Classical homological algebra takes place in additive categories. In homotopy theory such additive categories arise as homotopy categories of ``additive groupoid enriched categories'', in which a secondary analog of homological algebra can be performed. We introduce secondary chain complexes and secondary resolutions leading to the concept of secondary derived functors. As a main result we show that the E_3-term of the Adams spectral sequence can be expressed as a secondary derived functor. This result can be used to compute the E_3-term explicitly by an algorithm.
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.
We describe the dualization of the algebra of secondary cohomology operations in terms of generators extending the Milnor dual of the Steenrod algebra. In this way we obtain explicit formulae for the computation of the E_3-term of the Adams spectral sequence converging to the stable homotopy groups of spheres.
We introduce the compactness locus of a geometric functor between rigidly-compactly generated tensor-triangulated categories, and describe it for several examples arising in equivariant homotopy theory and algebraic geometry. It is a subset of the tensor-triangular spectrum of the target category which, crudely speaking, measures the failure of the functor to satisfy Grothendieck-Neeman duality (or equivalently, to admit a left adjoint). We prove that any geometric functor --- even one which does not admit a left adjoint --- gives rise to a Wirthmüller isomorphism once one passes to a colocali…
In a well generated triangulated category T, given a regular cardinal a, we consider the following problems: given a functor from the category of a-compact objects to abelian groups that preserves products of <a objects and takes exact triangles to exact sequences, is it the restriction of a representable functor in T? Is every natural transformation between two such restricted representable functors induced by a map between the representatives? If the answer to both questions is positive we say that T satisfies a-Adams representability. A classical result going back to Brown and Adams shows t…
We give a short and streamlined proof of the following statement recently proven by the author and M. Zeinalian: the cobar construction of the dg coassociative coalgebra of normalized singular chains on a path-connected pointed space is naturally quasi-isomorphic as a dg associative algebra to the singular chains on the based loop space. This extends a classical theorem of F. Adams originally proven for simply connected spaces. Our proof is based on relating the cobar functor to the left adjoint of the homotopy coherent nerve functor.
An algorithm is described giving effective determination of the second differential in the Adams spectral sequence. The algorithm is based on the notion of secondary derived functor, and on the explicit algebraic model of the groupoid enriched category of stable maps and stable tracks between Eilenberg-Mac Lane spaces.
Let T be a triangulated category with coproducts, C the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [Adams71]: All contravariant homological functors C --> Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this generalises to other triangulated categories. In [Neeman97], it was proved that Adams' theorem remains true as long as C is countable, but can fail in general. The failure exhibited was that…
For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by proving the existence of truncated versions of resolution or $E_2$-model structures. Examples of functors fitting in our framework are given by every generalized homology theory represented by a ring spectrum satisfying the Adams-Atiyah …
We use combinatorial group theory methods to extend the definition of a classical James-Hopf invariant to a simplicial group setting. This allow us to realize certain coalgebra idempotents at sSet -level and obtain a functorial decomposition of the spectral sequence, associated with the lower p-central series filtration on the free simplicial group.
In this paper we prove that $β_{p^2/p^2-1}$ survives to $E_\infty$ in the Adams-Novikov spectral sequence for $p\geqslant 5$. As an easy consequence we prove that $β_{sp^2/j}$ are perminent cycles for all $s\geqslant 1$, $j\leqslant p^2-1$. From the Thom map $Φ: Ext^{s,t}_{BP_*BP}(BP_*, BP_*)\longrightarrow Ext^{s,t}_A(\mathbb{Z}/p, \mathbb{Z}/p)$, we also see that $h_0h_3$ survives to $E_\infty$ in the classical Adams spectral sequence.
