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Hans-Joachim Baues

Publications and source records attributed to Hans-Joachim Baues.

At least 19 recordsLinked to original sources
arXiv2020

The DG-category of secondary cohomology operations

We study track categories (i.e., groupoid-enriched categories) endowed with additive structure similar to that of a 1-truncated DG-category, except that composition is not assumed right linear. We show that if such a track category is right linear up to suitably coherent correction tracks, then it is weakly equivalent to a 1-truncated DG-category. This generalizes work of the first author on the strictification of secondary cohomology operations. As an application, we show that the secondary integral Steenrod algebra is strictifiable.

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arXiv2019

Truncated Derived Functors and Spectral Sequences

The $E_2$ term of the Adams spectral sequence may be identified with certain derived functors, and this also holds for a number of other spectral sequences. Our goal is to show how the higher terms of such spectral sequences are determined by truncations of relative derived functors, defined in terms of certain simplicial functors called mapping algebras

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arXiv2017

Eilenberg-MacLane mapping algebras and higher distributivity up to homotopy

Primary cohomology operations, i.e., elements of the Steenrod algebra, are given by homotopy classes of maps between Eilenberg--MacLane spectra. Such maps (before taking homotopy classes) form the topological version of the Steenrod algebra. Composition of such maps is strictly linear in one variable and linear up to coherent homotopy in the other variable. To describe this structure, we introduce a hierarchy of higher distributivity laws, and prove that the topological Steenrod algebra satisfies all of them. We show that the higher distributivity laws are homotopy invariant in a suitable sense. As an application of $2$-distributivity, we provide a new construction of a derivation of degree $-2$ of the mod $2$ Steenrod algebra.

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arXiv2016

2-track algebras and the Adams spectral sequence

In previous work of the first author and Jibladze, the $E_3$-term of the Adams spectral sequence was described as a secondary derived functor, defined via secondary chain complexes in a groupoid-enriched category. This led to computations of the $E_3$-term using the algebra of secondary cohomology operations. In work with Blanc, an analogous description was provided for all higher terms $E_m$. In this paper, we introduce $2$-track algebras and tertiary chain complexes, and we show that the $E_4$-term of the Adams spectral sequence is a tertiary Ext group in this sense. This extends the work with Jibladze, while specializing the work with Blanc in a way that should be more amenable to computations.

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arXiv2015

Higher Toda brackets and Massey products

We provide a uniform definition of higher order Toda brackets in a general setting, covering the known cases of long Toda brackets for topological spaces and chain complexes and Massey products for differential graded algebras, among others.

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arXiv2014

The realizability of operations on homotopy groups concentrated in two degrees

The homotopy groups of a space are endowed with homotopy operations which define the Π-algebra of the space. An Eilenberg-MacLane space is the realization of a Π-algebra concentrated in one degree. In this paper, we provide necessary and sufficient conditions for the realizability of a Π-algebra concentrated in two degrees. We then specialize to the stable case, and list infinite families of such Π-algebras that are not realizable.

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arXiv2014

Higher order derived functors and the Adams spectral sequence

Classical homological algebra considers chain complexes, resolutions, and derived functors in additive categories. We describe "track algebras in dimension n", which generalize additive categories, and we define higher order chain complexes, resolutions, and derived functors. We show that higher order resolutions exist in higher track categories, and that they determine higher order Ext-groups. In particular, the E_m-term of the Adams spectral sequence (m<n+3) is a higher order Ext-group, which is determined by the track algebra of higher cohomology operations.

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arXiv2013

The Third Homotopy Group as a pi_1-Module

It is well-known how to compute the structure of the second homotopy group of a space, $X$, as a module over the fundamental group, $π_1X$, using the homology of the universal cover and the Hurewicz isomorphism. We describe a new method to compute the third homotopy group, $π_3 X$, as a module over $π_1 X$. Moreover, we determine $π_3 X$ as an extension of $π_1 X$-modules derived from Whitehead's Certain Exact Sequence. Our method is based on the theory of quadratic modules. Explicit computations are carried out for pseudo-projective 3-spaces $X = S^1 \cup e^2 \cup e^3$ consisting of exactly one cell in each dimension $\leq 3$.

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arXiv2010

Stems and Spectral Sequences

We introduce the category Pstem[n] of n-stems, with a functor P[n] from spaces to Pstem[n]. This can be thought of as the n-th order homotopy groups of a space. We show how to associate to each simplicial n-stem Q an (n+1)-truncated spectral sequence. Moreover, if Q=P[n]X is the Postnikov n-stem of a simplicial space X, the truncated spectral sequence for Q is the truncation of the usual homotopy spectral sequence of X. Similar results are also proven for cosimplicial n-stems. They are helpful for computations, since n-stems in low degrees have good algebraic models.

