The aim of this paper is to show that homotopy pro-nilpotent structured ring spectra are \({ \mathsf {TQ} }\)-local, where structured ring spectra are described as algebras over a spectral operad \({ \mathcal {O} }\). Here, \({ \mathsf {TQ} }\) is short for topological Quillen homology, which is weakly equivalent to \({ \mathcal {O} }\)-algebra stabilization. An \({ \mathcal {O} }\)-algebra is called

We study some properties of the coset poset associated with the family of subgroups of class \(\le 2\) of a nilpotent group of class \(\le 3\). We prove that under certain assumptions on the group the coset poset is simply-connected if and only if the group is 2-Engel, and 2-connected if and only if the group is nilpotent of class 2 or less. We determine the homotopy type of the coset poset for the

Hess and Shipley defined an invariant of coalgebra spectra called topological coHochschild homology, and Bohmann–Gerhardt–Høgenhaven–Shipley–Ziegenhagen developed a coBökstedt spectral sequence to compute the homology of \(\mathrm {coTHH}\) for coalgebras over the sphere spectrum. We construct a relative coBökstedt spectral sequence to study \(\mathrm {coTHH}\) of coalgebra spectra over any commutative

The usual iterated integral map given by Chen produces an equivalence between the two-sided bar complex on differential forms and the de Rham complex on the path space. This map fails to make sense when considering the curved differential graded algebra of bundle-valued forms with a covariant derivative induced by a connection. In this paper, we define a curved version of Chen’s iterated integral that

The goal of this paper is to establish an equivalence of profinite completions of pro-spaces in model category theory and in \(\infty \)-category theory. As an application, we show that the author’s comparison theorem for algebro-geometric objects in the setting of model categories recovers that of David Carchedi in the setting of \(\infty \)-categories.

For a unital commutative Banach algebra A and its closed ideal I, we study the relative Čech cohomology of the pair \((\mathrm {Max}(A),\mathrm {Max}(A/I))\) of maximal ideal spaces and show a relative version of the main theorem of Lupton et al. (Trans Amer Math Soc 361:267–296, 2009): \(\check{\mathrm {H}}^{j}(\mathrm {Max}(A),\mathrm {Max}(A/I));{\mathbb {Q}}) \cong \pi _{2n-j-1}(Lc_{n}(I))_{{\mathbb

The aim of this paper is twofold. In the first part, we consider twisted Rota–Baxter operators on associative algebras that were introduced by Uchino as a noncommutative analogue of twisted Poisson structures. We construct an \(L_\infty \)-algebra whose Maurer–Cartan elements are given by twisted Rota–Baxter operators. This leads to cohomology associated to a twisted Rota–Baxter operator. This cohomology

We compute the Lusternik–Schnirelmann category and all the higher topological complexities of non-k-equal manifolds \(M_d^{(k)}(n)\) for certain values of d, k and n. This includes instances where \(M_d^{(k)}(n)\) is known to be rationally non-formal. The key ingredient in our computations is the knowledge of the cohomology ring \(H^*(M_d^{(k)}(n))\) as described by Dobrinskaya and Turchin. A fine

The goal of this short paper is to study the convergence of the Taylor tower of the identity functor in the context of operadic algebras in spectra. Specifically, we show that if A is a \((-1)\)-connected \({ \mathcal {O} }\)-algebra with 0-connected \({ \mathsf {TQ} }\)-homology spectrum \({ \mathsf {TQ} }(A)\), then there is a natural weak equivalence \(P_\infty ({ \mathrm {id} })A \simeq A^\wedge

We introduce a construction that produces from each bialgebra H an operad \(\mathsf {Ass}_H\) controlling associative algebras in the monoidal category of H-modules or, briefly, H-algebras. When the underlying algebra of this bialgebra is Koszul, we give explicit formulas for the minimal model of this operad depending only on the coproduct of H and the Koszul model of H. This operad is seldom quadratic—and

For a simply connected rationally elliptic CW-complex X, we show that the cohomology and the homotopy Euler–Poincaré characteristics are related to two new numerical invariants namely \(\eta _{X}\) and \(\rho _{X}\) which we define using the Whitehead exact sequences of the Quillen and the Sullivan models of X.

We establish connectedness criteria for graphs associated to monomials in certain quotients of the mod 2 dual Steenrod algebra \(\mathscr {A}^*\). We also investigate questions about trees and Hamilton cycles in the context of these graphs. Finally, we improve upon a known connection between the graph theoretic interpretation of \(\mathscr {A}^*\) and its structure as a Hopf algebra.