We prove a general representation stability result for polynomial coefficient systems which lets us prove representation stability and secondary homological stability for many families of groups with polynomial coefficients. This gives two generalizations of classical homological stability theorems with twisted coefficients. We apply our results to prove homological stability for hyperelliptic mapping class groups with twisted coefficients, prove new representation stability results for congruence subgroups, establish secondary homological stability for groups of diffeomorphisms of surfaces vi…
In the early 2000's, Baues computed the secondary Steenrod algebra, the algebra of all secondary cohomology operations. Together with Jibladze, they showed that this gives an algorithm that computes all Adams $d_2$ differentials for the sphere. The goal of this paper is to reinterpret their results in the language of synthetic spectra in order to achieve stronger computational results. Using this, we obtain an algorithm that computes hidden extensions on the $E_3$ page that jump by one filtration, in addition to the $d_2$ differentials of Baues--Jibladze. We then implement and run this algorit…
Adams operations are the natural transformations of the representation ring functor on the category of finite groups, and they are one way to describe the usual lambda-ring structure on these rings. From the representation-theoretical point of view, they codify some of the symmetric monoidal structure of the representation category. We show that the monoidal structure on the category alone, regardless of the particular symmetry, determines all the odd Adams operations. On the other hand, we give examples to show that monoidal equivalences do not have to preserve the second Adams operations and…
We prove for a large family of rings R that their lambda-pure global dimension is greater than one for each infinite regular cardinal lambda. This answers in negative a problem posed by Rosicky. The derived categories of such rings then do not satisfy the Adams lambda-representability for morphisms for any lambda. Equivalently, they are examples of well generated triangulated categories whose lambda-abelianization in the sense of Neeman is not a full functor for any lambda. In particular we show that given a compactly generated triangulated category, one may not be able to find a Rosicky funct…
We give a natural construction and a direct proof of the Adams isomorphism for equivariant orthogonal spectra. More precisely, for any finite group G, any normal subgroup N of G, and any orthogonal G-spectrum X, we construct a natural map A of orthogonal G/N-spectra from the homotopy N-orbits of X to the derived N-fixed points of X, and we show that A is a stable weak equivalence if X is cofibrant and N-free. This recovers a theorem of Lewis, May, and Steinberger in the equivariant stable homotopy category, which in the case of suspension spectra was originally proved by Adams. We emphasize th…
Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a track functor on the track category of pointed spaces compatible with fiber sequences, suspensions and loop spaces. They also yield algebraic models of homotopy types with homotopy groups concentrated in two consecutive dimensions.
In previous work it is shown that there is an abelian category A(G) constructed to model rational G-equivariant cohomology theories, where G is a torus of rank r together with a homology functor \piA_* : Gspectra ---> A(G), and an Adams spectral sequence Ext_{A (G)} (\piA_*(X), \piA_*(Y)) ===> [X,Y]^G_* In joint work with Shipley (arxiv:1101.2511), it is shown that the Adams spectral sequence can be lifted to a Quillen equivalence Rational-Gspectra = DG-A (G). The purpose of the present paper is to prove that A(G) has injective dimension precisely r, and to construct certain torsion functors a…
We construct a smash product operation on secondary homotopy groups yielding the structure of a lax symmetric monoidal functor. Applications on cup-one products, Toda brackets and Whitehead products are considered. In particular we prove a formula for the crossed effect of the cup-one product operation on unstable homotopy groups of spheres which was claimed by Barratt-Jones-Mahowald.
The theory of secondary chomology operations leads to a conjecture concerning the algebra of higher cohomology operations in general. This conjecture is discussed here in detail and its connection with homotopy groups of spheres and the Adams spectral sequence is described.
For a finite Galois extension of fields L/k with Galois group G, we study a functor from the G-equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical Galois correspondence. We show that after completing at a prime and eta (the motivic Hopf map) this results in a full and faithful embedding whenever k is real closed and L = k[i]. It is a full and faithful embedding after eta-completion if a motivic version of Serre's finiteness theorem is valid. We produce strong necessary conditions on the field extension L/k for this functor to be full and…
These are notes, by Z. Fiedorowicz, from lectures given by J. Frank Adams at the University of Chicago in spring of 1973. They give an elegant axiomatic presentation of localization and completion in algebraic topology. The construction of localization and completion functors with respect to an arbitrary generalized homology theory is derived from the axioms by using the Brown representability theorem. These notes were never formally published, due to an apparent flaw in the proof. The relevant representable functors could not be shown to be set-valued, as opposed to class-valued. Subsequent w…
In previous work with Niles Johnson the author constructed a spectral sequence for computing homotopy groups of spaces of maps between structured objects such as G-spaces and E_n-ring spectra. In this paper we study special cases of this spectral sequence in detail. Under certain assumptions, we show that the Goerss-Hopkins spectral sequence and the T-algebra spectral sequence agree. Under further assumptions, we can apply a variation of an argument due to Jennifer French and show that these spectral sequences agree with the unstable Adams spectral sequence. From these equivalences we obtain i…
Let k be an algebraically closed field of characteristic zero. Let SH(k) denote the motivic stable homotopy category of T-spectra over k and SH the classical stable homotopy category. Let c:SH -> SH(k) be the functor induced by sending a space to the constant presheaf of spaces on Sm/k. We show that c is fully faithful. In particular, c induces an isomorphism π_n(E)-> π_{n,0}c(E) for all spectra E. Fix an embedding of k into the complex numbers and let Re:SH(k) -> SH be the associated Betti realization. We show that the slice tower for the motivic sphere spectrum has Betti realization which is…
In this paper, we develop a structure theory for generalized spectral sequences, which are derived from chain complexes that are filtered over arbitrary partially ordered sets. Also, a more general construction method reminiscent of exact couples is studied, together with examples where they arise naturally. As for ordinary spectral sequences we will see differentials and group extensions, however the real power comes from the appearance of natural isomorphims between pages of differing indices. The constructions reveal finer invariants than ordinary spectral sequences, and they connect to oth…
