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Homotopy pronilpotent structured ring spectra and topological Quillen localization J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20220916
Yu ZhangThe aim of this paper is to show that homotopy pronilpotent 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

On families of nilpotent subgroups and associated coset posets J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20220914
Simon Gritschacher, Bernardo VillarrealWe 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 simplyconnected if and only if the group is 2Engel, and 2connected if and only if the group is nilpotent of class 2 or less. We determine the homotopy type of the coset poset for the

Toward a minimal model for $$H_*(\overline{\mathcal {M}})$$ H ∗ ( M ¯ ) J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20220908
Benjamin C. Ward 
Unitary calculus: model categories and convergence J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20220809
Niall Taggart 
Computations of relative topological coHochschild homology J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20220805
Sarah KlandermanHess 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

Smashing localizations in equivariant stable homotopy J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20220715
Christian Carrick 
Modeling bundlevalued forms on the path space with a curved iterated integral J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20220713
Cheyne Glass, Corbett ReddenThe usual iterated integral map given by Chen produces an equivalence between the twosided 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 bundlevalued forms with a covariant derivative induced by a connection. In this paper, we define a curved version of Chen’s iterated integral that

An equivalence of profinite completions J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20220706
ChangYeon ChoughThe goal of this paper is to establish an equivalence of profinite completions of prospaces in model category theory and in \(\infty \)category theory. As an application, we show that the author’s comparison theorem for algebrogeometric objects in the setting of model categories recovers that of David Carchedi in the setting of \(\infty \)categories.

Rational stabilization and maximal ideal spaces of commutative Banach algebras J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20220701
Kazuhiro KawamuraFor 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 _{2nj1}(Lc_{n}(I))_{{\mathbb

Cohomology and deformations of twisted Rota–Baxter operators and NSalgebras J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20220505
Apurba DasThe 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

On Lusternik–Schnirelmann category and topological complexity of nonkequal manifolds J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20220425
Jesús González, José Luis LeónMedinaWe compute the Lusternik–Schnirelmann category and all the higher topological complexities of nonkequal 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 nonformal. 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

$${ \mathsf {TQ} }$$ TQ completion and the Taylor tower of the identity functor J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20220330
Nikolas SchonsheckThe 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 0connected \({ \mathsf {TQ} }\)homology spectrum \({ \mathsf {TQ} }(A)\), then there is a natural weak equivalence \(P_\infty ({ \mathrm {id} })A \simeq A^\wedge

Resolutions of operads via Koszul (bi)algebras J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20220303
Pedro TamaroffWe introduce a construction that produces from each bialgebra H an operad \(\mathsf {Ass}_H\) controlling associative algebras in the monoidal category of Hmodules or, briefly, Halgebras. 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

On the Euler–Poincaré characteristics of a simply connected rationally elliptic CWcomplex J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20220222
Mahmoud BenkhalifaFor a simply connected rationally elliptic CWcomplex 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.

Connectedness of graphs arising from the dual Steenrod algebra J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20220208
Donald M. LarsonWe 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.

The completion theorem in twisted equivariant Ktheory for proper actions J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20220131
Noé Bárcenas, Mario VelásquezWe compare different algebraic structures in twisted equivariant Ktheory for proper actions of discrete groups. After the construction of a module structure over untwisted equivariant Ktheory, we prove a completion Theorem of Atiyah–Segal type for twisted equivariant Ktheory. Using a universal coefficient theorem, we prove a cocompletion Theorem for twisted Borel Khomology for discrete groups.

On graded $${\mathbb {E}}_{\infty }$$ E ∞ rings and projective schemes in spectral algebraic geometry J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20220131
Mariko Ohara, Takeshi ToriiWe 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 quasicoherent sheaves over a spectral projective scheme

$$C_2$$ C 2 equivariant topological modular forms J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20220110
Chua, DexterWe 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.

Marked colimits and higher cofinality J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20211216
Abellán García, Fernando 
On the LScategory and topological complexity of projective product spaces J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20211108
Fişekci, Seher, Vandembroucq, LucileWe determine the LusternikSchnirelmann 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. TorresGiese, and M. Xicoténcatl.