We compare different algebraic structures in twisted equivariant K-theory for proper actions of discrete groups. After the construction of a module structure over untwisted equivariant K-theory, we prove a completion Theorem of Atiyah–Segal type for twisted equivariant K-theory. Using a universal coefficient theorem, we prove a cocompletion Theorem for twisted Borel K-homology for discrete groups.

We introduce graded \({\mathbb {E}}_{\infty }\)-rings and graded modules over them, and study their properties. We construct projective schemes associated to connective \({\mathbb {N}}\)-graded \({\mathbb {E}}_{\infty }\)-rings in spectral algebraic geometry. Under some finiteness conditions, we show that the \(\infty \)-category of almost perfect quasi-coherent sheaves over a spectral projective scheme

We compute the homotopy groups of the \(C_2\) fixed points of equivariant topological modular forms at the prime 2 using the descent spectral sequence. We then show that as a \({\mathrm {TMF}}\)-module, it is isomorphic to the tensor product of \({\mathrm {TMF}}\) with an explicit finite cell complex.

We determine the Lusternik-Schnirelmann category of the projective product spaces introduced by D. Davis. We also obtain an upper bound for the topological complexity of these spaces, which improves the estimate given by J. González, M. Grant, E. Torres-Giese, and M. Xicoténcatl.

Given an augmentation for a Legendrian surface in a 1-jet space, \(\Lambda \subset J^1(M)\), we explicitly construct an object, \(\mathcal {F} \in \mathbf {Sh}^\bullet _{\Lambda }(M\times \mathbb {R}, \mathbb {K})\), of the (derived) category from Viterbo (Shende, V., Treumann, D., Zaslow, E. Invent Math 207(3), 1031–1133 (2017)) of constructible sheaves on \(M\times \mathbb {R}\) with singular support

The string coproduct is a coproduct on the homology with field coefficients of the free loop space of a closed oriented manifold introduced by Sullivan in string topology. The coproduct and the Chas-Sullivan loop product give an infinitesimal bialgebra structure on the homology if the Euler characteristic is zero. The aim of this paper is to study the string coproduct using Sullivan models in rational

We explicitly construct and investigate a number of examples of \({\mathbb {Z}}/p^r\)-equivariant formal group laws and complex-oriented spectra, including those coming from elliptic curves and p-divisible groups, as well as some other related examples.

In this paper, we develop the \(A_\infty \)-analog of the Maurer-Cartan simplicial set associated to an \(L_\infty \)-algebra and show how we can use this to study the deformation theory of \(\infty \)-morphisms of algebras over non-symmetric operads. More precisely, we first recall and prove some of the main properties of \(A_\infty \)-algebras like the Maurer-Cartan equation and twist. One of our

We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: 1. On marked simplicial sets (due to Lurie [31]), 2. On bisimplicial spaces (due to deBrito [12]), 3. On bisimplicial sets, 4. On marked simplicial spaces. The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories and complete

In 1947, N.E. Steenrod defined the Steenrod Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cup-i products of cocycles. He later recast the construction in more general homological terms, using group homology and acyclic model methods, rather than explicit cochain formulae, to define mod p operations for all primes p. Steenrod’s student J. Adem applied the homological

Let R be a right noetherian ring. We introduce the concept of relative singularity category \(\Delta _{\mathcal {X} }(R)\) of R with respect to a contravariantly finite subcategory \(\mathcal {X} \) of \({\text {{mod{-}}}}R.\) Along with some finiteness conditions on \(\mathcal {X} \), we prove that \(\Delta _{\mathcal {X} }(R)\) is triangle equivalent to a subcategory of the homotopy category \(\mathbb

We describe two ways to define higher order Toda brackets in a pointed simplicial model category \({\mathcal {D}}\): one is a recursive definition using model categorical constructions, and the second uses the associated simplicial enrichment. We show that these two definitions agree, by providing a third, diagrammatic, description of the Toda bracket, and explain how it serves as the obstruction to

The equivalence between dg duality and Verdier duality has been established for cyclic operads earlier. We propose a generalization of this correspondence from cyclic operads and dg duality to twisted modular operads and the Feynman transform. Specifically, for each twisted modular operad \(\mathcal {P}\) (taking values in dg-vector spaces over a field k of characteristic 0), there is a certain sheaf

As a step towards understanding the \(\mathrm {tmf}\)-based Adams spectral sequence, we compute the K(1)-local homotopy of \(\mathrm {tmf}\wedge \mathrm {tmf}\), using a small presentation of \(L_{K(1)}\mathrm {tmf}\) due to Hopkins. We also describe the K(1)-local \(\mathrm {tmf}\)-based Adams spectral sequence.