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arXiv2010

Comparing cohomology obstructions

We show that three different kinds of cohomology - Baues-Wirsching cohomology, the (S,O)-cohomology of Dwyer-Kan, and the Andre-Quillen cohomology of a Pi-algebra - are isomorphic, under certain assumptions. This is then used to identify the cohomological obstructions in three general approaches to realizability problems: the track category version of Baues-Wirsching, the diagram rectifications of Dwyer-Kan-Smith, and the Pi-Algebra realization of Dwyer-Kan-Stover. Our main tool in this identification is the notion of a mapping algebra: a simplicially enriched version of an algebra over a theory.

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arXiv2010

Self-Maps of the Product of Two Spheres Fixing the Diagonal

We compute the monoid of essential self-maps of of the product of two n-spheres fixing the diagonal. More generally, we consider products S x S, where S is a suspension. Essential self-maps of S x S demonstrate the interplay between the pinching action for a mapping cone and the fundamental action on homotopy classes under a space. We compute examples with non-trivial fundamental actions.

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arXiv2010

Presentation of homotopy types under a space

We compare the structure of a mapping cone in the category Top^D of spaces under a space D with differentials in algebraic models like crossed complexes and quadratic complexes. Several subcategories of Top^D are identified with algebraic categories. As an application we show that there are exactly 16 essential self--maps of S^2 x S^2 fixing the diagonal.

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arXiv2009

Higher order track categories and the algebra of higher order cohomology operations

We describe a conjecture on the algebra of higher cohomology operations which leads to the computations of the differentials in the Adams spectral sequence. For this we introduce the notion of an n-th order track category which is suitable to study higher order Toda brackets and the differentials in spectral sequences. We describe various examples of higher order track categories which are topological, in particular the track category of higher cohomology operations. Also differential algebras give rise to higher order track categories.

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arXiv2008

Intersection of subgroups in free groups and homotopy groups

We show that the intersection of three subgroups in a free group is related to the computation of the third homotopy group $π_3$. This generalizes a result of Gutierrez-Ratcliffe who relate the intersection of two subgroups with the computation of $π_2$. Let $K$ be a two-dimensional CW-complex with subcomplexes $K_1,K_2,K_3$ such that $K=K_1\cup K_2\cup K_3$ and $K_1\cap K_2\cap K_3$ is the 1-skeleton $K^1$ of $K$. We construct a natural homomorphism of $π_1(K)$-modules $$ π_3(K)\to \frac{R_1\cap R_2\cap R_3}{[R_1,R_2\cap R_3][R_2,R_3\cap R_1][R_3,R_1\cap R_2]}, $$ where $R_i=ker\{π_1(K^1)\to π_1(K_i)\}, i=1,2,3$ and the action of $π_1(K)=F/R_1R_2R_3$ on the right hand abelian group is defined via conjugation in $F$. In certain cases, the defined map is an isomorphism. Finally, we discuss certain applications of the above map to group homology.

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arXiv2008

Homotopy types of reduced 2-nilpotent simplicial groups

We classify the homotopy types of reduced 2-nilpotent simplicial groups in terms of the homology an d boundary invariants $b,β$. This contains as special cases results of J.H.C. Whitehead on 1-connected 4-dimensional complexes and of Quillen on reduced 2-nilpotent rational simplicial groups. Moreover it yields for 1-nilpotent (or abelian) simplicial groups a classification due to Dold-Kan. Our result describes a new natural structure of the integral homology of any simply connected space. We also classify the homotopy types of connective spectra in the category of 2-nilpotent simplicial groups. Moreover we compute homotopy groups of spheres in the category of $m$-nilpotent groups for $m=2,3$ and partially for $m=4,5$.

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arXiv2007

The algebra of secondary homotopy operations in ring spectra

The primary algebraic model of a ring spectrum is the ring of homotopy groups. We introduce the secondary model which has the structure of a secondary analogue of a ring. This new algebraic model determines Massey products and cup-one squares. As an application we obtain new derivations of the homotopy ring.

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arXiv2006

Toda brackets and cup-one squares for ring spectra

In this paper we prove the laws of Toda brackets on the homotopy groups of a connective ring spectrum and the laws of the cup-one square in the homotopy groups of a commutative connective ring spectrum.

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