Mathematics

Using Dugger's construction of universal model categories, we produce replacements for simplicial and combinatorial symmetric monoidal model categories with better operadic properties. Namely, these replacements admit a model structure on algebras over any given colored operad. As an application, we show that such symmetric monoidal model categories are classified by commutative ring spectra when the monoidal unit is a compact generator. In other words, they are strong monoidally Quillen equivalent to modules over a uniquely determined commutative ring spectrum.

In this work we study the E ∞ -ring THH( F p ) from various perspectives. Following an identification at the level of E 2 -algebras of THH( F p ) with F p [Ω S 3 ] , the group ring of the E 1 -group Ω S 3 over F p , we use trace methods to compute its algebraic K -theory. We also show that as an E 2 H F p -ring, THH( F p ) is uniquely determined by its homotopy groups. These results hold in fact for THH(k) , where k is any perfect field of characteristic p . Along the way we expand on some of the methods used by Hesselholt-Madsen and later by Speirs to develop certain tools to study the THH of graded ring spectra and the algebraic K -theory of formal DGAs.

We consider orbit configuration spaces associated to finite groups acting freely by orientation preserving homeomorphisms on the 2 -sphere minus a finite number of points. Such action is equivalent to a homography action of a finite subgroup G⊂PGL( C 2 ) on the complex projective line P 1 minus a finite set Z stable under G . We compute the cohomology ring and the Poincaré series of the orbit configuration space C G n ( P 1 ∖Z) . This can be seen as a generalization of the work of Arnold for the classical configuration space C n (C) ( (G,Z)=({1},∞ )). It follows from the work that C G n ( P 1 ∖Z) is formal in the sense of rational homotopy theory. We also prove the existence of an LCS formula relating the Poincaré series of C G n ( P 1 ∖Z) to the ranks of quotients of successive terms of the lower central series of the fundamental group of C G n ( P 1 ∖Z) . The successive quotients correspond to homogenous elements of graded Lie algebras introduced by the author in an earlier work. Such formula is also known for classical configuration spaces of C , where fundamental groups are Artin braid groups and the ranks correspond to dimensions of homogenous elements of the Kohno-Drinfeld Lie algebras.

Greenlees-Shipley and Pol and the author have given an algebraic model for rational (co)free equivariant spectra. We give a model categorical argument showing that the induction-restriction-coinduction functors between categories of (co)free rational equivariant spectra correspond to functors between the algebraic models in the case of connected compact Lie groups.

For certain motivic spectra, we construct a square of spectral sequences relating the effective slice spectral sequence and the motivic Adams spectral sequence. We show the square can be constructed for connective algebraic K-theory, motivic Morava K-theory, and truncated motivic Brown-Peterson spectra. In these cases, we show that the R -motivic effective slice spectral sequence is completely determined by the ρ -Bockstein spectral sequence. Using results of Heard, we also obtain applications to the Hill-Hopkins-Ravenel slice spectral sequences for connective Real K-theory, Real Morava K-theory, and truncated Real Brown-Peterson spectra.

This document aims to give a self-contained account of the parts of abelian group theory that are most relevant for algebraic topology. It is almost purely expository, although there are some slightly unusual features in the treatment of tensor products, torsion products and Ext groups.

We present an algorithm to decompose filtered chain complexes into sums of interval spheres. The algorithm's correctness is proved through principled methods from homotopy theory. Its asymptotic runtime complexity is shown to be cubic in the number of generators, e.g. the simplices of a simplicial complex, as it is based on the row reduction of the boundary matrix by Gaussian elimination. Applying homology to a filtered chain complex, one obtains a persistence module. So our method also provides a new algorithm for the barcode decomposition of persistence modules. The key differences with respect to the state-of-the-art persistent homology algorithms are that our algorithm uses row rather than column reductions, it intrinsically adopts both the clear and compress optimisation strategies, and, finally, it can process rows according to any random order.

This paper is a continuation of Alternative to Morse-Novikov Theory for a closed 1-form(I), where the configurations δ ω and γ ω were defined, and establishes: -- a refinement of Poincaré duality to an equality between the configurations BM δ ω and δ ω resp. BM γ ω and γ ω in complementary dimensions, -- the stability property for the configurations δ ω r , -- a collection of additional results needed for the proof of Theorems 1.2 and 1.3 in the paper mentioned above.

We introduce and study the notion of \emph{semiadditive height} for higher semiadditive ∞ -categories, which generalizes the chromatic height. We show that the higher semiadditive structure trivializes above the height and prove a form of the redshift principle, in which categorification increases the height by one. In the stable setting, we show that a higher semiadditive ∞ -category decomposes into a product according to height, and relate the notion of height to semisimplicity properties of local systems. We place the study of higher semiadditivity and stability in the general framework of smashing localizations of P r L , which we call \emph{modes}. Using this theory, we introduce and study the universal stable ∞ -semiadditive ∞ -category of semiadditive height n , and give sufficient conditions for a stable 1 -semiadditive ∞ -category to be ∞ -semiadditive.

Motivated by potential applications in network theory, engineering and computer science, we study r -ample simplicial complexes. These complexes can be viewed as finite approximations to the Rado complex which has a remarkable property of {\it indestructibility,} in the sense that removing any finite number of its simplexes leaves a complex isomorphic to itself. We prove that an r -ample simplicial complex is simply connected and 2 -connected for r large. The number n of vertexes of an r -ample simplicial complex satisfies exp(Ω( 2 r r √ )) . We use the probabilistic method to establish the existence of r -ample simplicial complexes with n vertexes for any n>r 2 r 2 2 r . Finally, we introduce the iterated Paley simplicial complexes, which are explicitly constructed r -ample simplicial complexes with nearly optimal number of vertexes.