We show that the Cantor–Schröder–Bernstein Theorem for homotopy types, or \(\infty \)-groupoids, holds in the following form: For any two types, if each one is embedded into the other, then they are equivalent. The argument is developed in the language of homotopy type theory, or Voevodsky’s univalent foundations (HoTT/UF), and requires classical logic. It follows that the theorem holds in any boolean

We define a category parameterizing Calabi–Yau algebra objects in an infinity category of spans. Using this category, we prove that there are equivalences of infinity categories relating, firstly: 2-Segal simplicial objects in C to algebra objects in Span(C); and secondly: 2-Segal cyclic objects in C to Calabi–Yau algebra objects in Span(C).

Magnitude homology is a bigraded homology theory for finite graphs defined by Hepworth and Willerton, categorifying the power series invariant known as magnitude which was introduced by Leinster. We analyze the structure and implications of torsion in magnitude homology. We show that any finitely generated abelian group may appear as a subgroup of the magnitude homology of a graph, and, in particular

In this paper, we consider Leibniz algebras with derivations. A pair consisting of a Leibniz algebra and a distinguished derivation is called a LeibDer pair. We define a cohomology theory for LeibDer pair with coefficients in a representation. We study central extensions of a LeibDer pair. In the next, we generalize the formal deformation theory to LeibDer pairs in which we deform both the Leibniz

Simons and Sullivan constructed a model of differential K-theory, and showed that the differential K-theory functor fits into a hexagon diagram. They asked whether, like the case of differential characters, this hexagon diagram uniquely determines the differential K-theory functor. This article provides a partial affirmative answer to their question: For any fixed compact manifold, the differential

In this paper we study N-differential graded categories and their derived categories. First, we introduce modules over an N-differential graded category. Then we show that they form a Frobenius category and that its homotopy category is triangulated. Second, we study the properties of its derived category and give triangle equivalences of Morita type between derived categories of N-differential graded

We use derived localization of the bar and nerve constructions to provide simple proofs of a number of results in algebraic topology, both known and new. This includes a recent generalization of Adams’s cobar-construction to the non-simply connected case, and a new algebraic model for the homotopy theory of connected topological spaces as an \(\infty \)-category of discrete monoids.

We introduce the notion of cumulants as applied to linear maps between associative (or commutative) algebras that are not compatible with the algebraic product structure. These cumulants have a close relationship with \(A_{\infty }\) and \(C_{\infty }\) morphisms, which are the classical homotopical tools for analyzing deformations of algebraically compatible linear maps. We look at these two different

In the Lagrangian approach to 2-dimensional sigma models, B-fields and D-branes contribute topological terms to the action of worldsheets of both open and closed strings. We show that these terms naturally fit into a 2-dimensional, smooth open-closed functorial field theory (FFT) in the sense of Atiyah, Segal, and Stolz–Teichner. We give a detailed construction of this smooth FFT, based on the definition

We examine the relation between the gauge groups of \(\mathrm {SU}(n)\)- and \(\mathrm {PU}(n)\)-bundles over \(S^{2i}\), with \(2\le i\le n\), particularly when n is a prime. As special cases, for \(\mathrm {PU}(5)\)-bundles over \(S^4\), we show that there is a rational or p-local equivalence \(\mathcal {G}_{2,k}\simeq _{(p)}\mathcal {G}_{2,l}\) for any prime p if, and only if, \((120,k)=(120,l)\)

In the 1980’s, Ravenel introduced sequences of spectra X(n) and T(n) which played an important role in the proof of the Nilpotence Theorem of Devinatz–Hopkins–Smith. In the present paper, we solve the homotopy limit problem for topological Hochschild homology of X(n), which is a generalized version of the Segal Conjecture for the cyclic groups of prime order. This result is the first step towards computing

The purpose of this article is to record the computation of the homotopy groups of 3-local \(\mathrm {tmf}\) via the Adams spectral sequence.

