(1) A streamlined proof of Goodwillie's n-excisive approximation by Charles Rezk
We give a shorter proof of Goodwillie's, [Geom. Topol. 7 (2003) 645--711; Lemma 1.9], which is the key step in proving that the construction P_n F gives an n-excisive functor.
(2) Unstable splittings for real spectra by Nitu Kitchloo and W Stephen Wilson
We show that the unstable splittings of the spaces in the Omega spectra representing BP, BP<n> and E(n) from [Amer. J. Math. 97 (1975) 101--123] may be obtained for the real analogs of these spectra using techniques similar to those in [Progr. Math. 196 (2001) 35--45]. Explicit calculations for ER(2) are given.
(3) On the geometric realization and subdivisions of dihedral sets by Sho Saito
We extend to dihedral sets Drinfeld's filtered-colimit expressions of the geometric realization of simplicial and cyclic sets. We prove that the group of homeomorphisms of the circle continuously act on the geometric realization of a dihedral set. We also see how these expressions of geometric realization clarify subdivision operations on simplicial, cyclic and dihedral sets defined by Boekstedt, Hsiang and Madsen, and Spalinski.
(4) On the construction of functorial factorizations for model categories by Tobias Barthel and Emily Riehl
We present general techniques for constructing functorial factorizations appropriate for model structures that are not known to be cofibrantly generated. Our methods use `algebraic' characterizations of fibrations to produce factorizations that have the desired lifting properties in a completely categorical fashion. We illustrate these methods in the case of categories enriched, tensored and cotensored in spaces, proving the existence of Hurewicz-type model structures, thereby correcting an error in earlier attempts by others. Examples include the categories of (based) spaces, (based) G-spaces and diagram spectra among others.
(5) Bridge number and tangle products by Ryan Blair
We show that essential punctured spheres in the complement of links with distance three or greater bridge spheres have bounded complexity. We define the operation of tangle product, a generalization of both connected sum and Conway product. Finally, we use the bounded complexity of essential punctured spheres to show that the bridge number of a tangle product is at least the sum of the bridge numbers of the two factor links up to a constant error.
(6) Nonseparating spheres and twisted Heegaard Floer homology by Yi Ni
If a 3-manifold Y contains a nonseparating sphere, then some twisted Heegaard Floer homology of Y is zero. This simple fact allows us to prove several results about Dehn surgery on knots in such manifolds. Similar results have been proved for knots in L-spaces.
(7) Cosimplicial models for the limit of the Goodwillie tower by Rosona Eldred
We call attention to the intermediate constructions T_n F in Goodwillie's Calculus of homotopy functors, giving a new model which naturally gives rise to a family of towers filtering the Taylor tower of a functor. We also establish a surprising equivalence between the homotopy inverse limits of these towers and the homotopy inverse limits of certain cosimplicial resolutions. This equivalence gives a greatly simplified construction for the homotopy inverse limit of the Taylor tower of a functor F under general assumptions.
(8) Homology of moduli spaces of linkages in high-dimensional Euclidean space by Dirk Schutz
We study the topology of moduli spaces of closed linkages in R^d depending on a length vector l in R^n. In particular, we use equivariant Morse theory to obtain information on the homology groups of these spaces, which works best for odd d. In the case d = 5 we calculate the Poincare polynomial in terms of combinatorial information encoded in the length vector.
(9) The Kunneth Theorem in equivariant K-theory for actions of a cyclic group of order 2 by Jonathan Rosenberg
The Kuenneth Theorem for equivariant (complex) K-theory K^*_G, in the form developed by Hodgkin and others, fails dramatically when G is a finite group, and even when G is cyclic of order 2. We remedy this situation in this very simplest case G=Z/2 by using the power of RO(G)-graded equivariant K-theory.
(10) Parametrized ring-spectra and the nearby Lagrangian conjecture by Thomas Kragh Appendix: Mohammed Abouzaid
Let L be an embedded closed connected exact Lagrangian sub-manifold in a connected cotangent bundle T*N. In this paper we prove that such an embedding is, up to a finite covering space lift of T*N, a homology equivalence. We prove this by constructing a fibrant parametrized family of ring spectra FL parametrized by the manifold N. The homology of FL will be (twisted) symplectic cohomology of T*L. The fibrancy property will imply that there is a Serre spectral sequence converging to the homology of FL. The fiber-wise ring structure combined with the intersection product on N induces a product on this spectral sequence. This product structure and its relation to the intersection product on L is then used to obtain the result. Combining this result with work of Abouzaid we arrive at the conclusion that L --> N is always a homotopy equivalence.
(11) A universal characterization of higher algebraic K-theory by Andrew Blumberg, David Gepner and Gonalo Tabuada
In this paper we establish a universal characterization of higher algebraic K-theory in the setting of small stable infinity-categories. Specifically, we prove that connective algebraic K-theory is the universal additive invariant, ie the universal functor with values in spectra which inverts Morita equivalences, preserves filtered colimits, and satisfies Waldhausen's additivity theorem. Similarly, we prove that nonconnective algebraic K-theory is the universal localizing invariant, ie the universal functor that moreover satisfies the Thomason-Trobaugh-Neeman Localization Theorem.
To prove these results, we construct and study two stable infinity-categories of `noncommutative motives'; one associated to additivity and another to localization. In these stable infinity-categories, Waldhausen's S_dot-construction corresponds to the suspension functor and connective and nonconnective algebraic K-theory spectra become corepresentable by the noncommutative motive of the sphere spectrum. In particular, the algebraic K-theory of every scheme, stack, and ring spectrum can be recovered from these categories of noncommutative motives. In the case of a connective ring spectrum R, we prove moreover that its negative K-groups are isomorphic to the negative K-groups of the ordinary ring pi_0(R).
In order to work with these categories of noncommutative motives, we establish comparison theorems between the category of spectral categories localized at the Morita equivalences and the category of small idempotent-complete stable infinity-categories. We also explain in detail the comparison between the infinity-categorical version of Waldhausen K-theory and the classical definition.
As an application of our theory, we obtain a complete classification of the natural transformations from higher algebraic K-theory to topological Hochschild homology (THH) and topological cyclic homology (TC). Notably, we obtain an elegant conceptual description of the cyclotomic trace map.
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