n + 1], [m + 1]) = Cat([m], [n]) = ∆([m], [n]).

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