We introduce a notion of c-group, which is a group up to congruence relation and consider the corresponding category. Extensions, actions and crossed modules (c-crossed modules) are defined in this category and the semi-direct product is constructed. We prove that each categorical group gives rise to a c-group and to a c-crossed module, which is a connected, special and strict c-crossed module in the

In a previous paper we showed that, under some assumptions, the relative K-group in the Burns–Flach formulation of the equivariant Tamagawa number conjecture (ETNC) is canonically isomorphic to a K-group of locally compact equivariant modules. Our approach as well as the standard one both involve presentations: One due to Bass–Swan, applied to categories of finitely generated projective modules; and

We show that any homotopy Gerstenhaber algebra is naturally a strongly homotopy commutative (shc) algebra in the sense of Stasheff–Halperin with a homotopy associative structure map. In the presence of certain additional operations corresponding to a \(\mathbin {\cup _1}\)-product on the bar construction, the structure map becomes homotopy commutative, so that one obtains an shc algebra in the sense

We introduce a notion of categorical homotopic distance between functors by adapting the notion of homotopic distance in topological spaces, recently defined by the authors, to the context of small categories. Moreover, this notion generalizes the work on categorical LS-category of small categories by Tanaka.

We introduce a notion of oriented dialgebra and develop a cohomology theory for oriented dialgebras by mixing the standard chain complexes computing group cohomology and associative dialgebra cohomology. We also introduce a formal deformation theory for oriented dialgebras and show that cohomology of oriented dialgebras controls such deformations.

We provide a general definition of Toda brackets in a pointed model category, show how they serve as obstructions to rectification, and explain their relation to the classical stable operations.

The goal of the paper is to define Hochschild and cyclic homology for bornological coarse spaces, i.e., lax symmetric monoidal functors \({{\,\mathrm{\mathcal {X}HH}\,}}_{}^G\) and \({{\,\mathrm{\mathcal {X}HC}\,}}_{}^G\) from the category \(G\mathbf {BornCoarse}\) of equivariant bornological coarse spaces to the cocomplete stable \(\infty \)-category \(\mathbf {Ch}_\infty \) of chain complexes reminiscent

In a 2012 note in Comptes Rendus Mathématique, the author did try to answer a question of Jean-Pierre Serre; it has recently been announced that the scope of that answer needs an adjustment, and the details of this adjustment are given in the present paper. The original question is the following. Consider the ring of integers \(\mathcal {O}\) in an imaginary quadratic number field, and the Borel–Serre

The purpose of this paper is to develop a theory of \((\infty , 1)\)-stacks, in the sense of Hirschowitz–Simpson’s ‘Descent Pour Les n–Champs’, using the language of quasi-category theory and the author’s local Joyal model structure. The main result is a characterization of \((\infty , 1)\)-stacks in terms of mapping space presheaves. An important special case of this theorem gives a sufficient condition

In this paper we introduce concepts of higher equivariant and invariant topological complexities and study their properties. Then we compare them with equivariant LS-category. We give lower and upper bounds for these new invariants. We compute some of these invariants for moment angle complexes.

Let A be a finite abelian p-group of rank at least 2. We show that \(E^0(BA)/I_{tr}\), the quotient of the Morava E-cohomology of A by the ideal generated by the image of the transfers along all proper subgroups, contains p-torsion. The proof makes use of transchromatic character theory.

Let X be a simply connected space with finite-dimensional rational homotopy groups. Let \(p_\infty :UE \rightarrow B\mathrm {aut}_1(X)\) be the universal fibration of simply connected spaces with fibre X. We give a DG Lie algebra model for the evaluation map \( \omega :\mathrm {aut}_1(B\mathrm {aut}_1(X_\mathbb {Q})) \rightarrow B\mathrm {aut}_1(X_\mathbb {Q})\) expressed in terms of derivations of

We consider the décalage construction \({{\,\mathrm{Dec}\,}}\) and its right adjoint \(T\). These functors are induced on the category of simplicial objects valued in any bicomplete category \({\mathcal {C}}\) by the ordinal sum. We identify \(T{{\,\mathrm{Dec}\,}}X\) with the path object \(X^{\Delta [1]}\) for any simplicial object X. We then use this formula to produce an explicit retracting homotopy

We prove that in formal dimension \(\le 20\) the Hilali conjecture holds, i.e. that the total dimension of the rational homology bounds from above the total dimension of the rational homotopy for a simply connected rationally elliptic space.

Manifold calculus is a form of functor calculus that analyzes contravariant functors from some categories of manifolds to topological spaces by providing analytic approximations to them. In this paper, using the technique of the h-principle, we show that for a symplectic manifold N, the analytic approximation to the Lagrangian embeddings functor \(\mathrm {Emb}_{\mathrm {Lag}}(-,N)\) is the totally

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K. Borsuk in 1979, at the Topological Conference in Moscow, introduced the concept of capacity and depth of a compactum. In this paper we compute the capacity and depth of compact surfaces. We show that the capacity and depth of every compact orientable surface of genus \(g\ge 0\) is equal to \(g+2\). Also, we prove that the capacity and depth of a compact non-orientable surface of genus \(g>0\) is