Mathematics

We use model theory to study relative profinite rigidity of 3 -manifold groups and show that given any residually finite group ? with finite character variety and single-cusped finite volume hyperbolic 3 -manifold M , cofinitely many Dehn fillings M p/q are profinitely distinguishable from ? .

Let R be a (P.I.D) and let T(V),∂) be a free R -dga. The quasi-isomorphism type of (T(V),∂) is the set, denoted {(T(V),∂)} , of all free dgas which are quasi-isomorphic to (T(V),∂) . In this paper we investigate to characterize and to compute the set {(T(V),∂)} for a new class of free dgas called perfect (a special kind of a perfect dga is the Adams-Hilton model of simply connected CW-complex such that H ∗ (X,R) is free). We show that if (T(V),∂) and (T(W),δ) are two perfect dgas, then (T(W),δ)∈{(T(V),∂)} if and only if their Whitehead exact sequences are isomorphic. Moreover we show that every dga (T(V),∂) can be split to give a pair ((T(V), ∂ ˜ ),( π n ) n≥2 ) consisting with a perfect dga (T(V), ∂ ˜ ) and a family of extensions ( π n ) n≥2 and we establish that if (T(W), δ ˜ )∈{(T(V), ∂ ˜ )} and if the extensions ( π n ) n≥2 and ( π ′ n ) n≥2 are isomorphic (in a certain sense), then (T(W),δ)∈{(T(V),∂)} .

We study the spaces of embeddings of manifolds in a Euclidean space. More precisely we look at the homotopy fiber of the inclusion of these spaces to the spaces of immersions. As a main result we express the rational homotopy type of connected components of those embedding spaces through combinatorially defined L ∞ -algebras of diagrams.

Banagl's method of intersection spaces allows to modify certain types of stratified pseudomanifolds near the singular set in such a way that the rational Betti numbers of the modified spaces satisfy generalized Poincaré duality in analogy with Goresky-MacPherson's intersection homology. In the case of one isolated singularity, we show that the duality isomorphism comes from a nondegenerate intersection pairing which depends on the choice of a chain representative of the fundamental class of the regular stratum. On the technical side, we use piecewise linear polynomial differential forms due to Sullivan to define a suitable commutative cochain algebra model for intersection spaces. Our construction parallels Banagl's commutative cochain algebra of smooth differential forms modeling intersection space cohomology, and we show that both algebras are weakly equivalent.

We provide a new description of logarithmic topological André-Quillen homology in terms of the indecomposables of an augmented ring spectrum. The new description allows us to interpret logarithmic TAQ as an abstract cotangent complex, and leads to an étale descent formula for logarithmic topological Hochschild homology. The latter is analogous to results of Weibel-Geller for Hochschild homology of discrete rings, and of McCarthy-Minasian and Mathew for topological Hochschild homology. We also summarize and clarify analogous results relating notions of formal étaleness defined in terms of ordinary THH and TAQ.

Let G be a compact connected Lie group and n⩾1 an integer. Consider the space of ordered commuting n -tuples in G , Hom( Z n ,G) , and its quotient under the adjoint action, Rep( Z n ,G):=Hom( Z n ,G)/G . In this article we study and in many cases compute the homotopy groups π 2 (Hom( Z n ,G)) . For G simply--connected and simple we show that π 2 (Hom( Z 2 ,G))≅Z and π 2 (Rep( Z 2 ,G))≅Z , and that on these groups the quotient map Hom( Z 2 ,G)→Rep( Z 2 ,G) induces multiplication by the Dynkin index of G . More generally we show that if G is simple and Hom( Z 2 ,G ) 1 ⊆Hom( Z 2 ,G) is the path--component of the trivial homomorphism, then H 2 (Hom( Z 2 ,G ) 1 ;Z) is an extension of the Schur multiplier of π 1 (G ) 2 by Z . We apply our computations to prove that if B com G 1 is the classifying space for commutativity at the identity component, then π 4 ( B com G 1 )≅Z⊕Z , and we construct examples of non-trivial transitionally commutative structures on the trivial principal G -bundle over the sphere S 4 .

The sectional category of a subgroup inclusion H?�G can be defined as the sectional category of the corresponding map between Eilenberg--MacLane spaces. We extend a characterization of topological complexity of aspherical spaces given by Farber, Grant, Lupton and Oprea to the context of sectional category of subgroup inclusions and investigate it by means of Adamson cohomology theory.

We prove that the topological complexity TC(?) equals cd(???) for certain toral relatively hyperbolic groups ? .

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex K , we specify a necessary and sufficient combinatorial condition for the commutator subgroup R C ′ K of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex R K , to be a one-relator group; and for the Pontryagin algebra H ∗ (Ω Z K ) of the moment-angle complex to be a one-relator algebra. We also give a homological characterisation of these properties. For R C ′ K , it is given by a condition on the homology group H 2 ( R K ) , whereas for H ∗ (Ω Z K ) it is stated in terms of the bigrading of the homology groups of Z K .

We give a description of unital operads in a symmetric monoidal category as monoids in a monoidal category of unital Λ -sequences. This is a new variant of Kelly's old description of operads as monoids in the monoidal category of symmetric sequences. The monads associated to unital operads are the ones of interest in iterated loop space theory and factorization homology, among many other applications. Our new description of unital operads allows an illuminating comparison between the two-sided monadic bar constructions used in such applications and ``classical'' monoidal two-sided bar constructions.