Overcategories and undercategories of cofibrantly generated model categories J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20211013
Hirschhorn, Philip S. 
Sheaves via augmentations of Legendrian surfaces J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20211009
Rutherford, Dan, Sullivan, MichaelGiven an augmentation for a Legendrian surface in a 1jet 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

Rational model for the string coproduct of pure manifolds J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20211007
Naito, TakahitoThe 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 ChasSullivan 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

Equivariant formal group laws and complexoriented spectra over primary cyclic groups: elliptic curves, Barsotti–Tate groups, and other examples J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210927
Hu, Po, Kriz, Igor, Somberg, PetrWe explicitly construct and investigate a number of examples of \({\mathbb {Z}}/p^r\)equivariant formal group laws and complexoriented spectra, including those coming from elliptic curves and pdivisible groups, as well as some other related examples.

On the MaurerCartan simplicial set of a complete curved $$A_\infty $$ A ∞ algebra J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210925
de Kleijn, Niek, Wierstra, FelixIn this paper, we develop the \(A_\infty \)analog of the MaurerCartan 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 nonsymmetric operads. More precisely, we first recall and prove some of the main properties of \(A_\infty \)algebras like the MaurerCartan equation and twist. One of our

Quasicategories vs. Segal spaces: Cartesian edition J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210820
Rasekh, NimaWe 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 quasicategories and complete

A cochain level proof of Adem relations in the mod 2 Steenrod algebra J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210819
Brumfiel, Greg, MedinaMardones, Anibal, Morgan, JohnIn 1947, N.E. Steenrod defined the Steenrod Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cupi 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

Relative singularity categories and singular equivalences J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210818
Hafezi, RasoolLet 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

Higher order Toda brackets J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210727
Aziz KharoofWe 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 Feynman transform and Verdier duality J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210723
Hao YuThe 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 dgvector spaces over a field k of characteristic 0), there is a certain sheaf

On the K(1)local homotopy of $$\mathrm {tmf}\wedge \mathrm {tmf}$$ tmf ∧ tmf J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210720
Dominic Leon Culver, Paul VanKoughnettAs 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.

The Cantor–Schröder–Bernstein Theorem for $$\infty $$ ∞ groupoids J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210628
Martín Hötzel EscardóWe 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

2Segal objects and algebras in spans J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210517
Walker H. SternWe 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: 2Segal simplicial objects in C to algebra objects in Span(C); and secondly: 2Segal cyclic objects in C to Calabi–Yau algebra objects in Span(C).

Torsion in the magnitude homology of graphs J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210515
Radmila Sazdanovic, Victor SummersMagnitude 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

Leibniz algebras with derivations J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210409
Apurba DasIn 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

Uniqueness of differential characters and differential Ktheory via homological algebra J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210403
Ishan MataSimons and Sullivan constructed a model of differential Ktheory, and showed that the differential Ktheory functor fits into a hexagon diagram. They asked whether, like the case of differential characters, this hexagon diagram uniquely determines the differential Ktheory functor. This article provides a partial affirmative answer to their question: For any fixed compact manifold, the differential

Derived categories of N DG categories J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210331
Junichi Miyachi, Hiroshi NagaseIn this paper we study Ndifferential graded categories and their derived categories. First, we introduce modules over an Ndifferential 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 Ndifferential graded

Homotopy theory of monoids and derived localization J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210303
Joe Chuang, Julian Holstein, Andrey LazarevWe 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 cobarconstruction to the nonsimply connected case, and a new algebraic model for the homotopy theory of connected topological spaces as an \(\infty \)category of discrete monoids.

Homotopical perspective on statistical quantities J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210209
Nissim RanadeWe 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

Smooth functorial field theories from Bfields and Dbranes J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210123
Severin Bunk, Konrad WaldorfIn the Lagrangian approach to 2dimensional sigma models, Bfields and Dbranes contribute topological terms to the action of worldsheets of both open and closed strings. We show that these terms naturally fit into a 2dimensional, smooth openclosed 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

Homotopy types of gauge groups of $$\mathrm {PU}(p)$$ PU ( p ) bundles over spheres J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210121
Simon ReaWe 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 plocal equivalence \(\mathcal {G}_{2,k}\simeq _{(p)}\mathcal {G}_{2,l}\) for any prime p if, and only if, \((120,k)=(120,l)\)

The Segal conjecture for topological Hochschild homology of Ravenel spectra J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210119
Gabriel AngeliniKnoll, J. D. QuigleyIn 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 Adams spectral sequence for 3local $$\mathrm {tmf}$$ tmf J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20210106
D. CulverThe purpose of this article is to record the computation of the homotopy groups of 3local \(\mathrm {tmf}\) via the Adams spectral sequence.

Groups up to congruence relation and from categorical groups to ccrossed modules J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20201121
Tamar Datuashvili, Osman Mucuk, Tunçar ŞahanWe introduce a notion of cgroup, which is a group up to congruence relation and consider the corresponding category. Extensions, actions and crossed modules (ccrossed modules) are defined in this category and the semidirect product is constructed. We prove that each categorical group gives rise to a cgroup and to a ccrossed module, which is a connected, special and strict ccrossed module in the

On the relative K group in the ETNC, Part II J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20201106
Oliver BraunlingIn a previous paper we showed that, under some assumptions, the relative Kgroup in the Burns–Flach formulation of the equivariant Tamagawa number conjecture (ETNC) is canonically isomorphic to a Kgroup 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

Homotopy Gerstenhaber algebras are strongly homotopy commutative J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20201101
Matthias FranzWe 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

Homotopic distance between functors J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20201013
E. MacíasVirgós, D. MosqueraLoisWe 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 LScategory of small categories by Tanaka.

Cohomology and deformations of oriented dialgebras J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20200916
Ali N. A. Koam, Ripan SahaWe 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.

Note on Toda brackets J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20200828
Samik Basu, David Blanc, Debasis SenWe 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.

Cyclic homology for bornological coarse spaces J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20200724
Luigi CaputiThe 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

Bianchi’s additional symmetries J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20200720
Alexander D. RahmIn a 2012 note in Comptes Rendus Mathématique, the author did try to answer a question of JeanPierre 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

Descent theory and mapping spaces J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20200703
Nicholas J. MeadowsThe 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 quasicategory 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

Higher equivariant and invariant topological complexities J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20200621
Marzieh Bayeh, Soumen SarkarIn this paper we introduce concepts of higher equivariant and invariant topological complexities and study their properties. Then we compare them with equivariant LScategory. We give lower and upper bounds for these new invariants. We compute some of these invariants for moment angle complexes.

Transfer ideals and torsion in the Morava E theory of abelian groups J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20200523
Tobias Barthel, Nathaniel StapletonLet A be a finite abelian pgroup of rank at least 2. We show that \(E^0(BA)/I_{tr}\), the quotient of the Morava Ecohomology of A by the ideal generated by the image of the transfers along all proper subgroups, contains ptorsion. The proof makes use of transchromatic character theory.

The universal fibration with fibre X in rational homotopy theory J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20200402
Gregory Lupton, Samuel Bruce SmithLet X be a simply connected space with finitedimensional 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

The unit of the total décalage adjunction J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20200319
Viktoriya Ozornova, Martina RovelliWe 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

Verifying the Hilali conjecture up to formal dimension twenty J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20200312
Spencer Cattalani, Aleksandar MilivojevićWe 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.

An application of the h principle to manifold calculus J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20200311
Apurva NakadeManifold 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 hprinciple, we show that for a symplectic manifold N, the analytic approximation to the Lagrangian embeddings functor \(\mathrm {Emb}_{\mathrm {Lag}}(,N)\) is the totally

Correction to: Representations are adjoint to endomorphisms J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20200306
Gabriel C. DrummondCole, Joseph Hirsh, Damien LejayThe first equation under section “Remark 3” was processed and published incorrectly. The correct equation should read as follows:

On the capacity and depth of compact surfaces J. Homotopy Relat. Struct. (IF 0.419) Pub Date : 20200212
Mahboubeh Abbasi, Behrooz MashayekhyK. 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 nonorientable surface of genus \(g>0\) is